integrability properties of quad equations consistent on the cube · versity in october 2013 i knew...

UNIVERSITAgrave DEGLI STUDI

ROMA

TREcorso di dottorato di ricerca in matematica

xxix ciclo

I N T E G R A B I L I T Y P R O P E RT I E S O FQ U A D E Q U AT I O N S C O N S I S T E N T

O N T H E C U B E

CandidatoGiorgio Gubbiotti

Docente guidaTutorProf Decio Levi

Coordinatore del DottoratoProf Luigi Chierchia

Giorgio Gubbiotti Integrability properties of quad equations consistent onthe cube copy AA 2016ndash2017

A Valentina ed ad Arabella

A B S T R A C T

In this Thesis we present some results obtained recently about the in-tegrability properties of the multi-affine partial difference equationsConsistent Around the Cube classified by R Boll We review someknown result and we present for the first time the non-autonomousform on the lattice of some of these equations Using the so-called Al-gebraic Entropy test we conjecture that two sub-families of the equa-tions found by Boll namely the trapezoidal H4 equations and the H6

equations are linearizable By computing the Generalized Symmetriesof the trapezoidal H4 equations and the H6 equations we propose anon-autonomous generalization of the QV equation Finally we provethe linearizability of these equation by showing that they are Darbouxintegrable equations and we show how to use this property in orderto obtain general solutions

S O M M A R I O

In questa Tesi presentiamo alcuni risultati recentemente ottenuti sul-le proprietagrave di ingregrabilitagrave delle equazioni alle differenze parzialimultiaffini Consistenti sul Cubo classificate da R Boll Passiamo inrassegna alcuni risultati nuovi e presentiamo la forma nonautonomasul reticolo di alcune di queste equazioni per la prima volta Usan-do il cosiddetto metodo dellrsquoEntropia Algebrica congetturiamo che ledue sottofamiglie delle equazioni trovate da Boll ossia le equazioniH4 trapezoidali e le equazioni H6 siano linearizzabili Calcolando leSimmetrie Generalizzate delle equazioni H4 trapezoidali e delle equa-zioni H6 proponiamo una generalizzazione nonautonoma dellrsquoequa-zione QV Per ultima cosa dimostriamo come queste equazioni sianolinearizzabili mostrando che sono Darboux integrabili e che tramitequesta proprietagrave egrave possibile scrivere le soluzioni generali

v

A C K N O W L E D G M E N T S

First of all I would like to thank prof D Levi for accepting of beingmy supervisor In these past three years his guidance has been fun-damental for my growth as would-be researcher I would also like toheartfelt thank Christian Scimiterna for his patience in teaching mealmost everything I needed to know about Discrete Integrable Sys-tems and Generalized Symmetries At my arrival in Roma Tre Uni-versity in October 2013 I knew almost nothing about this subject To-day if I know something about these subject it is thanks to what Irsquovelearned in these years working with prof Levi and Christian Dur-ing my PhD we authored five papers together which are now thebare-bones of this Thesis A further thank goes to prof RI Yamilovfor two different but equally important reasons First during his stayin Roma Tre University in FebruaryndashApril 2016 we worked togetherand produced two papers which are going to be the last Chapter ofthis Thesis Second for his help and availability when I was writinga review on Generalized Symmetries I would like to thank also profC-M Viallet for his help and fruitful discussion when I was writing areview on Algebraic Entropy

Furthermore I would like to thank prof P Winternitz and Dr RRebelo for choosing me to give a cycle of lectures at ASIDE SchoolThe review material in this Thesis is based on these lectures andI would like to thank all the students who attended them I wouldalso like to thank prof R Quispel Dr I Marquette prof N Joshiprof M Radnovic and Dr Y Shi for their hospitality during my stayin Australia in NovemberndashDecember 2015 Moreover I would like tothank prof L Martina for his hospitality in Lecce in November 2016Finally I would like to thank prof MC Nucci for more than five yearsof fruitful scientific collaboration

Besides these scientific acknowledgments I would like to makesome personal ones First I would like to thank my family for sup-porting me during all these years Then I would like to my girlfriendValentina for tolerating and supporting me especially during thewriting of these Thesis I would like to thank my former classmateand housemate Davide especially because this Thesis is written onArch Linux we installed together I would like also to thank my newpseudo-housemate Francesco and the many discussion on the math-ematics behind the optimization in construction and managementsimulation games I would also like to thank my former classmatesin Perugia Davide (who is also my co-author in two publications)Francesco Simone and Luisa A special thanks goes to all the friendof the XLI Summer School on Mathematical Physics in Ravello par-

vii

ticularly to my housemates Andrea T Andrea G and Luigi FinallyI also would like to thank my longtime friends Rosangela Laura andLella for having tolerated me for such a long time

Many thanks to all of you

viii

R I N G R A Z I A M E N T I

Per prima cosa vorrei ringraziare il prof D Levi per aver accettato difarmi da relatore e di avermi guidato in questi tre anni di Dottoratofondamentali per il mio sviluppo come aspirante ricercatore Un altroprofondo ringraziamento va ovviamente a Christian Scimiterna peravermi pazientemente spiegato praticamente tutto quello che crsquoerada sapere sui sistemi integrabili discreti e le loro simmetrie Quandoarrivai a Roma Tre nellrsquoOttobre del 2013 erano un argomento di cuisapevo poco o nulla se oggi ne so qualcosa egrave merito di quanto ho ap-preso in questi anni a stretto contatto con loro Nella durata del mioDottorato abbiamo scritto cinque articoli scientifici insieme articoliche oggi sono lo scheletro di questa Tesi Un ulteriore ringraziamentova al prof RI Yamilov per due motivi differenti ma di eguale impor-tanza Il primo motivo egrave per il lavoro svolto insieme durante la suavisita allrsquoUniversitagrave degli Studi Roma Tre tra il Marzo e lrsquoAprile del2016 Il prodotto di questo lavoro sono stati due articoli scientifici cheche costituiscono lrsquoultimo Capitolo di questa Tesi Il secondo motivoegrave per la dispobilitagrave e lrsquoaiuto quando stavo scrivendo la parte della miarassegna sulle Simmetrie Generalizzate Voglio inoltre ringraziare ilprof C-M Viallet per lrsquoaiuto e le discussioni produttive quando stavoscrivendo la parte della mia rassegna sullrsquoEntropia Algebrica

Inoltre vorrei ringraziare il prof P Winternitz e il Dott R Rebelooer avermi scelto per tenere un ciclo di lezioni alla Scuola Estiva ASI-DE La parte compilativa di questa Tesi egrave anche in parte basata sulcontenuto di quelle lezioni e colgo lrsquooccasione per ringraziare tutti glistudenti che vi hanno assistito Voglio ringraziare il prof R Quispelil Dott I Marquette la profssa N Joshi la profssa M Radnovic ela Dottssa Y Shi per la loro ospitalitagrave durante la mia permanenza inAustralia tra Novembre e Dicembre 2015 Infine vorrei ringraziare laprofssa MC Nucci per i cinque anni di ininterrotta collaborazionescientifica

Oltre a questi ringraziamenti di natura scientifica vorrei farne altridi natura personale In primo luogo vorrei ringraziare la mia famigliaper avermi supportato in questi anni Quindi vorrei ringraziare la miaragazza Valentina per sopportarmi e supportarmi specialmente du-rante la stesura di questa Tesi Vorrei anche ringraziare il mio ex com-pagno di studi e poi coinquilino Davide specialmente percheacute la stesu-ra di questa Tesi egrave avvenuta sulrsquoArch Linux che installammo insiemeInoltre vorrei anche ringraziare il mio quasi-coinquilino Francesco ele innumerevoli discussioni sulla matematica dellrsquoottimizzazione neigiochi di simulazione manageriale Vorrei anche ringraziare i miei excompagni di studi di Perugia Davide (che egrave anche mio coautore in

ix

due pubblicazioni) Francesco Simone e Luisa Infine vorrei ringra-ziare anche i miei amici di vecchia data Rosangela Laura e Lella persopportarmi ancora dopo tanti anni

A tutti voi grazie

x

C O N T E N T S

introduction 1

1 integrability and the consistency around the

cube 5

11 The meaning of integrability 5

12 The Consistency Around the Cube 8

13 The Consistency Around the Cube as a search method 11

131 Original attempts 12

132 Bollrsquos classification non-autonomous lattices 14

133 The equations on the single cell 17

14 Construction of the 2D3D lattice in the general case 19

15 Classification tools on the 2D lattice 31

16 Independent equations on the 2D-lattice 36

2 algebraic entropy and direct linearization 41

21 Definition and basic properties 41

211 Algebraic entropy for ordinary difference equa-tions and differential-difference equations 43

212 Algebraic entropy for quad equations 49

213 Geometric meaning of the Algebraic Entropy 52

22 Computational tools 54

23 Algebraic Entropy test for (191 193) 59

24 Examples of direct linearization 66

241 The tHε1 equation 66

242 The 1D2 equation 71

3 generalized symmetries of the trapezoidal H4

and H6 equations and the non-autonomous ydkn

equation 79

31 Generalized Symmetries and integrability 80

32 Three-point Generalized Symmetries for quad equa-tion 83

33 Three-point generalized symmetries of the trapezoidalH4 and H6 equations 96

331 Trapezoidal H4 equations 96

332 H6 equations 99

34 The non-autonomous YdKN equation 102

35 Integrability properties of the non autonomous YdKNequation and its sub-cases 109

36 A non-autonomous generalization of theQV equation 116

4 darboux integrability and general solutions 125

41 Darboux integrability 125

42 Darboux integrability and Consistency Around the Cube 129

43 First integrals for the trapezoidal H4 and H6 equa-tions 131

xi

xii contents


4311 tHε1 equation 132



432 H6 equations 135

4321 1D2 equation 135



4324 D3 equation 139



44 General solutions through the first integrals 142

441 The iD2 equations i = 1 2 3 144




442 The D3 and the iD4 equations i = 1 2 164




443 The trapezoidal H4 equations 174




conclusions 190

a final part of the proof of the commutative dia-gram 1 9 195

b python programs for computing algebraic entropy 201

b1 Algebraic Entropy of systems of quad equations withae2dpy 201

b11 Description of the content of ae2dpy 201

b12 Code 206

b2 Algebraic Entropy for systems of ordinary differenceequations and differential difference equations of arbi-trary order with ae_differential_differencepy 216

b3 Analysis of the degree sequences 221

b31 The padepy module 221

b32 The pade_rat_gfuncpy module 223

b33 The z_transformpy module 224

c evolution matrices 227

c1 Trapezoidal H4 equations (191) 229

c11 tHε1 equation (191a) 229

c12 tHε2 equation (191b) 233

c13 tHε3 equation (191c) 237

c2 H6 equations (193) 241

c21 1D2 equation (193b) 241

contents xiii

c22 2D2 equation (193c) 245

c23 3D2 equation (193d) 249

c24 D3 equation (193e) 253

c25 1D4 equation (193f) 257

c26 2D4 equation (193g) 261

d examples of calculations of generalized symme-tries for equations with two-periodic coefficients 265

d1 Three-points generalized symmetries of the D3 equa-tion 265

d2 Three-points generalized symmetries of the tHε1 in di-rection m 267

e three-points generalized symmetries of the rhom-bic H4 equations 271

e1 The rhombic H4 equations 271

e2 Three-points generalized symmetries of the rhombicH4 equations 271

f connection formulaelig between the non-autonomous

QV and the non-autonomous ydkn in the m di-rection 273

g calculation of the first integrals of the tH1equation 275

h linearization of the tHε1 equation through the

first integral in the direction m 277

bibliography 279

L I S T O F F I G U R E S

Figure 11 A quad-graph 9

Figure 12 The extension of the 2D lattice to a 3D lat-tice when the equation on the edges are thesame 9

Figure 13 The purely geometric quad graph not embed-ded in any lattice 15

Figure 14 Equations on a Cube 15

Figure 15 The ldquofour colorsrdquo lattice 20

Figure 16 Rhombic Black and White lattice 20

Figure 17 Trapezoidal Black and White lattice 21

Figure 18 The extension of the consistency cube 22

Figure 19 The commutative diagram defining Mob4

32

Figure 21 The Heacutenon map in the plane (xy) with a =

14 and b = 03 and the initial conditions (x0y0) =(06 02) 47

Figure 22 Regular and non-regular staircases 49

Figure 23 Varius kinds of restricted initial conditions 50

Figure 24 The range covered by the initial conditions ∆(3)[minus21] 50

Figure 25 The four principal diagonals 51

Figure 26 A possible blow-down blow-up scheme in P3 53

Figure 31 The extended square lattice with 9 points 83

Figure 41 The six-point lattice 177

xiv

L I S T O F TA B L E S

Table 11 Multiplication rules for the functions fnm asgiven by (152) 35

Table 21 Sequences of growth for the trapezoidalH4 andH6 equations The first one is the principal se-quence while the second the secondary All se-quences Lj j = 0 middot middot middot 20 are presented in Table23 60

Table 22 Sequences of growth generating functions an-alytic expression of the degrees and entropyfor the trapezoidal H4and H6equations 63

Table 31 Identification of the coefficients in the sym-metries of the rhombic H4 equations trape-zoidal H4 equations and H6 equations withthose of the non autonomous YdKN equationHere the symmetries of tHε1 in the m directionare the subcase (3104) of (390) while those inthe n direction are the subcase (3102) of (389)The symmetries of the iD2 equations (392) arecombined with the point symmetries (393) ac-cording to the prescriptions (3105) and the co-efficients given by (3106) 106

Table 32 The value of the coefficients of the Master Sym-metry (3123) obtained by solving the system(3125) in the case of the trapezoidal H4 equa-tions (191) 113

Table 33 Identification of the coefficients of the non au-tonomous QV equation with those of the Bollrsquosequations Since a1 = a2i = 0 for all equa-tions considered in this Table these coefficientsabsent 124

xv

I N T R O D U C T I O N

The main object of this Thesis are the multi-affine partial difference equa-tions defined on a quad graph ie equations for an unknown functionunm of the two discrete variables (nm) isin Z2 which are multi-affinepolynomial in the unknown function and its shifts in a way that thecorresponding points on the Z2 plane lie on the vertices of a quadri-lateral figure The most known example of this kind of equations isthe discrete wave equation

un+1m+1 minus un+1m minus unm+1 + unm = 0

ie the lattice analogue of the wave equation in the light-cone coordi-nates uxt = 0 Indeed the partial difference equations defined on aquad graph are the lattice analogue of the hyperbolic partial differentialequations

The property of the multi-affine partial difference equations de-fined on a quad graph we are interested in is their integrability Themost accepted definition of integrability for nonlinear equations (ofany kind) is that of the existence of a Lax pair [97] A Lax pair isa linear representation of a nonlinear problem which yield throughthe method of the Inverse Scattering Transform the solution of theoriginal nonlinear equation

Finding a Lax pair for a nonlinear equation is a non-trivial andnon-algorithmic task so a preeminent rocircle in the theory of IntegrableSystems is given to the so-called integrability criteria [100 172] In-tegrability criteria are algorithmic procedure that permit to say if agiven equation is integrable or not Integrability criteria are thereforeparticularly useful in applications

In the case of the nonlinear multi-affine partial difference equa-tions defined on a quad graph a particularly useful integrability crite-rion has been discovered the so-called Consistency Around the CubeRoughly speaking a multi-affine partial difference equations definedon a quad graph is said to possess the Consistency Around the Cubeif it can be extended in an agreeing way on a three dimensional latticeThis means that a single equation is replaced by a sextuple of equa-tions assigned on the faces of a cube which explains the name seeChapter 1 for a more formal definition The usefulness of the Con-sistency Around the Cube relays in the fact that it provides for anequation possessing it Baumlcklund transforms and as a consequence aLax pair [120 122]

Beyond its rocircle as integrability criterion ie as a method to estab-lish if a given equation is or not integrable the Consistency Aroundthe Cube has been also used for classification purposes in a program

1

2 introduction

aimed to find all the equations possessing it up to some chosen groupof transformations Indeed since if an equation possess the Consis-tency Around the Cube we can find its Lax pair to classify all theequations with this property means to classify a subset of the in-tegrable multi-affine partial difference equations defined on a quadgraph The classification using the Consistency Around the Cube wasdone in [2] In [2] were used two technical assumptions that the givenequation possesses the discrete symmetries of the square and the tetrahe-dron property An equation possesses the discrete symmetries of thesquare if the multi-affine polynomial defining it is symmetric undera particular exchange of its entries while it possess the tetrahedronproperty if the equation on the top of the cube does not containsthe field unm In the following years many authors dealt with theproblem of weakening the hypothesis in [2] and extending the classi-fication therein made eg [81 82] Most notably the paper [3] openedthe way for a complete classification using as only technical hypoth-esis the tetrahedron property A complete classification of the quadgraph equations possessing the Consistency Around the Cube andthe tetrahedron property was then accomplished by R Boll in a se-ries of papers culminating in his PhD thesis [20ndash22] The result of theclassification made by Boll was that there exist three families of equa-tions possessing the Consistency Around the Cube the Q equationsthe H4 equations divided in rhombic and trapezoidal and the H6 equa-tions Let us notice that the Q family was introduced in [2] and theirintegrability properties are well established A detailed study of allthe lattice equations derived from the rhombic H4 family includingthe construction of their three-leg forms Lax pairs Baumlcklund trans-formations and infinite hierarchies of generalized symmetries waspresented in [166]

In this Thesis we deal with two families of equations of the Bollrsquosclassification possessing the Consistency Around the Cube whichwere still unstudied namely the trapezoidal H4 equations and theH6 equations The main result is that these two classes of equationsconsist of linearizable equations We will go through the various stepswhich were made to understand their deep structure in a series of pa-pers The plan of the Thesis is the following

chapter 1 We introduce the concept of integrability for finite andinfinite dimensional systems with particular attention to the par-tial difference equations case In particular we introduce the con-cept of integrability indicators and we present in a rigorous waythe concept of Consistency Around the Cube and its historicaldevelopment We present then the explicit construction of thelattice equations from the single-cell equations in the case whenthe equations on the side of the cube are different We discussthen how the classification at the single-cell level is preservedwhen passing to the lattice by introducing explicitly the group

introduction 3

Mob4

nm The original part of this Chapter is mainly based on[67]

chapter 2 We introduce the Algebraic Entropy to study the trape-zoidal H4 and the H6 equation The fact that the trapezoidal H4

and the H6 equations are non-autonomous requires a modifica-tion of the usual definition of Algebraic Entropy for lattice equa-tion [155] We then present the results of the calculation of theAlgebraic Entropy for the trapezoidal H4 and the H6 equationsTo support this evidence we present two examples of direct lin-earization and an example of the fact that Lax pairs obtainedfrom Consistency Around the Cube for linearizable equationscan be fake [27 77] The original part of this Chapter is mainlybased on [67 68]

chapter 3 We introduce the Generalized Symmetry method forpartial difference equations on the quad graph We then presentthe three-point Generalized Symmetries of the trapezoidal H4

and the H6 equations and put them in relation with particularcases of the Yamilov discretization of the Kriechever-Novikovequation [168] Stimulated by the results of these computationswe introduce a new partial difference equation which we con-jecture to be integrable due to Algebraic Entropy test We showthat this new partial difference equation is a non-autonomousgeneralization of the so-called QV equation [156] The originalpart of this Chapter is mainly based on [65 66]

chapter 4 We present a rigorous proof of the fact that the H4 andthe H6 equations are linearizable based on the concept of Dar-boux integrability for partial difference equations [5] This en-ables us also to construct the general solutions of these equa-tion by solving some linear non-autonomous ordinary differ-ence equations or some non-autonomous discrete Riccati equa-tions This fact can be taken to be a confirmation of the Alge-braic Entropy conjecture [84] The original part of this Chapteris mainly based on [70 71]

In the Conclusions we make some comments on the results ob-tained in the various Chapters of this Thesis and we discuss someopen problems and further developments

1I N T E G R A B I L I T Y A N D T H E C O N S I S T E N C YA R O U N D T H E C U B E

Here we introduce the main concepts we will deal with In particularwe will introduce the main ingredients in order to construct the latticerepresentation of the trapezoidal H4 and H6 equations object of thisThesis

We begin in Section 11 by introducing the concept of integrabilityThe in Section 12 we introduce the basis of one of the most pro-lific integrability indicators of the last years the Consistency Around theCube In Section 13 we will give an account of how the ConsistencyAround the Cube has been used in order to find and classify inte-grable systems In particular in Subsection 132 we will introduce theBollrsquos classification [20ndash22] and present and discuss in the followingSubsection 133 the equations yet not studied In Section 14 we willdiscuss the problem of the embedding in the 2D and 3D lattices InSection 15 we give the proof of how the classification at the level ofsingle cell is preserved once embedded in the full lattice given origi-nally in [67] Finally in Section 16 we will present the explicit latticeform of the trapezoidal H4 and H6 equations as given in [67] whichwill used everywhere throughout the Thesis

11 the meaning of integrability

In this Thesis we will focus mainly on two-dimensional partial differenceequations defined on a square lattice for an unknown function unm

with (nm) isin Z2 ie relations of the form

Q (unmun+1munm+1un+1m+1) = 0 (nm) isin Z2 (11)

If the function Q is a multi-affine irreducible polynomial of its argu-ments we say that (11) is a quad-equation This will be the kind oftwo-dimensional partial difference equation we will consider mostly

We wish to study the integrability of these equations The notionof integrability comes from Classical Mechanics and roughly meansthe existence of a ldquosufficientlyrdquo high number of first integrals Indeedgiven an Hamiltonian system with Hamiltonian H = H (pq) withN degrees of freedom we say that this system is integrable if thereexists N integrals of motions ie N functions Hi i = 1 N well de-fined on the phase space ie analytic and single-valued which Poisson-commutes with the Hamiltonian

HiH = 0 (12)

5

6 integrability and the consistency around the cube

The Hamiltonian which commutes with itself is included in the listas H1 = H These first integrals must be well defined functions on thephase space and in involution

HiHj= 0 i 6= j = 1 N (13)

and finally they should be functionally independent

rankpart(H1 Hn)

part(p1 pnq1 qn)= N (14)

Indeed the knowledge of these integrals provides the remainingNminus1

integrals and this is the content of the famous Liouville theorem [112161] When more than N independent integrals exists we say that thesystem is superintegrable [44 140] Note that in this case the additionalintegrals will not be in involution with the previous ones Integrabil-ity in Classical Mechanics means that the motion is constrained onsubspace of the full phase space With some additional assumptionson the geometric structure of the first integral it is possible to provethat the motion is quasi-periodic on some tori in the phase space [1213] Note that in the classical case is possible to have some regularityon the behaviour of the system even when there exists M lt N firstintegrals This situation is called partial integrability [49 50 119]

In the infinite dimensional case ie for partial differential equa-tions the notion of integrability have been studied and developed inthe second half of the XXth Century In this case one should be ableto find infinitely many conservation laws These infinitely many conser-vations laws can be obtained for example from the so-called Lax pair[97] that is an overdetermined linear problem whose compatibilityis ensured if and only if the nonlinear equation is satisfied A bonafide Lax pair can be used to produce the required conservations lawswith the associated generalized symmetries The form of the Lax pairwill be different depending on the kind of equation we are consider-ing It is worth to note that Lax pairs can be used also in ClassicalMechanics [40] and that there exist examples of Lax pairs which arenot bona fide [27 77 104 109 114 115 143 144] the so-called fakeLax pairs These fake Lax pairs cannot provide the infinite sequenceof first integrals so they are useless in proving integrability

In the case of the partial difference equation (11) a Lax Pair isthe overdetermined system of linear equations for a field vector functionΦnm isin RK

Φn+1m = Lnm (unmun+1m)Φnm (15a)

Φnm+1 =Mnm (unmunm+1)Φnm (15b)

11 the meaning of integrability 7

where Lnm and Mnm are Ktimes K matrices depending possibly onsome spectral parameter λ1 The compatibility of the two equations in(15) is

Lnm+1Mnm =Mn+1mLnm (16)

which must be satisfied if and only if we are on the solution of ournonlinear partial difference equation (11) for any value of λ Remarkthat in (16) we have to consider the shifts also in the dependentvariable upon which the matrices Lnm and Mnm depend That is ifLnm = Lnm (unmun+1m) then Lnm+1 = Lnm+1 (unm+1un+1m+1)

and ifMnm =Mnm (unmunm+1) thenMn+1m =Mn+1m (un+1mun+1m+1)This notation will be employed everywhere

The vector function Φnm is usually called for historical reasonsthe wave function This is because in the case of the Korteweg-deVriesequation [92] which is the first example of Lax pair ever producedthe spatial part of the Lax pair is a one-dimensional Schroumldinger equa-tion [97]

Example 111 Let us consider the following couple of equations from[76]

xn+1m+1

xnm+1+ ynm =

xn+1m+1

xn+1m (17a)

xn+1m

xnm+ yn+1m =

xnm+1

xnm (17b)

This is a couple of equations for the unknown functions xnm andynm defined on a Z2 lattice This is not a quad equation in the form(11) but written as a single equation it is defined on a rectangleHowever we can see that the following 2times 2 matrices

Lnm =

(xn+1mxnm λ

λ 0

) (18a)

Mnm =

(xnm+1xnm λ

λ ynm

) (18b)

yield the system (17) as compatibility condition (16) Indeed by com-puting (16) we obtain 0 λ

(xn+1m+1

xnm+1+ ynm minus

xn+1m+1

xn+1m

)minusλ

(xn+1m

xnm+ yn+1m minus

xnm+1

xnm

)0

=

(0 0

0 0

)

(19)

That is the couple of matrices (18) is a Lax pair for the system (17)

1 In the case of 2times 2 matrices it was proved that any Lax pair where a non-trivialspectral parameter cannot be inserted [104] is fake [75 76]


If by integrability we mean the existence of a Lax pair (15) then theprove of the integrability of an equation is a highly non-trivial taskIndeed there is no general algorithm to produce a Lax pair For thisreason in the years many Integrability detectors have been developedIntegrability detectors are algorithmic procedures which are sufficientconditions for integrability or in some cases alternative definition ofintegrability In the next Section we will introduce one of the mostfruitful integrability detector developed in the past years the Consis-tency Around the Cube Its primary importance comes from the factthat it can produces algorithmically a Lax pair In Chapter 2 we willintroduce another integrability detector the Algebraic Entropy whichcan also discriminate between integrable and linearizable equationsAlso Generalized Symmetries which we will introduce in Chapter 3 canbe used as a definition of integrability The key importance of Alge-braic Entropy and Generalized Symmetries is that they avoid the needof prove (or disprove) the existence of a Lax pair at difference fromthe Consistency Around the Cube technique

As a final important remark we want to stress that integrability is aproperty affecting the regularity of the solutions In the case of discreteequations given an initial condition we can compute the whole set ofsolutions with machine precision The question of regularity can seemfutile but in fact what we want is to understand the behavior of thegiven equation without having to compute a full sequence of iterates

It is worth to note that the concept of integrability and the conceptof solvability are very different This can be easily explained on theexample of the famous logistic map

un+1 = 4un (1minus un) (110)

Given the initial condition through u0 = sin2 (c02) the solution ofthis equation is given by [145]

un = sin2(c02

nminus1)

(111)

Equation (111) is sensitive to the initial condition Indeed computingthe derivative with respect to the initial condition we have

dundc0

= 2n sin(c02

nminus1)

cos(c02

nminus1)

(112)

Thus we see that the error grows exponentially with n one of theindicators of ldquochaosrdquo So the system (111) is solvable but not inte-grable

12 the consistency around the cube

Here we discuss the basic properties of the so-called ConsistencyAround the Cube technique following mainly the exposition in [83]

12 the consistency around the cube 9

Let us now to have a quad equation defined on the quad graph de-pending also on some parameters p and q

Q (unmun+1munm+1un+1m+1pq) = 0 (113)

as it is shown in Figure 11

unm

un+1m

unm+1

un+1m+1

p

p

qq

Figure 11 A quad-graph

We say that this equation is Consistent Around the Cube if it canbe extended to an equation defined on a three dimensional lattice ina coherent way To do so first add a third direction unm rarr unmp

and the new three dimensional lattice is depicted in Figure 12

unmp un+1mp

unm+1p

unmp+1

un+1m+1p

un+1mp+1

unm+1p+1 un+1m+1p+1

A

A

B B

C

C

Figure 12 The extension of the 2D lattice to a 3D lattice when the equationon the edges are the same

Then on this new lattice we consider the following equations

A = Q(unmpun+1mpunm+1pun+1m+1ppq

)= 0

(114a)

A = Q(unmp+1un+1mp+1unm+1p+1un+1m+1p+1pq

)= 0

(114b)

B = Q(unmpunm+1punmp+1unm+1p+1q r

)= 0

(114c)

B = Q(un+1mpun+1m+1pun+1mp+1un+1m+1p+1q r

)= 0

(114d)


C = Q(unmpun+1mpunmp+1un+1mp+1 rp

)= 0

(114e)

C = Q(unm+1pun+1m+1punm+1p+1un+1m+1p+1 rp

)= 0

(114f)

There is a natural consistency problem Given the values unmpun+1mp unm+1p unmp+1 as in Figure 12 we can compute val-ues at un+1m+1p un+1mp+1 unm+1p+1 uniquely using the equa-tions on the bottom front and left sides (114a114e114c) respec-tively Then un+1m+1p+1 can be computed from the equations onthe top right and back side equations (114b114f114d) and theyshould all agree This is the consistency condition

The Consistency Around the Cube technique represents a ratherhigh level of integrability Indeed it can be thought as three dimen-sional version of the multi-dimensional consistency introduced in[35] In fact it is a kind of Bianchi identity and was observed to holdfor a sequence of Moutard transforms [124] In its form it was pro-posed as a property of maps in [122]

The main consequence of the Consistency Around the Cube is thatit allows the immediate construction of Lax pair [123] via Baumlcklundtransformations We will now give this construction in the way as pre-sented in [120] The idea is to take the third direction as a spectral di-rection and generate the fieldΦnm in (15) from the shifts in the thirddirection To do so we first solve (114e) with respect to un+1mp+1

and (114c) with respect to unm+1p+1 and relabel the third direc-tion parameter r = λ We can rewrite un+1mp+1 unm+1p+1 andunmp+1 introducing the inhomogeneous projective variables

unmp+1 =fnm

gnm un+1mp+1 =

fn+1m

gn+1m

unm+1p+1 =fnm+1

gnm+1

(115)

Then due to the multi-linearity assumption we can write down (114e)and (114c) as

un+1mp+1 =l11unmp+1 + l12

l21unmp+1 + l22 (116a)

unm+1p+1 =m11unmp+1 +m12

m21unmp+1 +m22 (116b)

where lij = lij (unmun+1m) and mij = mij (unmunm+1) withi j = 1 2 are the coefficients with respect to unmp+1 Introducing(115) into (116) we then obtain the following couple of equations

fn+1m

gn+1m=l11fnm + l12gnm

l21fnm + l22gnm (117a)

fnm+1

gnm+1=m11fnm +m12gnm

m21fnm +m22gnm (117b)

13 the consistency around the cube as a search method 11

Then if we introduce the vector function

Φnm =

(fnm

gnm

) (118)

we can write interpret (117) as matrix relation of the form (15) withthe matrix L and M given by

Lnm = γnm

(l11 l12

l21 l22

) Mnm = γ primenm

(m11 m12

m21 m22

) (119)

where γnm = γnm (unmun+1m) and γ primenm = γ primenm (unmunm+1)

are the so-called separation constants These separation constants arisefrom the fact that the coefficients in (117) are defined in projectivesense ie up a common multiple The determination of the separationconstants is crucial in the application of the method [23] To deter-mine the separation constants it is always possible to impose somerestrictions on Lnm and Mnm eg they can be taken as matriceswith unit determinant (members of the special linear group) How-ever this kind of restrictions usually introduces square roots whichcan make the expressions unmanageable Therefore usually it is eas-ier to consider a single separation constant τnm From (119) writingLnm = γnmLnm and Mnm = γ primenmMnm and taking into accountthe compatibility condition (16) we can write

τnmLnm+1Mnm = Mn+1mLnm (120)

where we now have a unique separation constant

τnm =γnm+1γ

primenm

γnmγprimen+1m

(121)

Note that a priori τnm = τnm (unmun+1munm+1un+1m+1) Thiskind of reasoning is sufficient if we want to prove that (119) gives aLax pair for the quad equation (11) consistent on the cube On theother hand if we want to use the obtained Lax pair for constructingsoliton solutions or for the usual Inverse Scattering machinery weneed to have the precise form of the separation constants Separationconstants are needed also if we want to prove that a Lax pair is fakeas showed in [68] and will be discussed in Subsection 241

At the end we can state the following result given a quad equation(11) compatible around the cube given the matrices Lnm and Mnm con-structed according to (119) we can find a Lax pair together with the sep-aration constant τnm such that the compatibility condition (120) holdsidentically on the solutions of (11)

13 the consistency around the cube as a search method

The Consistency Around the Cube is an algorithmic tool given aquad equation (11) by a few calculations it is possible to tell if it is or


it is not consistent around the cube In the affirmative case it is possi-ble to construct its Lax pair using (119) However it is also possibleto reverse the reasoning and assume to be given a generic quad equa-tion (11) which possesses the Consistency Around the Cube propertyand the derive the form the equation must have This means to usethe Consistency Around the Cube as a search method and in principleobtain equations with the desired properties From the ConsistencyAround the Cube have been obtained two kinds of results In a first in-stance by the papers [2 81 82] were produced autonomous equationsIn a second instance by the papers [3 20 22] and in R Boll PhDThesis [21] a more general situation was studied and non-autonomousequations were produced We will now list the main findings of theseresearches especially we will discuss the generalization introducedin the second phase and following [67] we will give the explicit for-mulaelig for the construction of the lattice equations in Section 14

131 Original attempts

The first attempt of using the Consistency Around the Cube as aclassifying tool for quad equations was made in [2] The authors usedthe following hypothesis

1 The quad equation (113) possess the tetrahedron property un+1m+1p+1

is independent of unmp

2 The quad equation (113) possess the discrete symmetries of thesquare ie it has the following invariance property

Q (unmun+1munm+1un+1m+1pq)

= microQ (unmunm+1un+1mun+1m+1qp)

= micro primeQ (un+1munmun+1m+1unm+1pq)

(122)

where micromicro prime isin plusmn1

The outcome of this classification up to Moumlbius transformationsare the celebrated ABS equations namely two families of autonomousquad equations The first one is the H family with three members

H1 (unm minus un+1m+1)(unm+1 minus un+1m) minusα+β = 0

(123a)

H2 (unm minus un+1m+1)(unm+1 minus un+1m) minusα2 +β2

+ (βminusα)(unm + unm+1 + un+1m + un+1m+1) = 0(123b)

H3 α(unmunm+1 + un+1mun+1m+1)

minusβ(unmun+1m + unm+1un+1m+1) + δ(α2 minusβ2) = 0

(123c)


while the second one is the Q family consisting of four members

Q1 α(unm minus un+1m)(unm+1 minus un+1m+1)

minusβ(unm minus unm+1)(un+1m minus un+1m+1)

+δ2αβ(αminusβ) = 0

(124a)



+αβ(αminusβ)(unm + unm+1 + un+1m + un+1m+1)

minusαβ(αminusβ)(α2 minusαβ+β2) = 0(124b)

Q3 (β2 minusα2)(unmun+1m+1 + unm+1un+1m)

+β(α2 minus 1)(unmunm+1 + un+1mun+1m+1)

minusα(β2 minus 1)(unmun+1m + un+1m+1un+1m+1)

minusδ2(α2 minusβ2)(α2 minus 1)(β2 minus 1)

4αβ= 0

(124c)

Q4 k0unmun+1munm+1un+1m+1

minus k1(unmun+1munm+1 + un+1munm+1un+1m+1

+ unmunm+1un+1m+1 + unmun+1mun+1m+1)

+ k2(unmun+1m+1 + un+1munm+1)

minus k3(unmun+1m + unm+1un+1m+1)

minus k4(unmunm+1 + un+1mun+1m+1)

+ k5(unm + un+1m + unm+1 + un+1m+1) + k6 = 0

(124d)

with

k0 = α+β k1 = αν+βmicro k2 = αν2 +βmicro2

k3 =αβ(α+β)

2(νminus micro)minusαν2 +β(2micro2 minus

g24)

k4 =αβ(α+β)

2(microminus ν)minusβmicro2 +α(2ν2 minus

g24)

k5 =g32k0 +

g24k1 k6 =

g2216k0 + g3k1

(125)

where

α2 = r(micro) β2 = r(ν) r(z) = 4z3 minus g2zminus g3 (126)

Here α and β are the parameters associated to the edges of the cellpresented in Figure 11

The Q4 equation given by (124d) is in the so-called Adlerrsquos formfirst found in [1] This equation possesses two other reparametrizationsone in terms of points on a Weierstraszlig elliptic curve [120] called theNijhoffrsquos form and one in terms of Jacobi elliptic functions [82] calledthe Hietarintarsquos form


In [2] a third family of equations consisting of two members is pre-sented namely theA family However it was proved in the same paperthat such family is included in a sub-family of the Q one through anon-autonomous Moumlbius transformation From the discussion in Sec-tion 14 and in Section 15 it will be clear why we can safely omit suchfamily

After the introduction of the ABS equations J Hietarinta tried toweaken its hypotheses First in [81] he made a new classification withno assumption about the symmetry and the tetrahedron propertyTherein he found the following new equation

J1 unm + e2unm + e1

un+1m+1 + o2un+1m+1 + o1

=un+1m + e2un+1m + o1

unm+1 + o2unm+1 + e1

(127)

where ei and oi are constantsLater in [82] he released just the tetrahedron property maintaining

the symmetries of the square and he found the following equations

J2 unm + un+1m + unm+1 + un+1m+1 = 0 (128a)

J3 unmun+1m+1 + un+1munm+1 = 0 (128b)

J4 unmun+1munm+1 + unmun+1mun+1m+1

+ unmunm+1un+1m+1 + un+1munm+1un+1m+1

+ unm + un+1m + unm+1 + un+1m+1 = 0

(128c)

It is worth to note that all these J equations namely (127) and (128)are linear or linearizable [138] Furthermore it was proved in [69] thatthe J equations (128) are also Darboux integrable shedding light onthe origin of integrability This property will be discussed in Chapter4 and its discovery was a crucial step which led to a better under-standing of the relations between Consistency Around the Cube andlinearizability

132 Bollrsquos classification non-autonomous lattices

New classes of equations where introduced by a generalization of theConsistency Around the Cube technique given in [3] Let us astrayfrom the quad equation as equation embedded on a lattice and justthink it as multi-affine relation between some a priori independentfields x x1 x2 x12 with edge parameters α1 and α2

Q (x x1 x2 x12α1α2) = 0 (129)

The situation is that pictured in Figure 13 where the cell is singleand yet not embedded in any lattice

In [3] the authors considered then a more general perspective in theclassification problem They assumed that on faces of the consistencycube A B C and A B and C are different quad equations of the form(129) Furthermore they made no assumption either of the symmetry


x

x1

x2

x12

α1

α1

α2α2

Figure 13 The purely geometric quad graph not embedded in any lattice

of the square (122) nor of the tetrahedron property They consideredsix-tuples of (a priori different) quad equations assigned to the facesof a 3D cube

A (x x1 x2 x12α1α2) = 0 (130a)

A (x3 x13 x23 x123α1α2) = 0 (130b)

B (x x2 x3 x23α3α2) = 0 (130c)

B (x1 x12 x13 x123α3α2) = 0 (130d)

C (x x1 x3 x13α1α3) = 0 (130e)

C (x2 x12 x23 x123α1α3) = 0 (130f)

see Figure 14 Such a six-tuple is then defined to be 3D consistentif for arbitrary initial data x x1 x2 and x3 the three values for x123(calculated by using A = 0 B = 0 and C = 0) coincide As a resultin [3] the authors obtained the same Q family equations of [2] Inaddition some new quad equations of type H These new equationsturned out to be deformations of those present up above (123)

x x1

x2

x3

x12

x13

x23 x123

A

A

B B

C

C

Figure 14 Equations on a Cube

The new equations were the rhombic H4 equations which possess thesymmetries of the rhombus

Q(x x1 x2 x12α1α2) = microQ(x x2 x1 x12α2α1)

= micro primeQ(x12 x1 x2 xα2α1)(131)


with micromicro prime isin plusmn1 and are given by

rHε1 (xminus x12)(x1 minus x2) minus (α1 minusα2) (1+ εx1x2) (132a)

rHε2 (xminus x12)(x1 minus x2) + (α2 minusα1)(x+ x1 + x2 + x12)

+ ε(α2 minusα1)(2x1 +α1 +α2)(2x2 +α1 +α2)

+ ε(α2 minusα1)3 minusα21 +α

22

(132b)

rHε3 α1(xx1 + x2x12) minusα2(xx2 + x1x12)

+ (α21 minusα22)

(δ+

εx1x2α1α2

) (132c)

Note that as ε rarr 0 these equations reduce to the H family (123)discussed before

As these equations do not possess anymore the symmetry of thesquare (122) they cannot be embedded in a lattice with a fundamen-tal cell of size one but the size of the elementary cell should be biggerequal to two This implies that the corresponding equation on the lat-tice will be a non-autonomous equation [3] The explicit form of thenon-autonomous equation was displayed in [166] We will postponeto Section 14 the discussion on how this reasoning is carried out sincewe will present a more general case

In [20ndash22] Boll starting from [3] classified all the consistent equa-tions on the quad graph possessing the tetrahedron property onlyThe results were summarized by Boll in [21] in a set of theoremsfrom Theorem 39 to Theorem 314 listing all the consistent six-tuplesconfigurations (130) up to Mob8 the group of independent Moumlbiustransformations of the eight fields on the vertexes of the consistencythree dimensional cube see Figure 14 The essential tool used in theclassification where the bi-quadratics ie the expression

hij =partQ

partxk

partQ

partxlminusQ

part2Q

partxk partxl (133)

where the pair k l is the complement of the pair i j in 0 1 2 122A bi-quadratic is called degenerate if it contains linear factors of theform xi minus c with c isin C a constant otherwise a bi-quadratic is callednon-degenerate The three families are classified depending on howmany bi-quadratics are are degenerate

bull Q family all the bi-quadratics are non-degenerate

bull H4 family four bi-quadratics are degenerate

bull H6 family all of the six bi-quadratics are degenerate

Let us notice that the Q family is the same as presented in [2 3] TheH4 equations are divided into two subclasses the rhombic one which

2 Here x0 = x


we discussed above and the trapezoidal one The rhombic symmetry isgiven by (131) whereas the trapezoidal one is given by [3]

Q(x x1 x2 x12α1α2) = Q(x1 x x12 x2α1α2) (134)

The trapezoidal symmetry is an invariance with respect to the axisparallel to (x1 x12) There might be a trapezoidal symmetry also withrespect to the reflection around an axis parallel to (x2 x12) but thiscan be reduced to the previous one by a rotation So there is no needto treat such symmetry but it is sufficient to consider (134)

We remark that a simplest trapezoidal equation appeared withoutpurpose of classification already in [3]

133 The equations on the single cell

In Theorems 39 ndash 314 [21] Boll classified up to the action of thegroup Mob8 every consistent six-tuples of equations with the tetra-hedron property Here we consider all independent quad equationsdefined on a single cell not of type Q (Qε1 Qε2 Qε3 and Q4) or rhom-bic H4 (rHε1 rHε2 and rH

ε3) as these two families have been already

studied extensively [2 3 102 139 166] By independent we mean thatthe equations are defined up to the action of the group Mob4 on thefields rotations translations and inversions of the reference systemBy reference system we mean those two vectors applied on the pointx which define the two oriented directions i and j upon which theelementary square is constructed The vertex of the square lying ondirection i (j) is then indicated by xi (xj) The remaining vertex is thencalled xij In Fig 13 one can see an elementary square where i = 1

and j = 2 or viceversaThe list we present in the following expands the analogous one

given by Theorems 28ndash29 in [21] where the author does not distin-guish between different arrangements of the fields xi over the fourcorners of the elementary square Different choices reflects in differ-ent bi-quadratics patterns and for any system presented in Theorems28ndash29 in [21] it is easy to see that a maximum of three differentchoices may arise up to rotations translations and inversions

The list obtain consists of nine different representatives three ofH4-type and six of H6-type We list them with their quadruples ofdiscriminants as defined by equation (133) and we identify the six-tuple where the equation appears by the theorem number indicatedin [21] in the form 3ab where b is the order of the six-tuple into theTheorem 3a

The independent trapezoidal equations of type H4 are

tHε1 (ε2 ε2 0 0

) Eq B of 3101

(xminus x2) (x3 minus x23) minusα2(1+ ε2x3x23) = 0 (135a)


tHε2 (1 + 4εx 1 + 4εx2 1 1) Eq B of 3102

(xminus x2) (x3 minus x23) +α2(x+ x2 + x3 + x23)

+εα22

(2x3 + 2α3 +α2) (2x23 + 2α3 +α2)

+ (α2 +α3)2 minusα23 +

εα322

= 0

(135b)

tHε3 (x2 minus 4δ2ε2 x22 minus 4δ

2ε2 x23 x223)

Eq B of 3103

e2α2 (xx23 + x2x3) minus (xx3 + x2x23)

minus e2α3(e4α2 minus 1

)(δ2 +

ε2x3x23e4α3+2α2

)= 0

(135c)

The independent equations of type H6 are

D1 (0 0 0 0) Eq A of 3121 and 3131

x+ x1 + x2 + x12 = 0 (136a)

This equation is linear and invariant under any exchange of thefields

1D2 (δ21 (δ1δ2 + δ1 minus 1)

2 1 0)

Eq A of 3122

δ2x+ x1 + (1minus δ1) x2 + x12 (x+ δ1x2) = 0 (136b)

D3 (4x 1 1 1) Eq A of 3123

x+ x1x2 + x1x12 + x2x12 = 0 (136c)

This equation is invariant under the exchange x1 harr x2

1D4 (x2 + 4δ1δ2δ3 x21 x212 x22

) Eq A of 3124

xx12 + x1x2 + δ1x1x12 + δ2x2x12 + δ3 = 0 (136d)

This equation is invariant under the simultaneous exchangesx1 harr x2 and δ1 harr δ2

2D2 (δ21 0 1 (δ1δ2 + δ1 minus 1)

2)

Eq C of 3132

δ2x+(1minus δ1) x3+x13+x1 (x+ δ1x3 minus δ1λ)minusδ1δ2λ = 0 (136e)

3D2 (δ21 0 (δ1δ2 + δ1 minus 1)

2 1)

Eq C of 3133

δ2x+ x3 + (1minus δ1) x13 + x1 (x+ δ1x13 minus δ1λ) minus δ1δ2λ = 0

(136f)

14 construction of the 2d3d lattice in the general case 19

2D4 (x2 + 4δ1δ2δ3 x21 x22 x212

) Eq A of 3135

xx1 + δ2x1x2 + δ1x1x12 + x2x12 + δ3 = 0 (136g)


Let us note that at difference from the rhombic H4 equations (132)which as stated above are ε-deformations of the H equations in theABS classification [2] the trapezoidal H4 equations in the limit εrarr 0

keep their discrete symmetry Such class is then completely new withrespect the ABS classification and the ldquodeformedrdquo and the ldquounde-formedrdquo equations share the same discrete symmetry

Up to now we have written the result of the classification on asingle cell and no dynamical system over the entire lattice yet existsTherefore we pass now to the discussion of the embedding in the 2Dand 3D lattices

14 construction of the 2d3d lattice in the general

case

Let us assume to have a geometric quad equation in the form (129)We need to embed it into into a Z2 lattice with an elementary cell ofsize greater than one To do so we have to impose a lattice structurewhich preserves the properties of the quad equation (129) Following[20] one reflects the square with respect to the normal to its right andto the top and then complete a 2times 2 cell by reflecting again one ofthe obtained equation with respect to the other direction Let us notethat whatsoever side we reflect the result of the last reflection is thesame Such a procedure is graphically described in Figure 15 and atthe level of the quad equation this corresponds to constructing thethree equations obtained from (129) by flipping its arguments

Q = Q(x x1 x2 x12α1α2) = 0 (137a)

|Q = Q(x1 x x12 x2α1α2) = 0 (137b)

Q = Q(x2 x12 x x1α1α2) = 0 (137c)

|Q = Q(x12 x2 x1 xα1α2) = 0 (137d)

By paving the whole Z2 with such equations we get a partial dif-ference equation which we can in principle study with the knownmethods Since a priori Q 6= |Q 6= Q 6= |Q the obtained lattice will bea four stripe lattice ie an extension of the Black and White latticeconsidered for example in [3 85 166]

Let us notice that if the quad-equation Q possess the symmetriesof the square given by (122) one has

Q = |Q = Q = |Q (138)


x x1 x

x2x12

x2

x x1 x

Q |Q

Q |Q

Figure 15 The ldquofour colorsrdquo lattice

implying that the elementary cell is actually of size one and one fallsinto the case of the ABS classification

On the other hand in the case of the rhombic symmetry as given by(131) we can see from the explicit form of the rhombic equationsthemselves (132) that holds the relation

Q = |Q Q = |Q (139)

This means that in the case of the equations with rhombic symmetriesthis construction yields the Black and White lattice considered in [385 166] The geometric picture of this case is given in Figure 16

x x1 x

x2x12

x2

x x1 x

Q |Q

|Q Q

Figure 16 Rhombic Black and White lattice

Lastly in the case of quad equations invariant under trapezoidal sym-metry as given by (134) we have the following equality

Q = |Q Q = |Q (140)

This means that in the case of the equations with trapezoidal symme-tries this construction yields the Black and White lattice consideredas stated in [21] The geometric picture of this case is given in Figure17


x x1 x

x2x12

x2

x x1 x

Q Q

Q Q

Figure 17 Trapezoidal Black and White lattice

Now by paving the whole Z2 with the elementary cell defined inFigure 15 and choosing the origin on the Z2 lattice in the point x weobtain a lattice equation of the following form

Q [u] =

Q(unmun+1munm+1un+1m+1) n = 2km = 2k k isin Z

|Q(unmun+1munm+1un+1m+1) n = 2k+ 1m = 2k k isin Z

Q(unmun+1munm+1un+1m+1) n = 2km = 2k+ 1 k isin Z

|Q(unmun+1munm+1un+1m+1) n = 2k+ 1m = 2k+ 1 k isin Z

(141)

We could have constructed Q [u] starting from any other point in Fig-ure 15 as the origin but such equations would differ from each otheronly by a translation a rotation or a reflection So in the sense of thediscussion made in Subsection 133 they will be equivalent to (141)and we will not discuss them

Example 141 In the case of rHε1 (132a) we have

rHε1 =

(unm minus un+1m+1)(un+1m minus unm+1)

minus(α1 minusα2)(1+ ε2un+1munm+1)

|n|+ |m| = 2k k isin Z


minus(α1 minusα2)(1+ ε2unmun+1m+1)

|n|+ |m| = 2k+ 1 k isin Z

(142)

where we have used the symmetry properties of the equation rHε1

This result coincide with that presented in [166]

We shall now consider quad equations which possess the Consis-tency Around the Cube since we are concerned about integrabilitySo let us consider six-tuples of quad equations (130) assigned to thefaces of a 3D cube as displayed in Figure 14

First let us notice that without loss of generality we can assumethat if Q is the consistent quad equation we are interested in then


we may assume that Q is the bottom equation ie Q = A Indeed ifwe are interested in an equation on the side of the cube of Figure 14and these equations are different from A (once made the appropriatesubstitutions) we may just rotate it and re-label the vertices in anappropriate manner so that our side equation will become the bottomequation In this way following again [20] and taking into accountthe result stated above we may build an embedding in Z3 whosepoints we shall label by triples (nmp) of the consistency cube Tothis end we reflect the consistency cube with respect to the normalof the back and the right side and then complete again with anotherreflection just in the same way we did for the square Using the samenotations as in the planar case we see which are the proper equationswhich must be put on the sides of the ldquomulti-cuberdquo Their form cantherefore be described as in (141) As a result we end up with Figure

xx1

x2

x3

x12

x13

x23x123

x

x

x2

xx1

x3

x3

x23

x3x13

A |A

A |A

A |A

A |A

B B B

B B B

C

C

C

|C

|C

|C

Figure 18 The extension of the consistency cube

18 where the functions appearing on the top and on the bottom canbe defined as in (141)3 while on the sides we shall have

B(x x2 x3 x23) = B(x2 x x23 x3) (143a)

B(x1 x12 x13 x123) = B(x12 x1 x123 x13) (143b)

|C(x x1 x3 x13) = C(x1 x x13 x3) (143c)

|C(x2 x12 x23 x123) = C(x12 x2 x123 x23) (143d)

3 Obviously in the case of A one should traslate every point by one unit in the pdirection


From (143) we obtain the extension to Z3 of (141) We have a newconsistency cube see Figure 18 with equations given by4

˜A [u] =

A(unmp+1un+1mp+1unm+1p+1un+1m+1p+1)

|A(unmp+1un+1mp+1unm+1p+1un+1m+1p+1)


|A(unmp+1un+1mp+1unm+1p+1un+1m+1p+1)(144a)

B [u] =

B(unmpunm+1punmp+1unm+1p+1)




(144b)

˜B [u] =

B(un+1mpun+1m+1pun+1mp+1un+1m+1p+1)



B(un+1mpun+1m+1pun+1mp+1un+1m+1p+1)(144c)

C [u] =

C(unmpun+1mpunmp+1un+1mp+1)

|C(unmpun+1mpunmp+1un+1mp+1)


|C(unmpun+1mpunmp+1un+1mp+1)(144d)

˜C [u] =

C(unm+1pun+1m+1punm+1p+1un+1m+1p+1)

|C(unm+1pun+1m+1punm+1p+1un+1m+1p+1)

C(unmpun+1m+1punm+1p+1un+1m+1p+1)

|C(unm+1pun+1m+1punm+1p+1un+1m+1p+1)(144e)

Formula (144) means that the ldquomulti-cuberdquo of Figure 18 appears asthe usual consistency cube of Figure 12 with the following identifica-tions

A A A ˜A B B B ˜B C C C ˜C (145)

4 For the sake of the simplicity of the presentation we have left implied when n andm are even or odd integers They are recovered by comparing with (141)


Example 142 Let us consider again the equation rHε1 (132a) This

equation comes from the six-tuple [3]

A = (xminus x12)(x1 minus x2) + (α1 minusα2)(1+ εx1x2) (146a)

A = (x3 minus x123)(x13 minus x23) + (α1 minusα2)(1+ εx3x123) (146b)

B = (xminus x23)(x2 minus x3) + (α2 minusα3)(1+ εx2x3) (146c)

B = (x1 minus x123)(x12 minus x13) + (α2 minusα3)(1+ εx1x123) (146d)

C = (xminus x13)(x1 minus x3) + (α1 minusα3)(1+ εx1x3) (146e)

C = (x2 minus x123)(x12 minus x23) + (α1 minusα2)(1+ εx2x123) (146f)

therefore from (144 145) we get the following consistent system onthe ldquomulti-cuberdquo

A =

(unmp minus un+1m+1p)(un+1mp minus unm+1p)

minus(α1 minusα2)(1+ ε2un+1mpunm+1p)



minus(α1 minusα2)(1+ ε2unmpun+1m+1p)

|n|+ |m| = 2k+ 1 k isin Z

(147a)

A =

(unmp+1 minus un+1m+1p+1)(un+1mp+1 minus unm+1p+1)

minus(α1 minusα2)(1+ ε2unmp+1un+1m+1p+1)



minus(α1 minusα2)(1+ ε2un+1mp+1unm+1p+1)

|n|+ |m| = 2k+ 1 k isin Z

(147b)

B =

(unmp minus unm+1p+1)(unm+1p minus unmp+1)

minus(α2 minusα3)(1+ ε2unm+1punmp+1)



minus(α2 minusα3)(1+ ε2unmpunm+1p+1)

|n|+ |m| = 2k+ 1 k isin Z

(147c)

B =

(un+1mp minus unm+1p+1)(un+1m+1p minus un+1mp+1)

minus(α2 minusα3)(1+ ε2un+1mpun+1m+1p+1


(un+1mp minus un+1m+1p+1)(un+1m+1p minus un+1mp+1)

minus(α2 minusα3)(1+ ε2un+1m+1pun+1mp+1)

|n|+ |m| = 2k+ 1 k isin Z

(147d)

C =

(unmp minus un+1mp+1)(un+1mp minus unmp+1)

minus(α1 minusα3)(1+ ε2un+1mpunmp+1)



minus(α1 minusα3)(1+ ε2unmpun+1mp+1)

|n|+ |m| = 2k+ 1 k isin Z

(147e)

C =

(unm+1p minus un+1m+1p+1)(un+1m+1p minus unm+1p+1)

minus(α1 minusα3)(1+ ε2unm+1pun+1m+1p+1)



minus(α1 minusα2)(1+ ε2un+1m+1punm+1p+1)

|n|+ |m| = 2k+ 1 k isin Z

(147f)

The reader can easily check that the equations in (147) possess theConsistency Around the Cube in the form presented in Section 12


Up to now we showed how given a quad equation of the form (129)which possess the property of the Consistency Around the Cube itis possible to embed it into a quad equation in Z2 given by (141)Furthermore we showed that this procedure can be extended alongthe third dimension in such a way that the consistency is preservedThis have been done following [20] and filling the details (which aregoing to be important)

The quad difference equation (141) is not very manageable sincewe have to change equation according to the point of the lattice weare in It will be more efficient to have an expression which ldquoknowsrdquoby itself in which point we are This can obtained by going over tonon-autonomous equations as was done in the BW lattice case [166]

We shall present here briefly how from (141) it is possible to con-struct an equivalent non-autonomous system and moreover how toconstruct the non-autonomous version of CAC (144)

We take an equation Q constructed by a linear combination of theequations (137) with n and m depending coefficients

Q = fnmQ(unmun+1munm+1un+1m+1)+

+ |fnm |Q(unmun+1munm+1un+1m+1)+

+ fnmQ(unmun+1munm+1un+1m+1)+

+ |fnm |Q(unmun+1munm+1un+1m+1)

(148)

We require that it satisfies the following conditions

1 The coefficients are periodic of period 2 in both directions sincein the Z2 embedding the elementary cell is a 2times 2 one

2 The coefficients are such that they produce the right equationin a given lattice point as specified in (141)

Condition 1 implies that any function fnm in (148) ie either fnm

or |fnm or fnm or |fnm solves the two ordinary difference equations

fn+2m minus fnm = 0 fnm+2 minus fnm = 0 (149)

whose solution is

fnm = c0 + c1 (minus1)n + c2 (minus1)

m + c3 (minus1)n+m (150)

with ci constants to be determinedThe condition 2 depends on the choice of the equation in (141) and

will give some ldquoboundary conditionsrdquo for the function f allowing usto fix the coefficients ci For fnm for example we have substitutingthe appropriate lattice points the following conditions

f2k2k = 1 f2k+12k = f2k2k+1 = f2k+12k+1 = 0 (151)


which yield

fnm =1+ (minus1)n + (minus1)m + (minus1)n+m

4 (152a)

In an analogous manner we obtain the form of the other functions in(148)

|fnm =1minus (minus1)n + (minus1)m minus (minus1)n+m

4 (152b)

fnm =1+ (minus1)n minus (minus1)m minus (minus1)n+m

4 (152c)

|fnm =1minus (minus1)n minus (minus1)m + (minus1)n+m

4 (152d)

Then inserting (152) in (148) we obtain a non-autonomous equationwhich corresponds to (141) Note that finally this quad equation is aquad equation in the form (11) with non-autonomous coefficients Wecan say that these kind of equations are weakly non-autonomous sinceas we will see in Chapter 2 and in 4 this kind of non-autonomicitycan be easily removed by introducing more components

If the quad-equation Q possess some discrete symmetries the ex-pression (148) greatly simplify If an equation Q possess the sym-metries of the square we trivially have using (138) Q = Q Thisresult states that an equation with the symmetry (138) is defined ona monochromatic lattice as expected since we are in the case of theABS classification [2] If the equation Q has the symmetries of therhombus namely (139) we get

Q = (fnm + |fnm)Q+ (|fnm + fnm)|Q (153)

Using (152)

fnm + |fnm = F(+)n+m |fnm + fnm = F

(minus)n+m (154)

where

F(plusmn)k =

1plusmn (minus1)k

2 k isin Z (155)

we obtain

Q = F(+)n+mQ+ F

(minus)n+m|Q (156)

This obviosly match with the results in [166]In the case of trapezoidal symmetry (140) one obtains

Q = (fnm + |fnm)Q+ (fnm + |fnm)Q (157)

Therefore using that

fnm + |fnm = F(+)m fnm + |fnm = F

(minus)m (158)

we obtain

Q = F(+)m Q+ F

(minus)m Q (159)


Example 143 As an example of such construction let us consideragain rH

ε1 (132a) Since we are in the rhombic case [166] we use for-

mula (154) and get

rHε1 = (unm minus un+1m+1)(un+1m minus unm+1) minus (α1 minusα2)

+ (α1 minusα2)ε2(F(+)n+m un+1munm+1 + |F

(minus)n+m unmun+1m+1

)= 0

(160)

which corresponds to the case σ = 1 of [166] The discussion of theabsence of the parameter σ introduced in [166] from our theory ispostponed to the end this Section

The consistency of a generic system of quad equations is obtainedby considering the consistency of the tilded equations as displayed in(145) We now construct starting from (145) the non autonomouspartial difference equations in the (nm) variables using the weightsfnm as given in (148) applied to the relevant equations Carryingout such construction we end with the following six-tuple of equa-tions

A(unmpun+1mpunm+1pun+1m+1p) = fnmA+ |fnm|A

+ fnmA+ |fnm|A = 0(161a)A(unmp+1un+1mp+1unm+1p+1un+1m+1p+1) = fnmA+ |fnm|A+

+ fnmA+ |fnm|A = 0(161b)

B(unmpunm+1punmp+1unm+1p+1) = fnmB+ |fnm|B+

+ fnmB+ |fnm|B = 0(161c)B(un+1mpun+1m+1pun+1mp+1un+1m+1p+1) = fnmB+ |fnm|B+

+ fnmB+ |fnm|B = 0(161d)

C(unmpun+1mpunmp+1un+1mp+1) = fnmC+ |fnm|C+

+ fnmC+ |fnm|C = 0(161e)C(unm+1pun+1mpunm+1p+1un+1m+1p+1) = fnmC+ |fnm|C+

+ fnmC+ |fnm|C = 0(161f)

where all the functions on the right hand side of the equality signare evaluated on the point indicated on the left hand side

We note that a Lax pair obtained by making use of equations (161)will be effectively a pair since the couples (B B) and (C C) are relatedby translation so they are just two different solutions of the same


equation Indeed by using the properties of the functions fnm wehave

B = TnB C = TmC (162)

where Tn is the operator of translation in the n direction and Tmthe operator of translation in the m direction This allow us to con-struct Baumlcklund transformations and Lax pair in the way explainedin Section 12 [19 123]

Example 144 As a final example we shall derive the non-autonomousside equations forHε1 and its Lax pair We will then confront the resultwith that obtained in [166] Considering (146 161 162) and usingthe fact that the equation is rhombic (154) we get the following result

A = (unmp minus un+1m+1p)(un+1mp minus unm+1p) minus (α1 minusα2)middot

middot[1+ ε2

(F(+)n+m un+1mpunm+1p + F

(minus)n+m unmpun+1m+1p

)]= 0

(163a)A = (unmp+1 minus un+1m+1p+1)(un+1mp+1 minus unm+1p+1) minus (α1 minusα2)middot

middot[1+ ε2

(F(minus)n+m un+1mp+1unm+1p+1 + F

(+)n+m unmp+1un+1m+1p+1

)]= 0

(163b)

B = (unmp minus unm+1p+1)(unm+1p minus unmp+1) minus (α2 minusα3)middot

middot[1+ ε2

(F(+)n+m unm+1punmp+1 + F

(minus)n+m unmpunm+1p+1

)]= 0

(163c)

C = (unmp minus un+1mp+1)(un+1mp minus unmp+1) minus (α1 minusα3)middot

middot[1+ ε2

(F(+)n+m un+1mpunmp+1 + F

(minus)n+m unmpun+1mp+1

)]= 0

(163d)

From the equations B and C we find the following Lax pair

Lnm =

(unm α1 minusα3 minus unmun+1m

1 minusun+1m

)

+ (α1 minusα3)ε2

(F(+)n+mun+1m 0

0 minusF(minus)n+munm

) (164a)

Mnm =

(unm α2 minusα3 minus unmunm+1

1 minusunm+1

)

+ (α2 minusα3) ε2

(F(+)n+munm+1 0


)

(164b)


Lnm+1 =

(unm+1 α1 minusα3 minus unm+1un+1m+1

1 minusun+1m+1

)

+ (α1 minusα3)ε2

(F(minus)n+mun+1m+1 0

0 minusF(+)n+munm+1

)(164c)

Mn+1m =

(un+1m α2 minusα3 minus un+1mun+1m+1

1 minusun+1m+1

)



0 minusF(+)n+mun+1m

)

(164d)

with the following separation constant (121)

τ =1+ ε2

(F(minus)n+munm + F

(+)n+mun+1m

)1+ ε2

(F(minus)n+munm + F

(+)n+munm+1

) (165)

This is a Lax pair since L = TmL and M = TnM

Remark 141 The Lax pair (164) is gauge equivalent to that obtainedin [166] with gauge

G =

(0 1

minus1 0

) (166)

A calculation similar to that of Example 144 can be done for theother two rhombic equations Up to gauge transformations this cal-culation gives as expected the same Lax pairs as in [166] Indeed thegauge transformations (166) is needed for Hε1 and Hε2 whereas for Hε3we need the gauge

G =

(0 (minus1)n+m

minus(minus1)n+m 0

) (167)

Now notice that at the level of the non-autonomous equations thechoice of origin of Z2 in a point different from x in (141) would haveled to different initial conditions in (151) which ultimately lead tothe following form for the functions f

fσ1σ2nm =

1+ σ1 (minus1)n + σ2 (minus1)

m + σ1σ2 (minus1)n+m

4 (168a)

|fσ1σ2nm =

1minus σ1 (minus1)n + σ2 (minus1)

m minus σ1σ2 (minus1)n+m

4 (168b)

fσ1σ2nm =

1+ σ1 (minus1)n minus σ2 (minus1)


4 (168c)

|fσ1σ2nm =

1minus σ1 (minus1)n minus σ2 (minus1)


4 (168d)


where the two constants σi isin plusmn1 depends on the point chosenIndeed if the point is x we have σ1 = σ2 = 1 whereas if we choose x1we have σ1 = 1 σ2 = minus1 if we choose x2 then σ1 = minus1 and σ2 = 1

and finally if we choose x12 we shall put σ1 = σ2 = minus1 It is easy tosee that in the rhombic and in the trapezoidal case the functions (168)collapse to the σ version of the functions F(plusmn)k as given by (155)

F(plusmnσ)k =

1plusmn σ(minus1)k

2 (169)

The final equation will then depend on σ1 or σ2 only if rhombic ortrapezoidal

It was proved in [166] that the transformations

vnm = un+1m wnm = unm+1 (170)

map a rhombic equation with a certain σ into the same rhombic equa-tion with minusσ In [166] this fact was used to construct a Lax Pair andBaumlcklund transformations An analogous result can be easily provenfor trapezoidal equations using the transformation wnm = unm+1

we can send a trapezoidal equation with a certain σ into the sameequation with minusσ A similar transformation in the n direction wouldjust trivally leave invariant the trapezoidal equation since there is noexplicit dependence on n However in general if an equation doesnot posses discrete symmetries as it is the case for a H6 equationno trasformation like (170) would take the equation into itself withdifferent coefficents We can anyway construct a Lax pair with theprocedure explained above which is then slightly more general thanthe approach based on the transformations (170)

We end this Section by remarking that the choice of the embed-ding is crucial to determine the integrability properties of the quadequations Indeed this was proved in [85] where it was shown thatis possible to produce many Black and White lattices of consistentyet non-integrable equations Here we shall give just a very simpleexample to acquaint the reader about this fact Let us consider againthe (132a) equation Suppose one makes the trivial embedding of suchequation into a lattice given by the identification

xrarr unm x1 rarr un+1m x2 rarr unm+1 x12 rarr un+1m+1

(171)

Then equation (132a) with the identification (171) becomes the fol-lowing lattice equation

(unmminusun+1m+1)(un+1mminusunm+1)minus(α1minusα2)(1+ε2un+1munm+1) = 0

(172)

15 classification tools on the 2D lattice 31

We apply the algebraic entropy test5 to (172) and we find the follow-ing growth in the Nord-East (minus+) direction of the degrees

dminus+ = 1 2 4 9 21 50 120 289 (173)

This sequence has generating function

gminus+ =1minus sminus s2

s3 + s2 minus 3s+ 1(174)

and therefore has a non-zero algebraic entropy given by

ηminus+ = log(1+radic2)

(175)

corresponding to the entropy of a non-integrable lattice equation Formore complex examples the reader may refer to [85]

15 classification tools on the 2D lattice

We have addressed in the previous Section the problem of the con-struction of the lattices out of single cells equations In this Sectionwe present a proof of the fact that the classification carried out at thelevel of single cell is preserved when passing to the lattice

First of all recall that the classification of quad equations presentedin Section 132 has been carried out up to a Moumlbius transformationsin each vertex

M (x x1 x2 x12) 7rarr(a0x+ b0c0x+ d0

a1x1 + b1c1x1 + d1

a2x2 + b2c2x2 + d2

a12x12 + b12c12x12 + d12

)

(176)

As in the usual Moumlbius transformation we have here (aibi cidi) isinCP4 V (aidi minus bici) PGL(2 C) with i = 0 1 2 12 ie each setof parameters is defined up to to a multiplication by a number [36]Obviously as the usual Moumlbius transformations these transformationswill form a group under composition and we shall call such groupMob4 [14]

On the other hand when dealing with equations defined on the lat-tice we have to follow the prescription of Section 14 and use the rep-resentation given by (148) ie we will have non-autonomous latticeequations In this Section we prove a result which extends the groupMob4 to the level of the transformations of the non autonomous lat-tice equations and shows that the classification made at the levelof single cell equation is preserved up to the action of this groupfor the equation on the 2D lattice We will call such group the non-

autonomous lifting of Mob4 and denote it by Mob4

The group Mob4

5 For more details on degree of growth algebraic entropy and related subjects seeChapter 2 and references therein


Q

Q M(Q)

M(Q) equiv M(Q)M isin (Mob)

4

M isin (Mob)4

11


is the non-autonomous counterpart of the group Mob4 on the four-color lattice and its existence shows that the classification at the levelof single cell is preserved passing to the full lattice Our proof is con-structive we will first construct a candidate transformation and then wewill prove that this is object we are looking for What we will do is tofill the entries in the commutative diagram given by Figure (19) weprove that the result that we get by acting with a group of transfor-

mations Mob4

is the same of that we obtain if we first transform asingle cell equation using M isin Mob4 and then we construct the non-autonomous quad equation with the prescription of Section 14 orviceversa if we first construct the non-autonomous equation and thenwe transform it using the ldquonon-autonomous liftingrdquo of M M

Let us first construct a transformation which will be the candidateof the non-autonomous lifting of M isinMob4 Given M isinMob4 usingthe same ideas of Section 14 we can construct the following transfor-mation

Mnm isin Mob4 unm 7rarr fnm

a0unm + b0c0unm + d0

+ |fnma1unm + b1c1unm + d1

+fnma2unm + b2c2unm + d2


(177)

where the functions fnm are defined by (152)Equation (177) gives us a mappingΦ between the group Mob4 and

a set of non-autonomous transformations of the field unm Moreoverthere is a one to one correspondence between an element M isin Mob4


(176) and one Mnm isin Mob4

(177) This correspondence is given by

Φ

ax+ b

cx+ da1x1 + b1c1x1 + d1a2x2 + b2c2x2 + d2a12x12 + b12c12x12 + d12

T

7minusrarrfnm

aunm + b

cunm + d+ |fnm


+ fnma2unm + b2c2unm + d2


(178a)

and its inverse is given by

Φminus1

fnmα(0)unm +β(0)

γ(0)unm + δ(0)+ |fnm

α(1)unm +β(1)

γ(1)unm + δ(1)

+ fnmα(2)unm +β(2)


α(3)unm +β(3)

γ(3)unm + δ(3)

7minusrarr

α(0)x+β(0)

γ(0)x+ δ(0)

α(1)x1 +β(1)

γ(1)x1 + δ(1)

α(2)x2 +β(2)

γ(2)x2 + δ(2)

α(3)x12 +β(3)

γ(3)x12 + δ(3)

T

(178b)

We have now to prove that Mob4

is a group and that the mapping(178) is actually a group homomorphism

Using the computational rules given in Table 11 we obtain a new

representation of the generic element of the group Mob4

Mnm (unm) =

(a0fnm + a1|fnm + a2fnm + a12|fnm

)unm

+ b0fnm + b1|fnm + b2fnm + b12|fnm

(

c0fnm + c1|fnm + c2fnm + c12|fnm)unm

+ d0fnm + d1|fnm + d2fnm + d12|fnm

(179)

This form of the elements of Mob4

shows immediately that Mob4

isa subset of the general non-autonomous Moumlbius transformation

Wnm unm 7rarranmunm + bnm

cnmunm + dnm (180)

From the general rule of composition of two Moumlbius transformations(180) we get

W1nm

(W0nm (unm)

)=

(a0nma1nm + b1nmc

0nm)unm + a1nmb

0nm + b1nmd

0nm

(a0nmc1nm + c0nmd

1nm)unm + b0nmc

1nm + d0nmd

1nm

(181)


Its inverse is given by

Wminus1nm(unm) =

dnmunm minus bnm

minuscnmunm + anm (182)

Using the computational rules given in Table 11 the formula forthe composition (181) and the representation (179) we find that the

composition of two elements M(1)nmM(2)

nm isin Mob4

with parameters(a

(i)j b(i)j c(i)j d(i)j ) j = 0 1 2 12 and i = 1 2 gives

M2nm

(M1nm (unm)

)=

(a(2)0 fnm + a

(2)1 |fnm + a

(2)2 fnm + a

(2)12 |fnm

)unm

+ b(2)0 fnm + b

(2)1 |fnm + b

(2)2 fnm + b

(2)12 |fnm

(c(2)0 fnm + c

(2)1 |fnm + c

(2)2 fnm + c

(2)12 |fnm

)unm

+ d(2)0 fnm + d

(2)1 |fnm + d

(2)2 fnm + d

(2)12 |fnm

(183)

where the new coefficients (a(2)j b(2)j c(2)j d(2)j ) with j = 0 1 2 12 aregiven by

a(2)0 = a

(0)0 a

(1)0 + b

(1)0 c

(0)0 a

(2)1 = a

(0)1 a

(1)1 + b

(1)1 c

(0)1

a(2)2 = a

(0)2 a

(1)2 + b

(1)2 c

(0)2 a

(2)12 = a

(0)12 a

(1)12 + b

(1)12 c

(0)12

b(2)0 = a

(1)0 b

(0)0 + b

(1)0 d

(0)0 b

(2)1 = a

(1)1 b

(0)1 + b

(1)1 d

(0)1

b(2)2 = a

(1)2 b

(0)2 + b

(1)2 d

(0)2 b

(2)12 = a

(1)12 b

(0)12 + b

(1)12 d

(0)12

c(2)0 = a

(0)0 c

(1)0 + c

(0)0 d

(1)0 c

(2)1 = a

(0)1 c

(1)1 + c

(0)1 d

(1)1

c(2)2 = a

(0)2 c

(1)2 + c

(0)2 d

(1)2 c

(2)12 = a

(0)12 c

(1)12 + c

(0)12 d

(1)12

d(0)0 = b

(0)0 c

(1)0 + d

(0)0 d

(1)0 d

(0)1 = b

(0)1 c

(1)1 + d

(0)1 d

(1)1

d(0)2 = b

(0)2 c

(1)2 + d

(0)2 d

(1)2 d

(0)12 = b

(0)12 c

(1)12 + d

(0)12 d

(1)12

(184)

The inverse of (179) is given by

Mminus1nm(unm) =

(fnmd+ |fnmd1 + fnmd2 + |fnmd12)unm

minus (fnmb0 + |fnmb1 + fnmb2 + |fnmb12)

minus (fnmc0 + |fnmc1 + fnmc2 + |fnmc12)unm

+ (fnma0 + |fnma1 + fnma2 + |fnma12)

(185)

Thus one has proved that Mob4

is a group and in fact it is a subgroupof the general autonomous Moumlbius transformations (180)

Let us show that the maps Φ and Φminus1 given in (178) representa group homomorphism They preserve the identity and from theformula of composition of Moumlbius transformations in Mob4 (183) wederive the required result


middot fnm |fnm fnm |fnm

fnm fnm 0 0 0

|fnm 0 |fnm 0 0

fnm 0 0 fnm 0

|fnm 0 0 0 |fnm

Table 11 Multiplication rules for the functions fnm as given by (152)

Let us now check if the diagram of Figure 19 is satisfied Let usconsider (176) and (177) and a general multi-linear quad equation

Qgen (x x1 x2 x12) = A01212xx1x2x12

+B012xx1x2 +B0112xx1x12

+B0212xx2x12 +B1212x1x2x12

+C01xx1 +C02xx2 +C012xx12

+C12x1x2 +C112x1x12 +C212x2x12

+D0x+D1x1 +D2x2 +D12x12 +K

(186)

where A01212 Bijk Cij Di and K with i jk isin 0 1 2 12 arearbitrary complex constants The proof that Q(M(x x1 x2 x12)) =

Q(Mnm(unm)) where M isin Mob4 and Mnm = Φ(M) isin Mob4

isa computationally very heavy calculation due to the high number ofparameters involved (twelve in the transformation6 and fifteen in theequation (186) twenty-seven paramerters in total) and to the fact thatrational functions are involved To simplify the problem it is sufficientto recall that every Moumlbius

m(z) =az+ b

cz+ d z isin C (187)

transformation can be obtained as a superposition of a translation

Ta(z) = z+ a (188a)

dilatation

Da(z) = az (188b)

and inversion

I(z) =1

z (188c)

6 Using that Moumlbius transformations are projectively defined one can lower the num-ber of parameters from sixteen to twelve but this implies to impose that some pa-rameters are non-zero and thus all various different possibilities must be taken intoaccount


ie

m(z) =(Tac D(bcminusad)c2 I Tdc

)(z) (189)

As the group Mob4 is obtained by four copies of the Moumlbius groupeach acting on a different variable we can decompose each entryM isin Mob4 as in (189) Therefore we need to check 34 = 81 transfor-mations depending at most on four parametersWe can automatizesuch proof by making a specific computer program to generate allthe possible fundamental transformations in Mob4 and then checkthem one by one reducing the computational effort To this end weused the Computer Algebra System (CAS) sympy [150] This ends theproof of the preservation of the classification when we pass from thecell equation to the lattice equation The details of this calculation arecontained in Appendix A

16 independent equations on the 2D-lattice

In this last Section of this Chapter we present the explict form onthe 2D-lattice of the all the independent systems listed in Subsection133 The construction of these systems is carried out according tothe prescription given in Section 14 In light of the preceding Sec-tion independence is now understood to be up to the action of the

group Mob4

and rotations translations and inversions of the refer-ence system The transformations of the reference system are actingon the discrete indices rather than on the reference frame For sakeof compactness and as the equations are on the lattice we shall omitthe hats on the Moumlbius transformations when clear Obviously as inthe single lattice case we have nine non-autonomous representativesthree of trapezoidal H4-type and six of H6-type These equations arenon-autonomous with two-periodic coefficients which can be givenin terms of the functions

F(plusmn)k =

1plusmn (minus1)k

2 (190)

The H4 type equations are

tH1 (unm minus un+1m) middot (unm+1 minus un+1m+1)minus

minusα2ε2(F(+)m unm+1un+1m+1 + F

(minus)m unmun+1m

)minusα2 = 0

(191a)

16 independent equations on the 2D-lattice 37

tH2 (unm minus un+1m) (unm+1 minus un+1m+1)

+α2 (unm + un+1m + unm+1 + un+1m+1)

+εα22

(2F

(+)m unm+1 + 2α3 +α2

)(2F

(+)m un+1m+1 + 2α3 +α2

)+εα22

(2F

(minus)m unm + 2α3 +α2

)(2F

(minus)m un+1m + 2α3 +α2

)+ (α3 +α2)

2 minusα23 minus 2εα2α3 (α3 +α2) = 0

(191b)

tH3 α2 (unmun+1m+1 + un+1munm+1)

minus (unmunm+1 + un+1mun+1m+1) minusα3(α22 minus 1

)δ2+

minusε2(α22 minus 1)

α3α2

(F(+)m unm+1un+1m+1 + F

(minus)m unmun+1m

)= 0

(191c)

These equations arise from the B equation of the cases 3101 3102and 3103 in [21] respectively We remark that we have passed to therational form of the tHε3 (191c) by making the identification

e2α2 rarr α2 e2α3 rarr α3 (192)


D1 unm + un+1m + unm+1 + un+1m+1 = 0 (193a)

1D2 (F(minus)n+m minus δ1F

(+)n F

(minus)m + δ2F

(+)n F

(+)m

)unm

+(F(+)n+m minus δ1F

(minus)n F

(minus)m + δ2F

(minus)n F

(+)m

)un+1m+


(+)n F

(+)m + δ2F

(+)n F

(minus)m

)unm+1

+(F(minus)n+m minus δ1F

(minus)n F

(+)m + δ2F

(minus)n F

(minus)m

)un+1m+1+

+ δ1

(F(minus)m unmun+1m + F

(+)m unm+1un+1m+1

)+ F

(+)n+munmun+1m+1 + F

(minus)n+mun+1munm+1 = 0

(193b)

2D2 (F(minus)m minus δ1F

(+)n F

(minus)m + δ2F

(+)n F

(+)m minus δ1λF

(minus)n F

(+)m

)unm

+(F(minus)m minus δ1F

(minus)n F

(minus)m + δ2F

(minus)n F

(+)m minus δ1λF

(+)n F

(+)m

)un+1m

+(F(+)m minus δ1F

(+)n F

(+)m + δ2F

(+)n F

(minus)m minus δ1λF

(minus)n F

(minus)m

)unm+1

+(F(+)m minus δ1F

(minus)n F

(+)m + δ2F

(minus)n F


(+)n F

(minus)m

)un+1m+1

+ δ1

(F(minus)n+munmun+1m+1 + F

(+)n+mun+1munm+1

)+ F

(+)m unmun+1m + F

(minus)m unm+1un+1m+1 minus δ1δ2λ = 0

(193c)



(minus)n F

(minus)m + δ2F

(+)n F

(+)m minus δ1λF

(minus)n F

(+)m

)unm


(+)n F

(minus)m + δ2F

(minus)n F

(+)m minus δ1λF

(+)n F

(+)m

)un+1m

+(F(+)m minus δ1F

(minus)n F

(+)m + δ2F

(+)n F


(minus)n F

(minus)m

)unm+1

+(F(+)m minus δ1F

(+)n F

(+)m + δ2F

(minus)n F


(+)n F

(minus)m

)un+1m+1

+ δ1

(F(minus)n unmunm+1 + F

(+)n un+1mun+1m+1

)+ F

(minus)m unm+1un+1m+1 + F

(+)m unmun+1m minus δ1δ2λ = 0

(193d)

D3 F(+)n F

(+)m unm + F

(minus)n F

(+)m un+1m + F

(+)n F

(minus)m unm+1

+ F(minus)n F

(minus)m un+1m+1 + F

(minus)m unmun+1m

+ F(minus)n unmunm+1 + F

(minus)n+munmun+1m+1+

+ F(+)n+mun+1munm+1 + F

(+)n un+1mun+1m+1

+ F(+)m unm+1un+1m+1 = 0

(193e)

1D4 δ1


(+)n un+1mun+1m+1

)+

+ δ2


(+)m unm+1un+1m+1

)+

+ unmun+1m+1 + un+1munm+1 + δ3 = 0(193f)

2D4 δ1


(+)n un+1mun+1m+1

)+

+ δ2


(+)n+mun+1munm+1

)+

+ unmun+1m + unm+1un+1m+1 + δ3 = 0(193g)

The equations 1D2 1D4 2D4 and D3 arise from the A equation inthe cases 3122 3123 3124 and 3135 respectively The equations2D2 and 3D2 arise from the C equation in the cases 3132 and 3133respectively

To write down the explicit form of the equations in (193) we used(154 158) and the following identities

fnm = F(+)n F

(+)m |fnm = F

(minus)n F

(+)m

fnm = F(+)n F

(minus)m |fnm = F

(minus)n F

(minus)m

(194)

As mentioned in Section 14 if we apply this procedure to an equationof rhombic type we get a result consistent with [166]

We note that the explicit lattice form of these equations was firstgiven in [67] being absent in the works of Boll [20ndash22]

For the rest of the thesis we will study the integrability propertiesof the trapezoidal H4 and H6 equations in their lattice avatars given


by (191) and (193) The main result ie the proof of the fact thatthey are linearizable equation will be first stated by considering theheuristic test of the Algebraic Entropy in the Chapter 2 whereas inChapter 4 we will give a formal proof of the linearizability based onthe concept of Darboux integrability

2A L G E B R A I C E N T R O P Y A N D D I R E C TL I N E A R I Z AT I O N

In this Chapter we will focus on the integrability detector called Al-gebraic Entropy The first part of this Chapter consisting of Section21 and of Section 22 is mainly a review of known concepts andit is essentially based on the exposition given in [63] In Section 21we will expand the intuitive idea given in the example of the logis-tic map (110) we presented in 11 in order to explain the differencebetween solvability and integrability Motivated by the discussion ofthis example we will introduce the precise definition of Algebraic En-tropy for difference equation differential equations and quad equa-tions We note that the definition of Algebraic Entropy for quad equa-tions is taken from [67] which is a generalization of that given in [155]We will address the problem from the theoretical point of view dis-cussing mainly the setting Then we will derive the main properties ofAlgebraic Entropy and discuss briefly the geometric meaning of theAlgebraic Entropy In Section 22 we will discuss the computationaltools that allow us to extract the value of the Algebraic Entropy fromfinite sequences The implementation of such algorithm in python in-troduced in [64] is discussed in Appendix B The second part of theChapter consisting of Section 23 and of Section 24 is instead an orig-inal part based on [67 68] In Section 23 is presented and discussedthe Algebraic Entropy test applied to the trapezoidal H4 equations(191) and to the H6 equations (193) This result shows that the trape-zoidal H4 equations (191) and to the H6 equations (193) should belinearizable equations To support the statement made in Section 23 inSection 24 we present some examples of explicit linearization In par-ticular we will treat the tHε1 equation (191a) and the 1D2 equationIn the case of the tHε1 equation (191a) we also present a proof of thefact that its Lax pair obtained with procedure presented in Chapter 1

is fake according to the definition of [77 78] Hence the Lax obtainedfrom the Consistency Around the Cube is useless in discussing theintegrability of the tHε1 equation (191a) This proof was first given in[68]

21 definition and basic properties

As we saw in the example of the logistic equation (110) the notionof chaos is related with the exponential growth of the solution withrespect to the initial condition Equation (110) possess an explicit so-lution therefore it is easy to understand a posteriori its properties Let

41

42 algebraic entropy and direct linearization

us now consider the general logistic map with an arbitrary parameterr isin R

un+1 = run (1minus un) (21)

We wish to understand how this equation evolves therefore we fix aninitial condition u0 and we compute the iterates

u1 = minusru20 + ru0 (22a)

u2 = minusr3u40 + 2r3u30 minus

(r3 + r2

)u20 + r

2u0 (22b)


(6r7 minus 2r6

)u60 + lot (22c)

u4 = minusr15u160 + 8r15u150 minus (28r15 + 4r14)u140 + lot (22d)

where by lot we mean terms with lower powers in u0 Then it isclear that there is a regularity in how the degree of the polynomialun in u0 dn grows Indeed one can guess that dn = 2n and checkthis guess for successive iterations This is a clear indication that oursystem is chaotic as it was in the solvable case r = 4 This should beno surprise since the generic r case is even more general

This simple example shows how examining the iterates of a recur-rence relation can be a good way to extract information about integra-bility even if we cannot solve the equation explicitly However in morecomplicated examples it is usually impossible to calculate explicitlythese iterates by hand or even with any state-of-the-art formal calcu-lus software simply because the expressions one should manipulateare rational fractions of increasing degree of the various initial condi-tions The complexity and size of the calculation make it impossibleto calculate the iterates

It was nevertheless observed that ldquointegrablerdquo maps are not as com-plex as generic ones This was done primarily experimentally by anaccumulation of examples and later by the elaboration of the con-cept of Algebraic Entropy for difference equations [18 33 39 141 153]In [152 155] the method was developed in the case of quad equa-tions and then used as a classifying tool [84] Finally in [32] the sameconcept was introduced for differential-difference equation and later[157] to the very similar case of differential-delay equations In ourreview we will mainly follow [60]

The basic idea given a rational map which can be an ordinary dif-ference equation a differential difference equation or even a partialdifference equation is to examine the growth of the degree of its iter-ates as we did for (21) and extract a canonical quantity which is anindex of complexity of the map This will be the algebraic entropy (orits avatar the dynamical degree) We will now introduce formally thissubject by considering before the case of the ordinary difference equa-tions and of the differential-difference equations The case of the ordinarydifference equations here is mainly intended to be preparatory for thecase of the partial difference equations but in Chapter 3 we will also see

21 definition and basic properties 43

the application of the Algebraic Entropy to the differential-differencecase

211 Algebraic entropy for ordinary difference equations and differential-difference equations

In this Section we will consider ordinary difference equations ie expres-sions of the form

un+l = fn (un+lminus1 un+l prime) l prime ln isin Z l prime lt l (23)

where the unknown function un is a function of the discrete integervariable n isin Z The difference equation (23) is said to be of orderlminus l prime if partfnpartun+l prime 6= 0 identically We will also discuss differential-difference equations ie equations where the unknown is a functionun (t) of two variables one continuous t isin R and one discrete n isin ZA differential-difference equations is then given by an expression ofthe form

u(p)n = fn

(un+l un+l prime u primen u(pminus1)n

) (24)

where we used the prime notation for derivatives u(p)n = dpudtp Furthermore l prime ln isin Z with the conditions l prime lt l and finally p isinNp gt 1 otherwise we fall back into the case of ordinary differenceequations A differential-differential equation of the form (24) is saidto have differential order p and difference order lminus l prime provided that

partfn

partun+l

partfn

partun+l prime6= 0 (25)

Typical example of such equations are the so called Volterra-like equa-tions

u primen = fn (un+1unun+1) n isin Z (26)

or Toda-like equations

u primeprimen = fn(u primenun+1unun+1

) n isin Z (27)

Volterra-like equations are first order differential-difference equationsin the differential order while Toda-like equations are second orderdifferential-difference equations in the differential order Both theseclasses are second order in the difference order provided that thefunctions in the right hand side of (26) and (27) satisfy the condition(25)

For theoretical purposes it is usual to consider maps in a projectivespace rather than in the affine one One then transforms its recurrence


relation into a polynomial map in the homogeneous coordinates ofthe proper projective space over some closed field1

xi 7rarr ϕi (xk) (28)

with xi xk isin IN where IN is the space of the initial conditions Therecurrence is then obtained by iterating the polynomial map ϕi It isusual to ask that the map ϕi be bi-rational ie it possesses an inversewhich is again a rational map In the affine setting the bi-rationality ofthe map means that we can solve the relation also for the lower-indexvariable eg un+l prime in (23) and (24) This fact is of crucial theoreticalimportance as it will be explained at the end of this Subsection

Example 211 To clarify the concept of how we can translate a recur-rence relation into a projective map we present a very simple examplewhich however gives the flavor of the method Let us a consider themost general bilinear first order recurrence relation of the form (23)

un+1 =aun + b

cun + d (29)

where we are assuming ab cd constants such that ad minus bc 6= 0Since (29) is a first order difference equations we must convert it intoa bi-rational map of P1 into itself To this end we can introduce theprojective coordinates

un =x

y un+1 =

X

y (210)

From (29) we obtain then

X = yax+ by

cx+ dy (211)

Since the recurrence relation in the projective plane is given by

ϕ (xy) isin P1 7rarr (Xy) isin P1 (212)

using (211) we obtain

(Xy) =(yax+ by

cx+ dyy) (ax+ by cx+ dy) (213)

Therefore we have converted the recurrence relation (29) into thefollowing map of P1 into itself

ϕ (xy) 7rarr (ax+ by cx+ dy) (214)

This is a linear map The inverse is the clearly given by

ϕminus1 (xy) 7rarr (dxminus byminuscx+ ay) (215)

and can also be obtained directly from (29) using the substitution(210) and then solving with respect to x

1 The reader can think this field to be the complex one C but we will see in Subsection22 that in practice finite fields can be useful


Now to proceed further we need to specify the space of the initialconditions IN The space of the initial condition depends on whichtype of recurrence relation we are considering Let us enumerate thevarious cases

1 If we are dealing with a lminus l prime order difference equation in theform (23) the initial conditions will be just the starting lminus l prime-tuple

IN = ul prime ul prime+1 ulminus2ulminus1 (216)

To obtain the map we just need to pass to homogeneous coor-dinates in (23) and in (216) Note that the logistic map we con-sidered above (21) is a polynomial map but it is not bi-rationalsince its inverse is algebraic

2 If we are dealing with a differential-difference equation of thediscrete l minus l prime-th order and of the p-th continuous order thespace of initial conditions is infinite dimensional Indeed in thecase the order of the equation is lminus l prime we need the initial valueof lminus l prime-tuple as a function of the parameter t but also the valueof all its derivatives

IN =u(j)l prime (t)u

(j)l prime+1(t) u

(j)lminus2(t)u

(j)lminus1(t)

jisinN0

(217)

where by u(j)i (t) we mean the jndashderivative of ui(t) with respectto t We need all the derivatives of ui (t) and not just the firstp because at every iteration the order of the equation is raisedby p To obtain the map one just need to pass to homogeneouscoordinates in the equation and in (217)

For both kind of equations we are in the position to define theconcept of Algebraic Entropy Indeed if we factors out any commonpolynomial factors we can say that the degree with respect to theinitial conditions is well defined in a given system of coordinatesalthough it is not invariant with respect to changes of coordinatesWe can therefore form the sequence of degrees of the iterates of themap ϕ and call it dk = degϕk

1 1︸︷︷︸lminusl prime

d1d2d3d4d5 dk (218)

The degree of the bi-rational projective map ϕ have to be understoodas the maximum of the total polynomial degree in the initial conditionsIN of the entries of ϕ The same definition in the affine case justtranslates to the maximum between the degree of the numerator and of thedenominator of the kth iterate in terms of the affine initial conditionsDegrees in the projective and in the affine setting can be different but


the global behavior will be the same due to the properties of homog-enization and de-homogenization which is an invertible procedureThen the entropy of such sequence is defined to be

η = limkrarrinfin 1k logdk (219)

Such limit exists since from the elementary property of any pair ofbi-rational maps ϕ and ψ

deg (ψ ϕ) 6 degψdegϕ (220)

Furthermore the inequality (220) proves that there is an upper boundfor the Algebraic Entropy

η 6 degϕ (221)

An equation whose Algebraic Entropy is equal to degϕ is said to satu-rate the bound Indeed from (220) applied toϕk we obtain deg

(ϕk)6

kdegϕ from which (221) follows using the definition (219) We seethen that if η = 0 we must have

dk sim kν with ν isinN0 as krarrinfin (222)

We will then have the following classification of equations accordingto their Algebraic Entropy [84]

linear growth The equation is linearizable

polynomial growth The equation is integrable

exponential growth The equation is chaotic

Furthermore it is easy to see that the Algebraic Entropy is a bi-rational invariant of such kind of maps Indeed if we suppose that wehave two bi-rationally equivalent maps ϕ and ψ then there exists a bi-rational map χ such that

ψ = χminus1 ϕ χ (223)

implying

degψk 6MdegϕK (224)

where M = degχdeg(χminus1

)isin N Since we can obtain an analogous

equation as ϕ = χ ψ χminus1 we conclude that ηψ = ηϕTo clarify the theory we presented we now give three very simple

examples of calculation of the Algebraic Entropy of difference equa-tions2

2 We will consider the degrees always computed in the affine setting


Example 212 (Heacutenon map [79]) In this example we consider the socalled Heacutenon map of the plane [79] It relates the iterates of a twocomponent vector (xnyn) via the recurrence3

xn+1 = 1minusαx2n + yn (225a)

yn+1 = βxn (225b)

It can be written as a second order difference equation

un+1 = 1minusαu2n +βunminus1 (226)

Computing the degrees of the iterates for any non-zero value of thecoefficients α and β we get

1 1 2 4 8 16 32 64 128 256 512 (227)

The Heacutenon map is of degree 2 The sequence it is not only boundedby 2k but it saturates its bound The sequence is exactly fitted by 2k

and its entropy will be log 2

minus15 minus1 minus05 0 05 1 15

minus04

minus02

0

02

04

x

y

Figure 21 The Heacutenon map in the plane (xy) with a = 14 and b = 03 andthe initial conditions (x0y0) = (06 02)

The trajectory of the map (225) in the plane are displayed in Figure21 which clearly shows the ldquochaoticityrdquo of the system

3 In the original work Heacutenon used the particular choice of parameters α = 14 andβ = 03


Example 213 (Non saturating exponential difference equation) Con-sider the second order nonlinear difference equation

un+1 = αununminus1 +βun + γunminus1 (228)

where α β and γ are real constants Computing the degrees of theiterates for any non-zero choice of the parameters α β and γ we get

1 1 2 3 5 8 13 21 34 55 89 144 233 377 (229)

Also this map is of degree 2 but now its growth is different Wesee that from the second iterate there is a drop in the degree sodk lt 2k asymptotically The reader may recognize in this sequencethe Fibonacci sequence The Fibonacci sequence is known to solve thesecond order linear difference equation

dk+1 = dk + dkminus1 (230)

This means that we have asymptotically

dk sim

(1+radic5

2

)k(231)

and the algebraic entropy of the recurrence relation (228) will be

η = log

(1+radic5

2

) (232)

ie the logarithm of the Golden Ratio

Example 214 (Hirota-Kimura-Yahagi equation [86]) Consider the non-linear second order difference equation

un+1unminus1 = u2n +β2 (233)

This equation possess the first integral [86]

K (ununminus1) =2ununminus1

u2n + u2nminus1 +β2

(234)

ie

K (un+1un) minusK (ununminus1) = 0 (235)

along the solutions of (233) First integrals are a constraint to themotion of a system so in this case we expect a great drop in thedegrees Indeed we have for every value of the parameter β

1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 (236)

The degrees appear to grow linearly dk = 2k and the discrepancyfrom the saturation start from the third iterate


(1)

(2) (3)(4)

Figure 22 Regular and non-regular staircases

212 Algebraic entropy for quad equations

Now we introduce the concept of Algebraic Entropy for quad equa-tions which was introduced in [152] We will follow mainly the ex-position presented in [155] with some generalizations which weregiven in [67] These generalization where introduced explicitly withthe scope of calculating the Algebraic Entropy of the trapezoidal H4

(191) and H6 equations (193) due to the peculiar behavior of theseequations

In the case of quad equations the situation is more complicatedthan in the case of one-dimensional equations First of all in the one-dimensional case we have to worry only about the evolution in twoopposite directions This was the meaning of the bi-rationality con-dition as explained in the preceding Subsection In the two dimen-sional case we have to worry about four possible directions of evolu-tion corresponding to the four ways we can solve the quad equationIn general initial conditions can be given along straight lines in thefour direction However usually is preferred to give initial conditionson staircase configurations Examples of staircase-like arrangementsof initial values are displayed in Figure 22 The evolution from anystaircase-like arrangement of initial values is possible in the quadri-lateral lattice This in principle does not rules out configurations inwhich are present hook-like configurations (see (4) in Figure 22)These kind of configurations will require compatibility conditions onthe initial data since they give more than one way to calculate thesame value for the dependent variable which is not ensured to bethe equal The staircases need to go from (n = minusinfinm = minusinfin) to(n = infinm = infin) or from (n = minusinfinm = +infin) to (n = infinm = minusinfin)


because the space of initial conditions is infinite We will restrict our-selves to regular diagonals which are staircases with steps of constanthorizontal length and constant vertical height Figure 22 shows fourstaircase-like configurations The ones labeled (1) and (2) are regulardiagonals The one labeled (3) would be acceptable but we will notconsider such configurations Line (4) is excluded since it may leadto incompatibilities

Given a line of initial conditions it is possible to calculate the val-ues unm all over the two-dimensional lattice We have a well definedevolution since we restrict ourselves to regular diagonals Moreoverand this is a crucial point if we want to evaluate the transformation for-mula for a finite number of iterations we only need a regular diagonal ofinitial conditions with finite extent

For any positive integerN and each pair of relative integers [λ1 λ2]we denote by ∆(N)

[λ1λ2] a regular diagonal consisting of N steps each

having horizontal size l1 = |λ1| height l2 = |λ2| and going in thedirection of positive (resp negative) nk if λk gt 0 (resp λk lt 0) fork = 1 2 For examples see Figure 23

∆(3)[11]

∆(2)[minus13] ∆

(4)[minus3minus2]

∆(3)[2minus1

Figure 23 Varius kinds of restricted initial conditions

Suppose we fix the initial conditions on ∆(N)[λ1λ2]

We may calculateu over a rectangle of size (Nl1+ 1)times (Nl2+ 1) The diagonal cuts therectangle in two halves One of them uses all initial values and wewill calculate the evolution only on that part See Figure 24

Figure 24 The range covered by the initial conditions ∆(3)[minus21]


We are now in position to calculate ldquoiteratesrdquo of the evolutionChoose some restricted diagonal ∆(N)

[λ1λ2] The total number of ini-

tial points is q = N(l1 + l2) + 1 For such restricted initial data thenatural space where the evolution acts is the projective space Pq ofdimension q We may calculate the iterates and fill Figure 24 consid-ering the q initial values as inhomogeneous coordinates of Pq Eval-uating the degrees of the successive iterates we will produce doublesequences of degrees d(l)k The simplest possible choice is to applythis construction to the restricted diagonals ∆(N)

[plusmn1plusmn1] which we will

denote ∆(N)++ ∆(N)

+minus ∆(N)minus+ and ∆(N)

minusminus We will call them fundamental di-agonals (the upper index (N) is omitted for infinite lines) The fourprincipal diagonals are shown in Figure 25

∆minusminus

∆+minus ∆++

∆minus+

Figure 25 The four principal diagonals

Assuming for example that we are given the ∆+minus principal diago-nal the pattern of degrees is of the form

1 d(Nminus1)1 d

(Nminus2)2 d

(2)Nminus1 d

(1)N

1 1 d(Nminus2)1 d

(3)2 d

(1)Nminus1

1 1 d(3)1 d

(2)2

1 1 d(2)1 d

(1)2

1 1 d(1)1

1 1

(237)

Therefore to each choice of indices [plusmn1plusmn1] we associate a double se-quence of degrees d(l)kplusmnplusmn

We shall call the sequence

1d(1)1plusmnplusmnd(1)2plusmnplusmnd(1)3plusmnplusmnd(1)4plusmnplusmn (238)


the principal sequence of growth A sequence as

1d(i)1plusmnplusmnd(i)2plusmnplusmnd(i)3plusmnplusmnd(i)4plusmnplusmn (239)

with i = 2will be a secondary sequence of growth for i = 3 a third andso on The fundamental entropies of the lattice equation are given by

η(i)plusmnplusmn = lim

krarrinfin 1k logd(i)kplusmnplusmn (240)

The existence of this limit can be proved in an analogous ways as forone dimensional systems [18] We note that a priori the degrees alongthe diagonals of a quad equation do not need to be equal howeverin most cases they are In the cases in which the degrees along thediagonals are not equal it is important to isolate repeating patterns in(237) ie if there exists a positive integer κ isinN such that

d(i+κ)kplusmnplusmn = d

(i)kplusmnplusmn i = 1 κminus 1 (241)

If we can find such κ we can describe the rate of growth of the quadequation through a finite number of sequences and define its Alge-braic Entropy of the quad equation as

ηTOTplusmnplusmn = max

i=1κminus1η(i)plusmnplusmn (242)

The definition given in [155] corresponds to (242) with κ = 1 ie thedegrees are assumed a priori equal For the moment the only knownexamples of equations with κ gt 1 are the trapezoidal H4 equations(191) and the H6 equations (193) The discussion of the AlgebraicEntropy of these equation is postponed to Section 23

213 Geometric meaning of the Algebraic Entropy

One may wonder about the origin of the drop of the degrees whichis observed in integrable and linearizable equations It is actually geo-metrically very simple and comes from the singularity structure Wewill discuss this problem in the case of difference equations since theother cases are essentially the same but they are less intuitive Werecall that to a difference equation (23) of order n = lminus l prime we can as-sociate a map projective map ϕ Pn rarr Pn A point of homogeneouscoordinates [x] = [x0 x1 xn] is singular if all the homogeneouscoordinates of the image by ϕ vanish The set of these points is thusgiven by n + 1 homogeneous equations This set has co-dimensionat least 2 it will be points in P2 complex curves and points in P3and so on One important point is that as soon as the map is non-linear and this is the case we will be mainly interested in there arealways singular points Without singular points the drop of degreesjust cannot happen The vanishing of all homogeneous coordinates


means that there is no image point in Pn The mere vanishing of afew but not all coordinates means that the image ldquogoes to infinityrdquobut this is harmless contrary to what happens in affine space Thisis what projective space has been invented for to cope with pointsat infinity which are not to be forgotten when one consider algebraicvarieties and rational maps Moreover using projective space overclosed fields simplifies a lot the counting of intersection points byBezout theorem The maps we consider are almost invertible Theyare diffeomorphisms on a Zariski open set ie they are invertible ev-erywhere except on an algebraic variety Suppose the map ϕ and itsinverse ψ = ϕminus1 are written in terms of homogeneous coordinatesThe composed maps ϕ ψ and respectively ψ ϕ are then just multi-plication of all coordinates by some polynomial κϕ and respectivelyκψ

ϕ ψ ([x]) = κϕ ([x]) Id ([x]) ψ ϕ ([x]) = κψ ([x]) Id ([x]) (243)

The map ϕ is clearly not invertible on the image of the variety ofequation κϕ ([x]) What may happen is that further action of ϕ onthese points leads to images in the singular set of ϕ This meansthat κϕ ([x]) (or a piece of it if it is decomposable) has to factorizefrom all the components of the iterated map This is the origin ofthe drop of the degree This is the link between singularity (in theprojective sense) and the degree sequence A graphical explanationof this procedure is illustrated schematically in Figure 26 Figure 26can be explained as follows the equation of the surface Σ is κ = 0

and the factor κ appears anew in ϕ ϕ(4) The fifth iterate ϕ(5) isthen regular of κ = 0

Σ

Γ

Π Π prime

Γ prime

Σ prime

Figure 26 A possible blow-down blow-up scheme in P3

This is a link to the method of singularity confinement [61 89 137]If for all components of the variety κϕ ([x]) = 0 one encounters sin-gular points of ϕ in such a way that some finite order iterate of ϕdefine non ambiguously a proper image in Pn we have ldquosingularityconfinementrdquo


The relation of the notion of Algebraic Entropy with the structureof the singularities of maps and lattice equations have been used alsoas a computational tool Examples of such approach are given in [34142 151] and more recently in [158]

22 computational tools

Now that we have introduced the definition of Algebraic Entropy anddiscussed the origin of the dropping of the degrees it is time to turnto the computational tools which help us to calculate it

First of all it is clear from our first example with the logistic mapthat the complexity of the calculations grow more and more iterateswe compute A good way to reduce the computational complexity ofthe problem is to consider a particular set of initial condition given bystraight lines in the appropriate projective space Pq This correspondto the following choice of inhomogeneous coordinates

ui =αit+βiα0t+β0

ui isin IN (244)

where αi and βi are the constant parameters describing the straightline and t is the ldquotimerdquo parameter ie the curve parametrization Inthe case differential-difference equation we will assume that the pa-rameter t is the same as the evolution parameter of the problem

A useful simplification consists in using only integer numbers inthe computations Moreover we want to avoid accidental cancellationsie factorizations due numerical substitution Using generic integersto this end is not recommended since due to prime number factoriza-tion they can introduce such kinds of cancellations Eg let us assumethat we are given the rational function

R =α1t+β1α0t+β0

P(t)

Q(t) (245)

where P and Q are polynomials in t of degree p and q respectivelyand with no common factors and αi βi with i = 0 1 are real orcomplex numbers Therefore the degree of the numerator of R is p+ 1and the degree of its denominator is q+ 1 For generic values of theconstants αi and βi with i = 0 1 obviously there is no factorization Ifhowever to evaluate the polynomial R numerically we choose mayberandomly the αi and the βi with i = 0 1 as integer it may happenthat

α1 = λα0 β1 = λβ0 or α0 = λα1 β0 = λβ1 λ isin Z (246)

due to integer number factorization Then in this situation we have in(245)

R = λP(t)

Q(t) or R =

1

λ

P(t)

Q(t) (247)

22 computational tools 55

so in both cases the degrees of numerator and denominator drop byone This is the mechanism of an accidental cancellation So since wewant to avoid accidental cancellation as in equation (247) the bestway is to consider only prime numbers If we had chosen the αi andthe βi with i = 0 1 as prime numbers the situation displayed in equa-tion (246) couldnrsquot have happened This means that we must choosethe αiβi in (244) and the eventual parameters appearing in the equa-tion as prime numbers A final simplification which can speed up thecalculations is to consider the factorization of the iterates on a finitefield Kp with p prime This is particularly useful since otherwise weneed to perform factorization over the integer domain and the mostcommon algorithms for factorization over integer domain actually atfirst compute factorization over finite fields [159] Therefore using thefactorization over finite fields we can speed up the evaluation of theiterates

With these choices we should be able to avoid eventual accidentalcancellations and therefore to have a bona fide sequence of degreesSince the result is experimental it is still better to do it more thanonce with different initial data to be sure of the result In AppendixB a python implementation of these ideas is given including alsoother useful tools for the post evolution analysis Let us just mentionthe fact that the prime number p is chosen adaptively as the firstprime after the square plus one of the biggest prime in αi βi and theequation parameters This program is also presented in [64]

Now let us assume that we have computed our iterations and weare given the finite sequence

d0d1 dN (248)

To get the information on the asymptotic behavior of the sequence(248) we calculate its generating function [95] ie a function g =

g (s) such that its Taylor series coincides with the elements of theseries To look for such functions it is important to use the minimumnumber of dk possible It is reasonable to suppose that such generatingfunction is rational even if it is known that it is not alway the case[15] Such generating function can therefore be calculated by usingPadeacute approximants [17 135]

In the Padeacute approximant we represent a function as the ratio of twopolynomials ie as a rational function an idea which dates back tothe work of Frobenius [45] Let us assume that we are given a functionf = f (s) analytic in some domain D sub C containing s = 04 Thereforethe function f can be represented as power series centered at s = 0

f (s) =

infinsumk=0

dksk (249)

4 This is always possible to be done up to a translation


To find the Padeacute approximant of order [L M] for the function f

means to find a rational function

[L M] (s) =P[LM] (s)

Q[LM] (s)=a0 + a1s+ aLs

L

1+ b1s+ bMsM (250)

such that its Taylor series centered at s = 0 coincides with (249) asmuch as possible Notice that the choice of b0 = 1 into the denomina-tor (ie in the polynomial Q[LM] (s)) is purely conventional since theratio of the two polynomial P[LM] (s) and Q[LM] (s) is defined up toa common factor A Padeacute approximant of order [L M] has a prioriL+ 1 independent coefficients in the numerator and M independentcoefficients in denominator ie L +M + 1 independent coefficientsThis means that given a full Taylor series (249) and a Padeacute approx-imant of order [L M] we will have a precision of order sL+M+1

f (s) = [L M] (s) + O(sL+M+1

) (251)

In the same way as a polynomial is its own Taylor series a ratio-nal function is its own Padeacute approximant for the correct choice of Land M This is why the best approach in the search for rational gen-erating functions is the use of Padeacute approximants By computing asufficiently high number of terms if the generating function is ratio-nal we will eventually find it

Without going into the details of such beautiful theory we note thatthe simplest way to compute the Padeacute approximant of order [L M]

(250) is from Linear Algebra by computing two determinants

Q[LM] (s) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

dLminusM+1 dLminusM+2 dL dL+1

dLminusM+2 dLminusM+3 dL+1 dL+2

dLminus1 dL dL+Mminus2 dL+Mminus1

dL dL+1 dL+Mminus1 dL+M

sM sMminus1 s 1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(252a)

P[LM] (z) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

dLminusM+1 dLminusM+2 dL+1


dLminus1 dL dL+Mminus1

dL dL+1 dL+MLminusMsumk=0

dksM+k

LminusM+1sumk=0

dksM+kminus1

Lsumk=0

dksk

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(252b)


The Padeacute approximant of order [L M] is then given by [88]

[L M] (s) =P[LM] (s)

Q[LM] (s) (253)

The interested reader can consult the reference [17] for a completeexposition

Let us suppose now we have obtained a generating function usinga subset of the elements in (248) Such generating function is a pre-dictive tool since we can confront the next terms in (248) with thesuccessive terms of the Taylor expansion of the generating functionIf they agree we are almost surely on the right way This is the reasonwhy it is important to find the generating function with the minimumnumber of dk possible If we used all the terms in the sequence (248)then we need to compute more iterates in order to have a predictiveresult Notice that since the sequence (248) is a series of degrees ieof positive integers if the Taylor series of the generating functionsgive raise to rational numbers it can be immediately discarded asnon bona fide generated function A reasonable strategy for findingrational generating functions with Padeacute approximants is to use Padeacuteapproximants of equal order [j j] which as stated above need 2j+ 1point to be determined This kinds of computational issues are dis-cussed practically in Appendix B following [64] Let us remark thatthe Taylor coefficients of a rational generating function satisfy a finiteorder linear recurrence relation [37]

Once we have a generating function we need to calculate the asymp-totic behavior of the coefficients of its Taylor series To do this we willuse the inverse Z-transform [31 37 90] Indeed let f(τ) be a functionexpressible as Laurent series of negative powers of its argument

f(τ) =

infinsumk=0

dkτminusk (254)

The inverse Z-transform of f which we denote by Zminus1 is then definedto be [90]

Zminus1 [f(τ)]k =1

2πi

intC

f(τ)τkminus1dτ k isinN (255)

where C is a simple closed path outside of which f(τ) is analytic Iff is a rational function C can be taken as a circle of radius R in thecomplex τ plane enclosing all the singularities of f(τ) By the residuetheorem [28 162] this implies as the rational functions have only afinite numbers of poles τj isin C j isin 1 P

Zminus1 [f(τ)] =

Psumj=1

Resτ=τjf(τ)τkminus1

(256)


Given the generating function g which is expressed as the Taylorseries

g(s) =

infinsumk=0

dksk (257)

ie a series of positive powers of s we have by the definition (255)

dk = Zminus1[g(τminus1)

]k

(258)

Since we suppose our generating function to be rational g(τminus1

)is

again a rational function and we can easily compute it using theresidue approach (256)

Formula (256) will be valid asymptotically and for rational f (τ)we can estimate for which k it will be valid Assume that f(τ) =

P(τ)Q(τ) with P Q isin C[τ] ie polynomials in τ Indeed if τ = 0 is aroot of Q of order k0 we must distinguish the cases when k gt k0 + 1and k 6 k0 + 1 For k 6 k0 + 1 we will have a pole in τ = 0 of orderk0 + 1minus k So the general formula (256) will be valid only for n gtk0 + 1 In the case of rational generating functions g(s) = P(s)Q(s)with PQ isin C[s] we will introduce a spurious τ = 0 singularity ing(τminus1

)if we will have degP gt degQ

From the generating function we can get the Algebraic Entropy de-fined by formula (219) Recalling the notion of radius of convergenceR of a power series [162]

Rminus1 = limkrarrinfin |dk|

1k (259)

and the continuity of the logarithm function we have from (219) thatthe Algebraic Entropy can be always given by the logarithm of theinverse of the smallest root of the denominator of g

η = min sisinC |Q(s)=0

log |s|minus1 (260)

To conclude this Subsection we examine the growths of Examples212 213 and 214 using the generating functions in order to have amore rigorous proof of their growth

Example 221 (Growth of the Heacutenon map) The first and very triv-ial example is to consider the growth of the Heacutenon map (225) (orequation (226)) as given by (227) We see that computing the Padeacuteapproximant with the first three points ie [1 1](s) we obtain

[1 1](s) =1minus s

1minus 2s (261)

This first Padeacute approximant is already predictive since

[1 1](s) = 1+ s+ 2s2 + 4s3 + 8s4 + 16s5 + 32s5 + O(s6)

(262)

23 algebraic entropy test for (1 91 1 93) 59

So we can conclude g (s) = [1 1](s) This obviously gives the growthas dk = 2kminus1 for every k gt 15 The only pole of g(s) is then s0 = 12

which according to (260) yields η = log 2

Example 222 (Growth of the map (228)) We see that in this case thePadeacute approximant [1 1](s) gives exactly the same result as (261)therefore it does not describes the series The next one [2 2](s) in-stead gives

[2 2] (s) =1

1minus sminus s2 (263)

which is predictive We conclude that in this case g (s) = [2 2](s)This gives the growth

dk =5+ 3

radic5

10

(1+radic5

2

)k+5minus 3

radic5

10

(1minusradic5

2

)k (264)

which as we said before has the asymptotic behavior (231) Thepoles of (263) are then

splusmn =minus1plusmn

radic5

2(265)

and as |s+| gt |sminus| the algebraic entropy is given by (232)

Example 223 (Growth of the Hirota-Kimura-Yahagi equation) Wenow consider the very slow growth of equation (233) given by (236)Again the [1 1](s) is the same as in (261) and therefore does notdescribes (236) as well as [2 2](s) On the other hand [3 3](s) gives

[3 3](s) =s3 + s2 minus s+ 1

(1minus s)2(266)

and again this approximant is predictive So we conclude that g (s) =[3 3](s) and that dk = 2kminus 2 Since the only pole lays on the unitcircle we have that the entropy is zero

23 algebraic entropy test for (1 91 1 93)

Before considering the Algebraic Entropy of the equations presentedin Section 16 we discuss briefly the Algebraic Entropy for the rhom-bic H4 equations 132 which are well known to be integrable [166]Running the program ae2dpy from [64] which is contained in Ap-pendix B on these equations one finds that they possess only a prin-cipal sequence with the following isotropic sequence of degrees

1 2 4 7 11 16 22 29 37 (267)

5 We note that in terms of the iterations the first 1 in (227) have to be interpreted asdminus1 but this is just matter of notation


Equation Growth direction

minus+ ++ +minus minusminus

tHε1 L1L2 L1L2 L3L4 L3L4



D1 L0L0 L0L0 L0L0 L0L0

1D2 L9L10 L9L10 L9L10 L9L10

2D2 L14L15 L14L15 L15L14 L15L14

3D2 L16L17 L16L17 L17L16 L17L16D3 L11L12 L11L12 L11L12 L11L12

1D4 L13L8 L13L8 L13L8 L13L8

2D4 L18L8 L18L8 L19L20 L19L20

Table 21 Sequences of growth for the trapezoidalH4 andH6 equations Thefirst one is the principal sequence while the second the secondaryAll sequences Lj j = 0 middot middot middot 20 are presented in Table 23

To this sequence corresponds the generating function

g =s2 minus s+ 1

(1minus s)3 (268)

which gives through the application of the Z-transform (255) theasymptotic fit of the degrees

dk =k(k+ 1)

2+ 1 (269)

Since the growth is quadratic η = 0 or it can be seen directly from(267) using (260) This result is a confirmation by the Algebraic En-tropy approach of the integrability of the rhombic H4 equations Letus note that the growth sequences (267) are the same in the Rhom-bic H4 equations also when ε = 0 ie if we are in the case of the Hequations of the ABS classification (123)

For the trapezoidal H4 and H6 equations the situation is more com-plicated Indeed those equations have in every direction two differentsequence of growth the principal and the secondary one as the co-efficients of the evolution matrix (237) are two-periodic The mostsurprising feature is however that all the sequences of growth are lin-ear This means that all such equations are not only integrable due tothe fact that they possess the Consistency Around the Cube propertybut also linearizable

Instead of presenting here the full evolution matrices (237) whichwould be very lengthily and obscure we present two tables with the


relevant properties The interested reader will find the full matricesin Appendix C In Table 21 we present a summary of the sequencesof growth of both trapezoidal H4and H6 equations The explicit se-quence of the degrees of growth with generating functions asymp-totic fit of the degrees of growth and entropy is given in Table 23Following the convention in [84] we labeled the different sequencesby Lj with j isin 1 2 20

Observing Table 21 and Table 23 we may notice the followingfacts

bull The trapezoidal H4 equations are not isotropic the sequences inthe (minus+) and (++) directions are different from those in the(+minus) and (minusminus) directions These results reflect the symmetryof the equations

bull The H6 equations except from 2D4 which has the same be-haviour as the trapezoidal H4 are isotropic Equations 2D2 and3D2 exchange the principal and the secondary sequences fromthe (minus+) (++) directions and the (+minus) (minusminus) directions

bull All growths except L0 L3 L4 L7 L8 L12 and L17 exhibit ahighly oscillatory behaviour They have generating functions ofthe form

g(s) =P(s)

(sminus 1)2(s+ 1)2 (270)

with the polynomial P(s) isin Z[s] We may write

g(s) = P0(s) +P1(s)

(sminus 1)2(s+ 1)2 (271)

with P0(s) isin Z[s] of degree less than P and P1(s) = αs3+βs2+γs+ δ Expanding the second term in (271) in partial fractionswe obtain

g(s) = P0(s) +1

4

[β+ δminusαminus γ

(s+ 1)2+α+ γ+β+ δ

(sminus 1)2

+2α+βminus δ

sminus 1+2αminusβ+ δ

s+ 1

]

(272)

Expanding the term in square parentheses in Taylor series wefind that

g(s) = P0(s) +

infinsumk=0

[A0 +A1(minus1)

k +A2k+A3(minus1)kk]sk

(273)

with Ai = Ai(αβγ δ) constants This means the dk = A0 +

A1(minus1)k + A2k + A3(minus1)

kk for k gt degP0(s) and therefore


it asymptotically solves a fourth order difference equation Asfar as we know even if some example of behaviour containingterms like (minus1)k are known [84] this is the first time that weobserve patterns with oscillations given by k (minus1)k

We remark that the usage of the Algebraic Entropy as integrabilityindicator is actually justified by the existence of of finite order recur-rence relations between the degrees dk Indeed the existence of suchrecurrence relations means that from a local property (the sequenceof degrees) we may infer a global one (chaoticityintegrabilitylin-earizaribilty) [158]

We finally note that at the moment no other quad equation isknown to possess an analogous growth property as the trapezoidalH4 and H6 equations Furthermore is also unknown if more compli-cated behavior with higher order periodicities are possible We notethat the existence of two sequence of growth is coherent with the factthat the trapezoidal H4 and H6 equations have two-periodic coeffi-cients So we can conjecture that the existence of multiples growthpatterns is linked with the periodicity of the coefficients In literatureequations with higher periodicity have been introduced eg in [53]but are still unstudied from the point of view of Algebraic Entropy

23

al

ge

br

aic

en

tr

op

yt

es

tf

or

(1

91

19

3)6

3

Table 22 Sequences of growth generating functions analytic expression of the degrees and entropy for the trapezoidal H4and H6equations

Name Degrees Generating function Degree fit Entropy

dk g(s) dk η

L0 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1minus s1 0

L1 1 2 2 5 3 8 4 11 5 14 6 17 7 s3 + 2s+ 1

(sminus 1)2(s+ 1)2(minus1)k

4(minus2k+ 1) + k+

3

40

L2 1 2 4 3 7 4 10 5 13 6 16 7 19 minuss3 + 2s2 + 2s+ 1


4(2kminus 1) + k+

5

40

L3 1 2 2 3 3 4 4 5 5 6 6 7 7 minuss2 + s+ 1

s3 minus s2 minus s+ 1minus(minus1)k

4+k

2+5

40

L4 1 2 4 5 7 8 10 11 13 14 16 17 19 s2 + s+ 1

s3 minus s2 minus s+ 1

(minus1)k

4+3k

2+3

40

L5 1 2 4 6 11 10 19 14 27 18 35 22 43 s6 + 4s4 + 2s3 + 2s2 + 2s+ 1


(kminus

5

2

)+ 3kminus

5

20

L6 1 2 4 7 8 15 12 23 16 31 20 39 24 3s5 + s4 + 3s3 + 2s2 + 2s+ 1


(minusk+

5

2

)+ 3kminus

5

20

L7 1 2 4 7 11 15 19 23 27 31 35 39 43 s4 + s3 + s2 + 1

(sminus 1)24kminus 5 0

Continued on next page

64

al

ge

br

aic

en

tr

op

ya

nd

dir

ec

tl

in

ea

riz

at

io

n

Table 22 ndash Continued from previous page


dk g(s) dk η

L8 1 2 4 6 8 10 12 14 16 18 20 22 24 s2 + 1

(sminus 1)22k 0

L9 1 2 2 5 3 8 4 11 5 14 6 17 7 s3 + 2s+ 1

(sminus 1)2 (s+ 1)2(minus1)k

4(minus2k+ 1) + k+

3

40

L10 1 2 3 5 5 8 7 11 9 14 11 17 13 s3 + s2 + 2s+ 1


4(minusk+ 1) +

5k

4+3

40

L11 1 2 4 5 10 8 16 11 22 14 28 17 34 3s4 + s3 + 2s2 + 2s+ 1


4(3kminus 5) +

9k

4minus3

40

L12 1 2 4 5 7 8 10 11 13 14 16 17 19 s2 + s+ 1

(sminus 1)2 (s+ 1)

(minus1)k

4+3k

2+3

40

L13 1 2 4 6 11 10 18 14 25 18 32 22 39 4s4 + 2s3 + 2s2 + 2s+ 1

(sminus 1)2 (s+ 1)23 (minus1)k

4(kminus 2) +

11k

4minus3

20

L14 1 2 3 3 6 4 9 5 12 6 15 7 18 s4 minus s3 + s2 + 2s+ 1


4(2kminus 3) + k+

3

40

L15 1 1 3 3 6 5 9 7 12 9 15 11 18 s4 + s3 + s2 + s+ 1

(sminus 1)2 (s+ 1)2k

4

((minus1)k + 5

)0


23

al

ge

br

aic

en

tr

op

yt

es

tf

or

(1

91

19

3)6

5



dk g(s) dk η

L16 1 2 3 2 5 2 7 2 9 2 11 2 13 minus2s3 + s2 + 2s+ 1


2(kminus 1) +

k

2+3

20

L17 1 1 3 3 5 5 7 7 9 9 11 11 13 s2 + 1

(sminus 1)2 (s+ 1)

(minus1)k

2+ k+

1

20

L18 1 2 4 5 11 9 19 13 27 17 35 21 43 s6 + s5 + 4s4 + s3 + 2s2 + 2s+ 1

(sminus 1)2 (s+ 1)2(minus1)k (kminus 2) + 3kminus 3 0

L19 1 2 4 6 11 10 19 14 27 18 35 22 43 s6 + 4s4 + 2s3 + 2s2 + 2s+ 1


(kminus

5

2

)+ 3kminus

5

20

L20 1 1 3 2 6 3 9 4 12 5 15 6 18 s4 + s2 + s+ 1


4(2kminus 1) + k+

1

40


24 examples of direct linearization

In this Section we considers in detail the tHε1 (191a) and 1D2 (193b)equations and shows the explicit form of the quadruple of matricescoming from the Consistency Around the Cube the non-autonomousequations which give the consistency on Z3 and the effective Lax pairFinally we confirm the predictions of the algebraic entropy analysisshowing how they can be explicitly linearized looking directly at theequation

241 The tHε1 equation

To construct the Lax Pair for (191a) we have to deal with Case 3101in [21] The sextuple of equations we consider is

A = α2 (xminus x1) (x2 minus x12) minusα1 (xminus x2) (x1 minus x12)

+ ε2α1α2 (α1 minusα2)

(274a)

B = (xminus x2) (x3 minus x23) minusα2(1+ ε2x3x23

)= 0 (274b)

C = (xminus x1) (x3 minus x13) minusα1(1+ ε2x3x13

)= 0 (274c)

A = α2 (x13 minus x3) (x123 minus x23) minusα1 (x13 minus x123) (x3 minus x123) (274d)

B = (x1 minus x12) (x13 minus x123) minusα2(1+ ε2x13x123

)= 0 (274e)

C = (x2 minus x12) (x23 minus x123) minusα1(1+ ε2x23x123

)= 0 (274f)

In this sextuple (191a) originates from the B equationWe now make the following identifications

A xrarr upn x1 rarr up+1n x2 rarr upn+1 x12 rarr up+1n+1

B xrarr unm x2 rarr un+1m x3 rarr unm+1 x23 rarr un+1m+1

C xrarr upm x1 rarr up+1m x3 rarr upm+1 x13 rarr up+1m+1

(275)

so that in any equation we can suppress the dependence on the ap-propriate parametric variables On Z3 we get the following triplet ofequations

A = α2(upn minus up+1n

) (upn+1 minus up+1n+1

)minusα1

(upn minus upn+1

) (up+1n minus up+1n+1

)+ ε2α1α2 (α1 minusα2) F

(+)m

(276a)

B = (unm minus un+1m) (unm+1 minus un+1m+1) minusα2


(minus)m unmun+1m

)

(276b)

C =(upm minus up+1m

) (upm+1 minus up+1m+1

)minusα1

minusα1ε2(F(+)m upm+1up+1m+1 + F

(minus)m upmup+1m

)

(276c)

24 examples of direct linearization 67

Then with the method in Section 14 we find the following Lax Pair6

Lnm =

(unm+1 minusunmunm+1 +α1

1 minusunm

)

minus ε2α1

(minusF

(minus)m unm 0

0 F(+)m unm+1

)

(277a)

Mnm = α1 (unm minus un+1m)

(1 0

0 1

)

+α2

(un+1m minusunmun+1m

1 minusunm

)

minus ε2α1α2 (α1 minusα2) F(+)m

(0 1

0 0

)

(277b)

and the separation constant (121) is given by

τnm = α21+ ε2

(F(minus)m u2nm + F

(+)m u2nm+1

)(unm minus unm+1)

2 + ε2α22F(+)m

(278)

With a highly non-trivial computation it is however possible to showthat the separation constant (278) arise from the following rationalseparation constants

γnm =1

1+ iε(F(minus)m unm minus F

(+)m unm+1

) (279a)

γ primenm =1

unm minus un+1m + iα2F(minus)m

(279b)

Therefore the Lax pair for the tHε1 equation (191a) is completely

characterized from the Consistency Around the CubeLet us now turn to the linearization procedure In (191a) we must

have α2 6= 0 otherwise the equation becomes

(unm minus un+1m) (unm+1 minus un+1m+1) = 0 (280)

which is factorizable and therefore degenerate Let us define two newfields for the even and odd values of m in order to write the tH

ε1

equation (191a) as an autonomous system

un2k = wnk un2k+1 = znk (281)

Then the tHε1 equation (191a) becomes the following system of cou-pled autonomous difference equations

(wnk minuswn+1k) (znk minus zn+1k) = ε2α2znkzn+1k +α2

(282a)

6 In this case for simplicity of calculations the rocircle of the L and M matrices are in-verted


(wnk+1 minuswn+1k+1) (znk minus zn+1k) = ε2α2znkzn+1k +α2

(282b)

Subtracting (282b) to (282a) we obtain

(wnk minuswn+1k minuswnk+1 +wn+1k+1) (znk minus zn+1k) = 0 (283)

At this point the solution of the system bifurcates depending onwhich factor in (283) we choose to annihilate

case 1 zn k = fk Let us assume that znk = fk where fk is anarbitrary function of its argument Then the second factor in (283) issatisfied and from (282a) or (282b) we have that ε 6= 0 and solvingfor fk one gets

fk = plusmn iε

(284)

This is essentially a degenerate case since this solution holds for ev-ery value of wnk

case 2 wn k = gn + hk Let us assume that wnk = gn + hkwhere gn and hk are arbitrary functions of their argument and thatznk 6= fk Hence (282) reduces to the equation

ε2znkzn+1k + κn (znk minus zn+1k) + 1 = 0 (285)

where κn = (gn+1 minus gn)α2 Depending on the value of ε we candistinguish two different cases

case 2 1 ε = 0 If ε = 0 (285) implies that κn 6= 0 so that we have

zn+1k minus znk =1

κn (286)

Solving this equation we get

znk =

jk +

nminus1suml=n0

1

κl n gt n0 + 1

jk minus

n0minus1suml=n

1

κl n 6 n0 minus 1

(287)

where jk = zn0k is an arbitrary function of its argument

case 2 2 ε 6= 0 If ε 6= 0 then (285) is a discrete Riccati equationwhich can be linearized through the Moumlbius transformation

znk =iε

ynk minus 1

ynk + 1(288)


to

(iκn minus ε)yn+1k = (iκn + ε)ynk (289)

Note that (289) implies that κn 6= plusmniε because otherwise ynk =

0 and znk = minusiε Therefore equation (289) implies that wehave the following solution

ynk =

jk

nminus1prodl=n0

iκl + εiκl minus ε

n gt n0 + 1

jk

n0minus1prodl=n

iκl minus εiκl + ε

n 6 n0 minus 1

(290)

where jk = yn0k is another arbitrary function of its argument

In conclusion we have always completely integrated the originalsystem

Remark 241 Let us note that in the case ε = 0 the (191a) can belinearized also without the need to introduce the transformation sep-arating the even and the odd part in m (281) Indeed by putting ε in(191a) we obtain

(unm minus un+1m) (unm+1 minus un+1m+1) minusα2 = 0 (291)

and it is easy to see that the Moumlbius-like transformation dependingon the field and on its first order shift

un+1m minus unm =radicα21minuswnm

1+wnm (292)

brings (191a) into the following first order linear equation

wnm+1 +wnm = 0 (293)

Therefore we can write the solution to (291) as

unm =

km +

radicα2

l=nminus1suml=n0

1minus (minus1)mwl1+ (minus1)mwl

n gt n0 + 1

km minusradicα2

l=n0minus1suml=n


n 6 n0 minus 1

(294)

Here km = un0m and wn are two arbitrary integration functions

Now we prove that the Lax pair of the tHε1 equation (191a) givenby equation (277) is fake This was presented for the first time in[68] Indeed we recall that according to the definition in [77 78] aLax pair is called G-fake if on solutions of the equation appearing inits compatibility condition one can remove all dependent variablesin the associated nonlinear system from the Lax pair by applying


a gauge transformations A gauge transformation for a Lax pair in theform (15) is a KtimesK invertible matrix Gnm possibly dependent on theunknown unm and its shifts acting at the level of the wave functionΦnm in the following way

Φnm = GnmΨnm (295)

The transformation (295) yields a new Lax pair of the form (15) forthe new wave function Ψnm with new matrices

Lnm = Gminus1n+1mLnmGnm (296a)

Mnm = Gminus1nm+1MnmGnm (296b)

We note that a Gauge transformation in the sense of equation (295)is a particular kind of symmetry of the Lax pair

Now that we have stated the required definition let us return tothe Lax pair for the tHε1 equation (191a) as given by equation (277)First we separate the even and odd part in m in (277) defining

Ξnk = Ψn2k Θnk = Ψn2k+1 (297)

Substituting m odd into the spectral problem defined from (277) weobtain the following constraint

Θnk =1

i minus εznk

(znk α1 minuswnkznk

1 wnk minus ε2α1znk

)Ξnk (298)

Then using (298) and its difference consequences we can write downa Lax pair involving only the field wnk and the wave function Ξnk

Ξnk+1 = L(1)nkΞnk Ξn+1k =M

(1)nkΞnk (299)

where

L(1)nk = minusα1

(1 wnk+1 minuswnk

0 1

) (2100a)

M(1)nk =

α1 (wnk minuswn+1k)

wnk minuswnk+1 + iεα2

(1 0

0 1

)

+α2


(wn+1k minuswnkwn+1k

1 minuswnk

)

minusε2α1α2 (α1 minusα2)


(0 1

0 0

)

(2100b)

The compatibility conditions of (299) yields the linear equation

wn+1k+1 minuswnk+1 minuswn+1k +wnk = 0 (2101)


Therefore we have incidentally given another proof of the lineariza-tion using the Lax pair

Finally we see that applying the following gauge transformation

Gnk = (minusα1)k αn1

(1 wnk

0 1

)(2102)

yields the new matrices obtained from (296)

L(1)nk =

(1 0

0 1

) (2103a)

M(1)nk =

wnk minuswn+1k

wnk minuswn+1k + iεα2

(1 wnk minuswn+1k

α2α1 1

)

+α2


(0 minusε2 (α1 minusα2)

1α1 0

)

(2103b)

Since (2103a) is independent on the field wnk we conclude that theLax pair (2100) is a G-fake Lax pair according to the above definition

We recall that a fake Lax pair do not provide the infinite number ofconservation laws needed by the definition of integrability [27 77 78]Since we proved that the tHε1 is linearizable this have to be expectedIndeed linear equations possess a finite number of conservation laws[169] and therefore any Lax pair for a linear equation must be fakeUltimately this example shows that a Lax pair produced from theConsistency Around the Cube property is not bona fide in advancebut its properties must be thoroughly checked

242 The 1D2 equation

Now we consider the equation 1D2 (193b) The 1D2 equation (193b)is given by the sextuple of equations given by Case 3122 in [21]

A = δ2x+ x1 + (1minus δ1) x2 + x12 (x+ δ1x2) (2104a)

B = (xminus x3) (x2 minus x23)

+ λ [x+ x3 minus δ1 (x2 + x23)] + δ1λ

(2104b)

C = (xminus x3) (x1 minus x13) + δ1 (δ1δ2 + δ1 minus 1) λ2

minus λ [(δ1δ2 + δ1 minus 1) (x+ x3) + δ1 (x1 + x13)]

(2104c)

A = δ2x3 + x13 + (1minus δ1) x23 + x123 (x+ δ1x23) (2104d)

B= (x1 minus x13) (x12 minus x123)

+ λ [2δ2 (δ1 minus 1) + (δ1 minus 1minus δ1δ2) minus 2δ1x12x123] (2104e)

C = (x1 minus x23) (x12 minus x123) minus λ (2δ2 + x12 + x123) (2104f)


The triplet of consistent dynamical systems on the 3D-lattice is

A =(F(minus)p+n minus δ1F

(+)p F

(minus)n + δ2F

(+)p F

(+)n

)upn

+(F(+)p+n minus δ1F

(minus)p F

(minus)n + δ2F

(minus)p F

(+)n

)up+1n


(+)p F

(+)n + δ2F

(+)p F

(minus)n

)upn+1

+(F(minus)p+n minus δ1F

(minus)p F

(+)n + δ2F

(minus)p F

(minus)n

)up+1n+1

+ δ1

(F(minus)n upnup+1n + F

(+)n upn+1up+1n+1

)+ F

(+)p+nupnup+1n+1 + F

(minus)p+nup+1nupn+1

(2105)

B = λ[

(δ1 minus 1) F(+)n minus δ1

]F(+)p + (δ1 minus 1minus δ1δ2) F

(minus)p F

(minus)n

(unm + unm+1)

+ λ[

(δ1 minus 1) F(minus)n minus δ1


(minus)p F

(+)n

(un+1m + un+1m+1)

minus 2δ1λF(minus)p


(+)n un+1mun+1m+1

)+ (unm minus unm+1) (un+1m minus un+1m+1)

+ δ1λ2F

(+)p + 2 (δ1 minus 1) δ2λF

(minus)p

(2106)

C = λ[

(1minus δ1δ2) F(+)p minus δ1

]F(+)n + F

(minus)p F

(minus)n

(upm + upm+1

)+ λ[

(1minus δ1δ2) F(minus)p minus δ1

]F(+)n + F

(+)p F

(minus)n

(up+1m + up+1m+1

)+(upm minus upm+1

) (up+1m minus up+1m+1

)+ 2δ2λF

(minus)n + δ1 (δ1 minus 1+ δ1δ2) λ

2F(+)n

(2107)

We leave out the Lax pair for 1D2 as it is too complicate to writedown and not worth while the effort for the reader If necessary onecan always write it down using the standard procedure outlined inSection 14

Let us now turn to the linearization procedure Notice that thereis no combination of the parameters δ1 and δ2 such that (193b) be-comes non-autonomous So we are naturally induced to introduce thefollowing four fields

wst = u2s2t yst = u2s+12t

vst = u2s2t+1 zst = u2s+12t+1(2108)

which transform (193b) into the following system of four coupledautonomous difference equations

(1minus δ1) vst + δ2wst + yst + (δ1vst +wst) zst = 0 (2109a)

(1minus δ1) vs+1t + δ2ws+1t + yst

+ (δ1vs+1t +ws+1t) zst = 0(2109b)

(1minus δ1) vst + δ2wst+1 + yst+1

+ (δ1vst +wst+1) zst = 0(2109c)


(1minus δ1) vs+1t + δ2ws+1t+1 + yst+1

+ (δ1vs+1t +ws+1t+1) zst = 0(2109d)

Let us solve (2109a) with respect to yst

yst = minus(1minus δ1) vst minus δ2wst minus (δ1vst +wst) zst (2110)

and let us insert yst into (2109b) in order to get an equation solvablefor zst This is possible iff δ1vst +wst 6= ft with ft an arbitraryfunction of t since in this case the coefficient of zst is zero Then thesolution of the system (2109) bifurcates

case 1 δ1vs t + ws t 6= ft Assume that δ1vst +wst 6= ft thenwe can solve with respect to zst the expression obtained inserting(2110) into (2109b) We get

zst = minus(1minus δ1) (vs+1t minus vst) + δ2 (ws+1t minuswst)

δ1 (vs+1t minus vst) +ws+1t minuswst (2111)

Now we can substitute (2110) and (2111) together with their differ-ence consequences into (2109c) and (2109d) and we get two equa-tions for wst and vst

(δ1δ2 + δ1 minus 1)[ws+1t+1vst+1wst + vs+1t+1wst+1ws+1t

minus vs+1t+1wst+1wst +wst+1vstws+1t+1

+ vstws+1t+1ws+1t minuswst+1vs+1tws+1t+1

minusw2st+1vst +w2st+1vs+1t

minusws+1t+1vst+1ws+1t minus vstws+1t+1wst

minus vstwst+1ws+1t + vstwst+1wst

minus δ1(vstvst+1ws+1t +wst+1vstvst+1

minusws+1t+1vst+1vst +ws+1t+1vst+1vs+1t

+ vstvs+1t+1wst minus vstvs+1t+1ws+1t

minus vstvst+1wst minuswst+1vs+1tvst+1)]

(2112a)

(δ1δ2 + δ1 minus 1)[w2s+1t+1vs+1t +ws+1t+1vst+1ws+1t

minusws+1t+1vst+1wst minusw2s+1t+1vst

minuswst+1vs+1tws+1t+1 +wst+1vstws+1t+1

minus vs+1t+1wst+1ws+1t + vs+1twst+1ws+1t

+ vs+1t+1wst+1wst minus vs+1tws+1t+1ws+1t

+ vs+1tws+1t+1wst minus vs+1twst+1wst+

δ1(minusvs+1tvs+1t+1ws+1t + vs+1tvs+1t+1wst

minusws+1t+1vstvs+1t+1 minus vs+1tvst+1wst

minus vs+1t+1wst+1vs+1t +ws+1t+1vs+1tvs+1t+1

+ vs+1tvst+1ws+1t + vs+1t+1wst+1vst)]

(2112b)


If

δ1(1+ δ2) = 1 (2113)

then (2112) are identically satisfied If δ1 (1+ δ2) 6= 1 adding (2112a)and (2112b) we obtain

(wst minusws+1t minuswst+1 +ws+1t+1) middot (vs+1t minus vst) (δ1 + δ1δ2 minus 1) = 0

(2114)

Therefore we are in front of a new factorization Since δ1 + δ1δ2 6= 1

we can divide by δ1 + δ1δ2 minus 1 and annihilate alternatively the firstor the second factor

If set equal to zero the second factor in (2114) we get vst = gt withgt arbitrary function of t alone Substituting this result into(2112a)or (2112b) we obtain gt = g0 with g0 constant Since we do nothave any other condition we have that wst is an arbitrary functionsatisfying wst 6= ft minus g0 while from (2111) we obtain zst = minusδ2 andfinally from (2115) we have

yst = minus(1minus δ1 minus δ1δ2)g0 (2115)

Therefore the only non-trivial case is when δ1 + δ1δ2 6= 1 andvst 6= gt In this case we have that wst solves the discrete waveequation ie wst = hs + lt Substituting wst into (2112) we get asingle equation for vst

(hs minus hs+1)[(vs+1t+1 minus vs+1t) (hs + lt+1)

minus (vst+1 minus vst) (hs+1 + lt+1)

+ δ1 (vstvs+1t+1 minus vs+1tvst+1)]= 0

(2116)

which is identically satisfied if hs = h0 with h0 a constant Thereforewe have a non-trivial cases only if hs 6= h0

case 1 1 δ1 = 0 We have a great simplification if in addition tohs 6= h0 we have δ1 = 0 In this case (2116) is linear

(vs+1t+1 minus vs+1t) (hs + lt+1)

minus (vst+1 minus vst) (hs+1 + lt+1) = 0(2117)

Eq (2117) can be easily integrated twice to give

vst =

js +sumtminus1k=t0

(hs + lk+1) ik t gt t0 + 1

js minussumt0minus1k=t (hs + lk+1) ik t 6 t0 minus 1

(2118)

with it and js = vst0 arbitrary integration functions


case 1 2 δ1 6= 0 but lt = l0 Now let us suppose hs 6= h0 δ1 6= 0but let us choose lt = l0 with l0 a constant Performing thetranslation θst = vst + (hs + l) δ1 from (2116) we get

θstθs+1t+1 minus θs+1tθst+1 = 0 (2119)

Eq (2119) is linearizable via a Cole-Hopf transformation Θst =

θs+1tθst as vst cannot be identically zero This linearizationyields the general solution θst = SsTt with Ss and Tt arbitraryfunctions of their argument

case 1 3 δ1 6= 0 lt 6= l0 Finally if hs 6= h δ1 6= 0 and lt 6= l0 weperform the transformation

θst =1

δ1[(lt minus lt+1) vst minus hs minus lt+1] (2120)

Then from (2116) we get

θst (1+ θs+1t+1) minus θs+1t (1+ θst+1) = 0 (2121)

which as vst cannot be identically zero is easily linearizedvia the Cole-Hopf transformation Θst = (1 + θst+1)θst toΘs+1s minusΘst = 0 which yields for θst the linear equation

θst+1 minus ptθst + 1 = 0 (2122)

where pt is an arbitrary integration function Then the generalsolution is given by

θst =

us minus tminus1suml=t0

lprodj=t0

pminus1j

tminus1prodk=t0

pk t gt t0 + 1

us

t0minus1prodk=t

pminus1k +

t0minus1suml=t

lprodj=t

pminus1j t 6 t0 minus 1

(2123)

where us = θst0 is an arbitrary integration function

case 2 δ1vs t + ws t = ft Let us suppose

wst = ft minus δ1vst (2124)

where ft is an arbitrary function of its argument Inserting (2110) and(2124) and their difference consequences into (2109) we get

(vs+1t minus vst) (δ1 + δ1δ2 minus 1) = 0 (2125)

and the two relations

ft+1zst+1 + [δ1 (vst minus vst+1) minus ft+1] zst

+ (δ1 minus 1) (vst minus vst+1) = 0(2126a)


ft+1zst+1 + [δ1 (vs+1t minus vs+1t+1) minus ft+1] zst

+ (δ1 minus 1) (vs+1t minus vst+1)

+ δ1δ2 (vs+1t+1 minus vst+1) = 0

(2126b)

Hence in (2125) we have a bifurcation depending on the factor wechoose to annihilate

case 2 1 δ1 (1 + δ2) = 1 If we annihilate the second factor in (2125)we get δ1 (1+ δ2) = 1 ie δ1 6= 0 Then adding (2126a) and(2126b) we obtain

(vst minus vs+1t minus vst+1 + vs+1t+1) (1minus δ1 + δ1zst) = 0 (2127)

It seems that we are facing a new bifurcation However anni-hilating the second factor ie setting zst = 1minus 1δ1 is trivialsince (2126) are identically satisfied

Therefore we may assume that zst 6= 1minus 1δ1 This implies thatvst solves a discrete wave equation whose solution is given by

vst = hs + kt (2128)

where hs and kt are generic integration functions of their ar-gument Inserting (2128) in (2126) we can obtain the followinglinear equation for zst

ft+1zst+1 + (δ1jt minus ft+1) zst + (δ1 minus 1) jt = 0 (2129)

with jt = kt+1 minus kt This equation can is solved by

zst = (minus1)t (δ1 minus 1)

tminus1prodt prime=0

δ1jt prime minus ft prime+1ft prime+1

middottminus1sumt primeprime=0

jt primeprime (minus1)t primeprime

ft primeprime+1

t primeprimeprodt prime=0


+ (minus1)t zs0



(2130)

case 2 2 vs t = lt Now we annihilate the first factor in (2125) ieδ1 (1+ δ2) 6= 1 and vst = lt where lt is an arbitrary function ofits argument From (2126) we obtain (2129) with jt = lt+1 minus lt

In conclusion we have always integrated the original system us-ing an explicit linearization through a series of transformations andbifurcations

As a final remark we observe that every transformation used in thelinearization procedure both for the tHε1 (191a) equation and for the


1D2 (193b) equation is bi-rational in the fields and their shifts (likeCole-Hopf-type transformations) This in fact has to be expectedsince the Algebraic Entropy test is valid only if we allow transfor-mations which preserve the algebrogeometric structure underlyingthe evolution procedure [158] Indeed there are examples on one-dimensional lattice of equations chaotic according to the AlgebraicEntropy but linearizable using some transcendental transformations[62] So exhibiting the explicit linearization and showing that it canbe attained by bi-rational transformations is indeed a very strong con-firmation of the Algebraic Entropy conjecture [84]

Indeed this does not prevent the fact that in some cases such equa-tions can be linearized through some transcendental transformationsIn fact if ε = 0 the 1Hε1 equation (291) can be linearized through thetranscendental contact transformation

unm minus un+1m =radicα2e

znm (2131)

ie

znm = logunm minus un+1mradic

α2 (2132)

with log standing for the principal value of the complex logarithm(the principal value is intended for the square root too) The transfor-mation (2131) brings (191a) into the following family of first orderlinear equations

znm+1 + znm = 2iπκ κ = 0 1 (2133)

However this kind of transformation does not prove the result of theAlgebraic Entropy as it involves exponential transformations There-fore the method explained in Subsection 241 should be consideredthe correct one

3G E N E R A L I Z E D S Y M M E T R I E S O F T H ET R A P E Z O I D A L H 4 A N D H 6 E Q U AT I O N S A N D T H EN O N - A U T O N O M O U S Y D K N E Q U AT I O N

In this Chapter we introduce the concept of Generalized Symmetriesfor quad equations We present the main computational tools for find-ing a specific class of Generalized Symmetries Then we will usethe developed tools to compute the Generalized Symmetries of thetrapezoidal H4 equations (191) and of the H6 equations (193) Themain result will then be the fact that all the symmetries we computedare related to a well-known differential-difference equation the non-autonomous Yamilov discretization of the Krichever-Novikov (YdKN)equation This identification led us to conjecture and prove using theAlgebraic Entropy test the existence of a new non-autonomous inte-grable quad equation which is a generalization of the QV equation[156]

The material is structured as follows In the first part of the Chap-ter we will introduce the concept needed in particular in Section 31we will discuss the relationship of the Generalized Symmetries withintegrability properties which is one of the reason for interest in theGeneralized Symmetries To do so we will treat the well known exam-ple of the Korteweg-deVries equation In Section 32 we will constructthree-point generalized symmetries for quad equations We will givein particular a method for finding three-point generalized symme-tries in the case of equation with two-periodic coefficients In thissecond part of the Chapter we present some original results bases on[65 66 68] In particular in Section 33 we will present the symme-tries of the trapezoidal H4 equations (191) and of the H6 equation(193) constructed with the method explained in 32 This Section ismainly based on the exposition in [66 68] Then in Section 34 weidentify the three-point generalized symmetries computed in Section(33) with some particular sub-cases of the non-autonomous YdKNequation Since an analogous result which for sake of completeness isstated in Appendix E was known for the rhombic H4 equation [166]we conclude that the three-point Generalized Symmetries of all theequations belonging to the classification of Boll [3 20ndash22] discussedin Chapter 1 are all sub-cases of the YdKN equation This result ex-tends what was previously known for three-point Generalized Sym-metries of the ABS equations [2] which were known to be particularsub-cases of the autonomous YdKN equation [102] In Section 35 wediscuss the integrability properties of the obtained three-point gener-alized symmetries seen as differential-difference equations both with

79

80 generalized symmetries and the na ydkn equation

the Master Symmetries approach and with the Algebraic Entropy testThese two last Sections are is essentially based on the results given in[66] In the last Section 36 based on the considerations made in Sec-tion 34 and on the results of [165] about the QV equation [156] andits relation to the autonomous YdKN equation we conjecture the ex-istence of a non-autonomous generalization of this equation We thenfind the appropriate candidate for such an extension by providinga non-autonomous extension of the Klein symmetries used in [156]to find the QV equation We then prove the integrability using theAlgebraic Entropy test and finally we discuss the three-point general-ized symmetries of the obtained equation and their relation with thenon-autonomous YdKN equation This last Section is based on [65]

31 generalized symmetries and integrability

At the time of the Franco-Prussian war in 1870 the Norwegian math-ematician Sophus Lie considered the question of the invariance ofdifferential equations with respect to continuous infinitesimal trans-formation ie transformations which can be seen as continuous de-formations of the identity In the successive 30 years Lie developeda theory which includes all the implications of such invariance Thesumma of the work of Sophus Lie on the subject is contained in [111]

Liersquos works on differential equation had a brief moment of successbut soon they were forgotten due to the complexity of the calcula-tions They were subsequently rediscovered around the middle ofthe XXth century by Russian mathematicians under the leadershipof Ovsiannikov [133 134] On the other hand Lie work on continu-ous groups was a powerful tool in the development of Quantum Me-chanics where they were used to show unexpected results about theatomic spectra [160] However the ldquophysicistsrdquo theory of Lie groupsturns out to be completely separated from the application of continu-ous groups to differential equations in such a way that it was possiblethat somebody knew much of one and completely ignore the other

The original idea of Sophus Lie was to unify the various methods ofsolution for differential equations uncovering the underlying geomet-rical structure in the same spirit as it was done by Evariste Galois foralgebraic equations Liersquos theory of infinitesimal point transformationis linear and one is able to reconstruct the full group of transforma-tion by solving some differential equations

In his work Lie [110] considered infinitesimal transformations de-pending also on the first derivatives of the dependent variables theso-called contact transformations [11 87] Later Baumlcklund consideredtransformations depending on finitely many derivatives of the depen-dent variables adding some closure relations in order to preserve thegeometrical structure of the transformations [16] The first to recog-nize the possibility of considering also transformations depending

31 generalized symmetries and integrability 81

on higher order derivatives without additional conditions was EmmyNoether in her fundamental paper [125] The mathematical objectsNoether considered was what now we call Generalized Symmetries

In modern times due to their algorithmic nature generalized sym-metry have been used both as tools for the study of given systems[11 41 48 87 91 93 94 130 132] and as a tool to classify integrableequations The symmetry approach to integrability has mainly beendeveloped by a group of researchers belonging to the scientific schoolof AB Shabat in Russia It has been developed in the continuum case[4 73 116ndash118 146 147] in the differential-difference case [57 105167 168] and more recently also in the completely discrete case [106107]

Let us start discussing the relation between integrability and gener-alized symmetries in the case of partial differential equations in twoindependent variables and one dependent variable Let us assume weare given a partial differential equation (PDE) of order k in evolution-ary form for an unknown function u = u (x t)

ut = f(uu1 uk) (31)

where ut = partupartt and uj = partjupartxj for any j gt 0A generalized symmetry of order m for (31) is an equation of the

form

uτ = g(uu1u2 um) (32)

compatible with (31) Here τ plays the role of the group parameter andof course we are assuming u = u (x t τ) The compatibility conditionbetween (31) and (32) implies the following PDE for the functions fand g

part2u

parttpartτminuspart2u

partτpartt= DtgminusDτf = 0 (33)

where Dt Dτ are the operators of total differentiation correspondingto (31 32) defined together with the operator of total x-derivativeD by

D =part

partx+sumigt0

ui+1part

partui (34a)

Dt =part

partt+sumigt0

Difpart

partui (34b)

Dτ =part

partτ+sumigt0

Digpart

partui (34c)

When (33) is satisfied we say that (32) is a generalized symmetry oforder m for (31)


We will now see in a concrete example how from the integrabil-ity properties is possible to obtain an infinite sequence of general-ized symmetries of the form (32) Let us consider the well knownKorteweg-de Vries equation (KdV equation) [92]

ut = 6uu1 + u3 (35)

It is known [97] that the KdV equation (35) possess the followinglinear representation

Lt = [LM] (36)

where L and M are differential operators given by

L = minuspartxx + u (37a)

M = 4partxxx minus 6upartx minus 3partx(u) (37b)

and by [ ] we mean the commutator of two differential operatorsWe look for a chain of differential operators Mj which correspond todifferent equations associated with the same L

Ltj =[LMj

] (38)

Since Lt = ut is a scalar operator one must have [LM] = V with Vscalar operator If there exists another equation associated to L thenthere must exists another M such that[

L M]= V (39)

where V is another scalar operatorNow one can relate M and M Noting that the equation ut = ux

can be written in the form (38) with Mj = partx we see that the opera-tors Mj are characterized from the power of the operator partx and thatthe relation between one and the other is of order two as the orderof the operator L We can therefore make the general assumption

M = LM+ Fpartx +G (310)

where F and G are scalar operators Using the ansatz (310) and im-posing to the coefficients of the various differential operators in (39)we find that

V = LV + ux (311)

where

LV = minus1

4Vxx + uV minus

1

2ux

intinfinx

V(y)dy (312)

This yields an infinite family of equations for any entire function F(z)

ut = F (L)ux (313)

32 three-point generalized symmetries for quad equation 83

The equations (313) are associated to the same L operator and bysolving the Spectral Problem associated to it [26] they are shown tobe commuting with the original KdV equation (35) The operator L

is called recursion operator or Leacutenard operator [51 98 129] In particularfor F (L) = Ln we find a whole hierarchy of generalized symmetriesfor the KdV equation (35) The operator (312) is called the recursionoperator for the KdV equation (35)

In general any evolution equation possessing a recursion operatorhas an infinite hierarchy of equations and if they commute they withthe evolution equation are generalized symmetries for the evolutionequation and then in an appropriate sense an evolution equationis integrable We therefore have a symmetry-based definition of in-tegrability An evolution equation is called integrable if it possesses non-constant generalized symmetries of any order m [131] Note that thisdefinition a priori does not distinguish between C-integrable and S-integrable equations Indeed both C-integrable and S-integrable equa-tions possess the property of having recursion operators like (312)but C-integrable equations are linearizable [25] The most famous C-integrable equation the Burgers equation also possesses a recursionoperator The difference between C-integrable and S-integrable equa-tions is in the fact that the S-integrable equations possess infinitelymany conservation laws of any order whereas the C-integrable onesonly up to the order of the equation itself Finally we note that sinceintegrable equations in this sense comes in hierarchies and the ele-ments of a hierarchy commute between themselves also the general-ized symmetries of an integrable equation are integrable

32 three-point generalized symmetries for quad equa-tion

unminus1mminus1 unmminus1 un+1mminus1

unminus1munm

un+1m

unminus1m+1 unm+1 un+1m+1

Figure 31 The extended square lattice with 9 points


In this Section we discuss the construction of the most simple gener-alized symmetries for quad equations (11)1 Symmetries of this kindwere first considered in [139] The exposition here is mainly based onthe original papers [52 106 107] and on the review in [63] wherea different approach to that used in [139] was developed The mostsimple Generalized Symmetries for a quad equation in the form (11)are those depending on nine points defined on a square of verticesunminus1mminus1 unminus1m+1 un+1m+1 and un+1mminus1 as depicted in Fig-ure 31 A priori the symmetry generator can depend on all thesepoints however by taking into account the difference equation (11)we can express the extremal points unminus1mminus1 unminus1m+1 un+1m+1

and un+1mminus1 in terms of the remaining five points unminus1m un+1munm unmminus1 and unm+1 In general this means that we are tak-ing as independent variables those laying on the coordinate axes ieun+jm and unm+k with jk isin Z This choice of independent vari-able is acceptable since these points do not lie on squares In thisway the most general nine points generalized symmetry generator isrepresented by the infinitesimal symmetry generator

X = g(unminus1mun+1munmunmminus1unm+1)partunm (314)

As in the case of the differential difference equation the function g iscalled the characteristic of the symmetry We note that this is not the onlypossible choice for the independent variables Another viable choiceof independent variables is given by an appropriate restriction of aninfinite staircase eg to consider the points un+1mminus1 unmminus1 unmunminus1m and unminus1m+1 This was the choice we made in the case of theAlgebraic Entropy but for the calculation of symmetries the choice ofthe independent variables on the axes is more convenient

Now we need to prolong the operator (314) in order to apply iton the quad equation (11) and construct the determining equationsThis prolongation is naturally given by

prX = g(unminus1mun+1munmunmminus1unm+1)partunm

+ g(unmun+2mun+1mun+1mminus1un+1m+1)partun+1m

+ g(unminus1m+1un+1m+1unm+1unmminus1+1unm+2)partunm+1

+ g(unm+1un+2m+1un+1m+1un+1mun+1m+2)partun+1m+1

(315)

Applying the prolonged vector field to equation (11) we get

gpartQ

partunm+ [Tng]

partQ

partun+1m

+ [Tmg]partQ

partunm+1+ [TnTmg]

partQ

partun+1m+1= 0

(316)

1 This kind of reasoning in fact can hold for every kind of partial difference equationson the square which are solvable with respect to all its variables

32 three-point generalized symmetries theory 85

where Tnfnm = fn+1m and Tmfnm = fnm+1 A priori (316) con-tains un+im+j with i = minus1 0 1 2 j = minus1 0 1 2

The invariance condition requires that (316) be satisfied on the so-lutions of (11) To be consistent with our choice of independent vari-ables which now include also unm+2 and un+2m we must use (11)and its shifted consequences to express

bull un+2m+1 = un+2m+1(un+2mun+1m+1un+1m)

bull un+1m+1 = un+1m+1(un+1munm+1unm)

bull unminus1m+1 = unminus1m+1(unminus1munm+1unm)

Doing so we reduce the determining equation (316) to an equationwritten just in terms of independent variables which thus must beidentically satisfied Differentiating (316) with respect to unm+2 andto un+2m we get

part2TnTmg

partun+2m+1partun+1m+2= TnTm

part2g

partun+1mpartunm+1= 0 (317)

Consequently the symmetry coefficient g is the sum of two simplerfunctions

g = g0(unminus1mun+1munmunmminus1)

+ g1(unminus1munmunmminus1unm+1)(318)

Introducing this result into the determining equation (316) and dif-ferentiating it with respect to unm+2 and to unminus1m we have that g1reduces to

g1 = g10(unminus1munmunmminus1) + g11(unmunmminus1unm+1)

(319)

In a similar way if we differentiate the resulting determining equationwith respect to un+2m and to unmminus1 we have that g0 reduces to

g0 = g00(unminus1munmunmminus1) + g01(unminus1mun+1munm)

(320)

Combining these results and taking into account the property sym-metrical to (317) ie part2g(partunminus1mpartunmminus1) = 0 we obtain the fol-lowing form for g

g = g0(unmminus1unmunm+1)+g1(unminus1munmun+1m) (321)

So the infinitesimal symmetry coefficient is the sum of functions that eitherinvolve shifts only in n with m fixed or only in m with n fixed [106 139]

Let us consider the subcase when the symmetry generator is givenby

dunm

dε= g1(unminus1munmun+1m) (322)


This is a differential-difference equation depending parametrically onm Setting unm = un and unm+1 = un the compatible partial dif-ference equation (11) turns out to be an ordinary difference equationrelating un and un

Q(unun+1 un un+1) = 0 (323)

ie a Baumlcklund transformation [102] for (322) A similar result is ob-tained in the case of g0 This result was first presented in [139]

We note that higher order symmetries ie symmetries dependingon more lattice points have the same splitting in simpler symmetriesin the two directions [52] However such higher order symmetrieswill no longer yield Baumlcklund transformations

To find the specific form of g1 we have to differentiate the deter-mining equation (316) with respect to the independent variables andget from these some further necessary conditions on its shape Let usdiscuss the case in which the symmetry is given in terms of shifts inthe n direction since the m direction shift case can be treated analo-gously2

X = g (un+1munmunminus1m)partunm (324)

We donrsquot impose restrictions on the explicit dependence of g on thelattice variables so in principle g = gnm

To obtain gwe have to solve the equation (316) which is a functionalequation since it must be evaluated on the four-tuples (unmun+1munm+1un+1m+1)

such that the quad equation (11) holds The best way to so is to getsome consequences of (316) and convert them into a system of lin-ear partial differential equation which we can solve This will imposerestrictions on the form of g and will allow us to solve the functionalequation (316) Using the assumption that Q is multi-linear we canexpress un+1m+1 as

un+1m+1 = f (un+1munmunm+1) (325)

therefore the determining equation takes the form

TnTmg = Tngpartf

partun+1m+ Tmg

partf

partunm+1+ g

partf

partunm(326)

Let us start to differentiate (326) with g given by (324) with respectto un+2m which is the higher order shift appearing in (326)

TnTmpartg

partun+1mTn

partf

partun+1m=

partf

partun+1mTn

partg

partun+1m (327)

2 In fact the most convenient way for treating the symmetries in the m direction isto consider the transformation n harr m and make the computations in the new n

direction Performing again the same transformation in the obtained symmetry willyield the result


Define

z = logpartg

partun+1m(328)

then we can rewrite (327) in the form of a conservation law

Tmz = z+(Tminus1n minus Id

)log

partf

partun+1m(329)

We also have another representation for (326) which will be usefulin deriving another relation similar to (329) Let us apply Tminus1n to(326) then solving the resulting equation for Tminus1n Tmg we obtain aequivalent representation for (326)

Tminus1n Tmg =minus Tminus1n

(partf

partunm

partf

partunm+1

)Tminus1n g

+ Tminus1n

(1

partf

partunm+1

)Tmg

minus Tminus1n

(partf

partun+1m

partf

partunm+1

)g

(330)

Differentiating (330) with respect to unminus2m we obtain

Tminus1n Tmpartg

partunminus1m

partunminus2m+1

partunminus2m= minusTminus1n

(partf

partunm

partf

partunm+1

)Tminus1n

partg

partunminus1m

(331)

From the implicit function theorem we get

partunminus2m+1

partunminus2m= Tminus1n

partunminus1m+1


(partf

partunm

partf

partunm+1

) (332)

So introducing

v = logpartg

partunminus1m(333)

we obtain from (331) a new equation written in conservation lawform

Tmv = v+ (Tn minus Id) log(

partf

partunm

partf

partunm+1

) (334)

The equations (329) and (334) are still functional equations butwe can derive from them a system of first order PDEs Indeed let s bea function such that s = s (un+1munmunminus1m) Then the functionTms = s (un+1m+1unm+1unminus1m+1) can be annihilated applyingthe differential operator

Yminus1 =part

partunmminus

partfpartunm

partfpartunm+1

part

partun+1m

minus Tminus1n

(partfpartun+1m

partfpartunm

)part

partunminus1m

(335)


On the other hand the function Tminus1m s = s (un+1mminus1unmminus1unminus1mminus1)

is annihilated by the operator

Y1 =part

partunmminus Tminus1m

partf

partunm+1

part

partun+1m

minus Tminus1n Tminus1m

(partf

partunm+1

)minus1part

partunminus1m

(336)

Therefore we can apply the operator Yminus1 as given by (335) to (329)and this will give us the linear PDE

Yminus1z = minusYminus1(Tminus1n minus Id

)log

partf

partun+1m (337)

Analogously by applying Tminus1m to (329) we can write

Tminus1m z = zminus Tminus1m(Tminus1n minus Id

)log

partf

partun+1m(338)

which applying Y1 gives us the linear PDE

Y1z = Y1Tminus1m

(Tminus1n minus Id

)log

partf

partun+1m (339)

Since z is independent from unmplusmn1 whereas the coefficients in (337339)may depend on it we may write down a system for the function z

Y1z = Y1Tminus1m

(Tminus1n minus Id

)log

partf

partun+1m (340a)


)log

partf

partun+1m (340b)

partz

partunm+1=

partz

partunmminus1= 0 (340c)

The system (340) may not be closed in the general case If the system(340) happens to be not closed it is possible to add an extra equationusing Lie brackets

[Y1 Yminus1] z = minus[Y1Yminus1T

minus1m minus Yminus1Y1

] (Tminus1n minus Id

)log

partf

partun+1m (341)

Applying the same line of reasoning we deduce a system of equa-tions also for v

Y1v = Y1Tminus1m (Tn minus Id) log

(partf

partunm

partf

partunm+1

) (342a)

Yminus1v = minusYminus1(Tn minus Id) log(

partf

partunm

partf

partunm+1

) (342b)

partv

partunm+1=

partv



As before if the system is not closed we may add the equation ob-tained using the Lie bracket




)log(

partf

partunm

partf

partunm+1

)

(343)

Once we have solved the systems (340) and (342) (eventually withthe aid of the auxiliary equations (341) and (343)) we insert the ob-tained values of z and v into (329) and (334) In this way we can fixthe dependency on the explicit functions of the lattice variables n mWe can then solve the potential-like equation

partg

partun+1m= ez

partg

partunminus1m= ev (344)

Therefore if the compatibility condition

partez

partunminus1m=

partev

partun+1m (345)

is satisfied we can write

g =

intezdun+1m + g(1) (unm) (346)

It only remains to determine the function g(1) (unm) This can be eas-ily done by plugging g as defined by (346) into the determining equa-tions (326) This determining equation will still be a functional equa-tion but the only implicit dependence will be in g(1) (f (un+1munmunm+1))

and can be annihilated by applying the operator

S =part

partunmminus

partfpartunm

partfpartun+1m

part

partun+1m (347)

Differentiating in an appropriate way the resulting equation we candetermine g(1) (unm) and check its functional form by plugging itback into (326)

To conclude this discussion we present two examples of calcula-tions of three-point generalized symmetries using the procedure pre-sented above

Example 321 (The H1 equation (123a)) In this example we will con-sider the H1 equation as given by (123a) We will compute its au-tonomous three-point symmetries in the n direction using the methodwe outlined above These symmetries were first derived with a differ-ent technique in [163 164] These symmetries appeared also in manyother papers see [139] and references therein for a complete list

We first have to find the function z = logpartgpartun+1m Using thedefinition we can write down (340a) with f given by solving (123a)

(unmminus1 minus un+1m)2partz

partun+1m+ (α1 minusα2)

partz

partunm

+ (unminus1m minus unmminus1)2 partz

partunminus1m= 2 (2unmminus1 minus un+1m minus unminus1m)


(348)

(340c) implies that we can take the coefficients with respect to unmminus1

in (348)

u2n+1mpartz


partz

partunm

+ u2nminus1mpartz

partunminus1m= minus2 (un+1m + 2unminus1m)

(349a)

un+1mpartz

partun+1m+ unminus1m

partz

partunminus1m= minus2 (349b)

partz

partunminus1m+

partz

partun+1m= 0 (349c)

This equations can be easily solved to give the form of z

z = log[C1 (un+1m minus unminus1m)minus2

] (350)

being C1 a constantWe do the same computations for v we obtain that v have to solve

exactly the same equations as z therefore we conclude that

v = log[C2 (un+1m minus unminus1m)minus2

] (351)

being C2 a new constant of integration Using the compatibility con-dition (345) we obtain that C2 = minusC1 and integrating (344) we have

g =C1

unminus1m minus un+1m+ g(1) (unm) (352)

Inserting this form of g into the determining equations (326) andapplying the operator S (347) we obtain the following equation

unm+1dg(1)

dunm+1(un+1m) minus unm+1

dg(1)

dunm(unm)

+ 2g(1) (un+1m) minus un+1mdg(1)

dun+1m(un+1m)

+ un+1mdg(1)

dunm(unm) minus 2g(1) (unm+1) = 0

(353)

Differentiating it with respect to unm we obtain d2g(1) (unm)

du2nm =

0 which implies g(1) = C3unm +C4 Substituting this result in (353)we obtain the restriction C3 = 0

In conclusion we have found

g =C1

unminus1m minus un+1m+C4 (354)

which satisfy identically the determining equations (326) This showsthat (354) is the most general three-point symmetry in the n direction


Note that whereas the coefficient of C1 is a genuine generalized sym-metry the coefficient of C4 is in fact a point symmetry Since the H1equation (123a) is invariant under the exchange of variables n harr m

we have the symmetry in the m direction

g =C1

unmminus1 minus unm+1 (355)

Therefore equations (354 355) represent the most general autonomousfive point symmetries of the H1 equation (123a) [139]

It is worth to note that the differential difference equation definedby the equations (354) and (355) ie

dukdt

=1

uk+1 minus ukminus1 k isin Z (356)

is a spatial discretization of the KdV equation [101 121]We will return to the symmetry (354) in Example 323 concerning

symmetry reduction

Example 322 (Autonomous equation with non-autonomous general-ized symmetries) Let us consider the quad equation

un+1m+1unm(un+1m minus 1)(unm+1 + 1)

+(un+1m + 1)(unm+1 minus 1) = 0(357)

It was proved in [107] that (357) does not admit any autonomousthree point generalized symmetry but it was conjectured there that itmight have a non-autonomous symmetry This conjecture was provedin [52] and we will shall give here such a proof

We first have to construct the function z = logpartgpartun+1m Usingits definition we can write down (340) with f obtained from (357) as

u2nminus1m minus 1

2unm

partz

partunminus1m+

partz

partunm

+2un+1m

u2nm minus 1

partz

partun+1m=u2nm minus 2unm minus 1

unm(u2nm minus 1)

(358a)

2unminus1m

u2nm minus 1

partz

partunminus1mminus

partz

partunm+u2n+1m minus 1

2unm

partz

partun+1m

=u2nmun+1m + 2u2nm minus un+1m

unm(1minus u2nm)

(358b)

Since in (358) there is no dependence on unmplusmn1 we can omit theequations concerning these variables The system (358) is not closedTo close this system one has to add the equation for the Lie bracket(341) Solving the obtained system of three equations with respect tothe partial derivatives of z we have

partz

partunminus1m= 0 (359a)


partz

partun+1m= minus

2(unm + 1)

un+1m(unm + 1) + 1minus unm (359b)

partz

partunm=

1

unm+

1

unm + 1+

1

unm minus 1

minus2(un+1m minus 1)

un+1m(unm + 1) + 1minus unm

(359c)

These three equations form an overdetermined system of equationsfor z which is consistent because is closed Hence its general solutionz is easily found It contains arbitrary function C1nm depending onboth discrete variables

z = logC1nmunm(u2nm minus 1)

(unmun+1m + un+1m minus unm + 1)2 (360)

Substituting (360) into the conservation law (329) we get

logminusC1nm+1

C1nm= 0 (361)

The last equation is solved by C1nm = (minus1)mC2n where C2m is anarbitrary function of one discrete variable

Therefore by solving equation z = logpartgpartun+1m we find

g =minus(minus1)nC2nunm(unm minus 1)

unmun+1m + un+1m minus unm + 1+g2(unminus1munm) (362)

with g2 possibly dependent on the lattice variables nm For thefurther specification consider v(unminus1munm) = logpartgpartunminus1m =

logpartg2partunminus1m which putted into (342) for v gives

u2nminus1m minus 1

2unm

partv

partunminus1m+

partv

partunm=2u2nm minus 2unmunminus1m + unminus1m

unm(u2nm minus 1)

(363a)

2unminus1m

u2nm minus 1

partv

partunminus1mminus

partv

partunm= minus

1minus u2nm minus 2unm

unm(u2nm minus 1) (363b)

The solution of the system (363) is given by

v = logC3nmunm(u2nm minus 1)

(unmunminus1m minus unminus1m + unm + 1)2 (364)

Substituting (364) in the conservation law (334) we obtain the rela-tion

logminusC3nm+1

C3nm= 0 (365)

whose solution is C3nm = (minus1)mC4n where C4n is an arbitrary func-tion of n


As a result function g takes the form

g =minus(minus1)mC2nunm(unm minus 1)

unmun+1m + un+1m minus unm + 1

+minus(minus1)mC4nunm(unm + 1)

unmunminus1m minus unminus1m + unm + 1+ g(1)(unm)

(366)

Substituting (366) into (326) applying the operator S defined by(347) then dividing by the factor

2unm(u2n+1m minus 1

) (u2nm+1 minus 1

)(un+1m + 1+ unmun+1m minus unm)

(367)

and finally differentiating with respect to unm we obtain

d2g(1)

du2nmminus

1

unm

dg(1)

dunm+g(1)

u2nm=

(minus1)m(u2n+1m + 1

) (C2n minusC4n+1

)(unmun+1m minus unm + un+1m + 1)2

(368)

This equation implies

d2g(1)

du2nmminus

1

unm

dg(1)

dunm+g(1)

u2nm= 0 C2n = C4n+1 (369)

Therefore g(1) = C5nmunm + C6nmunm logunm where the coeffi-cients do not depend on unm but might depend on the lattice vari-ables nm Substituting this result again into the determining equa-tions applying the operator S and taking the coefficients with respectto the independent functions unm+1 unm logun+1m un+1m weobtain the three equations

C6n+1m = 0 2C5n+1m +(C4n+2 minusC

4n+1

)(minus1)m = 0

2C5n+1m minus(3C4n+1 +C

4n+2

)(minus1)m = 0

(370)

which when solved give us

C4nm = C (minus1)n C5nm = C (minus1)n+m C6nm = 0 (371)

By plugging these results into (366) we obtain

g =(minus1)m+nCunm(u2nm minus 1)(un+1munminus1m + 1)

(unmun+1m + un+1m minus unm + 1)(unmunminus1m minus unminus1m + unm + 1)

(372)

From the direct substitution into (326) we obtain that (372) is a sym-metry


Generalized symmetries can be used to provide symmetry reductionsSuppose we have a three-point generalized symmetry in the form(324) We can consider its flux which we recall is the solution of thedifferential difference equation

dunm

dε= g (un+1munmunminus1m) (373)

and we can consider its stationary solutions ie the solutions such thatdunmdε equiv 0

g (un+1munmunminus1m) = 0 (374)

In this equation m plays the rocircle of a parameter and we can considercontemporaneous solutions of the stationary equation (374) and ofthe original quad equation (325) This will give raise to families ofparticular solutions which are known as symmetry solutions

We conclude this Subsection presenting an example of symmetryreduction

Example 323 (Reduction of the H1 equation (123a)) Consider the H1equation as given by formula (123a) In Example 323 we derived itsthree point generalized symmetries in both directions Here we willuse the symmetry (354) to derive a family of symmetry solutions [24136] First we start by observing that if we assume C4 equiv 0 we cannothave any stationary solution to the equation

C1

unminus1m minus un+1m+C4 = 0 (375)

so we must assume C4 6= 0 and then we can take without loss of gen-erality C4 = 1 This means that the we will have the linear stationaryequation

un+1m minus unminus1m = C1 (376)

This equation has solution

unm = U(0)m (minus1)n +U

(1)m +

1

4C1((minus1)n minus 1+ 2n) (377)

which substituted into (123a) gives us two equations for the coeffi-cients of (minus1)n

U(0)m+1 +U

(0)m +

C1

2= 0 (378a)(

U(0)m+1 +U

(0)m

)2minus(U

(1)m+1 minusU

(1)m

)2+

(C1)2

2+C1U

(0)m +C1U

(0)m+1 = α1 minusα2

(378b)

The solution of (378a) is given by

U(0)m = K1(minus1)

m minus1

4(1+ (minus1)m)C1 (379)


which substituted in (378b) gives to the equation

U(1)m+1 minusU

(1)m = plusmn1

2

radicminus4α1 + 4α2 + (C1)2 (380)

whose solution is given by

U(1)m = K2 plusmn

m

2

radicminus4α1 + 4α2 + (C1)2 (381)

So inserting (379) and (381) into (377) we obtain

unm = (minus1)n+m(K1 +

C1

4

)+K2

plusmnradic(C1)2 minus 4α1 + 4α2mminus

C1

4+C1

4n

(382)

This is our symmetry solution

To end this Section we present a convenient way to compute thethree-point generalized symmetries in the case of quad equationswith two-periodic coefficients

F(+)n F

(+)m Q(++) + F

(+)n F

(minus)m Q(+minus)

+F(minus)n F

(+)m Q(++) + F

(minus)n F

(minus)m Q(minusminus) = 0

(383)

where Q(plusmnplusmn) = Q(plusmnplusmn) (unmun+1munm+1un+1m+1) and F(plusmn)k

is given by (190) Whereas the method outlined above in principleapply to quad equations with two-periodic coefficients (383) the com-putations readily become very cumbersome So we will give a briefaccount on how these difficulties can be avoided following the expo-sition in [63] The necessity of treating equations like (383) as simplyas possible comes from the fact that we wish to study the trapezoidalH4 and H6 equations which have two-periodic coefficients as shownby their explicit form given in Section 16 The three-point generalizedsymmetries of these equations were first presented in [66] and werecomputed with the ideas presented above We recall that the three-point generalized symmetries of the rhombic H4 equations whichagain have two-periodic coefficients were studied in [166]

Now let us consider a quad equation with two-periodic coefficientsof the form (383) Since any point (nm) on the lattice have coordi-nates which can be even or odd we can derive the equations (340342)for the functions z and v and then use the decomposition

z = F(+)n F

(+)m z(++) + F

(+)n F

(minus)m z(+minus)

+ F(minus)n F

(+)m z(++) + F

(minus)n F

(minus)m z(minusminus)

(384a)

v = F(+)n F

(+)m v(++) + F

(+)n F

(minus)m v(+minus)

+ F(minus)n F

(+)m v(++) + F

(minus)n F

(minus)m v(minusminus)

(384b)


The decomposition (384) reduces the problem to the solution of fourdecoupled systems for the functions z(plusmnplusmn) = z(plusmnplusmn) (un+1munmunminus1m)

and v(plusmnplusmn) = v(plusmnplusmn) (un+1munmunminus1m) by considering the evenoddcombinations of discrete variables

The same decomposition can be used for the function g

g = F(+)n F

(+)m g(++) + F

(+)n F

(minus)m g(+minus)

+ F(minus)n F

(+)m g(++) + F

(minus)n F

(minus)m g(minusminus)

(385)

with g(plusmnplusmn) = g(plusmnplusmn) (un+1munmunminus1m) and the relative com-patibility conditions (345) This will yield the following form for g

g = Ωnm (un+1munmunminus1m)

+ F(+)n F

(+)m ϕ(++) + F

(+)n F

(minus)m ϕ(+minus)

+ F(minus)n F

(+)m ϕ(minus+) + F

(minus)n F

(minus)m ϕ(minusminus)

(386)

where Ωnm is a known function derived from z and v and ϕ(plusmnplusmn) =

ϕ(plusmnplusmn) (unm) These functions can be found rewriting the determin-ing equations with the same decomposition

33 three-point generalized symmetries of the trape-zoidal H4 and H6 equations

In this Section we apply the method discussed in Section 32 espe-cially in its final part to compute the three-point generalized sym-metries of the trapezoidal H4 equations as given by (191) and H6

equations as given by (193) These three-point generalized symme-tries were first computed in [66] except for the tHε1 equation whosesymmetries were first presented in [68] Since the computations arevery long and tedious we omit them and we present only the final re-sult The interested reader may refer to Appendix D where we presenttwo examples carried out in the details the tHε1 in them direction andthe D3 equation The three-point generalized symmetries of the othertrapezoidal H4 and H6 equation can be computed in an analogousway

331 Trapezoidal H4 equations

We now consider the trapezoidal H4 equations as given by equation(191)

33 three-point generalized symmetries (1 91 1 93) case 97

We present first the three-point generalized symmetries of the equa-tions tHε2 (191b) and of equations tHε3 (191c) are found to be

XtHε2n =

[(unm + εα22F

(+)m )(un+1m + unminus1m) minus un+1munminus1m

un+1m minus unminus1mminus

minusu2nm minus 2εF

(+)m α22unm minusα22 + 4εF

(+)m α32 + 8εF

(+)m α22α3 + ε

2F(+)m α42

un+1m minus unminus1m

]partunm

(387a)

XtHε2m =

[1

2minus ε(α2 +α3)F

(+)m

](unm+1 + unmminus1) minus εF

(+)m unm+1unmminus1

unm+1 minus unmminus1minus

minusεF

(minus)m u2nm minus

[1minus 2ε(α2 +α3)F

(minus)m

]unm +α3 + ε (α2 +α3)

2

unm+1 minus unmminus1

partunm

(387b)

XtHε3n =

12α2(1+α22)unm(un+1m + unminus1m) minusα22un+1munminus1m


minusα22u

2nm + ε2δ2(1minusα22)

2F(+)m


]partunm

(387c)

XtHε3m =

12α3unm(unm+1 + unmminus1) minus ε2F

(+)m unm+1unmminus1


minusε2F

(minus)m u2nm +α23δ

2


]partunm

(387d)

The symmetries in the n and m directions and the linearization ofthe tHε1 equation (191a) were first presented in [68] Their peculiarityis that they are written in term of two arbitrary functions of one con-tinuous variable and one discrete index and by arbitrary functions ofthe lattice variables This fact is related to the property of the tHε1 ofbeing Darboux integrable with integrals of the first and second order[5] as it was first found in [69] and we will discuss in Chapter 4


In the direction n we have that the tHε1 possess the following three-point generalized symmetries

XtHε1n1 =

F(+)m

(un+1m minus 2unm + unminus1m) (un+1m minus unminus1m)

middot

[((unm minus unminus1m)2 + ε2α22

)Bn

(α2

un+1m minus unm

)

minus((un+1m minus unm)2 + ε2α22

)Bnminus1

(α2

unm minus unminus1m

)]

+ F(minus)m

(un+1m minus unm)2 (unm minus unminus1m)2(2minus 2un+1mε

2unminus1m)unm

+ε2 (unminus1m + un+1m)u2nm

minusun+1m minus unminus1m

(unminus1m minus un+1m) (1+ unm2ε2)

middot[Bn

(un+1m minus unm

1+ ε2un+1munm

)minusBn

(unm minus unminus1m

1+ ε2unmunminus1m

)]partunm

(388a)

XtHε1n2 =

F(+)m

[unm minus

((un+1m minus unm)2 + ε2α2)(unm minus unminus1m)

(un+1m minus 2unm + unminus1m)(un+1m minus unminus1m)

]

minusF(minus)m

[unmunminus1m + un+1munm minus unm

2 minus un+1munminus1m

(unm2ε2 + 1) (minus2un+1m + 2unminus1m)

+(un+1m minus unm) (unm minus unminus1m)(2ε2 (unminus1m + un+1m)u2nm

+(4minus 4un+1mε

2unminus1m)unm minus 2un+1m minus 2unminus1m

)]partunm

(388b)

where Bn = Bn (ξ) is an arbitrary function of its argument There-fore the most general symmetry in the n direction is given by thecombination

XtHε1n = Xt

Hε1n1 +αXt

Hε1n2 (389)

where α is an arbitrary constantThe general symmetry in the m direction is

XtHε1m =

[F(+)m Bm

(unm+1 minus unmminus1

1+ ε2unm+1unmminus1

)

+ F(minus)m

(1+ ε2u2nm

)Cm (unm+1 minus unmminus1)

]partunm

(390)

where Bm = Bm (ξ) and Cm = Cm (ξ) are arbitrary functions of thelattice variable m and of their arguments

Furthermore the tHε1 equation (191a) possesses the following pointsymmetry

YtHε1 =

[F(+)m κm + F

(minus)m λm

(1+ ε2u2nm

)]partunm (391)


where κm and λm are arbitrary functions of the lattice variable mOn the contrary both the tHε2 equation (191b) and the tHε3 equation(191c) do not possess point symmetries

332 H6 equations

In this Subsection we consider the H6 equations as given by formula(193)

The three forms of the equation D2 (193b193c193d) which wewill collectively call iD2 assuming i in 1 2 3 possess the follow-ing three points generalized symmetries in the n direction and three-points generalized symmetries in the m direction

X1D2n =

(F(+)n F

(minus)m minus δ1F

(+)n F

(+)m

)(unm+1 + unminus1m)

un+1m minus unminus1m+

+

(F(+)n F

(minus)m minus δ1F

(+)n F

(+)m minus δ1δ2

)F(minus)n F

(+)m unminus1m


+(F

(+)n+m minus δ1F

(+)m minus δ1δ2F

(+)n F

(+)m )unm + δ2F

(minus)m


]partunm

(392a)

X1D2m =

[δ1F

(minus)n F

(+)m unm+1unmminus1 + F

(minus)n+m(unm+1 + unmminus1)

unm+1 minus unmminus1+

+δ1F

(+)m unm+1 + δ1δ2F

(minus)n F

(+)m unmminus1 + δ1F

(minus)n F

(minus)m u2nm


+

[F(+)n+m + δ1

(F(+)n F

(minus)m minus F

(minus)n F

(minus)m

)+ δ1δ2F

(minus)n F

(minus)m

]unm


minusδ2(δ1 minus 1)F

(minus)n


]partunm

(392b)

X2D2n =

(F(minus)n F

(minus)m δ1 + F

(minus)n F

(minus)m δ1δ2 minus F

(minus)n F

(minus)m

)un+1m


+

(F(+)n F

(+)m δ1 minus F

(+)n F

(minus)m

)unminus1m


+

(δ1F

(minus)n+m minus F

(minus)m + δ1δ2F

(+)n F

(minus)m

)unm minus (δ1 minus 1)F

(+)m


partunm

(392c)


X2D2m

F(minus)n F

(minus)m δ1unm+1unmminus1 +

(δ1δ2F

(minus)n F

(minus)m + F

(+)n F

(minus)m

)unm+1


++(δ1F

(+)n F

(+)m + F

(minus)n F

(minus)m minus δ1F

(minus)n F

(minus)m

)unmminus1


+δ1F

(minus)n F

(+)m u2nm +

[F(+)n F

(minus)m + (δ2 minus 1)F

(minus)n F

(+)m + F

(+)m

]unm


+δ2(1minus δ2)F

(minus)n minus δ1λF

(+)n


]partunm

(392d)

X3D2n =

(δ1F

(+)n F

(minus)m + δ1δ2F

(+)n F

(minus)m minus F

(+)n F

(minus)m

)un+1m


+

(F(+)n F

(+)m δ1 minus F

(minus)n F

(minus)m

)unminus1m


+

(δ1F

(minus)n F

(minus)m δ2 + F

(minus)n δ1 minus F

(minus)m

)unm + (1minus δ1)F

(+)m


partunm

(392e)

X3D2m =

[(1minus δ1 minus δ1δ2) F

(+)n F

(minus)m unm+1


+

(F(minus)n F

(minus)m minus F

(+)n F

(+)m δ1

)unmminus1 + δ2F

(minus)n


+

(F(+)m minus δ1F

(+)n minus δ1δ2F

(+)n F

(+)m

)unm


minusλδ1(1minus δ1 minus δ1δ2)F

(+)n


]partunm

(392f)

Furthermore the equations iD2 possess also the following pointsymmetries

Y1D21 =(F(+)n F

(+)m + F

(+)n F

(minus)m + F

(minus)n F

(+)m

)unmpartunm (393a)

Y1D22 =[δ1F

(+)n F

(+)m + [1minus δ1(1+ δ2)]F

(minus)n F

(+)m + F

(+)n F

(minus)m

]partunm

(393b)

Y2D21 =[(F(+)n F

(+)m + F

(+)n F

(minus)m + F

(minus)n F

(+)m

)unmminus

minusλF(+)n F

(minus)m + λ[1minus δ1(1+ δ2)]F

(minus)n F

(minus)m

]partunm

(393c)

Y2D22 =[δ1F

(+)n F

(+)m + F

(+)n F

(minus)m [1minus δ1(1+ δ2)]F

(minus)n F

(minus)m

]partunm

(393d)


Y3D21 =[(F(+)n F

(+)m + F

(+)n F

(minus)m + F

(minus)n F

(+)m

)unmminus

minusλF(minus)n F


(minus)n F

(minus)m

]partunm

(393e)

Y3D22 =[δ1F

(+)n F

(+)m + [1minus δ1(1+ δ2)]F

(+)n F

(minus)m minus F

(minus)n F

(minus)m

]partunm

(393f)

The D3 equation (193e) admits the following three-point general-ized symmetries

XD3n =

F(+)n F

(+)m un+1munminus1m +

1

2

(F(minus)m minus F

(minus)n F

(+)m

)unm(un+1m + unminus1m)


+F(minus)n F

(+)m u2nm +

(F(minus)m minus F

(+)n F

(+)m

)unm


partunm

(394a)

XD3m =

F(+)n F

(+)m unm+1unmminus1 +

1

2

(F(minus)n minus F

(+)n F

(minus)m

)unm(unm+1 + unmminus1)


+F(+)n F

(minus)m u2nm +

(F(minus)n minus F

(+)n F

(+)m

)unm


partunm

(394b)

and the point symmetry

YD3 =[F(+)n

(2F

(+)m + F

(minus)m

)+ F

(minus)n

]unmpartunm (395)

Remark 331 The equation D3 (193e) is invariant under the exchangen harr m so the symmetry XD3m (394b) can be obtained from the sym-metry XD3n (394b) performing such exchange

Finally the two forms of D4 possess the following three-point gen-eralized symmetries

X1D4n =

minusδ1F(+)n un+1munminus1m minus

1

2unm(un+1m + unminus1m)


+minusδ1F

(minus)n u2nm + δ2δ3F

(+)m


]partunm

(396a)


X1D4m =

F(minus)m unm+1unmminus1 +

1

2unm(unm+1 + unmminus1)


+δ2F

(+)m u2nm minus δ1δ3F

(+)n


]partunm

(396b)

X2D4n =

minusδ1δ2F(+)n F


1



+minusδ1δ2F

(minus)n F

(+)m u2nm + δ3


]partunm

(396c)

X2D4m =

δ2F(+)n+munm+1unmminus1 +

1



+δ2F

(minus)n+mu

2nm minus δ1δ3F

(+)n


]partunm

(396d)

and no point symmetries

34 the non-autonomous ydkn equation

In 1983 Ravil I Yamilov [167] classified all differential-difference equa-tions of the Volterra class (26) using the generalized symmetry methodFrom the generalized symmetry method one obtains integrability con-ditions which allow to check whether a given equation is integrableMoreover in many cases these conditions enable us to classify equa-tions ie to obtain complete lists of integrable equations belongingto a certain class As integrability conditions are only necessary con-ditions for the existence of generalized symmetries andor conserva-tion laws one then has to prove that the equations of the resultinglist really possess generalized symmetries and conservation laws ofsufficiently high order One mainly construct them using Miura-typetransformations finding Master Symmetries (see Section 35 or prov-ing the existence of a Lax pair [167ndash169] The result of Yamilov clas-sification up to Miura transformation is the Toda equation and theso called Yamilov discretization of the Krichever-Novikov equation(YdKN) a differential-difference equation for the unknown functionqk = qk (t) with k isin Z and t isin R depending on six arbitrary coeffi-cients

dqkdt

=A(qk)qk+1qkminus1 +B(qk)(qk+1 + qkminus1) +C(qk)

qk+1 minus qkminus1 (397)

34 the non-autonomous ydkn equation 103

where

A(qk) = aq2k + 2bqk + c (398a)

B(qk) = bq2k + dqk + e (398b)

C(qk) = cq2k + 2eqk + f (398c)

The integrability of (397) is proven by the existence of point symme-tries [108] and of a Master Symmetries [169] from which one is able toexplicitly write down an infinite hierarchy of generalized symmetriesThe problem of finding the Baumlcklund transformation and Lax pair inthe general case where all the six parameters are different from zeroseems to be still open Some partial results are contained in [6]

In [105] the authors constructed a set of five conditions necessaryfor the existence of generalized symmetries for a class of differential-difference equations depending only on nearest neighbouring inter-action They used the conditions to propose the integrability of thefollowing non-autonomous generalization of the YdKN

dqkdt

=Ak(qk)qk+1qkminus1 +Bk(qk)(qk+1 + qkminus1) +Ck(qk)


where the now k-dependent coefficients are given by

Ak(qk) = aq2k + 2bkqk + ck (3100a)

Bk(qk) = bk+1q2k + dqk + ek+1 (3100b)

Ck(qk) = ck+1q2k + 2ekqk + f (3100c)

with bk ck and ek 2-periodic functions The equation (399) wasshown to possess non-trivial conservation laws of second and thirdorder and two generalized local symmetries of order i and i+ 1 withi lt 4

As it was discussed in Section 32 the symmetry generator associ-ated to a three-point symmetry of a quad equation is a differential-difference equation depending parametrically on the other lattice vari-able see equation (322) Therefore once the generalized symmetriesof an equation or of a class of equations are computed it is nat-ural to ask if the associated differential-difference equations are ofa known type [105 167ndash169] For example let us consider the equa-tions belonging to the ABS classification [2] ie the H equations (123)and the Q equations (124) whose three-point generalized symme-tries where computed systematically in [139] In [102] it was provedthat these symmetries are all particular cases of the YdKN equation(397) Moreover it was showed in [166] that the three-point gener-alized symmetries of the rhombic H4 equations (132) are particularinstances of the non-autonomous YdKN equation (399)

In the previous Section we presented following [66] the three-pointgeneralized symmetries of the trapezoidal H4 (191) and of the H6

equations which were previously unknown It is then natural to ask


if there is any relation between these three-point generalized symme-tries and the non-autonomous YdKN equation (399) The answer isthat all the three-point generalized symmetries are somehow relatedto the non-autonomous YdKN (399) For the rest of this Section wewill discuss the kind of identification needed for the various equa-tions

In the case of the tHε1 equation we have symmetries depending onarbitrary functions given by (389) and by (390) However it can bereadily showed that in (389) if and only if

Bn (ξ) = minus1

ξ α = 0 (3101)

we get a symmetry of YdKN type (399)

XtHε1n =

(un+1m minus unm) (unm minus unminus1m) minus F(+)m ε2α22

un+1m minus unminus1mpartunm (3102)

In the same way we can notice that the symmetry in the direction m(390) if and only if

Bm(ξ) =1

ξ Cm(ξ) =

1

ξ (3103)

becomes a symmetry of the non-autonomous YdKN type (399)

XtHε1m =

F(+)m

(1+ ε2unm+1unmminus1

)+ F

(minus)m

(1+ ε2u2nm

)unm+1 minus unmminus1

partunm

(3104)

On the other hand we have that the symmetries of the tHε2 equation(191b) and of the tHε3 equation (191c) as given by equations (387)are naturally identified as particular cases of the non-autonomousYdKN equations (399) It is worth to note that since the only non-autonomous coefficients in the trapezoidal H4 equations are depend-ing on the lattice variable m the symmetries in the n direction ofthese equations actually are sub-cases of the autonomous YdKN equa-tion (397)

Turning to the H6 equations it is easy to check that the symmetriesof the iD2 equations (193b193c193d) as given by equation (392)are not in the form of the YdKN equation (399) However a linearcombination with point symmetries (393)

ZiD2j = XiD2j +KiD21j YiD21 +KiD22j Y

iD22 j = nm i = 1 2 3 (3105)

for prescribed values of KiD21j and KiD22j is in the non-autonomousYdKN form (399) ie

K1D21j = K2D21j = minusK3D21j =1

2

K1D22j = K2D22j = K3D22j = 0(3106)


We notice that we are allowed to consider (3105) as bona fide three-point generalized symmetries since choosing them as new generatorsjust corresponds to choose a new basis for the linear space of thethree-point generalized symmetries On the other hand the symme-tries of the D3 equation (193e) as given by equation (394) are alreadysub-cases of the non-autonomous YdKN equation (399) Finally wehave that also the symmetries of the iD4 equations (193f193g) aresub-cases of the non-autonomous YdKN equation (399)

Taking into account the result presented in [166] reproduced forsake of completeness in Appendix E we can then state that all thethree-point generalized symmetries of the equations belonging to the clas-sification made by R Boll [20ndash22] are sub-cases of the non-autonomousYdKN equation (399) This result extend what was obtained in [102]for the ABS class of equations We can then build Table 31 where wecan show the explicit form of the coefficients of the non-autonomousYdKN equation corresponding to the aforementioned three-point gen-eralized symmetries

10

6g

en

er

al

iz

ed

sy

mm

et

rie

sa

nd

th

en

ay

dk

ne

qu

at

io

n

Table 31 Identification of the coefficients in the symmetries of the rhombic H4 equations trapezoidal H4 equations and H6 equations with those ofthe non autonomous YdKN equation Here the symmetries of tHε1 in the m direction are the subcase (3104) of (390) while those in then direction are the subcase (3102) of (389) The symmetries of the iD2 equations (392) are combined with the point symmetries (393)according to the prescriptions (3105) and the coefficients given by (3106)

Eq k a bk ck d ek f

rHε1

n 0 0 minusεF(+)n+m 0 0 1

m 0 0 minusεF(+)n+m 0 0 1

rHε2

n 0 0 minus4εF(+)n+m 0 1minus 4εαF

(minus)n+m 2αminus 4εα2

m 0 0 minus4εF(+)n+m 0 1minus 4εβF

(minus)n+m 2βminus 4εβ2

rHε3

n 0 0 minusεF

(+)n+m

α

1

20 δα

m 0 0 minusεF

(+)n+m

β

1

20 δβ

tHε1

n 0 0 minus1 1 0 minusε2α22F(+)m

m 0 0 ε2F(+)m 0 0 2

tHε2

n 0 0 minus1 1 εα22F(+)m α22 minus εα

22

(4α2 + 8α3 + εα

22

)F(+)m

m 0 0 minusεF(+)m 0

1


(minus)m minusα3 minus ε (α2 +α3)

2


34

th

en

on

-a

ut

on

om

ou

sy

dk

ne

qu

at

io

n1

07


Eq k a bk ck d ek f

tHε3

n 0 0 minusα221

2α2(1+α

22) 0 minusε2δ2F

(+)m (1minusα22)

2

m 0 0 minusε2F(+)m

1

2α3 0 minusα23δ

2

1D2n 0 0 0 0

1

2[δ1(1+ δ2) minus 1]F

(+)n F

(+)m

+1

2F(minus)n F

(+)m minus

1

2F(minus)n F

(minus)m

minusδ2F(minus)m

m 0 0 minusF(minus)n F

(+)m δ1 0

1

2(δ1(1minus δ2 minus 1)F

(minus)n F

(minus)m

minus1

2F(+)n F

(+)m minus

1

2δ1F

(minus)n F

(+)m

δ2(δ1 minus 1)F(minus)n

2D2n 0 0 0 0

1

2[1minus δ1(1+ δ2)]F

(+)n F

(minus)m

+1

2F(minus)n F

(minus)m minus

1

2δ1F

(minus)n F

(+)m

(δ1 minus 1)F(+)m

m 0 0 minusδ1F(minus)n F

(minus)m 0

1

2[δ1(1minus δ2) minus 1]F

(minus)n F

(+)m

minus1

2F(+)n F

(+)m minus

1

2δ1F

(minus)n F

(+)m

δ2 [δ1 minus 1] F(minus)n + λδ1F

(+)n


10

8g

en

er

al

iz

ed

sy

mm

et

rie

sa

nd

th

en

ay

dk

ne

qu

at

io

n


Eq k a bk ck d ek f

3D2n 0 0 0 0

1

2[δ1(1+ δ2) minus 1]F

(minus)n F

(minus)m

+1

2F(+)n F

(minus)m +

1

2δ1F

(minus)n F

(+)m

(1minus δ1)F(+)m

m 0 0 0 0

1


(+)n F

(+)m

minus1

2F(+)n F

(minus)m +

1

2δ1F

(minus)n F

(+)m

δ1λ[minusδ1(1+ δ2)]F(+)n minus δ2F

(minus)n

D3n 0 0 F

(+)n F

(+)m 0

1

2

(F(+)n F

(minus)m + F

(minus)n F

(minus)m minus F

(+)n F

(+)m

)0

m 0 0 F(+)n F

(+)m 0

1

2

(F(minus)n F

(+)m + F

(minus)n F

(minus)m minus F

(+)n F

(+)m

)0

1D4n 0 0 minusδ1

(F(+)n F

(+)m + F

(+)n F

(minus)m

)minus1

20 δ2δ3F

(+)m

m 0 0 δ2

(F(+)n F

(+)m + F

(minus)n F

(+)m

) 1

20 minusδ1δ3F

(+)n

2D4n 0 0 minusF

(+)n F

(+)m δ1δ2

1

20 δ3

m 0 0 δ2

(F(+)n F

(+)m + F

(minus)n F

(minus)m

) 1

20 minusδ1δ3F

(+)n

35 integrability properties of the non autonomous ydkn equation and its sub-cases 109

35 integrability properties of the non autonomous ydkn

equation and its sub-cases

In the previous Section we saw that the fluxes of all the generalizedthree point symmetries of the H4 and H6 equations are eventuallyrelated either to the YdKN (397) or to the non autonomous YdKNequation (399) In this Section we extend those results by consideringthe ideas in [102] We will use the Master Symmetries approach Mas-ter Symmetries are particular kind of symmetries which can generatethe whole hierarchy of symmetries of a given equation starting fromthe given symmetry The notion of master symmetry has been intro-duced in [43] see also [42 46 47 126] In the continuous case mastersymmetries usually are non-local ie they contain terms involvingintegrations In the semi-discrete and discrete case this would corre-spond to the presence of operators like (T minus Id)minus1 where T is a latticetranslation operator which give raise to infinite summations Fortu-nately enough in the semi-discrete and discrete case there are manylocal master symmetries [4 7 102 127 128 139 166 170 171 173] iemaster symmetries with no such dependence

Here we will not discuss the general method to find Master Sym-metries for differential-difference or quad equation but we will justpresent the method we can use to construct the Master Symmetries ofthe non-autonomous YdKN equation and its sub-cases following [454 102 169] Let us consider differential-difference equations of theform

unτ = ϕn(τun+kun+kminus1 un+k prime+1un+k prime) k prime lt k (3107)

Now let us suppose we have two such equations (3107) namely twofunctions ϕ(1)

n and ϕ(2)n depending on different ldquotimesrdquo τ1 and τ2

and depending on uα for n+ k gt α gt n+ k prime and n+ l gt α gt n+ l prime

respectively We can then require that the two equations commute

unτ1τ2 minus unτ2τ1 = Dτ2ϕ(1)n minusDτ1ϕ

(2)n (3108)

where Dτi is the total derivative with respect to τi and it is defined as

Dτi =part

partτi+sumjisinZ

ϕ(i)n+j

part

partun+j (3109)

Let us define a Lie algebra structure on the set of functions ϕn of theform (3107) For any functions ϕ(1)

n and ϕ(1)n we introduce the equa-

tions unτ1 = ϕ(1)n and unτ2 = ϕ

(2)n and the corresponding total

derivatives Dτi as in (3109) Then we can define the following opera-tion on the functions ϕ(1)

n and ϕ(2)n

[ϕ(1)n ϕ(2)

n ] = Dτ2ϕ(1)n minusDτ1ϕ

(2)n (3110)


The result of this operation is a new function ϕ(3)n depending on a

finite number of shifts of un We prove that the operation [ ] is aLie bracket ie that is bilinear skew-symmetric and satisfies the Jacobiidentity

[[ϕ(1)n ϕ(2)

n ]ϕ(3)n ] = [[ϕ

(1)n ϕ(3)

n ]ϕ(2)n ] + [ϕ

(1)n [ϕ(2)

n ϕ(3)n ]] (3111)

Bilinearity follows from the fact that the total derivative operators(3109) are linear It is obviously skew-symmetric

[ϕ(1)n ϕ(2)

n ] = minus(Dτ1ϕ

(2)n minusDτ2ϕ

(1)n

)= minus[ϕ

(2)n ϕ(1)

n ] (3112)

Finally Jacobi identity (3111) can be checked by direct computationA differential-difference equation of the form

unτ = gn(un+kun+kminus1 un+k prime+1un+k prime) k prime lt k (3113)

is called a generalized symmetry for a Volterra-like equation (26) if itsright hand side commutes with the right hand side of the Volterra-like equation ie [gn fn] = 0 This generalized symmetry is callednon-trivial if k gt 1 and k prime lt minus1

A differential-difference of the first differential-order of the form

unτ = ϕn(τun+1ununminus1) (3114)

where the dependence on τ and on the lattice variable n may beexplicit is called a Master Symmetry for a Volterra-like equation (26)if the function

gn = [ϕn fn] (3115)

is the right hand side of a generalized symmetry This generalizedsymmetry must be nontrivial ie in (3113) k gt 1 and k prime lt minus1 Thefunction ϕn satisfies then the following equation

[[ϕn fn] fn] = 0 (3116)

Any generalized symmetry (3113) has a trivial solution ϕn = gnThe Master Symmetry corresponds to a nontrivial solution of (3116)

A practical way of computing Master Symmetries is as follows Sup-pose that we are given a Volterra-like equation (26) Suppose thatthere exists a Master Symmetry of the form

unτ = nfn (un+1ununminus1) (3117)

where fn is the right hand side of (26) Then assume that fn dependson some constants say ki i = 1 K let us replace such constantsby functions of the master symmetry ldquotimerdquo τ

ki rarr κi = κi (τ) i = 1 M (3118)

35 integrability properties 111

We can impose the conditions (3115) and (3116) so that (3117) isactually a Master Symmetry When imposing (3115) due to the def-inition of the total derivative Dτ (3109) we get a set of first orderdifferential equations for the κi functions with the initial conditionsgiven by the original constants

κ primei (τ) = Gi (κ1 (τ) κM (τ)) i = 1 M (3119a)

κi (0) = ki (3119b)

Then we can derive the symmetries for the original equation (26) atany order from the master symmetry (3117) by putting τ = 0 in theresulting symmetry

Let us construct the Master Symmetries of the non-autonomousYdKN equation (399) Then we can proceed with the method wediscussed above Using the fact that bk ck and ek are two periodic

bk = b+ (minus1)kβ ck = c+ (minus1)kγ ek = e+ (minus1)kε (3120)

and substituting the coefficients a b c γ d e ε and f with functionof τ we obtain the following expression for (3100)

Ak(qk τ) = a(τ)q2k + 2[b(τ) + (minus1)kβ(τ)

]qk

+ c(τ) + (minus1)kγ(τ)

(3121a)

Bk(qk τ) =[b(τ) minus (minus1)kβ(τ)

]q2k + d(τ)qk

+ e(τ) minus (minus1)kε(τ)

(3121b)

Ck(qk τ) =[c(τ) minus (minus1)kγ(τ)

]q2k

+ 2[e(τ) + (minus1)kε(τ)

]qk + f(τ)

(3121c)

From (3121) we build up the τ-dependent version of (399)

dqkdt

=Ak(qk τ)qk+1qkminus1 +Bk(qk τ)(qk+1 + qkminus1) +Ck(qk τ)

qk+1 minus qkminus1

(3122)

We make the ansatz (3117) for the Master Symmetry

dqkdτ

= nAk(qk τ)qk+1qkminus1 +Bk(qk τ)(qk+1 + qkminus1) +Ck(qk τ)

qk+1 minus qkminus1

(3123)

Commuting the flows of Dt and Dτ we must obtain a five point sym-metry g(1)n according to (3115) This obtained function has a purelythree point part which is of the same form as (3122) but with differ-ent coefficients Annihilating this part we obtain some equations forthe coefficients a b β c γ d e ε f

dadτ

(τ) = a (τ)d (τ) + 2β2 (τ) minus 2b2 (τ) (3124a)


dbdτ

(τ) = minusb (τ) c (τ) +β (τ)γ (τ) + a (τ) e (τ) (3124b)

dβdτ

(τ) = c (τ)β (τ) minus b (τ)γ (τ) minus a (τ) ε (τ) (3124c)

dcdτ

(τ) = 2b (τ) e (τ) minus d (τ) c (τ) minus 2β (τ) ε (τ) (3124d)

dγdτ

(τ) = 2β (τ) e (τ) minus d (τ)γ (τ) minus 2ε (τ)b (τ) (3124e)

dddτ

(τ) = minusc2 (τ) + a (τ) f (τ) + γ2 (τ) (3124f)

dedτ

(τ) = minusc (τ) e (τ) minus γ (τ) ε (τ) + b (τ) f (τ) (3124g)

dεdτ

(τ) = γ (τ) e (τ) minusβ (τ) f (τ) + c (τ) ε (τ) (3124h)

dfdτ

(τ) = f (τ)d (τ) minus 2e2 (τ) + 2ε2 (τ) (3124i)

If the coefficients satisfy the system (3124) then it is easy to showthat the obtained g(1)n is a generalized symmetry depending on five-points We remark that the system (3124) in its generality it is impos-sible to solve but since the right hand is polynomial we are ensuredby Cauchyrsquos theorem [162] that such solution always exists in a neigh-bourhood of τ = 0

The solutions with the initial conditions given by Table 31 willthen yield explicit form of the Master Symmetries in all the relevantsub-cases By using the master symmetry constructed above we canconstruct infinite hierarchies of generalized symmetries of the H4 andH6 equations in both directions Furthermore since for every H4 andH6 equation we have a = b = β = 0 we can in fact use the simplersystem where a (τ) = b (τ) = β (τ) = 0

dcdτ

(τ) = minusd (τ) c (τ) (3125a)

dγdτ

(τ) = minusd (τ)γ (τ) (3125b)

dddτ

(τ) = minusc2 (τ) + γ2 (τ) (3125c)

dedτ

(τ) = minusc (τ) e (τ) minus γ (τ) ε (τ) (3125d)

dεdτ

(τ) = γ (τ) e (τ) + c (τ) ε (τ) (3125e)

dfdτ

(τ) = f (τ)d (τ) minus 2e2 (τ) + 2ε2 (τ) (3125f)

As an example in Table 32 we list the form of the τ-dependentcoefficients in the case of the trapezoidal H4 equations (191) Theremaining cases can be computed analogously

35

in

te

gr

ab

il

it

yp

ro

pe

rt

ie

s1

13

Eq Dir c (τ) γ (τ) d (τ)

tHε1

nminus1

τ+ 10

1

τ+ 1

mε2

2

ε2

20

tHε2

nminus1

τ+ 10

1

τ+ 1

m minusε

2minusε

20

tHε3

nα22(α22 minus 1

)eminus

12τα2(α

22minus1)

eminusτα2(α22minus1) minusα22

01

2

α2(1minusα22

) [eminusτα2(α

22minus1) +α22

]+eminusτα2(α

22minus1) minusα22

m minusε2eminus

12α3τ

2minusε2eminus

12α3τ

2

α32

Eq Dir e (τ) η (τ) f (τ)

tHε1

n 0 0 minusε2α22F(+)m (τ+ 1)

m 0 0 2

tHε2

n εα22F

(+)m (τ+ 1) 0 minusα22

[minus1+α2

2F(+)m (τ+ 1)2 ε2 + 4F

(+)m (α2 + 2α3) ε

](τ+ 1)

m1

2+1

4(τminus 2α2 minus 2α3) ε minus

1


1

4(τminus 2α2 minus 2α3)

2 εminusα3 minus1

2τ

tHε3

n 0 0ε2δ2F

(+)m

(α22 minus 1

) (eminusτα2(α

22minus1) minusα2

2)

eminus12τα2(α22minus1)

m 0 0 minusα23δ2e

12α3τ

Table 32 The value of the coefficients of the Master Symmetry (3123) obtained by solving the system (3125) in the case of the trapezoidal H4 equations (191)


Now it is a known fact that Master Symmetries can exists also forlinearizable (C-integrable) equations eg the Burgers equation [103]as infinite hierarchies of generalized symmetries exists both for C andS integrable equations So to end this Section we apply the AlgebraicEntropy test as discussed in Chapter 2 which shows heuristicallythat the relevant sub-cases of the non-autonomous YdKN equation(399) have quadratic growth and therefore are genuine integrableequations

We look for the sequence of degrees of the iterate map for the non-autonomous YdKN equation (399) and in its particular cases foundin the previous section We find for all the cases except for rHε1 iefor the symmetries (3102 3104) of tHε1 and for iD2 equations thefollowing values

1 1 3 7 13 21 31 43 57 73 91 111 133 157 (3126)

This sequence has the following generating function

g(z) =1minus 2z+ 3z2

(1minus z)3(3127)

which gives the quadratic fit for the sequence (3126)

dl = l(lminus 1) + 1 (3128)

Therefore the Algebraic Entropy is zeroFor the three-point generalized symmetry in the n direction (E1a)

of the equation rHε1 we have the somehow different situation that thesequence growth is different depending if we consider the even orodd values of the m variable

m = 2k 1 1 3 7 10 17 23 33 42 55 67 83 98 117 (3129a)

m = 2k+ 1 1 1 3 4 9 13 21 28 39 49 63 76 93 109 (3129b)

These sequences have the following generating functions and asymp-totic fits

m = 2kg(z) =

2z5 minus 3z4 + 3z3 + z2 minus z+ 1

(1minus z)3(z+ 1)

dl =3

4l2 minus lminus

5(minus1)l minus 21

8

(3130a)

m = 2k+ 1g(z) =

(z2 + z+ 1)(2z2 minus 2z+ 1)

(1minus z)3(z+ 1)

dl =3

4l2 minus

3

2lminus

5(minus1)l minus 19

8

(3130b)

The symmetry in the m direction (E2b) of the equation rHε1 has the

same behavior obtained by exchanging m with n in formulaelig (3129-3130)


The three-point generalized symmetry of the equation tHε1 in the

n direction has almost the same growth as that obtained for m odd(3129b 3130b) with fit given by

dl =3

4l2 minus

5

4l+ (minus1)l

l

4+

(minus1)l + 15

8 (3131)

Notice the presence of a highly oscillatory term l(minus1)l which at ourknowledge it is observed for the first time For m even we have thesame growth as (3126) The three-point generalized symmetry of theequation tHε1 in the m direction has the same growth as the even oneof tHε1 (3129a 3130a)

For the three-point generalized symmetries (392) of the iD2 equa-tion we have different growth according to the even or odd values ofthe m and n variables These sequences are slightly lower than in thecase of equations Hε1 however always corresponding to a quadraticasymptotic fit

This shows that the whole family of the non autonomous YdKN isintegrable (S-integrable) according to the Algebraic Entropy test andthey should not be linearizable (C-integrable) even if the equations ofwhich they are symmetries are linearizable

We conclude this Section by giving an example on how the gen-eralized symmetries criterion of integrability [169] and the AlgebraicEntropy give the same result on a non-integrable equation ie anexample of how two definitions of integrability coincide As the sym-metries of the tHε1 equation depend on arbitrary functions not all ofthem will produce an integrable flux Let us consider the case of theflux (389) when ε = 0 Bn(ξ) = minus1ξ and α = 1 This corresponds tothe following symmetry

XPn =un+1munminus1m minus u2nm

un+1m minus 2unm + unminus1mpartunm (3132)

Following [169] necessary condition for the flux of (3132) dunmdt =

XPnunm to be integrable is that given

p1 = logpartfn

partun+1= 2 log


un+1m minus 2unm + unminus1m

)(3133)

we must have

dp1dt

=minus2u2nm + 4unmun+1m minus 2u2n+1m

(un+2m minus 2un+1m + unm) (minusun+1m + 2unm minus unminus1m)

minus2 (unminus1m minus un+1m) (minusun+1m + unm)

(un+1m minus 2unm + unminus1m)2

+2u2nm minus 2unmun+1m minus 2unminus1munm + 2unminus1mun+1m

(unm minus 2unminus1m + unminus2m) (minusun+1m + 2unm minus unminus1m)

= (T minus 1)gn


(3134)

for any function gn defined on a finite portion of the lattice for ex-ample such that gn = gn(un+1munmunminus1munminus2m) We searchthe function gn using the partial sum method [169] and we find an ob-struction at the third passage Then the function gn does not existsand therefore we conclude that the flux of (3132) is a non integrabledifferential-difference equation

Using the algebraic entropy test on the flux of (3132) we find thefollowing values for the degrees of the iterates

1 1 3 9 27 81 273 729 (3135)

which gives us the following generating function

g(s) =1minus 2s

1minus 3s (3136)

and the entropy is clearly non-vanishing η = log 3 So the non-integrabilityresult obtained by the Algebraic Entropy test agrees with those ob-tained by applying the formal generalized symmetry method

36 a non-autonomous generalization of the QV equa-tion

Up to now we constructed the three-point generalized symmetriesof the trapezoidal H4 equations (191) and of the H6 equation (193)We then showed that these three-point generalized symmetries arerelated to some sub-cases of the non-autonomous YdKN equation[105] Taking into account the results about the rhombic H4 equa-tions in [166] we were able to produce Table 31 where it shownthat the three-point generalized symmetries of any quad equationcoming from the Bollrsquos classification [20ndash22] is in the form of thenon-autonomous YdKN equation [105] We then showed how usingknown methods we can construct the Master Symmetries of theseequations generating a hierarchy of equations and that accordingto the Algebraic Entropy test these are genuine integrable equationssince their growth is quadratic

We finally note that as was proven in [102] for the YdKN (397) noequation belonging to the Boll classification has a generalized symme-try which corresponds to the general non-autonomous YdKN equa-tion (399) In all the cases of the Boll classification one has a = bk = 0

as is displayed by Table 31

36 a non-autonomous generalization of the QV equation 117

In [166] it was shown that the QV equation introduced in [156]

QV =a1unmun+1munm+1un+1m+1

+a20 (unmunm+1un+1m+1 + un+1munm+1un+1m+1

+unmun+1mun+1m+1 + unmun+1munm+1)

+a30 (unmun+1m + unm+1un+1m+1)

+a40 (unmun+1m+1 + un+1munm+1)

+a50 (un+1mun+1m+1 + unmunm+1)

+a60 (unm + un+1m + unm+1 + un+1m+1)

+a7 = 0

(3137)

admits a symmetry in the direction n

dunm

dt=

hn

un+1m minus unminus1mminus1

2partun+1mhn (3138)

where

hn(unmun+1m) = QVpartunm+1partun+1m+1

QV

minus(partunm+1

QV) (partun+1m+1

QV) (3139)

and a symmetry in the direction m

dunm

dt=

hm

unm+1 minus unmminus1minus1

2partunm+1

hm (3140)

where

hm(unmunm+1) = QVpartun+1mpartun+1m+1QV

minus(partun+1mQV

) (partun+1m+1

QV) (3141)

which are of the same form as the YdKN (397)The connection formulaelig ie the relations between the coefficient

of QV and the coefficients of its YdKN generalized symmetry (397)in the n direction YdKN are

a = a30a1 minus a220 (3142a)

b =1

2[a20(a30 minus a50 minus a40) + a60a1] (3142b)

c = a20a60 minus a40a50 (3142c)

d =1

2[a230 minus a

240 minus a

250 + a1a7] (3142d)

e =1

2[a60(a30 minus a40 minus a50) + a20a7] (3142e)

f = a30a7 minus a260 (3142f)

The connection formulaelig between the coefficient of QV and the mdirection YdKN (397) are

a = a50a1 minus a220 (3143a)


b =1


c = a20a60 minus a40a30 (3143c)

d =1

2[a250 minus a

240 minus a

230 + a1a7] (3143d)

e =1


f = a50a7 minus a260 (3143f)

The connection formulaelig (31423143) can be seen as a set of cou-pled nonlinear algebraic equations between the seven parameters a1a20 a30 a40 a50 a60 and a7 of the QV (3137) and the six onesa b c d e and f of the YdKN equation (3104) This tells us thatthe YdKN equation (3104) with coefficients given by (31423143) isa three-point generalized symmetry of the QV equation (3137) If asolution of (31423143) exists ie one is able to express the a1 a20a30 a40 a50 a60 and a7 in term of a b c d and f then the QV

equation (3137) maybe after a reparametrization turns out to be aBaumlcklund transformation of the YdKN [6 99] as explained in Section32 We furthermore remark that in general a and b will be non-zeroso that the three-point generalized symmetries of the QV equation(3137) belong to the class of the general YdKN equation (397)

From the results obtained in Section 34 and the above remarks weare lead to conjecture the existence a non autonomous generalizationof the QV equation To obtain such generalizations we first have torecall how the QV equation (3137) was obtained in [156] In [156] wasshowed that the QV equation (3137) is the most general multi-linearequation on a quad graph possessing Klein discrete symmetries iesuch that

Q (un+1munmun+1m+1unm+1) =

τQ (unmun+1munm+1un+1m+1)

Q (unm+1un+1m+1unmunm+1) =

τprimeQ (unmun+1munm+1un+1m+1)

(3144)

where τ τprime = plusmn1We have many possible ways of searching for a generalization of

the QV equation (3137) Since we want to generalize the QV equation(3137) to a non-autonomous equation let us consider the most gen-


eral multi-linear equation in the lattice variables with two-periodiccoefficients

p1unmun+1munm+1un+1m+1

+p2unmunm+1un+1m+1 + p3un+1munm+1un+1m+1

+p4unmun+1mun+1m+1 + p5unmun+1munm+1

+p6unmun+1m + p7unm+1un+1m+1 + p8unmun+1m+1

+p9un+1munm+1 + p10un+1mun+1m+1 + p11unmunm+1

+p12unm + p13un+1m + p14unm+1 + p15un+1m+1 + p16 = 0

(3145)

ie such that the coefficients pi have the following expression

pi = pi0 + pi1(minus1)n + pi2(minus1)

m+ pi3(minus1)n+m i = 1 16

(3146)

We will denote the left hand side of (3145) byQ (unmun+1munm+1un+1m+1 (minus1)n (minus1)m)A possibility is to require that the functionQ (unmun+1munm+1un+1m+1 (minus1)n (minus1)m)

to respect a strict discrete Klein symmetry (3144) and to require thatthe connection forumulaelig provide a non-autonomous YdKN equation(399) The requirement of the Klein symmetries gives us

p1 equiv a1 p2 = p3 = p4 = p5 equiv a2

p6 = p7 equiv a3 p8 = p9 equiv a4 p10 = p11 equiv a5

p12 = p13 = p14 = p15 equiv a6 p16 = a7

(3147)

where the ai are still two-periodic functions of the lattice variablesChoosing the coefficients for example as

a1 = 1+ (minus1)n a2 = (minus1)n

a3 = minus1+ (minus1)n a5 = (minus1)n a4 = 1+ 2(minus1)n

a6 = 1+ (minus1)n a7 = 4+ 2(minus1)n

(3148)

the we have such properties but this is not the only possible choiceIn this case performing the Algebraic Entropy test the equation turnsout to be integrable Its generalized symmetries however are not nec-essarily in the form of a non-autonomous YdKN equation (399) Adifferent non-autonomous choice of the coefficients of (3145) suchthat (3142) is satisfied for the coefficients of the non-autonomousYdKN (399) gives by the algebraic entropy test a non integrableequation

A further possibility is to generalize the original Klein symmetryby considering the discrete symmetries possessed by all the equa-tions belonging to the Bollrsquos classification It is easy to see that all the


equations belonging to the Bollrsquos classification possess the followingdiscrete symmetry

Q (un+1munmun+1m+1unm+1 (minus1)n (minus1)m) =

τQ (unmun+1munm+1un+1m+1minus(minus1)n (minus1)m)

Q (unm+1un+1m+1unmun+1m (minus1)n (minus1)m) =

τ primeQ (unmun+1munm+1un+1m+1 (minus1)nminus(minus1)m)

(3149)

when τ = τprime = 1 Furthermore it is possible to show that if the quadequation Q is autonomous then the condition (3149) reduces to theusual Klein discrete symmetry (3144) Therefore we will say that ifa non-autonomous quad equation Q of the form (3145) satisfies thediscrete symmetries (3149) that Q admits a non-autonomous discreteKlein symmetry

If we impose the non autonomous Klein symmetry condition (3149)with τ = τ prime = 1 the 64 coefficients of (3145) turn out to be relatedamong themselves and we can choose among them 16 independentcoefficients In term of the 16 independent coefficients (3145) reads

a1unmun+1munm+1un+1m+1

+[a20 minus (minus1)n a21 minus (minus1)m a22 + (minus1)n+m a23

]unmunm+1un+1m+1

+[a20 + (minus1)n a21 minus (minus1)m a22 minus (minus1)n+m a23

]un+1munm+1un+1m+1

+[a20 + (minus1)n a21 + (minus1)m a22 + (minus1)n+m a23

]unmun+1mun+1m+1

+[a20 minus (minus1)n a21 + (minus1)m a22 minus (minus1)n+m a23

]unmun+1munm+1

+ [a30 minus (minus1)m a32]unmun+1m

+ [a30 + (minus1)m a32]unm+1un+1m+1

+[a40 minus (minus1)n+m a43

]unmun+1m+1

+[a40 + (minus1)n+m a43

]un+1munm+1

+ [a50 minus (minus1)n a51]un+1mun+1m+1

+ [a50 + (minus1)n a51]unmunm+1


]unm


]un+1m


]unm+1


]un+1m+1

+a7 = 0

(3150)

Upon the substitution a21 = a22 = a23 = a32 = a43 = a51 =

a61 = a62 = a63 = 0 (3150) reduces to the QV equation (3137)Therefore we will call (3150) the non autonomous QV equation

If we impose the non-autonomous Klein symmetry condition (3149)with the choice τ = 1 and τ prime = minus1 we get an expression which re-duces to (3150) by multiplying it by (minus1)n and redefining the coef-ficients In an analogous manner the two remaining cases τ = minus1


τ prime = 1 and τ = τ prime = minus1 can be identified with the case τ = τ prime = 1

multiplying by (minus1)n and (minus1)n+m respectively and redefining thecoefficients Therefore the only equation belonging to the class of thelattice equation possessing the non autonomous Klein symmetries isjust the non-autonomous QV equation (3150)

We note that the non-autonomous QV equation contains as partic-ular cases the rhombic H4 equations the trapezoidal H4 equationsand the H6 equations The explicit identification of the coefficients ofsuch equations is given in Table 33

In order to establish if (3150) is integrable we use the Algebraic En-tropy integrability test as explained in Chapter 2 using the programae2dpy see Appendix B Applying it to the non autonomous QV

equation we find in all directions the following sequence of degrees

1 3 7 13 21 31 43 57 73 91 111 133 (3151)

which is the same as for the autonomous QV equation [156] Thegenerating function for the sequence (3151) is

g(z) =1+ z2

(1minus z)3 (3152)

which implies that we have the following quadratic fit for the growth

dk = k(k+ 1) + 1 (3153)

and thus the Algebraic Entropy is zero This is a strong indication ofthe integrability of the non autonomous QV equation (3150)

Using (3138 3139) or (3140 3141) with QV substituted by its non-autonomous version we get a version of the non-autonomous YdKN(399) however the proof that this is effectively a generalized sym-metry of the non-autonomous QV encounters serious computationaldifficulties

We can prove by a direct computation its validity for the followingsub-cases

bull When Qv equation is non autonomous with respect to one di-rection only either n or m All the trapezoidal tH4 equationsbelong to these two sub-classes

bull For all the H6 equations which are non autonomous in bothdirections

Its validity for the autonomous QV and for all the rhombic rH4 equa-tions was already showed respectively in [165] and [166] Howeverwe cannot prove its validity for the general case (3150)

Here in the following we compute the connection formulaelig for thegeneral non autonomous case (3150) For the n directional symmetrywe have

a = a1a30 minus a220 + a

221 minus a

222 + a

223

minus (minus1)m(2a20a22 minus 2a21a23 + a1a32)

(3154a)


b =1

2a20(a30 minus a50 minus a40)

+1

2(a1a60 + a22a32 minus a23a43 minus a21a51)

minus1

2(minus1)ma22(a50 + a30 + a40)

+1

2(minus1)m(a23a51 + a1a62 + a20a32 + a21a43)

(3154b)

β =1

2a21(a30 minus a40 + a50)

+1

2(a23a32 minus a22a43 + a20a51 minus a1a61)

+1

2(minus1)ma1a63 minus a23(a30 + a40 minus a50)

minus1

2(minus1)m(a21a32 + a20a43 minus a22a51)

(3154c)

c = a20a60 minus a40a50 minus a21a61 minus a23a63 + a22a62

minus (minus1)m[a22a60 minus a43a51 minus a23a61 + a20a62 minus a21a63](3154d)

γ = a40a51 + a21a60 minus a20a61 + a23a62 minus a22a63

+ (minus1)m[a22a61 minus a43a50 minus a23a60 minus a21a62 + a20a63](3154e)

d =1

2(a230 minus a

240 minus a

250 + a1a7 minus a

232 + a

243 + a

251)

minus 2(minus1)m(a22a60 + a23a61 + a20a62 + a21a63]

(3154f)

e =1


+1

2(a20a7 + a51a61 minus a32a62 + a43a63)

+1

2(minus1)m(a32a60 + a43a61 + a51a63 minus a22a7)

minus1

2(minus1)ma62(a30 + a40 + a50)

(3154g)

ε =1

2a61(a30 minus a40 + a50)

minus1

2(a51a60 minus a43a62 + a32a63 + a21a7)

+1

2(minus1)m[a43a60 + a32a61 minus a51a62 + a23a7

+1

2(minus1)ma63(a50 minus a30 minus a40)

(3154h)

f = a30a7 minus a260 minus a

262 + a

263 + a

261

minus (minus1)m(2a60a62 minus 2a61a63 minus a32a7)

(3154i)

The non-autonomous QV is not symmetric in the exchange of n andm so its symmetries in the m direction are different from those in then direction and so are the connection formulaelig However for the sakeof simplicity as they are not essentially different from those presented


in equation (3154) we do not write them here but present them inAppendix F

Table 33 Identification of the coefficients of the non autonomous QV equation with those of the Bollrsquos equations Since a1 = a2i = 0 for all equationsconsidered in this Table these coefficients absent

Eq a30 a32 a40 a43 a50 a51 a60 a61 a62 a63 a7

rHε1 1 0 1

2 ε(αminusβ) 12 ε(αminusβ) minus1 0 0 0 0 0 βminusα

rHε2 1 0 2ε (βminusα) 2ε (βminusα) minus1 0 minus(αminusβ) (εα+ 1+ εβ) 0 0 ε

(β2 minusα2

)minus(αminusβ)

(2εα2 +α+ 2εβ2 +β

)rHε3 α 0 1

2

ε(β2minusα2

)αβ

12

ε(β2minusα2

)αβ minusβ 0 0 0 0 0 δ

(α2 minusβ2

)tHε1 minus 12α2ε

2 minus 12α2ε2 minus1 0 1 0 0 0 0 0 minusα2

tHε2 εα2 εα2 minus1 0 1 0 1

2α2 [2+ ε(2α2 +α3)] 0 εα2α3 + 12 εα2

2 0 α2

[α2 + 2α3 + ε (α2 +α3)

2]

tHε3

12

ε2(1minusα2

2)

α3α2

12

ε2(1minusα2

2)

α3α2α2 0 minus1 0 0 0 0 0 δ2α3

(1minusα22

)1D2

12 δ1

12 δ1

12 minus 12 0 0 1

2 minus 14 (δ1 minus δ2)

14 (δ2 minus δ1) minus 14 (δ1 + δ2)

12 minus 1

4 (δ1 + δ2) 0

2D212 minus 12

12 δ1

12 δ1 0 0 1

2 minus 14 (δ1 minus δ2 + δ1λ)

14 (δ1λminus δ1 + δ2)

12 minus 1

4 (δ1 minus δ1λ+ δ2) minus 14 (δ1 + δ1λ+ δ2) minusδ1δ2λ

3D212 minus 12 0 0 1

2 δ1 minus 12 δ112 minus 1

4 (δ1 minus δ2 + δ1λ)14 (δ1 + δ1λ+ δ2)

12 minus 1

4 (δ1 minus δ1λ+ δ2)14 (δ1 minus δ1λminus δ2) minusδ1δ2λ

D312

12

12

12

12 minus 12

14

14 minus 14 minus 14 0

1D412 δ2

12 δ2 1 0 1

2 δ1 minus 12 δ1 0 0 0 0 δ3

2D4 1 0 12 δ2

12 δ2

12 δ1 minus 12 δ1 0 0 0 0 δ3

4D A R B O U X I N T E G R A B I L I T Y A N D G E N E R A LS O L U T I O N S

In this Chapter we will discuss the concept of Darboux integrabilityand its relation with the Consisency Around the Cube In particu-lar in Section 41 we will introduce a definition of Darboux integra-bility based on the existence of the first integrals We will state thedefinition using the analogy with the continuous case Then we willdiscuss a practical method for finding first integrals in the case of non-autonomous equations with two periodic coefficients proposed in theoriginal work [71] as a generalization of the method proposed in [5556 72] In Section 42 we prove that the three equations found by JHietarinta in [82] which are linearizable [138] solving the problemof the Consistency Around the Cube are actually Darboux integrableequations This was observed in [69] and it was the starting pointof the research on the relation between the Consistency Around theCube and the Darboux integrability Then in Section 43 we show themain result obtained in [71] ie that the trapezoidal H4 equations(191) and the H6 equations (193) are Darboux integrable by report-ing their first integrals We underline that this result can be thought asa formal proof of the heuristic result obtained in Chapter 2 using theAlgebraic Entropy since as it will be discussed in Section 41 Darbouxintegrable equations are naturally linearizable Finally in Section 44we show how to use the first integrals obtained in Section 41 in orderto obtain the general solution of the trapezoidal H4 (191) and of theH6 equations (193) This material is based on the discussion made in[70]

41 darboux integrability

A nonlinear hyperbolic partial differential equation (PDE) in two vari-ables

uxt = f (x tuutux) (41)

is said to be Darboux integrable if it possesses two independent firstintegrals depending only on derivatives with respect to one variable

T = T (x tuut unt) stdTdx

∣∣∣∣uxt=f

equiv 0

X = X (x tuux umx) stdXdt

∣∣∣∣uxt=f

equiv 0(42)

125

126 darboux integrability and general solutions

where ukt = partkuparttk and ukx = partkupartxk for every k isin N Themethod is based on the linear theory developed by Euler and Laplace[38 96] extended to the nonlinear case in the 19th and early 20thcentury [29 30 58 59 154] The method was then used at the endof the 20th century mainly by Russian mathematicians as a source ofnew exactly solvable PDEs in two variable [148 174ndash179]

We note that there exists an alternative definition of Darboux in-tegrability This definition is based on the stabilization to zero of theLaplace chain of the linearized equation However it can be provedthat the two definitions are equivalent [8ndash10 179]

The most famous Darboux integrable equation is the Liouville equa-tion [113]

uxt = eu (43)

which possesses the first integrals

X = uxx minus1

2u2x T = utt minus

1

2u2t (44)

We remark that the first integrals (44) defines two Riccati equationsIn the discrete setting Darboux integrability was introduced in [5]

and used to obtain a discrete analogue of the Liouville equation (43)namely the Adler-Startsev discretization of the Liouville equation

unmun+1m+1

(1+

1

un+1m

)(1+

1

unm+1

)= 1 (45)

Similarly as in the continuous case we shall say that an equation onthe quad graph possibly non-autonomous ie

Qnm (unmun+1munm+1un+1m+1) = 0 (46)

is Darboux integrable if there exist two independent first integrals onecontaining only shifts in the first direction and the other containingonly shift in the second direction such that

(Tn minus Id)W2 = 0 (47a)

(Tm minus Id)W1 = 0 (47b)

where

W1 =W1nm(un+l1mun+l1+1m un+k1m) (48a)

W2 =W2nm(unm+l2 unm+l2+1 unm+k2) (48b)

holds true on the solution of (46) Here l1 l2k1k2 are integers suchthat l1 lt k1 l2 lt k2 and Tn Tm are the shifts operators in thefirst and second direction respectively Tnhnm = hn+1m Tmhnm =

hnm+1 Finally Id denotes the identity operator Idhnm = hnm

41 darboux integrability 127

We notice that the existence of first integrals implies that the fol-lowing two transformations

unm rarr unm =W1nm (49a)

unm rarr unm =W2nm (49b)

bring the quad-equation (46) into trivial linear equations given by (47)[5] namely

unm+1 minus unm = 0 (410a)

un+1m minus unm = 0 (410b)

Therefore any Darboux integrable equation is linearizable in twodifferent ways This is the relationship between the Darboux integra-bility and linearization

The transformations (49) along with the relations (410) imply thefollowing relations

W1 = ξn W2 = λm (411)

where ξn and λm are arbitrary functions of the lattice variables nand m respectively The relations (411) can be seen as ordinary differ-ence equations which must be satisfied by any solution unm of (46)The transformations (49) and the ordinary difference equations (411)may be quite complicated For this reason we define the order of thefirst integral to be the difference equation obtained from it using(411)

Example 411 (Discrete Liouville equation (45)) In the case of thediscrete Liouville equation (45) the first integrals presented in [5] are

W1 =

[1+

unm (1+ unminus1m)

unminus1m

] [1+

unm

un+1m (1+ unm)

]

(412a)

W1 =

[1+

unm (1+ unmminus1)

unmminus1

] [1+

unm

unm+1 (1+ unm)

]

(412b)

These two first integrals are both two-points second order first inte-grals

From its introduction in [5] various papers were devoted to thestudy of Darboux integrability for quad-equations [55 56 72 74 149]In particular in [55 56 72] were developed computational methodsto compute the first integrals In [72] was presented a method tocompute the first integrals with fixed li ki of a given autonomousequation Then in [55] this method was slightly modified and ap-plied to autonomous equations with non-autonomous first integralsFinally in [56] it was applied to equations with two-periodic coeffi-cients For the rest of this Section we present a further modification of


this method which allows to treat in a simpler way non-autonomousequations with two periodic coefficients This new modification wasfirst presented in [71]

If we consider the operator

Yminus1 = Tmpart

partunmminus1Tminus1m (413)

and apply it to the first integral in the n direction (47b) we obtain

Yminus1W1 equiv 0 (414)

The application of the operator Yminus1 is to be understood in the fol-lowing sense first we must apply Tminus1m and then we should expressusing equation (46) un+imminus1 in terms of the variables un+jm andunmminus1 which for this problem are independent variables Then wecan differentiate with respect to unmminus1 and safely apply Tm [55]

Taking in (414) the coefficients at powers of unm+1 we obtain asystem of PDEs for W1 If this is sufficient to determine W1 up toarbitrary functions of a single variable then we are done otherwisewe can add similar equations by considering the ldquohigher-orderrdquo op-erators

Yminusk = Tkmpart

partunmminus1Tminuskm k isinN (415)

which annihilate the difference consequence of (47b) given by TkmW1 =W1

YminuskW1 equiv 0 k isinN (416)

with the same computational prescriptions as above We can addequations until we find a non-constant function1 W1 which dependson a single combination of the variables unm+j1 unm+k1

If we find a non-constant solution of the equations generated by(413) and possibly (415) we can insert such solution into (47b) Infact given a non-constant W1 which satisfies (416) but does not sat-isfy (47b) we can always construct a first integral out of it In orderto construct a bona fide first integral in [55] it was proposed to look ifthe first integral and its shifted version can be related through someMoumlbius transformation

TmW1 =aW1 + b

cW1 + d (417)

This ansatz was applied with success on all the examples therein con-tained

In the same way the m-integral W2 can be found by consideringthe operators

Zminusk = Tknpart

partunminus1mTminuskn k isinN (418)

1 Obviously constant functions are trivial first integrals

42 darboux integrability and consistency around the cube 129

which are such that

ZminuskW2 equiv 0 k isinN (419)

In the case of non-autonomous equations with two-periodic coeffi-cients we can assume the decomposition

Wi = F(+)n F

(+)m W

(++)i + F

(minus)n F

(+)m W

(minus+)i

+ F(+)n F

(minus)m W

(+minus)i + F

(minus)n F

(minus)m W

(minusminus)i

(420)

with

F(plusmn)k =

1plusmn (minus1)k

2(421)

and derive from (416419) a set of equations for the function W(plusmnplusmn)i

by considering the evenodd points on the lattice The final form ofthe function Wi will be then fixed explicitly by substituting in (47)and separating again

We note that when successful the above procedure gives first inte-grals depending on arbitrary functions However this fact has to beunderstood as a restatement of the trivial property that any functionof a first integral is again a first integral So in general one doesnot need first integrals depending on arbitrary functions Thereforewe can take as the first integral simply a linear function of the argu-ments of the functions we obtain from a successful application of theabove procedure If there is more than one arbitrary function in thesame direction then it sufficient for our purposes to consider linearcombination of the arguments of such functions The reader can findin Appendix G an example carried out in the details

42 darboux integrability and consistency around the

cube

Before discussing the Darboux integrability of the trapezoidal H4

equations (191) and of the H6 equations (193) in this Section wewould like to underline that previously known linearizable quad equa-tions possessing the Consistency Around the Cube are Darboux inte-grable by showing their first integrals In this Section we will con-sider the three quad equations found by J Hietarinta in [82] givenin formula (128) We recall that these equations were found to belinearizable in [138] and were found to be Darboux integrable in [69]

Indeed let us consider the J2 equation as given by (128a) (which isindeed the same as theD1 equation) is linear and Darboux integrablebeing its first integrals given by

W1 = (minus1)m (un+1m + unm) (422a)

W2 = (minus1)n (unm+1 + unm) (422b)


These two first integrals are first-order two-point meaning that theequation itself a first integral

In the same way the J3 equation (128b) is linearizable and Darbouxintegrable being its first integrals given by

W1 = (minus1)mun+1m

unm (423a)

W2 = (minus1)nunm+1

unm (423b)


Finally the J4 equation (128b) is linearizable and Darboux inte-grable being its first integrals given by

W1 = (minus1)mun+1m + unm

1+ unmun+1m (424a)

W2 = (minus1)nunm+1 + unm

1+ unmunm+1 (424b)


The idea that these equations should have been Darboux integrablecomes from the fact that they admit Generalized Symmetries depend-ing on arbitrary functions [69] like the tHε1 equation (191a) Indeedit was shown in [5] that Darboux integrable equations always possessGeneralized Symmetries depending on arbitrary functions of orderat least equal to the first integrals Therefore following [69] we triedto reverse the reasoning and prove that these equations were Dar-boux integrable knowing their Generalized Symmetries In this wayin [69] it was proved that also the tHε1 equation is Darboux integrableby showing its first integrals

This observations together with the result of Algebraic Entropy asreported in Chapter 2 led us to conjecture that every quad equationpossessing the Consistency Around the Cube and linear growth is Darbouxintegrable This encouraged us to check the Darboux integrability ofthe remaining trapezoidal H4 and H6 equations In Section 43 weshow how this intuition has been proved by showing all the firstintegrals of the H4 equations (191) and of the H6 equations

We conclude this Section noting that no result about the Darbouxintegrability of the J1 equation (127) is known Indeed it is possibleto prove using the procedure outlined in Section 41 that a candidatetwo-point first integral in the n direction for the J1 equation is givenby

W1 = Fm

((unm + e2)(un+1m + o1)

(unm + e1)(un+1m + e2)

) (425)

43 first integrals for the trapezoidal h4 and h6 equations 131

However plugging (425) into the definition of first integral (47b) weobtain that the function Fm becomes a trivial constant Defining

In =(unm + e2)(un+1m + o1)

(unm + e1)(un+1m + e2)(426)

we note that the candidate three-point first integral is given by

W1 = Fm (In Inminus1) (427)

However also in this case it is possible to show that plugging (427)into (47b) the function Fm becomes a trivial constant

Due to computational issues we were not able to prove or disprovethe existence of a first integral in the n direction for the J1 equation(127) In fact we conjecture that the solution of hierarchy of equations(416) for the J1 equation (127) is given by

W1 = Fm ( In+1 In Inminus1 ) (428)

and that there exists an order N at which this relation defines a firstintegral for the J1 equation (127) but this kind of computations aretoo demanding and we were not able to conclude it for technicalreasons The only estimate we know is that N gt 1 However we notethat the dynamical system given constructed for the J1 equation (127)do not respect the prescription of Chapter 1

43 first integrals for the trapezoidal h4 and h6 equa-tions

In this Section we show the explicit form of the first integrals for thetrapezoidal H4 equations (191) and the H6 equations (193) This inte-grals are computed with the method presented in Section 41 We willnot present the details of the calculations since they are algorithmicand they can be implemented in any Computer Algebra System avail-able (we have implemented these conditions in Maple) The interestedreader can find a worked out example in Appendix G

We have the following general result

bull The H4 equations (191) possess one first integral in the n direc-tion and two first integrals in the m direction

bull The H6 equations (193) possess two first integrals in the n di-rection and two first integrals in the m direction

We believe that these general properties reflect the fact that the H4

equations (191) are two-periodic only in the m direction whereas theH6 equations (193) are two-periodic in both directions


We now present the first integrals of the trapezoidal H4 equations(191) in both directions


4311 tHε1 equation

Consider the tHε1 equation as given by (191a) It has a two-point firstorder integral in the n direction

W1 = F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm

1+ ε2unmun+1m (429a)

and a three-point second order integral in the m-direction

W2 = F(+)m α



+ F(minus)m β (unm+1 minus unmminus1)

(429b)

The integrals (G8) were first found in [69] from direct computationand later re-derived in [71] using the method explained in Section41 The interested reader will find all the details of its derivation inAppendix G

4312 tHε2 equation

Consider the tHε2 equation as given by (191b) It has a four-point

third order integral in the n direction

W1 = F(+)m

(minusun+1m + unminus1m) (unm minus un+2m)

ε2α42 + 4εα32 + [(8α3 minus 2unm minus 2un+1m) εminus 1]α22 + (unm minus un+1m)2

minus F(minus)m


(minusunminus1m + unm +α2) (un+1m +α2 minus un+2m)

(430a)

and a five-point fourth order integral in the m-direction

W2 = F(+)m α

(unmminus1 minus unm+1)2 (unm+2 minus unm) (unm minus unmminus2)[

(α2 +α3 + unmminus1)2 εminus unmminus1 +α3 minus unm

]middot[

(α3 +α2 + unm+1)2 εminus unm+1 +α3 minus unm

]

+ F(minus)m β

(unmminus1 minus unm+2 minus unm+1 + unmminus2)unm

minusε (unmminus2 minus unm+2)u2nm

minus(α3 +α2)2 (unmminus2 minus unm+2) ε

minus2 (unmminus2 minus unm+2) (α3 +α2) εunm

+(minusα3 + unm+1)unmminus2 + unm+2 (α3 minus unmminus1)

(unm+2 minus unm) (unm+1 minus unmminus1) (unm minus unmminus2)

(430b)


Remark 431 We note that tHε2 equation possesses an autonomoussub-case when ε = 0 We denote this sub-case by tHε=02 In this sub-case the equations defining the first integrals become singular and sothe first integrals become simpler

Wε=01 = (minus1)m

2α2 minus unminus1m + 2unm minus un+1m

unminus1m minus un+1m (431a)

Wε=02 =

(unm+1 minus unmminus1) (unm+2 minus unm)

unm + unm+1 minusα3 (431b)

The first integral in the n-direction (431a) is still non-autonomousalthough the equation is autonomous but now it is a three-pointsecond order first integral On the contrary the first integral in them-direction (431b) is a four-point third order first integral

Finally we note that the tHε=02 equation is related to the equation(1) from List 3 in [55]

(un+1m+1 minus un+1m) (unm minus unm+1)+ unm+ un+1m+ unm+1+ un+1m+1 = 0

(432)

through the transformation

unm = minusα2umn +1

4α2 +

1

2α3 (433)

Note that in this formula (433) the two lattice variables are exchangedSo it was already known in the literature that the tHε=02 equation wasDarboux integrable

4313 tHε3 equation

Consider the tHε2 equation as given by (191c) It has a four-point


W1 = F(+)m

(unminus1m minus un+1m) (minusun+2m + unm)α24ε2δ2 minusα2

3un+1munm

+(unm

2 + u2n+1m minus 2ε2δ2)α22

minusα2unmun+1m + ε2δ2

+ F

(minus)m

(un+1m minus unminus1m) (un+2m minus unm)

α2 (un+2m minusα2un+1m) (α2unm minus unminus1m)

(434a)



W2 = F(+)m α

(unm+2 minus unm) (unm+1 minus unmminus1)2 (unm minus unmminus2)(

δ2α23 + u2nmminus1ε

2 minusα3unmminus1unm)middot(

δ2α23 + u2nm+1ε

2 minusα3unmunm+1

)

minus F(minus)m β

α3u

2nm (unm+1 minus unmminus1)

minusε2u2nm (unm+2 minus unmminus2)

+α3unm (unm+2unmminus1 minus unm+1unmminus2)

minusδ2α32unm+2 + δ

2α32unmminus2


(434b)

Remark 432 With the same notation as in Remark 431 we notethat also the tHε3 equation has an autonomous sub-case of the equa-tion tH

ε3 if ε = 0 namely the tHε=03 equation In this sub-case the

equations defining the first integrals become singular and so the firstintegrals become simpler

Wε=01 = (minus1)m

[α2unm minus unminus1m

un+1m minus unminus1m+1

2

] (435a)

Wε=02 =

(unm+2 minus unm)unmminus1 minus unm+1unm+2 +α3δ2

α3δ2 minus unmunm+1

(435b)


Finally we note that the tHε=03 equation is related to equation (2)from List 3 in [55]

un+1m+1 (unm + b2unm+1)+ un+1m (b2unm + unm+1)+ c4 = 0

(436)

through the inversion of two lattice parameters unm = umn and thechoice of parameters

b2 = minus1

α2 c4 =

δ2α3(1minusα22

)α2

(437)

So it was already known in the literature that the tHε=03 equation wasDarboux integrable

Remark 433 As a final remark we can say that the tHε2 equation

(191b) and tHε3 equation (191c) possess first integrals of the same or-der in each direction Furthermore the first integrals of all the trape-zoidalH4 equations share the important property that in the directionm which is the direction of the non-autonomous factors F(plusmn)m the W2integrals are built up from two different ldquosubrdquo-integrals


432 H6 equations

In this Subsection we present the first integrals of the H6 equations(193) in both direction

4321 1D2 equation

In the case of the 1D2 equation (193b) we have the following secondorder three-point integrals

W1 = F(+)n F

(+)m α

[(1+ δ2)unm + un+1m] δ1 minus unm

δ1 [(1+ δ2)unm + unminus1m] δ1 minus unm

+ F(+)n F

(minus)m α

1+ (un+1m minus 1) δ1(1+ (unminus1m minus 1) δ1) δ1

+ F(minus)n F

(+)m β (un+1m minus unminus1m)

minus F(minus)n F

(minus)m β

(un+1m minus unminus1m) [1minus (1minus unm) δ1]

δ2 + unm

(438a)

W2 = F(+)n F

(+)m α


unm + δ1unmminus1

+ F(+)n F

(minus)m β (unm+1 minus unmminus1)

minus F(minus)n F

(+)m α


1+ (δ1 minus 1)unm+1

minus F(minus)n F

(minus)m β


δ2 + unm

(438b)

Remark 434 We remark that when δ1 rarr 0 the first integral (438a)is singular since the coefficient at α approaches a constant In thisparticular case it can be shown that the first integrals are given by

W(0δ2)1 = F

(+)n F

(+)m α


unm

minus F(+)n F

(minus)m α (un+1m minus unminus1m)

+ F(minus)n F


+ F(minus)n F

(minus)m β

unminus1m minus un+1m

δ2 + unm

(439a)

W(0δ2)2 = F

(+)n F

(+)m α


unm

+ F(+)n F


minus F(minus)n F

(+)m α (unm+1 minus unmminus1)

minus F(minus)n F

(minus)m β


δ2 + unm

(439b)

Note that as the limit in (438b) is not singular then (439b) can beobtained directly from (438b)


If δ1 rarr (1+ δ2)minus1 then the limit of the first integral (438b) is not

singular However it can be seen from equations defining the firstintegral in the m-direction that in this case there might exist a two-point first order first integral Carrying out the computations weobtain that if δ1 = (1+ δ2)

minus1 the first integrals are given by

W((1+δ2)

minus1δ2)1 = F

(+)n F

(+)m α

un+1m

unminus1m

+ F(+)n F

(minus)m α

δ2 + un+1m

δ2 + unminus1m

+ F(minus)n F


minus F(minus)n F

(minus)m β

(un+1m minus unminus1m)

δ2 + 1

(440a)

W((1+δ2)minus1δ2)2 = F

(+)n F

(+)m α [(1+ δ2)unm + unm+1]

+ F(+)n F

(minus)m β

unm + (1+ δ2)unm+1

δ2 + 1

minus F(minus)n F

(+)m α

(δ2 + 1)unm

δ2 + unm+1

minus F(minus)n F

(minus)m β

unm+1

δ2 + unm

(440b)

So unlike the case δ1 6= (1+ δ2)minus1 the first integral in the m-

direction (440b) is a two-point first order first integral On the con-trary the first integral in the n-direction (440a) is still a three-pointsecond order first integral which can be obtained from the completeform (438a) by substituting δ1 = (1+ δ2)

minus1

4322 2D2 equation

In the case of the 2D2 equation (193c) we have the following secondorder three-point integrals

W1 = F(+)n F

(+)m α

δ2 + un+1m

δ2 + unminus1m

+ F(+)n F

(minus)m α

(1minus (1+ δ2) δ1)unm + un+1m

(1minus (1+ δ2) δ1)unm + unminus1m

+ F(minus)n F

(+)m β

(un+1m minus unminus1m) (unm + δ2)

1+ (minus1+ unm) δ1

minus F(minus)n F

(minus)m β (un+1m minus unminus1m)

(441a)

W2 = F(+)n F


minus F(+)n F

(minus)m β


(λminus unm) δ1 minus unmminus1

minus F(minus)n F

(+)m α



minus F(minus)n F

(minus)m β


unm+1 + δ2

(441b)


Remark 435 We remark that if δ1 rarr 0 the first integral (441a) isnot singular However it can be seen from equations defining the firstintegral in the n-direction that in this case there might exist a two-point first order first integral Carrying out the computations weobtain that if δ1 = 0 the first integrals are given by

W(0δ2)1 = F

(+)n F

(+)m α (δ2 + un+1m)unm

minus F(+)n F

(minus)m α (un+1m + unm)

+ F(minus)n F

(+)m β (δ2 + unm)un+1m

minus F(minus)n F

(minus)m β (un+1m + unm)

(442a)

W(0δ2)2 = F

(+)n F


+ F(+)n F

(minus)m β

unm+1

unmminus1

minus F(minus)n F


minus F(minus)n F

(minus)m β


unm+1 + δ2

(442b)

So differently from the case δ1 6= 0 the first integral in the n-direction (442a) is a two-point first order first integral On the con-trary the first integral in the m-direction (442b) is still a three-pointsecond order first integral which can be obtained from the completeform (441b) by substituting δ1 = 0

Moreover we note that if δ1 rarr (1+ δ2)minus1 the limit of the first

integral (441b) is not singular However it can be seen from equationsdefining the first integral in the m-direction that in this case theremight exist a two-point first order first integral Carrying out thecomputations we obtain that if δ1 = (1+ δ2)

minus1 the first integrals aregiven by


(+)n F

(+)m α

δ2 + un+1m

δ2 + unminus1m

+ F(+)n F

(minus)m α

un+1m

unminus1m

minus F(minus)n F

(+)m β (unminus1m minus un+1m)

minus F(minus)n F

(minus)m β


1+ δ2

(443a)


(+)n F

(+)m α [(1+ δ2)unm + unm+1]

+ F(+)n F

(minus)m β [unm + (1+ δ2)unm+1]

+ F(minus)n F

(+)m α

δ2λminus (1+ δ2)unm+1 + λunm

unm + δ2

+ F(minus)n F

(minus)m β

δ2λminus (1+ δ2)unm + λunm+1

unm+1 + δ2

(443b)

So differently from the case δ1 6= (1+ δ2)minus1 the first integral in

the m-direction (443b) is a two-point first order first integral On


the contrary the first integral in the n-direction (443a) is still a three-point second order first integral which can be obtained from thecomplete form (441a) by substituting δ1 = (1+ δ2)

minus1

4323 3D2 equation

In the case of the 3D2 equation (193d) we have the following secondorder three-point integrals

W1 = F(+)n F

(+)m α

(unminus1m + δ2) [1+ (un+1m minus 1) δ1]

(un+1m + δ2) [1+ (unminus1m minus 1) δ1]

+ F(+)n F

(minus)m α

unm + (1minus δ1 minus δ1δ2)unminus1m

unm + (1minus δ1 minus δ1δ2)un+1m

+ F(minus)n F

(+)m β (un+1m minus unminus1m) (δ2 + unm)

minus F(minus)n F


(444a)

W2 = F(+)n F


minus F(+)n F

(minus)m β


λ (1+ δ2) δ12 minus [(1+ δ2)unmminus1 + unm + λ] δ1 + unmminus1

+ F(minus)n F

(+)m α (unmminus1 minus unm+1) [1+ (unm minus 1) δ1]

+ F(minus)n F

(minus)m β


(δ2 + unm+1) [1+ (1minus δ1)unmminus1]

(444b)

Remark 436 We remark that if δ1 rarr 0 the first integral (444a) isnot singular However it can be seen from equations defining thefirst integral in the n-direction that in this case there might exist atwo-point first order first integral Carrying out the computationswe obtain that if δ1 = 0 the first integrals are given by

W(0δ2)1 = F

(+)n F

(+)m αunm (δ2 + un+1m)

minus F(+)n F


+ F(minus)n F

(+)m βun+1m (δ2 + unm)

minus F(minus)n F


(445a)

W(0δ2)2 = F

(+)n F


+ F(+)n F

(minus)m β

unm+1

unmminus1

minus F(minus)n F


+ F(minus)n F

(minus)m β

δ2 + unmminus1

δ2 + unm+1

(445b)

So differently from the case δ1 6= 0 the first integral in the n-direction (445a) is a two-point first order first integral On the con-trary the first integral in the m-direction (445b) is still a three-point


second order first integral which can be obtained from the completeform (444b) by substituting δ1 = 0

Moreover we remark that if δ1 rarr (1+ δ2)minus1 the first integral

(444a) is singular since the coefficient at α approaches a constantIn this particular case the first integrals are given by


(+)n F

(+)m α


(δ2 + un+1m) (δ2 + unminus1m)

+ F(+)n F

(minus)m α


(δ2 + 1)unm

minus F(minus)n F

(+)m β (unminus1m minus un+1m) (δ2 + unm)

minus F(minus)n F


(446a)


(+)n F


+ F(+)n F

(minus)m β


unm

minus F(minus)n F

(+)m α

(unm+1 minus unmminus1) (δ2 + unm)

δ2 + 1

+ F(minus)n F

(minus)m β


(δ2 + unm+1) (unmminus1 + δ2)

(446b)

We point out that the first integral in the m-direction (446b) can beobtained from the complete form (444b) in the limit δ1 rarr (1+ δ2)

minus1

4324 D3 equation

In the case of the D3 equation as given by (193e) we have the follow-ing third order four-point integrals

W1 = F(+)n F

(+)m α


u2n+1m minus unm

+ F(+)n F

(minus)m α

(un+1m minus unminus1m)(un+2mminusunm

)unm + unminus1m

minus F(minus)n F

(+)m β


un+1m minus u2nm

+ F(minus)n F

(minus)m β


un+1m + un+2m

(447a)

W2 = F(+)n F

(+)m α


u2nm+1 minus unm

minus F(+)n F

(minus)m β


unm+1 minus u2nm

+ F(minus)n F

(+)m α


unm + unmminus1

+ F(minus)n F

(minus)m β


unm+1 + unm+2

(447b)


Remark 437 The equation D3 is invariant under the exchange oflattice variables n harr m Therefore its W2 integral (447b) can be ob-tained from theW1 one (447a) simply by exchanging the index n andm

4325 1D4 equation

In the case of the 1D4 equation as given by (193f) we have the follow-ing third order four-point integrals

W1 = F(+)n F

(+)m α

u2n+1mδ1 + un+1mun+2m + unminus1m (unm minus un+2m) minus δ2δ3

un+1m (δ1 + unm) minus δ2δ3

+ F(+)n F

(minus)m α

(unm minus un+2m + δ1un+1m)unminus1m + un+1mun+2m

(unm + δ1unminus1m)un+1m

+ F(minus)n F

(+)m β


u2nmδ1 + un+1munm minus δ2δ3

+ F(minus)n F

(minus)m β


unm (un+2mδ1 + un+1m)

(448a)

W2 = F(+)n F

(+)m α

(unm+2 minus unm)unmminus1 + δ1δ3 minus δ2u2nm+1 minus unm+1unm+2

δ1δ3 minus unmunm+1 minus δ2u2nm+1

minus F(+)n F

(minus)m β


δ1δ3 minus δ2unm2 minus unmunm+1

+ F(minus)n F

(+)m α

(unm minus unm+2 + δ2unm+1)unmminus1 + unm+1unm+2

(unm + δ2unmminus1)unm+1

+ F(minus)n F

(minus)m β


unm (unm+2δ2 + unm+1)

(448b)

Remark 438 The equation 1D4 possesses an autonomous sub-casewhen δ1 = δ2 = 0 In this case the equations defining the first inte-grals become singular and the first integrals become simpler If δ3 6= 0we have

W(00δ3)1 = (minus1)m


unm (449a)

W(00δ3)2 = (minus1)n


unm (449b)

They are both non-autonomous three-point second order first inte-grals Notice that since this sub-case is such that the equation has thediscrete symmetry nharr m the first integral W(00δ3)

2 can be obtainedfrom W

(00δ3)1 by using such transformation

Moreover we notice that the case δ1 = δ2 = 0 δ3 6= 0 is linked tothe equation (4) with b3 = 1 of List 3 in [55]

un+1m+1unm + un+1munm+1 + 1 = 0 (450)


through the transformation unm =radicδ3unm

We finally notice that the sub-case where δ1 = δ2 = δ3 = 0 pos-sesses two two-point first order non-autonomous first integrals

W(000)1 = (minus1)m

un+1m

unm W

(000)2 = (minus1)n

unm+1

unm (451)

The resulting equation is linked to one of the linearizable and Dar-boux integrable cases presented in [69 82]

4326 2D4 equation

In the case of the 2D4 equation as given by (193g) we have the fol-lowing four-point integrals

W1 = F(+)n F

(+)m α

[(unm minus un+2m minus δ1δ2unminus1m)u2n+1m

+un+1mun+2munminus1m + δ3unminus1m

](δ2u

2n+1mδ1 minus δ3 minus unmun+1m

)unminus1m

minus F(+)n F

(minus)m α

un+2munminus1m + (minusun+2m + unm)un+1m + δ3unminus1munm + δ3

minus F(minus)n F

(+)m β

(un+1m minus unminus1m) (un+2m minus unm)unm

un+2m (δ2δ1unm2 minus unmun+1m minus δ3)

+ F(minus)n F

(minus)m β


un+1mun+2m + δ3(452a)

W2 = F(+)n F

(+)m α


δ1δ3 minus δ2u2nm+1 minus unmunm+1

minus F(+)n F

(minus)m β


δ1δ3 minus δ2u2nm minus unmunm+1

+ F(minus)n F

(+)m α

unm+2δ2unm + unmminus1unm + unm+1unm+2 minus unmunm+1

unm+2 (δ2unm + unmminus1)

+ F(minus)n F

(minus)m β


(δ2unm+1 + unm+2)unmminus1

(452b)

Remark 439 The equation 2D4 possesses an autonomous sub-casewhen δ1 = δ2 = 0 In this case the equations defining the first inte-grals become singular and the first integrals become simpler


(unmun+1m +

δ32

) W

(00δ3)2 =

(unm+1

unmminus1

)(minus1)n

(453)

Therefore this particular sub-case possesses a two-point first orderintegral W1 and a three-point second order integral W2 Both inte-grals are non-autonomous despite the equation is autonomous The


limit δ3 rarr 0 is in this case regular both in the first integrals and inthe equations defining them

Finally we have that the case δ1 = δ2 = 0 corresponds to the equa-tion (9) of List 4 in [55]

unmun+1m + unm+1un+1m+1 + c4 = 0 (454)

with the identification unm = unm and c4 = δ3

Remark 4310 Besides the general remarks on the structure of the firstintegrals for the H6 equations which were given at the beginning ofthe this Section we would like to note that the first integrals of the H6

equations are of the same order in both directions at difference of thetrapezoidal H4 equations (except that in the degenerate cases) Fur-thermore we note that the three forms of the iD2 equations possessfirst integrals of the second order In the same way we have that theD3 and the iD4 equations possess first integrals of the fourth orderThis different properties will be important when looking for generalsolution in Section 44

We conclude this Section noting that together with the results ob-tained in Section 42 we can state that any quad equation possessingthe Consistency Around the Cube with linear growth constructed accordingto the prescriptions of Chapter 1 is Darboux integrable As pointed out inSection 42 the J1 equation (127) is not constructed using the prescrip-tions of Chapter 1 therefore it is different from all the other equationsconsidered Since no definitive result about the Darboux integrabilityof the J1 equation (127) is known we cannot say if this result canexpand to a wider class of equations including the J1 equation (127)

44 general solutions through the first integrals

In Section 43 we showed that the trapezoidal H4 equations (191)and the H6 equations (193) are Darboux integrable in the sense of lat-tice equations see Section 41 In this Section we show that from theknowledge of the first integrals and from the properties of the equa-tions it is possible to construct maybe after some complicate algebrathe general solutions of the trapezoidal H4 equations (191) and of theH6 equations (193) By general solution we mean a representation ofthe solution of any of the equations in (191) and (193) in terms ofthe right number of arbitrary functions of one lattice variable n or mSince the trapezoidal H4 equations (191) and the H6 equations (193)are quad equations ie the discrete analogue of second order hyper-bolic partial differential equations the general solution must containan arbitrary function in the n direction and another one in the mdirection

To obtain the desired solution we will need only the W1 integralswe presented in Section 43 and the relation (411) ie W1 = ξn with

44 general solutions through the first integrals 143

ξn an arbitrary function of n The equation W1 = ξn can be inter-preted as an ordinary difference equation in the n direction depend-ing parametrically on m Then from every W1 integral we can derivetwo different ordinary difference equations one corresponding to meven and one corresponding to m odd In both the resulting equationwe can get rid of the two-periodic terms by considering the cases neven and n odd and using the general transformation

u2k2l = vkl u2k+12l = wkl (455a)

u2k2l+1 = ykl u2k+12l+1 = zkl (455b)

This transformation brings both equations to a system of coupled differ-ence equations This reduction to a system is the key ingredient in theconstruction of the general solutions for the trapezoidal H4 equations(191) and for the H6 equations (193)

We note that the transformation (455) can be applied to the trape-zoidal H4 equations2 and H6 equations themselves This casts thesenon-autonomous equations with two-periodic coefficients into autonomoussystems of four equations We recall that in this way some examples ofdirect linearization (ie without the knowledge of the first integrals)were produced in [67] Finally we note that if we apply the evenoddsplitting of the lattice variables given by equation (455) to describe ageneral solution we will need two arbitrary functions in both direc-tions ie we will need a total of four arbitrary functions

In practice to construct these general solutions we need to solveRiccati equations and non-autonomous linear equations which ingeneral cannot be solved in closed form Using the fact that theseequations contain arbitrary functions we introduce new arbitrary func-tions so that we can solve these equations This is usually done reduc-ing to total difference ie to ordinary difference equations which canbe trivially solved Let us assume we are given the difference equation

un+1m minus unm = vn (456)

depending parametrically on another discrete index m Then if I canexpress the function vn as a discrete derivative

vn = wn+1 minuswn (457)

then the solution of equation (456) is simply

unm = wn + υm (458)

where υm is an arbitrary function of the discrete variable m This isthe simplest possible example of reduction to total difference The

2 In fact in the case of the trapezoidal H4 equations (191) we use a simpler transfor-mation instead of (455) see Section 443


general solutions will then be expressed in terms of these new arbi-trary functions obtained reducing to total differences and in terms ofa finite number of discrete integrations ie the solutions of the simpleordinary difference equation

un+1 minus un = vn (459)

where un is the unknown and vn is an assigned function We notethat the discrete integration (459) is the discrete analogue of the dif-ferential equation u prime (x) = v (x)

To summarize in this Section we prove the following result

The trapezoidal H4 equations (191) and H6 equations (193)are exactly solvable and we can represent the solution in termsof a finite number of discrete integration (459)

The Section is structured in Subsections In Subsection 441 wetreat the 1D2 2D2 and 3D2 equations In this case the constructionof the general solution is carried out from the sole knowledge of thefirst integral The equation acts only as a compatibility condition forthe arbitrary functions obtained in the procedure The solution ob-tained is explicit In Subsection 442 we treat the D3 1D4 and 2D4equations In this case the construction of the general solution is car-ried out through a series of manipulations in the equation itself andfrom the knowledge of the first integral The key point will be that theequations of the first integrals can be reduced to a single linear equa-tion The solution is obtained up to two discrete integrations one inevery direction In Subsection 443 we treat the trapezoidal H4 equa-tions (191) The tHε1 equation (191a) is trivial since in the directionn it possess the first integral (429a) which is two-point second orderThis is trivial since possessing a first integral of order one means thatthe equation itself is a first integral For the tHε2 equation (191b) andthe tHε3 equation (191c) the construction of the general solution iscarried out reducing the equation to a partial difference equation de-fined on six points and then using the equations defined by the firstintegrals The equations defined by the first integrals are reduced todiscrete Riccati equations The solution will be be given in terms offour discrete integrations

441 The iD2 equations i = 1 2 3

In this Subsection we construct the general solution of the three formsof the D2 equation which we denote collectively as iD2 with i =

1 2 3 and are given by equations (193b) (193c) and (193d)

4411 1D2 equation

From Section 43 we know that the 1D2 equation (193b) possessesa three-point second order first integral W1 (438a) As stated at the


beginning of this Section from the relation W1 = ξn this integral de-fines a three-point second order ordinary difference equation in then direction which depends parametrically onm From the parametricdependence we find two different three-point non-autonomous ordi-nary difference equations corresponding to m even and m odd Wetreat them separately

case m = 2l If m = 2l we have the following non-autonomous or-dinary difference equation

F(+)n

[(1+ δ2)un2l + un+12l] δ1 minus un2l

[(1+ δ2)un2l + unminus12l] δ1 minus un2l

+ F(minus)n (un+12l minus unminus12l) = ξn

(460)

where without loss of generality α = δ1 and β = 1 We caneasily see that once solved for un+12l the equation is linear

un+12l +F(+)m (δ1 + δ1δ2 minus 1minus ξnδ1 minus ξnδ1δ2 + ξnδ1)un2l

δ1

minus(F(+)m ξn + F

(minus)m

)unminus12l minus F

(minus)m ξn = 0

(461)

Tackling this equation directly is very difficult but we can sepa-rate again the cases when n is even and odd and convert (461)into a system using the standard transformation (455a)

wkl minus ξ2kwkminus1l =δ

δ1(1minus ξ2k) vkl (462a)

vk+1l minus vkl = ξ2k+1 (462b)

where

δ = 1minus δ1δ2 minus δ1 (463)

Now we have two first order ordinary difference equations Equa-tion (462b) is uncoupled from equation (462a) Furthermoresince ξ2k and ξ2k+1 are independent functions we can writeξ2k+1 = ak+1 minus ak So the second equation possesses the triv-ial solution3

vkl = αl + ak (464)

Now introduce (464) into (462a) and solve the equation forwkl


δ1(1minus ξ2k) (αl + ak) (465)

3 From now on we use the convention of naming the arbitrary functions dependingon k with Latin letters and the functions depending on l by Greek ones


We define ξ2k = bkbkminus1 and perform the change of dependentvariable wkl = bkWkl Then Wkl solves the equation

Wkl minusWkminus1l =δ

δ1

(1

bkminus

1

bkminus1

)(αl + ak) (466)

The solution of this difference equation is given by

Wkl = βl +δ

δ1

αlbk

+ ck (467)

where ck is such that

ck minus ckminus1 =δ

δ1

(1

bkminus

1

bkminus1

)ak (468)

Equation (468) is not a total difference but it can be used todefine ak in terms of the arbitrary functions bk and ck

ak = minusδ1δ

bkbkminus1bk minus bkminus1

(ck minus ckminus1) (469)

This means that we have the following solution for the system(462)

vkl = αl minusδ1δ


(ck minus ckminus1) (470a)

wkl = bk (βl + ck) +δ

δ1αl (470b)

case m = 2l + 1 Ifm = 2l+1we have the following non-autonomousordinary difference equation

F(+)m1+ (un+12l+1 minus 1) δ11+ (unminus12l+1 minus 1) δ1

+ F(minus)m

(unminus12l+1 minus un+12l+1) [1minus δ1 (1minus un2l+1)]

δ2 + un2l+1= ξn

(471)

We can easily see that the equation is genuinely nonlinear How-ever we can separate the cases when n is even and odd andconvert (471) into a system using the standard transformation(455b)

zkl minus ξ2kzkminus1l =

(1minus

1

δ1

)(1minus ξ2k) (472a)

yk+1l minus ykl =ξ2k+1δ1

δminus 1+ δ1 (1minus zkl)

1minus δ1 (1minus zkl) (472b)

where we used the definition (463) This is a system of twofirst order difference equation and equation (472a) is linear and


uncoupled from (472b) As ξ2k = bkbkminus1 we have that (472a)is a total difference

zkl

bkminuszkminus1l

bkminus1=

(1minus

1

δ1

)(1

bkminus

1

bkminus1

) (473)

Hence the solution of (473) is given by

zkl = 1minus1

δ1+ bkγl (474)

Inserting (474) into (472b) and using the definition of ξ2k+1 interms of ak ie ξ2k+1 = ak+1 minus ak we obtain

yk+1l minus ykl = minus

(1

δ1+

δ

δ21bkγl

)(ak+1 minus ak) (475)

We can then represent the solution of (473) as

ykl = δl +δdk

δ21γlminusakδ1

(476)

where dk satisfies the first order linear difference equation

dk+1 minus dk =ak+1 minus ak

bk (477)

Inserting the value of ak given by (469) inside (477) we obtainthat this equation is a total difference Then dk is given by

dk = minus(ck minus ckminus1)bkminus1δ1

(bk minus bkminus1)δminusδ1ckδ

(478)

This means that finally we have the following solutions for thefields zkl and ykl

zkl = 1minus1

δ1+ bkγl (479a)

ykl = δl minus(ck minus ckminus1)bkminus1(bk minus bkminus1)δ1γl

minusckδ1γl

+1

δ

bkbkminus1 (ck minus ckminus1)

bk minus bkminus1

(479b)

Equations (470479) provide the value of the four fields but wehave too many arbitrary functions in the m direction namely αlβl γl and δl Introducing (470479) into (193b) and separating theterms even and odd in n andmwe obtain two independent equations

(αl + δ1δl)γl +βl = 0 (αl+1 + δ1δl)γl +βl+1 = 0 (480)

which allow us to reduce by two the number of independent func-tions in the m direction Solving equations (480) with respect to γland δl we obtain

γl = minusβl+1 minusβlαl+1 minusαl

δl =1

δ1

βlαl+1 minusβl+1αlβl+1 minusβl

(481)


Therefore the general solution for the 1D2 equation (193b) is givenby





δ1αl (482b)

zkl = 1minus1

δ1minus bk

βl+1 minusβlαl+1 minusαl

(482c)

ykl =1

δ1


+1

δ


bk minus bkminus1

+

[(ck minus ckminus1)bkminus1(bk minus bkminus1)δ1

+ckδ1

]αl+1 minusαlβl+1 minusβl

(482d)

Remark 441 It is easy to see that the solution (482) is ill-defined ifδ1 = 0 or δ = 0 We will treat these two particular cases separately

case δ = 0 If δ = 0 we can solve (463) with respect to δ1

δ1 =1

1+ δ2 (483)

The first integral (438a) is not singular for δ1 given by (483) Theprocedure of solution will become different only when we arrive tothe systems of ordinary difference equations (462) and (472) So wewill present the solution of the systems in this case

case m = 2k If δ1 is given by equation (483) the system (462) be-comes

wkl minus ξ2kwkminus1l = 0 (484a)


The system (484) is uncoupled and imposing ξ2k = akakminus1and ξ2k+1 = bk+1 minus bk it is readily solved to give

wkl = akαl (485a)

vkl = bk +βl (485b)

case m = 2k + 1 If δ1 is given by equation (483) the system (472)becomes

δ2 + zkl

ak=δ2 + zkminus1m

akminus1 (486a)

minusyk+1l minus ykl

1+ δ2= bk+1 minus bk (486b)


where we used the fact that ξ2k = akakminus1 and ξ2k+1 =

bk+1 minus bk The solution to this system is immediate and it isgiven by

zkl = akγl minus δ2 (487a)

ykl = (1+ δ2) (δl minus bk) (487b)

As in the general case we obtained the expressions of the four fieldsbut we have too many arbitrary functions in the l direction namelyαl βl γl and δl Substituting the obtained expressions (485487) inthe equation 1D2 (193b) with δ1 given by equation (483) separatingthe even and odd terms we obtain two compatibility conditions

αl + γlβl + γlδl = 0 αl+1 + γlβl+1 + γlδl = 0 (488)

We can solve this equation with respect to γl and δl and we obtain

γl = minusαl+1 minusαlβl+1 minusβl

δl = minusβlαl+1 minusβl+1αl

αl+1 minusαl (489)

Therefore the general solution for the 1D2 equation (193b) if δ1 isgiven by (483) is

wkl = akαl (490a)


zkl = minusakαl+1 minusαlβl+1 minusβl

minus δ2 (490c)

ykl = minus(1+ δ2)

(βlαl+1 minusβl+1αl

αl+1 minusαl+ bk

) (490d)

case δ1 = 0 If δ1 = 0 the first integral (438a) is singular Thenfollowing Remark 434 the 1D2 equation (193b) possesses in the di-rection n three-point second order integral W(0δ2)

1 (439a) In orderto solve the 1D2 in this case equation (193b) we use the first integral(439a) We start separating the cases even and odd in m

case m = 2k If m = 2k we obtain from the first integral (439a)

F(+)nun+12l minus unminus12l

un2l+ F

(minus)n (un+12l minus unminus12l) = ξn (491)

where we have chosen without loss of generality α = β = 1This equation is non-linear Applying the transformation (455a)we transform equation (491) into the system

wkl minuswkminus1l = ξ2kvkl (492a)



The system (492) is linear and equation (492b) is uncoupledfrom equation (492a) If we put ξ2k+1 = ak+1 minus ak then equa-tion (492b) has the solution

vkl = ak +αl (493)

Substituting into (492a) we obtain

wkl minuswkminus1l = ξ2k(ak +αl) (494)

Equation (494) becomes a total difference if we set

ξ2k = bk minus bkminus1 ak =ck minus ckminus1bk minus bkminus1

(495)

and then the solution of the system (484) is given by

vkl =ck minus ckminus1bk minus bkminus1

+αl (496a)

wkl = ck +αlbk +βl (496b)

case m = 2k + 1 If m = 2k + 1 we obtain from the first integral(439a)

F(+)n (unminus12l+1 minus un+12l+1)minus F

(minus)n

(un+12l+1 minus unminus12l+1)

δ2 + un2l+1= ξn

(497)

where we have chosen without loss of generality α = β = 1This equation is non-linear Applying the transformation (455b)we transform equation (497) into the system

zkminus1l minus zkl = bk minus bkminus1 (498a)

ykl minus yk+1l = (ak+1 minus ak)(δ2 + zkl) (498b)

where we used the values of ξ2k and ξ2k+1 The system is nowlinear and equation (498a) can be already solved to give

zkl = γl minus bk (499)

Substituting into (498b) we obtain

ykl minus yk+1l = (ak+1 minus ak)(δ2 + γl minus bk) (4100)

Then we have that ykl is given by

ykl = δl minus (γl + δ2)ak + dk (4101)

where dk solves the ordinary difference equation

dk+1 minus dk = bk+1ck+1 minus ckbk+1 minus bk

minus ck+1 minus bkck minus ckminus1bk minus bkminus1

+ ck


(4102)

In (4102) we inserted the value of ak according to (495) Equa-tion (4102) is a total difference and then dk is given by

dk = bkck minus ckminus1bk minus bkminus1

minus ck (4103)

Then the solution of the system (498) is

zkl = γl minus bk (4104a)

ykl = δl minus (γl + δ2)ck minus ckminus1bk minus bkminus1

+ bkck minus ckminus1bk minus bkminus1

minus ck

(4104b)

As in the general case we obtained the expressions of the four fieldsbut we have too many arbitrary functions in the l direction namelyαl βl γl and δl Substituting the obtained expressions (4964104) inthe equation 1D2 (193b) with δ1 = 0 separating the even and oddterms we obtain two compatibility conditions

(γl+ δ2)αl+ δl+βl = 0 (γl+ δ2)αl+1+βl+1+ δl = 0 (4105)

We can solve equation (4105) with respect to γl and δl and to obtain

γl = minusδ2 minusβl+1 minusβlαl+1 minusαl

δl =βl+1αl minusαl+1βl


Therefore the general solution for the 1D2 equation (193b) if δ1 = 0

is


+αl (4107a)


zkl = minusδ2 minusβl+1 minusβlαl+1 minusαl

minus bk (4107c)

ykl =βl+1αl minusαl+1βl

αl+1 minusαl+βl+1 minusβlαl+1 minusαl

ck minus ckminus1bk minus bkminus1


minus ck

(4107d)

This discussion exhausts the possible cases For any value of theparameters we have the general solution of the 1D2 equation (193b)

4412 2D2 equation

From Section 43 we know that the 2D2 equation (193c) possesses athree-point second order first integral W1 (441a) As stated at thebeginning of this Section from the relation W1 = ξn this integral de-fines a three-point second order ordinary difference equation in the n


direction which depends parametrically on m From this parametricdependence we find two different three-point non-autonomous ordi-nary difference equations corresponding to m even and m odd Wetreat them separately

case m = 2l If m = 2l we have the following non-autonomousnonlinear ordinary difference equation

F(+)n α

δ2 + un+12l

δ2 + unminus12l

minus F(minus)n β

(unminus12l minus un+12l) (un2l + δ2)

1+ (un2l minus 1) δ1= ξn

(4108)

Without loss of generality we set α = 1 and β = δ1 Then mak-ing the transformation

un2l = Un2l minus δ2 (4109)

and putting

δ =1minus δ1 minus δ1δ2

δ1(4110)

equation (4108) is mapped to

F(+)nUn+12l

Unminus12lminus F

(minus)n

(Unminus12l minusUn+12l)Un2l

Un2l + δ= ξn (4111)

From the definition (455a) applied to Un2l instead of un2l4

we can separate again the even and the odd part in (4111) Weobtain the following system of two coupled first-order ordinarydifference equations

Wkl minus ξ2kWkminus1l = 0 (4112a)

Vk+1l minus Vkl = ξ2k+1

(1+

δ

Wkl

) (4112b)

Putting ξ2k = akakminus1 the solution to (4112a) is immediatelygiven by

Wkl = akαl (4113)

Inserting the value of Wkl from (4113) into (4112b) we obtain


(1+

δ

akαl

) (4114)

If we define

ξ2k+1 = bk+1 minus bk ak =bk+1 minus bkck+1 minus ck

(4115)

4 We will denote the corresponding fields with capital letters


then (4114) becomes a total difference So we obtain the follow-ing solutions for the Wkl and the Vkl fields

Wkl = αlbk+1 minus bkck+1 minus ck

(4116a)

Vkl = bk +βl + δckαl

(4116b)

Inverting the transformation (4109) we obtain for the fieldswkl and vkl

wkl = αlbk+1 minus bkck+1 minus ck

+1

δ1minus 1minus δ (4117a)

vkl = bk +βl + δckαl

+1

δ1minus 1minus δ (4117b)


F(+)nδδ1un2l+1 + un+12l+1

δδ1un2l+1 + unminus12l+1+F

(minus)n δ1 (unminus12l+1 minus un+12l+1) = ξn

(4118)

where we already substituted δ as defined in (4110) Using thestandard transformation to get rid of the two-periodic factors(455b) we obtain

δδ1ykl + zkl

δδ1ykl + zkminus1l= ξ2k (4119a)

δ1 (ykl minus yk+1l) = ξ2k+1 (4119b)

Both equations in (4119) are linear in zkl ykl and their shiftsAs ξ2k+1 = bk+1 minus bk we have that the solution of equation(4119b) is given by

ykl = γl minusbkδ1

(4120)

As ξ2k = akakminus1 and ykl given by (4120) we obtain

zkl

akminuszkminus1l

akminus1=

(1

akminus

1

akminus1

)(δbk minus δδ1γl) (4121)

Recalling the definition of ak in (4115) we represent zkl as

zkl = δbk +bk+1 minus bkck+1 minus ck

(δl minus δck) minus δδ1γl (4122)

Equations (411741204122) provide the value of the four fieldsbut we have too many arbitrary functions in the m direction namely


αl βl γl and δl Inserting (411741204122) into (193c) and separat-ing the terms even and odd in n and m we obtain two independentequations

δ1δl +αlδ21γl minus δ1

2λαl +βlαlδ1 + δαlδ1 minusαl +αlδ1 = 0(4123a)

δ1δl +αl+1δ21γl minus δ

21λαl+1 +βl+1αl+1δ1 + δαl+1δ1 minusαl+1 +αl+1δ1 = 0

(4123b)

which allow us to reduce by two the number of independent func-tions in the m direction Solving (4123) with respect to γl and δl wefind

γl = minusβl+1δ1 minus 1minus δ

21λ+ δδ1 + δ1

δ21minusαl (βl+1 minusβl)

(αl+1 minusαl) δ1 (4124a)

δl = αl (βl+1 minusβl) +α2l (βl+1 minusβl)

αl+1 minusαl(4124b)

Therefore the general solution of the 2D2 equation (193c) is givenby


+1



+1


zkl = δbk + δβl+1δ1 minus 1minus δ

21λ+ δδ1 + δ1

δ1

+bk+1 minus bkck+1 minus ck

[αl (βl+1 minusβl) +

α2l (βl+1 minusβl)

αl+1 minusαlminus δck

]+δαl (βl+1 minusβl)

αl+1 minusαl

(4125c)

ykl = minusβl+1δ1 minus 1minus δ

21λ+ δδ1 + δ1

δ21

minusαl (βl+1 minusβl)

(αl+1 minusαl) δ1minusbkδ1

(4125d)

Remark 442 It is easy to see that the solution of the 2D2 equation(193c) given by (4125) is ill-defined if δ1 = 0 Therefore we have tothreat this case separately Following Remark 435 we have the 2D2equation (193c) with δ1 = 0 possesses the following two-point firstorder first integral in the direction n W(0δ2)

1 (442a) To solve the 2D2equation (193c) with δ1 = 0 we use the first integral (442a) Againwe start separating the cases m even and odd in (442a)


F(+)n (δ2 + un+12l)un2l + F

(minus)n (δ2 + un2l)un+12l = ξn


(4126)

where we have chosen without loss of generality α = β = 1Applying the transformation (455a) equation (4126) becomesthe system

vkl(δ2 +wkl) = ξ2k (4127a)

vk+1l(δ2 +wkl) = ξ2k+1 (4127b)

In this case the system (4127) do not consist of purely differ-ence equations Indeed from (4127a) we can derive immediatelythe value of the field wkl

wkl = minusδ2 +ξ2kvkl

(4128)

Inserting (4128) into (4127b) we obtain that vkl solves the equa-tion

vk+1l minusξ2k+1ξ2k

vkl = 0 (4129)

Defining

ξ2k+1 =ak+1ak

ξ2k (4130)

we have that (4129) becomes a total difference So we have thatthe system (4127) is solved by

vkl = akαl (4131a)

wkl = minusδ2 +ξ2kakαl

(4131b)


F(+)n (un+12l+1 + un2l+1)+ F

(minus)n (un+12l+1 + un2l+1) = minusξn

(4132)

Applying the transformation (455b) equation (4132) becomesthe system

ykl + zkl = minusξ2k (4133a)

zkl + yk+1l = minusak+1ak

ξ2k (4133b)

where ξ2k+1 is given by (4130) Equation (4133a) is not a dif-ference equation and can be solved to give

zkl = minusξ2k minus ykl (4134)


which inserted in (4133b) gives

yk+1l minus ykl =

(1minus

ak+1ak

)ξ2k (4135)

Defining

ξ2k = minusakbk+1 minus bkak+1 minus ak

(4136)

equation (4135) becomes a total difference Therefore we canwrite the solution of the system (4133) as

ykl = bk +βl (4137a)

zkl = akbk+1 minus bkak+1 minus ak

minus bk minusβl (4137b)

In this case we have the right number of arbitrary functions in bothdirections So the solution of the 2D2 equation with δ1 = 0 is givenby

vkl = akαl (4138a)

wkl = minusδ2 minus1

αl

bk+1 minus bkak+1 minus ak

(4138b)

ykl = bk +βl (4138c)


minus bk minusβl (4138d)

Inserting (4138) into the 2D2 equation (193c) and separating theeven and odd terms we verify that it is a solution

4413 3D2 equation

From Section 43 we know that the 2D2 equation (193c) possesses athree-point second order first integral W1 (441a) As stated at thebeginning of this Section from the relation W1 = ξn this integral de-fines a three-point second order ordinary difference equation in the ndirection which depends parametrically on m From this parametricdependence we find two different three-point non-autonomous ordi-nary difference equations corresponding to m even and m odd Wetreat them separately


F(+)n

(unminus12l + δ2) [1+ (un+12l minus 1) δ1]

(un+12l + δ2) [1+ (unminus12l minus 1) δ1]

+ F(minus)n (un+12l minus unminus12l) (δ2 + un2l) = ξn

(4139)


where we have chosen without loss of generality α = β = 1 Wecan apply the usual transformation (455a) in order to separatethe even and odd part in (4139)

1+ (wkl minus 1) δ1wkl + δ2

= ξ2k1+ (wkminus1l minus 1) δ1

wkminus1l + δ2 (4140a)

vk+1l minus vkl =ξ2k+1

(δ2 +wkl) (4140b)

This system of equations is still non-linear but the equation(4140a) is uncoupled from (4140b) Moreover equation (4140a)is a discrete Riccati equation which can be linearized throughthe Moumlbius transformation

wkl = minusδ2 +1

Wkl (4141)

into

Wkl minus ξ2kWkminus1l =δ1δ

(ξ2k minus 1) (4142a)

vk+1l minus vkl = ξ2k+1Wkl (4142b)

where δ is given by equation (463)Putting ξ2k = akakminus1 wehave the following solution for (4142a)

Wkl = akαl minusδ1δ

(4143)

Plugging (4143) into equation (4142b) and defining

ξ2k+1 = bk+1 minus bk ak =ck+1 minus ckbk+1 minus bk

(4144)

we have that equation (4142b) becomes a total difference Thenthe solution of (4142b) can be written as

vkl = minusδ1δbk + ckαl +βl (4145)

So using (4141) we obtain the following solution for the originalsystem (4140)

wkl =δ1 minus 1+ δ

δ1+

δ(bk+1 minus bk)

δαm(ck+1 minus ck) minus (bk+1 minus bk)δ1

(4146a)

vkl = minusδ1δbk + ckαl +βl (4146b)


F(+)nun2l+1 + δunminus12l+1

un2l+1 + δun+12l+1minus F

(minus)n (un+12l+1 minus unminus12l+1) = ξn


(4147)

where without loss of generality α = β = 1 and δ is given by(463) Solving with respect to un+12l+1 it is immediate to seethat the resulting equation is linear Then separating the evenand the odd part using the transformation (455b) we obtainthe following system of linear first-order ordinary differenceequations

zkl minus1

ξ2kzkminus1l =

1

δ

(1minus

1

ξ2k

)ykl (4148a)

yk+1 minus yk = minusξ2k+1 (4148b)

As ξ2k+1 = bk+1 minus bk we obtain immediately the solution ofequation (4148b) as

ykl = minusbk + γl (4149)

Substituting ykl given by (4149) into equation (4148a) beingξ2k = akakminus1 we obtain

akzkl minus akminus1zkminus1l =ak minus akminus1

δ(bk minus γl) (4150)

Then in the usual way we can represent the solution as

zkl =bk minus γlδ


(δl minus

ckδ

) (4151)

where we have used the explicit definition of ak given in (4144)So we have the explicit expression for both fields ykl and zkl

Equations (414641494151) provide the value of the four fieldsbut we have too many arbitrary functions in the m direction namelyαl βl γl and δl Inserting (414641494151) into (193d) and sepa-rating the terms even and odd in n and m we obtain we obtain twoequations

δlδ2αl + (βl minus δ1λ) δminus δ1γl = 0 (4152a)

δlδ2αl+1 + (βl+1 minus δ1λ) δminus δ1γl = 0 (4152b)

which allow us to reduce by two the number of independent func-tions in the m direction Indeed solving (4152) with respect to γl andδl we find

γl =δ

δ1

(βl minus λδ1 minusαl


) (4153a)

δl = minus1

δ


(4153b)


Therefore the general solution of the 3D2 equation (193d) is givenby

vkl = minusδ1δbk + ckαl (4154a)


δ1+

δ(bk+1 minus bk)

δαl(ck+1 minus ck) minus (bk+1 minus bk)δ1 (4154b)

zkl =bkδ

minus1

δ1



)+1

δ

bk+1 minus bkck+1 minus ck

(βl+1 minusβlαl+1 minusαl

+ ck

)

(4154c)

ykl = minusbk +δ

δ1



) (4154d)

Remark 443 It is easy to see that the solution (482) is ill-defined ifδ1 = 0 and if δ = 0 We will treat these two particular cases separately

case δ = 0 If δ = 0 we have that δ1 is given by equation (483)In this case the first integral (444a) is singular since the coefficient ofα goes to a constant Following Remark 436 we have that the 3D2equation with δ1 given by (483) the first integral in the direction n is

given by W((1+δ2)minus1δ2)1 (446a) This first integral is a three-point sec-

ond order first integral As in the general case we consider separatelythe m even and odd cases

case m = 2k If m = 2k then the first integral (446a) becomes thefollowing non-linear three-point second order difference equa-tion

F(+)n

un+12l minus unminus12l

(δ2 + un+12l) (δ2 + unminus12l)

minus F(minus)n (unminus12l minus un+12l) (δ2 + un2l) = ξn

(4155)

where without loss of generality α = β = 1 If we separate theeven and the odd part using the general transformation givenby (455a) we obtain the system

wkl minuswkminus1l

(wkl + δ2)(wkminus1l + δ2)= ξ2k (4156a)

(vk+1l minus vkl)(wkl + δ2) = ξ2k+1 (4156b)

This is a system of first order non-linear difference equationsHowever (4156a) is uncoupled from (4140b) and it is a discreteRiccati equation which can be linearized through the Moumlbiustransformation (4141) This linearize the system (4156) to

Wkl minusWkminus1l = ξ2k (4157a)



Defining ξ2k = ak minus akminus1 equation (4157a) is solved by

Wkl = ak +βl (4158)

Introducing (4158) into equation (4157b) we have

vk+1l minus vkl = ξ2k+1 (ak +αl) (4159)

Equation (4159) becomes a total difference if


(4160)

This yields the following solution of the system (4156)

vkl = ck + bkαl +βl (4161a)

wkl = minusδ2 +bk+1 minus bk

ck+1 minus ck +αl (bk+1 minus bk) (4161b)

case m = 2k + 1 If m = 2k + 1 the first integral (446a) becomesthe following nonlinear three-point second order differenceequation

F(+)nun+12l+1 minus unminus12l+1

(δ2 + 1)un2l+1minus F


(4162)

where without loss of generality α = β = 1 As usual we canseparate the even and odd part in n using the transformation(455b) This transformation brings equation (4162) into the fol-lowing linear system

zkl minus zkminus1l = (δ2 + 1)ykl

(ck minus ckminus1bk minus bkminus1

minusck+1 minus ckbk+1 minus bk

)

(4163a)

yk+1l minus ykl = minusbk+1 + bk (4163b)

where we used (4160) and the definition ξ2k+1 = ak+1 minus akEquation (4163b) is readily be solved and gives

ykl = γl minus bk (4164)

Inserting (4164) into (4163a) we obtain

zklminus zkminus1l = (δ2+ 1) (γl minus bk)



)

(4165)

We can then write for zkl the following expression

zkl = minus(δ2 + 1)

(γlck+1 minus ckbk+1 minus bk

+ dk

)+ δl (4166)


where dk solves the equation

dk minus dkminus1 = minusbk



) (4167)

Equation (4167) is a total difference with dk given by

dk = bk+1ck+1 minus ckbk+1 minus bk

minus ck+1 (4168)

Therefore we have the following solution to the system (4163)

ykl = γl minus bk (4169a)


[(γl + bk+1)

ck+1 minus ckbk+1 minus bk

minus ck+1

]+ δl

(4169b)

Equations (41614169) provide the value of the four fields but wehave too many arbitrary functions in the m direction namely αl βlγl and δl Inserting (41614169) into (193d) with δ1 given by (483)and separating the terms even and odd in n and m we obtain weobtain two equations

γl(δ2 + 1)αl minus λ+ δl +βl(δ2 + 1) = 0 (4170a)

γl(δ2 + 1)αl+1 minus λ+ δl +βl+1(δ2 + 1) = 0 (4170b)

Solving this compatibility condition with respect to γl and δl weobtain


(4171a)

δl = (δ2 + 1)βl+1αl minusαl+1βl

αl+1 minusαl+ λ (4171b)

Inserting then (4171) into (41614169) we obtain the following ex-pression for the solution of the 3D2 when δ1 is given by (483)




ykl = minusβl+1 minusβlαl+1 minusαl

minus bk (4172c)


[(bk+1 minus


)ck+1 minus ckbk+1 minus bk

minus ck+1

]+ (δ2 + 1)

βl+1αl minusαl+1βlαl+1 minusαl

+ λ

(4172d)


case δ1 = 0 The first integral (444a) is not singular when insert-ing δ1 = 0 Therefore the procedure of solution will become differentonly when we arrive to the systems of ordinary difference equations(4140) and (4148) So we will present the solution of the systems inthis case

case m = 2k If δ1 = 0 the system (4140) becomes

wkl + δ2 =wkminus1l + δ2

ξ2k (4173a)


(δ2 +wkl) (4173b)

The system (4173) is nonlinear but equation (4173a) is uncou-pled from equation (4173a) Defining ξ2k = akminus1ak equation(4173a) is solved by

wkl = minusδ2 + akαl (4174)

Substituting wkl given by (4174) into equation (4173b)

vk+1l minus vkl =ξ2k+1akαl

(4175)

Defining

ξ2k+1 = minusak (bk+1 minus bk) (4176)

we have that equation (4175) is a total difference Therefore wehave the following solution of the system (4173)

vkl = βl +bkαl

(4177a)

wkl = minusδ2 + akαl (4177b)

case m = 2k + 1 If δ1 = 0 the system (472) becomes

zkl minusakakminus1

zkminus1l =

(akakminus1

minus 1

)ykl (4178a)

yk+1l minus ykl = minusak (bk+1 minus bk) (4178b)

where we used (4176) and ξ2k = akminus1ak The system is linearand equation (4178b) is uncoupled from (4178a) If we put

ak = minusck+1 minus ckbk+1 minus bk

(4179)

then equation (4178a) becomes a total difference whose solu-tion is

ykl = ck + γl (4180)


Substituting ykl given by (4180) into equation (4178a) we ob-tain


zklminusbk minus bkminus1ck minus ckminus1

zkminus1l =

(bk minus bkminus1ck minus ckminus1

minusbk+1 minus bkck+1 minus ck

)(ck + γl)

(4181)

We can therefore represent the solution as

zkl =ck+1 minus ckbk+1 minus bk

(dk + δl) minus γl (4182)


dkminusdkminus1 = bk+1minusbk+1 minus bkck+1 minus ck

ck+1minusbkminus1+bk minus bk=1ck minus ckminus1

ckminus1

(4183)

Equation (4183) is a total difference and dk is given by

dk = bk minusbk+1 minus bkck+1 minus ck

ck (4184)

Therefore we have that the solution of the system (4178) is givenby

ykl = ck + γl (4185a)


δl minus γl +bkck+1 minus ckbk+1

bk+1 minus bk (4185b)

Equations (41774185) we have the value of the four fields but wehave too many arbitrary functions in the m direction namely αl βlγl and δl Inserting (41774185) into (193d) with δ1 = 0 and sepa-rating the terms even and odd in n and m we obtain we obtain twoequations

βlαl minus δl = 0 βl+1αl+1 minus δl = 0 (4186a)

We can solve this compatibility conditions with respect to βl and δlwe obtain

βl =δ0αl

δl = δ0 (4187a)

where δ0 is a constant Inserting then (4187) into (41774185) weobtain the following expression for the solution of the 3D2 when δ1 =0

vkl =bk + δ0αl

(4188a)

wkl = minusδ2 minusαlck+1 minus ckbk+1 minus bk

(4188b)


ykl = ck + γl (4188c)


δ0 minus γl +bkck+1 minus ckbk+1

bk+1 minus bk (4188d)

This discussion exhausts the possible cases So for any value of theparameters we have the general solution of the 3D2 equation (193d)

442 The D3 and the iD4 equations i = 1 2

In this Subsection construct the general solution of the D3 equationand of the two forms of theD4 equation which we denote collectivelyas iD4 with i = 1 2 These equations are given by (193e) (193f) and(193g) The procedure we will follow will make use of the first inte-grals but in a different way with respect to that used in Subsection441 for the iD2 equations since as explained in the beginning of thisSection we will need to extract some information from the equations

4421 D3 equation

From Section 43 we know that the first integrals of the D3 equa-tion (193e) in the n direction is a four-point third order first integral(447a) Therefore as stated at the beginning of Section this integraldefines a three-point third order ordinary difference equation fromthe relation W1 = ξn However to tackle the problem of finding thegeneral solution in this case we do not start directly from the firstintegral W1 (447a) but instead we start by looking to the equation it-self (193e) written as a system Applying the general transformation(455) we obtain the following system

vkl +wklykl +wklzkl + yklzkl = 0 (4189a)

vkl+1 + yklwkl+1 + zklwkl+1 + yklzkl = 0 (4189b)

vk+1l +wklyk+1l +wklzkl + zklyk+1l = 0 (4189c)

vk+1l+1 + yk+1lwkl+1 + zklwkl+1 + zklyk+1l = 0 (4189d)

From the system (4189) we have four different way for calculatingzkl This means that we have some compatibility conditions Indeedfrom (4189a) and (4189c) we obtain the following equation for vk+1l

vk+1l =(wkl + yk+1l)vkl

wkl + ykl+

(ykl minus yk+1l)w2kl

wkl + ykl (4190)

while from (4189b) and (4189d) we obtain the following equation forvk+1l+1

vk+1l+1 =(wkl+1 + yk+1l)vkl+1

wkl+1 + ykl+

(ykl minus yk+1l)w2kl+1

wkl+1 + ykl (4191)


Equations (4190) and (4191) give rise to a compatibility conditionbetween vk+1l and its shift in the l direction vk+1l+1 which is givenby (

yklwkl+1 + yk+1l+1wkl+1 + yk+1l+1ykl

minus ykl+1wkl+1 minus yk+1lwkl+1 minus yk+1lykl+1

)middot

(vkl+1 minusw2kl+1) = 0

(4192)

Discarding the trivial solution ykl = w2kl we obtain the followingvalue for the field wkl

wkl = minusyk+1lyklminus1 minus yk+1lminus1ykl

yk+1l + yklminus1 minus yk+1lminus1 minus ykl (4193)

which makes (4190) and (4191) compatible Then we have to solvethe equation with respect to vk+1l

vk+1l =yk+1l minus yk+1lminus1

ykl minus yklminus1vkl

+(yk+1lminus1ykl minus yk+1lyklminus1)

2

(yk+1lminus1 + ykl minus yk+1l minus yklminus1)(ykl minus yklminus1)

(4194)

Making the transformation

vkl = (ykl minus yklminus1)Vkl + y2klminus1 (4195)

we can reduce (4194) to the equation

Vk+1l = Vkl +(yklminus1 minus yk+1lminus1)

2

yk+1lminus1 + ykl minus yk+1l minus yklminus1 (4196)

To go further we need to specify the form of the field ykl Thiscan be extracted from the first integrals Consider the equation W1 =ξn with W1 given as in (447a) This relation defines a third orderfour-point ordinary difference equation in the n direction dependingparametrically on m In particular if we choose the case when m =

2l+ 1 we have the equation

F(+)n

(un+12l+1 minus unminus12l+1) (un+22l+1 minus un2l+1)

un2l+1 + unminus12l+1

+ F(minus)n


un+12l+1 + un+22l+1= ξn

(4197)

where we have chosen without loss of generality α = β = 1 Using thetransformation (455b) equation (4197) is converted into the system

(yk+1l minus ykl)(zkl minus zkminus1l) = ξ2k(ykl + zkminus1l) (4198a)

(yk+1l minus ykl)(zk+1l minus zkl) = ξ2k+1(yk+1l + zk+1l) (4198b)

This system is nonlinear but if we solve (4198b) with respect tozk+1l and we substitute it together with its shift in the k direction


into (4198a) we obtain a linear second order ordinary difference equationinvolving only the field ykl

ξ2kminus1yk+1m minus (ξ2k + ξ2kminus1)ykm

+ ξ2kykminus1m + ξ2kξ2kminus1 = 0(4199)

First we can lower the order of this equation by one with the potentialtransformation

Ykl = yk+1l minus ykl (4200)

Indeed we have that Ykl solves the equation

Ykl minusξ2kξ2kminus1

Ykminus1l + ξ2k = 0 (4201)

Defining

ξ2k = minusak (bk minus bkminus1) ξ2kminus1 = minusakminus1 (bk minus bkminus1) (4202)

we obtain that Ykl can be expressed as

Ykl = ak (bk +αl) (4203)

From (4200) we have that

yk+1l minus ykl = ak (bk +αl) (4204)

Setting

ak = ck+1 minus ck bk =dk+1 minus dkck+1 minus ck

(4205)

we have

ykl = αlck + dk +βl (4206)

Inserting now the obtained value of ykl from (4206) into the equa-tion (4196) we obtain

Vk+1l = Vkl minus(dk +αlminus1ck minus dk+1 minusαlminus1ck+1)

2

(αl minusαlminus1)(ck+1 minus ck)(4207)

So we get the following solution for Vkl

Vkl = γl minusαlminus1αlminus1ck + 2dkαl minusαlminus1

+ek

αl minusαlminus1(4208)

up to a quadrature for the function ek

ek+1 = ek minus(dk+1 minus dk)

2

ck+1 minus ck (4209)


Plugging the obtained value of vkl we can compute wkl from (4193)and finally zkl from the original system (4189) In this case we obtaina single compatibility condition given by

(αl+1 minusαl)γl+1 minus (αl minusαlminus1)γl

minusαlminus1βlminus1 +αlβl +αlβlminus1 minusαlminus1βl = 0(4210)

which can be expressed as

γl = minusαlminus1βlminus1 + δlαl minusαlminus1

(4211)

with δl given by the following first order difference equation

δl+1 = δl +αlβlminus1 minusαlminus1βl (4212)

We underline that this first order difference equation is the discreteanalogue of a quadrature A prime(x) = f(x)

So the function Vkl is given by (4208-4212) where ek and δl aredefined implicitly and can be found by discrete integration Then thegeneral solution of (193e) is constructed explicitly by successive sub-stitution in (4206) (4195) (4193) and (4189a)

4422 1D4 equation

From Section 43 we know that the first integrals of the 1D4 equa-tion (193f) in the n direction is a four-point third order first integral(448a) Therefore this integral defines a three-point third order or-dinary difference equation from the relation W1 = ξn However tofind the general solution in this case again we do not start directlyfrom the first integral W1 (448a) but instead we start by looking tothe equation itself (193f) written as a system Applying the generaltransformation (455) to (193f) we obtain the following system

vklzkl +wklykl + δ1wklzkl + δ2yklzkl + δ3 = 0 (4213a)

yklwkl+1 + zklvkl+1

+ δ1zklwkl+1 + δ2yklzkl + δ3 = 0(4213b)

wklyk+1l + vk+1lzkl

+ δ1wklzkl + δ2zklyk+1l + δ3 = 0(4213c)

zklvk+1l+1 + yk+1lwkl+1

+ δ1zklwkl+1 + δ2zklyk+1l + δ3 = 0(4213d)

From the equations (4213) we have four different way for calcu-lating zkl This means that we have some compatibility conditionsIndeed from (4213a) and (4213c) we obtain the following equationfor vk+1l

vk+1l =δ3 +wklyk+1l

δ3 +wklyklvkl+

(yk+1l minus ykl)(δ1w2kl minus δ2δ3)

δ3 +wklykl(4214)



vk+1l+1 =δ3 + yk+1lwkl+1

δ3 + yklwkl+1vkl+1

+(yk+1l minus ykl)(δ1w

2kl+1 minus δ2δ3

(yklwkl+1 + δ3

(4215)

Equations (4214) and (4215) give rise to a compatibility conditionbetween vk+1l and its shift in the l direction vk+1l+1 given by[

(yk+1l+1ykl minus yk+1lykl+1)wkl+1

+ δ3 (yk+1l+1 + ykl minus ykl+1 minus yk+1l)

]middot

(vkl+1wkl+1 minus δ2δ3 + δ1w2kl+1) = 0

(4216)

Discarding the trivial solution

vkl = minusδ1wkl +δ2δ3wkl

we obtain for wkl

wkl = δ3yk+1lminus1 minus yk+1l minus yklminus1 + ykl

yk+1lyklminus1 minus yk+1lminus1ykl(4217)

which makes (4214) and (4215) compatible Then we have to solvethe following equation for vkl



+δ1δ3(yk+1lminus1 minus yklminus1 minus yk+1l + ykl)

2

(yk+1lminus1ykl minus yk+1lyklminus1)(ykl minus yklminus1)

minusδ2(yk+1lminus1ykl minus yk+1lyklminus1)

ykl minus yklminus1

(4218)


vkl = (ykl minus yklminus1)Vkl +δ1δ3yklminus1

minus δ2yklminus1 (4219)

we obtain that Vkl satisfied the difference equation

Vk+1l = Vkl+δ1δ3(yklminus1 minus yk+1lminus1)

2

yklminus1yk+1lminus1(yk+1lminus1ykl minus yk+1lyklminus1)(4220)

At this point without any knowledge of the field ykl we cannot gofurther However as in the D3 case we can recover the missing infor-mation using the first integral W1 as given in (448a) This integralprovide us the relation W1 = ξn which is a third order four-point


ordinary difference equation in the n direction depending parametri-cally on m In particular if we choose the case when m = 2l+ 1 wehave the equation

F(+)n

(un2l+1 minus un+22l+1 + δ1un+12l+1)unminus12l+1 + un+12l+1un+22l+1

(un2l+1 + δ1unminus12l+1)un+12l+1

+ F(minus)n


un2l+1 (un+22l+1δ1 + un+12l+1)= ξn

(4221)

where we set without loss of generality α = β = 1 Using the trans-formation (455b) then (4221) is converted into the system

(ykl minus yk+1l + δ1zkl)zkminus1l + yk+1lzkl

(ykl + δ1zkminus1l)zkl= ξ2k (4222a)

(ykl minus yk+1l)(zkl minus zk+1l)

zkl(zk+1lδ1 + yk+1l)= ξ2k+1 (4222b)

It is quite easy to see that if we solve (4222b) with respect to zk+1l

and then substitute the result into (4222a) we obtain a linear secondorder ordinary difference equation for ykl

ξ2kminus1yk+1l+(1minusξ2kminusξ2kξ2kminus1)yklminus(1minusξ2k)ykminus1l = 0 (4223)

We can solve this equation as in the case of the D3 equation First ofall let us introduce Ykl = akykl + bkykminus1l such that Yk+1l minus Ykl isequal to the left hand side of (4223) Then we define

ξ2k = minusbk+1 minus bk minus ak

ak+1 (4224a)

ξ2kminus1 =minusbk + ak+1 + bk+1 minus ak

bk (4224b)

Therefore ykl must solve the first order equation

akykl + bkykminus1l = αl (4225)

The equation (4225) can be solved if we choose

ak =1

ck

1

dk minus dkminus1 bk = minus

1

ckminus1

1

dk minus dkminus1 (4226)

Then the solution of (4225) is

ykl = ck (αldk +βl) (4227)

Inserting (4227) into (4220) we obtain

Vk+1l = Vkl minusδ1δ3

βlminus1αl minusβlαlminus1

(ck minus ck+1)2

c2kc2k+1(dk+1 minus dk)

+δ1δ3αlminus1


[1

(αlminus1dk+1 +βlminus1)c2k+1

minus1

(αlminus1dk +βlminus1)c2k

]

(4228)


This means that we can represent the solution of Vkl as

Vkl = γl +δ1δ3


[αlminus1

c2k(αlminus1dk +βlminus1)+ ek

](4229)

where ek is defined from the quadrature

ek+1 = ek minus(ck minus ck+1)

2


(4230)

Using the obtained value of vkl we can compute wkl from (4217)and finally zkl from the original system (4213) In this case we obtaina single compatibility condition for the arbitrary functions given by

(βlαl+1 minusβl+1αl)γl+1minus(βlminus1αl minusβlαlminus1)γl = (βlminus1αl minusβlαlminus1) δ2

(4231)

This condition can be expressed also as

γl =δlδ2

βlminus1αl minusβlαlminus1 (4232)

with δl given by solving the following quadrature


Let us note that δl is the same as in (4212)So the auxiliary function Vkl is given by (4229-4233) where ek

and δl are defined implicitly and can be found by discrete integra-tion Then the general solution of (193f) is constructed explicitly bysuccessive substitution in (4227) (4219) (4217) and (4213a)

4423 2D4 equation

From Section 43 we know that the first integrals of the 2D4 equa-tion (193g) in the n direction is a four-point third order first integral(452a) Therefore this integral defines a three-point third order ordi-nary difference equation from the relation W1 = ξn However to findthe general solution in this case we do not start directly from the firstintegral W1 (452a) but we start instead by looking to the equation it-self (193g) written as a system Applying the general transformation(455) to (193g) we obtain the following system

vklwkl + δ2wklykl + δ1wklzkl + yklzkl + δ3 = 0 (4234a)

vkl+1wkl+1 + δ2yklwkl+1

+ δ1zklwkl+1 + yklzkl + δ3 = 0(4234b)

wklvk+1l + δ2wklyk+1l

+ δ1wklzkl + zklyk+1l + δ3 = 0(4234c)

wkl+1vk+1l+1 + δ2yk+1lwkl+1

+ δ1zklwkl+1 + zklyk+1l + δ3 = 0(4234d)


From the equations (4234) we have in principle four different wayfor calculating zkl This means that we have some compatibility con-ditions Indeed from (4234a) and (4234c) we obtain the followingequation for vk+1l

vk+1l =δ1wkl + yk+1l

δ1wkl + yklvkl

minus(yk+1l minus ykl)(δ2w

2klδ1 minus δ3)

(δ1wkl + ykl)wkl

(4235)


vk+1l+1 =δ1wkl+1 + yk+1l

δ1wkl+1 + yklvkl+1


2kl+1δ1 minus δ3)

(δ1wkl+1 + ykl)wkl+1

(4236)

Equations (4235) and (4236) give rise to a compatibility conditionbetween vk+1l and vk+1l+1 is given by[

yk+1l+1ykl + δ1 (yklwkl+1 + yk+1l+1wkl+1)

minus yk+1lykl+1 minus δ1 (yk+1lwkl+1 minus ykl+1wkl+1)

]middot(

vkl+1wkl+1 + δ3 minus δ1δ2w2kl+1

)= 0

(4237)


vkl = δ1δ2wkl minusδ3wkl

we obtain

wkl =1

δ1

yk+1lminus1ykl minus yk+1lyklminus1

yklminus1 + yk+1l minus ykl minus yk+1lminus1(4238)



ykl minus yklminus1vkl minus

δ1δ3(yklminus1 minus yk+1lminus1)2(ykl minus yklminus1)

(yk+1lyklminus1 minus yk+1lminus1ykl)y2klminus1

+y2klminus1yk+1lδ2 minus δ3yk+1lδ1 + δ1δ3yk+1lminus1 minus yk+1lminus1δ2y

2klminus1

(ykl minus yklminus1)yklminus1

minusyk+1lminus1δ2y

2klminus1 + δ1δ3yk+1lminus1 minus 2yklminus1δ3δ1)

y2klminus1

(4239)


vkl = (ykl minus yklminus1)Vkl +δ1δ3ykl

minus δ2ykl (4240)


we obtain that Vkl satisfied the first order difference equation

Vk+1l = Vkl minusδ1δ3(ykl minus yk+1l)

2

yklyk+1l(yk+1lminus1ykl minus yk+1lyklminus1) (4241)

At this point without any knowledge of the field ykl we cannotgo further However as in the previous two cases we can recover themissing information using the first integral W1 as given in (452a)This integral defines the relation W1 = ξn a third order four-pointordinary difference equation in the n direction depending parametri-cally on m In particular if we choose the case when m = 2l+ 1 wehave the equation

minus F(+)nun+22l+1unminus12l+1 + (minusun+22l+1 + un2l+1)un+12l+1 + δ3

unminus12l+1un2l+1 + δ3

+ F(minus)n


un+12l+1un+22l+1 + δ3= ξn

(4242)

where we choose without loss of generality α = β = 1 Using thetransformation (455b) the equation (4242) is converted into the sys-tem

(yk+1l minus ykl)zkl minus yk+1lzk+1l minus δ3(zk+1lykl + δ3)

= ξ2k (4243a)

(yk+1l minus ykl)(zk+1l minus zkl)

(yk+1lzk+1l + δ3)= ξ2k+1 (4243b)

Now if we solve (4243b) with respect to zk+1l and then substituteinto (4243a) we obtain a linear second order ordinary difference equa-tion for ykl

ξ2kminus1yk+1lminus(1+ξ2kminusξ2kξ2kminus1)ykl+(1+ξ2k)ykminus1l = 0 (4244)

We solve this equation using the freedom provided by the functionsξ2k and ξ2k+1 similarly as we did in the case of the 1D4 equationIndeed let us introduce the field Ykl = akykl+bkykminus1l and assumethat Yk+1lminus Ykl equals the left hand side of (4244) Then we choose

ξ2k =bk+1 minus bk +minusak

ak+1 (4245a)

ξ2kminus1 = minusbk+1 minus bk + ak+1 minus ak

bk (4245b)

so that ykl solves the first order equation


If we define

ak =1

ck

1


1

ckminus1

1



(4246) is solved by



Vk+1l = Vkl +δ3δ1(ck+1 minus ck)

2

(dk+1 minus dk)(βlαlminus1 minusβlminus1αl)c2kc2k+1

minusαlδ1δ3

βlαlminus1 minusβlminus1αl

[1

(αldk+1 +βl)c2k+1

minus1

(αldk +βl)c2k

]

(4249)

whose solution is

Vkl = γl minusαlδ1δ3

(αldk +βl)c2k(βlαlminus1 minusβlminus1αl)

+δ3δ1ek


(4250)

where ek satisfies the quadrature

ek+1 = ek +(ck+1 minus ck)

2

(dk+1 minus dk)c2kc2k+1

(4251)


(βlαl+1minusβl+1αl)γl+1minus(βlminus1αlminusβlαlminus1)γl = (βlαl+1minusβl+1αl)δ2

(4252)

ie

γl =δlδ2


with δl satisfying the following quadrature

δl+1 = δl +αl+1βl minusαlβl+1 (4254)

So the auxiliary function Vkl is given by (4250-4254) where ekand δl are defined implicitly by (4251) and (4254) and can be foundby discrete integration Then the general solution of (193g) is thenconstructed explicitly by successive substitution in (4248) (4240)(4238) and (4234a)


443 The trapezoidal H4 equations

In this Subsection we construct the general solution of the trapezoidalH4 equations (191) The procedure we will follow will make use ofthe first integrals as in the cases presented in Section 442 The maindifference is that since the H4 are non-autonomous only in the di-rection m with the two-periodic non-autonomous factors F(plusmn)m givenby (190) instead of the general transformation (455) we can use thesimplified transformation

un2l = pnl un2l+1 = qnl (4255)

Then to describe the general solution of a H4 we only need threearbitrary functions one in the n direction and two in the m direction


Let us start from the first integral W1 (429a) which is a two-pointfirst order first integral This implies that the tHε1 equation (191a) canbe written as a conservation law Indeed we can carefully rearrangethe terms in (191a) and use the properties of the functions F(plusmn)m torewrite (191a) as

(Tm minus Id)(F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm

1+ ε2unmun+1m

)= 0

(4256)

ie as a conservation law in the form (47b) From (4256) we canderive the general solution of (191a) itself In fact (4256) implies

F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm

1+ ε2unmun+1m= λn (4257)

where λn is an arbitrary function of n This is a first order differenceequation in the n direction in which m plays the rocircle of a parameterFor this reason we can safely separate the two cases m even and modd

case m = 2k In this case (4257) reduces to the first order linearequation

un+12k minus un2k =α2λn

(4258)

which has solution

un2k = θ2k +ωn (4259)

where θ2k is an arbitrary function and ωn is the solution of thesimple ordinary difference equation

ωn+1 minusωn =α2λn

ω0 = 0 (4260)


case m = 2k + 1 In this case (4257) reduces to the discrete Riccatiequation

λnε2un2k+1un+12k+1minusun+12k+1+un2k+1+ λn (4261)

Using the Moumlbius transformation

un2k+1 =iε

1minus vn2k+1

1+ vn2k+1 (4262)

this equation reduce to the linear equation

(i + ελn) vn+12k+1 minus (i minus ελn) vn2k+1 = 0 (4263)

If we define

λn =iε

κn minus κn+1κn + κn+1

(4264)

then we have that the general solution of (4263) is expressed as

vn2k+1 = κnθ2k+1 (4265)

Using (4262) we then obtain

un2k+1 =iε

1minus κnθ2k+11+ κnθ2k+1

(4266)

So we have the general solution to (191a) in the form

unm = F(+)m (θm +ωn) + F

(minus)m

iε

1minus κnθm1+ κnθm

(4267)

where ωn is defined by (4260) and λn is defined by (4264) This isanother proof different from that given in Subsection 241 following[67 69] of the linearization of the equation (191a)

It is worth to note that in the case of the tHε1 equation (191a) ispossible to give simple proof of the linearization also using the firstintegral in the m direction (429b) This kind of linearization was pre-sented in [71] and it is important since it was the first example oflinearization obtained using a higher-order fist integral We leave thisdiscussion to Appendix H


From Section 43 we know that the first integrals of the tHε2 equation(191b) in the n direction is four-point third order first integral (430a)Therefore this integral defines a three-point third order ordinary dif-ference equation W1 = ξn However this integral is particularly com-plex so we start first by inspecting the tH

ε2 equation (191b) itself


If we apply the transformation (4255) we can write down the tHε2equation (191b) as the following system of two coupled equations

(pnl minus pn+1l)(qnl minus qn+1l)

minusα2(pnl + pn+1l + qnl + qn+1l)

+εα22

(2qnl + 2α3 +α2)(2qn+1l + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2

minusα23 minus 2εα2α3(α2 +α3) = 0

(4268a)

(qnl minus qn+1l)(pnl+1 minus pn+1l+1)

minusα2(qnl + qn+1l + pnl+1 + pn+1l+1)

+εα22

(2qnl + 2α3 +α2)(2qn+1l + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2 minusα23

minus 2εα2α3(α2 +α3) = 0

(4268b)

We have that equation (4268a) depends on pnl and pn+1l and thatequation (4268b) depends on pnl+1 and pn+1l+1 So we apply thetranslation operator Tl to (4268a) to obtain two equations in terms ofpnl+1 and pn+1l+1

(pnl+1 minus pn+1l+1)(qnl+1 minus qn+1l+1)

minusα2(pnl+1 + pn+1l+1 + qnl+1 + qn+1l+1)

+εα22

(2qnl+1 + 2α3 +α2)(2qn+1l+1 + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2


(4269a)



+εα22

(2qnl + 2α3 +α2)(2qn+1l + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2 minusα23


(4269b)

The system (4269) is equivalent to the original system (4268) Wecan solve (4269) with respect to pnl+1 and pn+1l+1

pnl+1 =

(qnl+1 minusα3)qn+1l minus (qnl minusα3)qn+1l+1

minus(α2 +α3) (qnl+1 minus qnl) minus εα23 (qnl+1 minus qnl)

+ε[α32 + 2α3 (qnl +α2) minus (qnl+1 minus qnl)qn+1l

]qn+1l+1

+ε (α2 + qnl+1) (α2 + qnl)qn+1l+1

minusεqn+1l[α32 minus 2 (qnl+1 +α2)α3 + (α2 + qnl+1) (α2 + qnl)

]

qn+1l+1 minus qnl+1 + qnl minus qn+1l

(4270a)


pn+1l+1 =

(qn+1l minusα3)qnl+1 + (α3 minus qn+1l+1)qnl

minus(α2 +α3) (qn+1l minus qn+1l+1) + εα23 (qn+1l+1 minus qn+1l)

+εqnl+1[(qn+1l+1 minus qn+1l)qnl minusα

23 minus 2α3 (qn+1l +α2)

]minusεqnl+1 (α2 + qn+1l+1) (α2 + qn+1l)

+εqnl[α23 + 2 (α2 + qn+1l+1)α3 + (α2 + qn+1l+1) (α2 + qn+1l)

]


(4270b)

We see that the right hand sides of (4270) are functions only of qnlqn+1l qnl+1 and qn+1l+1 Moreover (4270a) and (4270b) must becompatible Therefore applying Tminus1l to (4270b) and imposing to theobtained expression to be equal to (4270a) we find that qnl mustsolve the following equation

α2 (qnminus1l+1 minus qnminus1l minus qn+1l+1 + qn+1l)

+ (qnl minus qn+1l)qnminus1l+1

minus(qnl minus qnminus1l)qn+1l+1

+qnl+1 (qn+1l minus qnminus1l)

+εα22 (qn+1l+1 minus qn+1l + qnminus1l minus qnminus1l+1)

+εα2 (qn+1l+1 minus qn+1l + qnminus1l minus qnminus1l+1) (qnl+1 + 2α3 + qnl)

+ε (qn+1l minus qnminus1l)qn+1l+1qnminus1l+1

+ε (qnl+1 minus qnl + qn+1l)qnminus1lqnminus1l+1

+ε [2α3qn+1l minus (qnl+1 + 2α3)qnl]qnminus1l+1

+ε (qnl+1 + 2α3)qnlqn+1l+1

minusε (2α3 + qn+1l)qnminus1lqn+1l+1

minusε (qnl+1 minus qnl)qn+1lqn+1l+1

minusε (2α3 + qnl) (qn+1l minus qnminus1l)qnl+1 = 0(4271)

This partial difference equation for qnl is not defined on a quadgraph but it is defined on the six-point lattice shown in Figure 41

(nminus 1 l)

(nminus 1 l+ 1)

(n l)

(n l+ 1) (n+ 1 l+ 1)

(n+ 1 l)

Figure 41 The six-point lattice


Remark 444 Let us note that if the field qnl satisfies the followingequation known as the discrete wave equation

qn+1l+1 + qnl minus qn+1l + qnl+1 = 0 (4272)

we cannot define pnl+1 and pn+1l+1 as given in (4270) In this casewe have first to solve equation (4272) with respect to qnl and thenuse the system (4268) to specify pnl The discrete wave equation(4272) is a trivial linear Darboux integrable equation since it pos-sesses the following two-point first order first integrals

W1 = qn+1l minus qnl (4273a)

W2 = qnl+1 minus qnl (4273b)

As remarked in the Introduction the existence of a two-point first or-der first integral means that the equation is itself a first integral There-fore the discrete wave equation (4272) can be alternatively written as(Tl minus Id)W1 or (Tn minus Id)W2 withW1 andW2 given by (4273) The so-lution is readily obtained from (4273a) which implies qn+1lminusqnl =

ξn This equation becomes a total difference setting ξn = an+1 minus anand we get the general solution of equation (4272)

qnl = an +αl (4274)

where both an and αl are arbitrary functions of their argument Thisis the discrete analogue of drsquoAlembert method of solution of the con-tinuous wave equation Substituting (4274) into (4269) we obtain thecompatibility condition

αl+1 minusαl = 0 (4275)

ie αl = α0 = constant and the system (4268) is now consistentTherefore we are left with one equation for pnl eg (4268a) There-fore inserting (4274) with αl = α0 in (4268a) and solving with re-spect to pn+1l we obtain

pn+1l =an+1 minus an +α2an+1 minus an minusα2

pnl +α2 (α2 minus an + 2α3 minus 2α0 minus an+1)

α2 + an minus an+1

+

α2ε

[α22 + (2α0 + an+1 + 2α3 + an)α2

+ 2 (an+1 +α0 +α3) (α3 + an +α0)

]α2 + an minus an+1

(4276)

Defining through discrete integration a new function bn such that

an+1 minus an +α2an+1 minus an minusα2

=bn+1bn

(4277)


we have that pnl solves the equation

pn+1l

bn+1=pnl

bn+

(an +α0 minusα3)bn+1 + bn (α2 +α3 minusα0 minus an)

bnbn+1

minus ε

[(α2 +α0 + an +α3)

2 bn+1 minus bn (α3 + an +α0)2]

bnbn+1

(4278)

Equation (4278) is solved by

pnl = bn (βl + cn) (4279)

where cn is given by the discrete integration

cn+1 = cn +(an +α0 minusα3)bn+1 + bn (α2 +α3 minusα0 minus an)

bnbn+1

minus ε

[(α2 +α0 + an +α3)


bnbn+1

(4280)

This yields the solution of the tHε2 equation (191b) when qnl satisfythe discrete wave equation (4272)

In the general case we have proved that the tHε2 equation (191b)is equivalent to the system (4268) which in turn is equivalent to thesolution of equations (4270a) and (4271) However (4270a) merelydefines pnl+1 in terms of qnl and its shifts Therefore if we find thegeneral solution of equation (4271) the value of pnl will follow Tofind such solution we turn to the first integrals Like in the case of theH6 equations (193) we will find an expression for qnl using the firstintegrals and then we will insert it into (4271) to reduce the numberof arbitrary functions to the right one

We consider the equation W1 = ξn where W1 is given by (430a)with k = 2l+ 1

(unminus12l+1 minus un+12l+1) (un+22l+1 minus un2l+1)

(un2l+1 minus unminus12l+1 +α2) (un+12l+1 minus un+22l+1 +α2)= ξn

(4281)

Using the substitutions (4255) we have

(qnminus1l minus qn+1l) (qn+2l minus qnl)

(qnl minus qnminus1l +α2) (qn+1l minus qn+2l +α2)= ξn (4282)

This equation contains only qnl and its shifts From equation (4282)it is very simple to obtain a discrete Riccati equation Indeed the trans-formation

Qnl =qnl minus qnminus1l +α2qn+1l minus qnminus1l

(4283)


brings (4282) into

Qn+1l +1

ξnQnl= 1 (4284)

which is a discrete Riccati equation Let us assume an to be a partic-ular solution of (4284) then we express ξn as

ξn =1

an (1minus an+1) (4285)

Using the standard linearization of the discrete Riccati equation

Qnl = an +1

Znl(4286)

from (4285) we obtain the following equation for Znl

Zn+1l =anZnl + 1

1minus an+1 (4287)

Introducing

an =bnminus1

bn + bnminus1(4288)

we obtain

Zn+1lminusbnminus1bn+1 + bnminus1bnbnbn+1 + bnminus1bn+1

Znl =bnminus1bn+1 + bnminus1bn + bnbn+1 + b

2n

bnbn+1 + bnminus1bn+1

(4289)

If we assume that (4289) can be written as a total difference ie

(Tn minus Id) (dnZnl minus cn) = 0 (4290)

we obtain

bn = cn+1 minus cn dn =(cn+1 minus cn)(cn minus cnminus1)

cn+1 minus cnminus1 (4291)

So bn must be a total difference and therefore we can represent Znl

as

Znl =(cn+1 minus cnminus1)(cn +αl)

(cn+1 minus cn)(cn minus cnminus1) (4292)

From (4286) and (4288) we obtain the form of Qnl

Qnl =(cn minus cnminus1)(cn+1 +αl)

(cn +αl)(cn+1 minus cnminus1) (4293)

Introducing the value of Qnl from (4293) into (4283) we obtain thefollowing equation for qnl

qn+1l minus qnminus1l

qnl minus qnminus1l +α2=

(cn +αl)(cn+1 minus cnminus1)

(cn minus cnminus1)(cn+1 +αl) (4294)


Performing the transformation

Rnl = (cn +αl)qnl (4295)

we obtain the following second order ordinary difference equationfor the field Rnl

Rn+1l minuscn+1 minus cnminus1cn minus cnminus1

Rnl

+cn+1 minus cncn minus cnminus1

Rnminus1l minusα2(cn +αl)cn+1 minus cnminus1cn minus cnminus1

= 0(4296)

Then we can represent the solutions of the equation (4296) as

Rnl = Pnl +αlen + fn (4297)

where en and fn are particular solutions of

en+1 minuscn+1 minus cnminus1cn minus cnminus1

en


enminus1 = α2cn+1 minus cnminus1cn minus cnminus1

(4298a)

fn+1 minuscn+1 minus cnminus1cn minus cnminus1

fn


fnminus1 = α2cncn+1 minus cnminus1cn minus cnminus1

(4298b)

Pnl will be then solve the following equation

Pn+1l minuscn+1 minus cnminus1cn minus cnminus1

Pnl +cn+1 minus cncn minus cnminus1

Pnminus1l = 0 (4299)

The equations (4298a) and (4298b) are not independent Indeeddefining

An =encnminus1 minus enminus1cn minus fn + fnminus1

cn minus cnminus1 (4300)

and using (4298) it is possible to show that the function An lies in thekernel of the operator Tn minus Id This implies that An = A0 = constantWe can without loss of generality assume the constant A0 to be zerosince if we perform the transformation

en = en minusA0 (4301)

the equation (4300) is mapped into

encnminus1 minus enminus1cn minus fn + fnminus1cn minus cnminus1

= 0 (4302)

Furthermore since (4298a) is invariant under the transformation (4301)we can safely drop the tilde in (4302) and assume that the functionsen and fn are solutions of the equations


en +cn+1 minus cncn minus cnminus1

enminus1 minusα2cn+1 minus cnminus1cn minus cnminus1

= 0

(4303a)


fn minus fnminus1 = encnminus1 minus cnenminus1 (4303b)

Finally we note that the function en can be obtained from (4303a) bytwo discrete integrations Indeed defining

En =en+1 minus encn+1 minus cn

(4304)

and substituting in (4303a) we obtain that En must solve the equation

En minus Enminus1 = α2

(1

cn+1 minus cn+

1

cn minus cnminus1

) (4305)

Note that the right hand side of (4305) is not a total difference So thefunction en can be obtained by integrating (4305) and subsequentlyintegrating (4304) This provides the value of en The obtained valuecan be plugged in (4303b) to give fn after discrete integration Thisreasoning shows that we can obtain the non-arbitrary functions enand fn as result of a finite number of discrete integrations

Now we turn to the solution of the homogeneous equation (4299)We can reduce (4299) to a total difference using the potential substi-tution Tnl = Pnl minus Pnminus1l

Tn+1l

cn+1 minus cnminus

Tnl

cn minus cnminus1= 0 (4306)

This clearly implies

Pnl minus Pnminus1l

cn minus cnminus1= βl (4307)

where βl is an arbitrary function The solution to this equation isgiven by5

Pnl = (cn +αl)βl + γl (4308)

where γl is an arbitrary function Using (429542974308) we obtainthen the following expression for qnl

qnl = βl +γl +αlen + fn

cn +αl (4309)

where en and fn are solutions of (4303)Now since the solution we obtained in (4309) depends on three ar-

bitrary functions in the l direction namely αl βl and γl there mustbe a constraint between these functions This constraint is readily ob-tained by plugging (4309) into (4271) Factorizing the n dependentpart away we are left with

γl+1 minus γl = minus(αl+1 minusαl)

(α2 +βl+1 +βl + 2α3 minus

1

ε

) (4310)

5 The arbitrary functions are taken in a convenient way


This equation tells us that the function γl can be expressed after adiscrete integration in terms of the two arbitrary functions αl andβl So the function qnl is defined by (430343094310) where thefunctions en fn and γl are defined implicitly and can be found bydiscrete integration The value of pnl now can be recovered by substi-tuting (430343094310) into (4270a) and applying Tminus1l This yieldsthe general solution of the tHε2 equation (191b)

Remark 445 Let us notice that in the case ε = 0 formula (4310)is singular However in this case we just express the compatibilitycondition as αl+1minusαl = 0 ie αl = α0 = constant It is easy to checkthat the obtained value of qnl through formula (4309) is consistentwith the substitution of ε = 0 in (4268) This means that in the caseε = 0 the value of qnl is given by

qnl = βl +γl +α0en + fn

cn +α0 (4311)

where the functions en and fn are defined implicitly and can befound by discrete integration from (4303) As in the general case thevalue of pnl now can be recovered by substituting (43034309) into(4270a) and applying Tminus1l This yields the general solution of the tHε2equation (191b) if ε = 0


From Section 43 we know that the first integrals of the tHε3 equa-

tion (191c) in the n direction is a four-point third order first integral(434a) Therefore this integral defines a four-point third order ordi-nary difference equation from W1 = ξn This integral is particularlycomplex so we start first by inspecting the tHε3 equation (191c) itselfIf we apply the transformation (4255) we can write the tHε3 equation(191c) as the following system of two coupled equations

α2(pnlqn+1l + pn+1lqnl)

minus pnlqnl minus pn+1lqn+1l

minusα3(α22 minus 1)

(δ2 + ε2

qnlqn+1l

α23α2

)= 0

(4312a)

α2(qnlpn+1l+1 + qn+1lpnl+1)

minus qnlpnl+1 minus qn+1lpn+1l+1


(δ2 + ε2

qnlqn+1l

α23α2

)= 0

(4312b)

As in the case of the tHε2 equation (191b) we have that equation

(4312a) depends on pnl and pn+1l and that equation (4312b) de-pends on pnl+1 and pn+1l+1 So we can apply the translation op-


erator Tl to (4312a) to obtain two equations in terms of pnl+1 andpn+1l+1

α2(pnl+1qn+1l+1 + pn+1l+1qnl+1)

minus pnl+1qnl+1 minus pn+1l+1qn+1l+1


(δ2 + ε2

qnl+1qn+1l+1

α23α2

)= 0

(4313a)




(δ2 + ε2

qnlqn+1l

α23α2

)= 0

(4313b)

The system (4313) is equivalent to the original system (4312) Thensince we can assume α2α3 6= 06 we can solve (4313) with respect topnl+1 and pn+1l+1

pnl+1 =

α2 (qn+1l+1 minus qn+1l)

(ε2qnlqnl+1 + δ

2α32)

+δ2α22α23 (qnl minus qnl+1)

+ε2qn+1l+1qn+1l (qnl minus qnl+1)

(qn+1l+1qnl minus qn+1lqnl+1)α3α2

(4314a)

pn+1l+1 =

α2(ε2qn+1l+1qn+1l + δ

2α32)(qnl minus qnl+1)

+δ2α22α23 (qn+1l+1 minus qn+1l)

+ε2qnlqnl+1 (qn+1l+1 minus qn+1l)


(4314b)


δ2α22α23 [qn+1l+1qnl minus qnlqnminus1l+1 + qnl+1 (qnminus1l minus qn+1l)]

minusα2(ε2qnlqnl+1 + δ

2α23)(qn+1l+1qnminus1l minus qnminus1l+1qn+1l)

+ε2 [qnlqn+1l+1 (qnminus1l minus qn+1l) minus qnl+1qnminus1lqn+1l]qnminus1l+1

+ε2qn+1l+1qn+1lqnl+1qnminus1l = 0(4315)

As in the case of the tHε2 equation (191b) the partial difference equa-tion for qnl is not defined on a quad graph but it is defined on thesix-point lattice shown in Figure 41

6 If α2 = 0 or α3 = 0 in (4312) we have that the system becomes trivially equivalentto qnl = 0 and pnl is left unspecified Therefore we can discard this trivial case


Remark 446 Let us note that if

qn+1l+1qnl minus qn+1lqnl+1 = 0 (4316)

we cannot define pnl+1 and pn+1l+1 as in (4314) In this case wefirst solve equation (4316) with respect to qnl and then use the sys-tem (4312) to specify pnl Indeed equation (4316) is a trivial Dar-boux integrable equation since it possesses the following two-pointfirst order first integrals

W1 =qn+1l

qnl (4317a)

W2 =qnl+1

qnl (4317b)

As remarked in the Introduction the existence of a two-point first or-der first integral means that the equation is itself a first integral There-fore the equation (4316) can be alternatively written as (Tl minus Id)W1or (Tn minus Id)W2 with W1 and W2 given by (4317) From (4317a) weobtain qn+1lqnl = ξn This equation is immediately solved bydefining ξn = an+1an and we get the general solution of equation(4316)

qnl = anαl (4318)

where both an and αl are arbitrary functions of their argument Letus note that equation (4316) is the logarithmic discrete wave equationsince it can be mapped into the discrete wave equation (4272) expo-nentiating (4272) and then taking qnl rarr eqnl and it is a discretiza-tion of the hyperbolic partial differential equation

uuxt minus uxut = 0 (4319)

which is obtained from the wave equation vxt = 0 using the trans-formation v = logu Since the transformation connecting (4272) and(4316) is not bi-rational it does not preserves a priori its properties[62] (in this case linearization and Darboux integrability) Substitut-ing (4318) into (4313) we obtain the compatibility condition

αl+1 minusαl = 0 (4320)


pn+1l =α2an+1 minus anan+1 minusα2an

pnl+(α22minus1)δ2α23α2 + ε

2anα20an+1

α3α2α0(α2an minus an+1) (4321)

Defining

α2an+1 minus anan+1 minusα2an

=bn+1bn

(4322)



pn+1l

bn+1=pnl

bn+δ2α23α

22bn minus bn+1

(δ2α23 + ε

2a2nα20

)α2 + ε

2a2nα20bn

bnanα0α2α3bn+1

(4323)

Note that bn in (4322) is given in terms of an and an+1 throughdiscrete integration Equation (4323) is solved by

pnl = bn (βl + cn) (4324)


cn+1 = cn+δ2α23α

22bn minus bn+1

(δ2α23 + ε

2a2nα20

)α2 + ε

2a2nα20bn

bnanα0α2α3bn+1

(4325)

This yields the solution of the tHε3 equation (191c) when qnl satisfyequation (4316)

In the general case we have proved that the tHε3 equation (191c)is equivalent to the system (4312) which in turn is equivalent to thesolution of equations (4314a) and (4315) However equation (4314a)merely defines pnl+1 in terms of qnl and its shifts Therefore if wefind the general solution of (4315) the value of pnl will follow Tofind such solution we turn to the first integrals Like in the case of thetHε2 equation (191b) we will find an expression for qnl using the first

integrals and then we will insert it into (4315) to reduce the numberof arbitrary functions to the right one

We consider the equation W1 = ξnα27 where W1 is given by

(434a) with k = 2l+ 1


(α2un2l+1 minus unminus12l+1) (un+22l+1 minusα2un+12l+1)= ξn (4326)


(qn+1l minus qnminus1l) (qn+2l minus qnl)

(α2qnl minus qnminus1l) (qn+2l minusα2qn+1l)= ξn (4327)

This equation contains only qnl and its shifts By the transformation

Qnl =α2qnl minus qnminus1l

qn+1l minus qnminus1l(4328)

equation (4327) becomes

Qn+1l +1

ξnQnl= 1 (4329)

7 The extra α2 is due to the arbitrariness of ξn and is inserted in order to simplify theforumulaelig


which is the same discrete Riccati equation as in (4284) This meansthat the solution of (4329) is given by (4293) with the appropriate def-initions (428542884291) We can substitute into (4328) the solution(4293)


α2qnl minus qnminus1l=


(cn+1 +αl)(cn minus cnminus1)(4330)

and we obtain an equation for qnl Introducing

Pnl = (cn +αl)qnl (4331)

we obtain that Pnl solves the equation

Pn+1l minusα2cn+1 minus cnminus1cn minus cnminus1



Using the transformation

Pnl =Rnl

Rnminus1l (4333)

we can cast equation (4332) in discrete Riccati equation form

Rn+1l +cn+1 minus cncn minus cnminus1

1

Rnl= α2

cn+1 minus cnminus1cn minus cnminus1

(4334)

Let dn be a particular solution of equation (4334)

dn+1 +cn+1 minus cncn minus cnminus1

1

dn= α2


(4335)

Assuming dn as the new arbitrary function we can express cn as theresult of two discrete integrations Indeed introducing zn = cnminus cnminus1in (4335) we have

zn+1zn

=(dn+1 minusα2)dnα2dn minus 1

(4336)

Equation (4336) represents the first discrete integration whereas thesecond one is given by the definition

cn minus cnminus1 = zn (4337)

Now we can linearize the discrete Riccati equation (4334) by the trans-formation

Rnl = dn +1

Snl (4338)

and we get the following linear equation for Snl

Sn+1l minusd2n(cn minus cnminus1)

cn+1 minus cnSnl =

dn(cn minus cnminus1)

cn+1 minus cn (4339)


Defining

dn =en

enminus1 (4340a)

fn minus fnminus1 =cn minus cnminus1enenminus1

(4340b)

the solution of (4339) is

Snl =(fnminus1 +βl)e

2nminus1


Inserting (4341) and (4340) into (4338) we obtain

Rnl =en(fn +βl)

enminus1(fnminus1 +βl) (4342)

Inserting the definition of Rnl (4333) into (4342) we obtain

Pnl

en(fn +βl)=

Pnminus1l


ie

Pnl = γlen(fn +βl) (4344)

Introducing (4344) into (4331) we obtain

qnl =γlen(fn +βl)

cn +αl (4345)

where fn is defined by (4340b) and cn is given by (4336) and (4337)ie cn is the solution of the equation

cn+1 minus cncn minus cnminus1

=en+1 minusα2enα2en minus enminus1

(4346)

and en is an arbitrary functionNow since the solution we obtained in (4345) depends on three ar-


αl+1 minusαl =ε2

α2δ2α23

γl+1γl(βl+1 minusβl) (4347)

This equation tells us that the function αl can be expressed after adiscrete integration in terms of the two arbitrary functions βl andγl So the function qnl is defined by (4340b4345ndash4347) where thefunctions cn fn and αl are defined implicitly and can be found bydiscrete integration The value of pnl now can be recovered by substi-tuting (4340b4345ndash4347) into (4314a) and applying Tminus1l This yieldsthe general solution of the tHε3 equation (191c)


Remark 447 Let us notice that when δ = 0 equation (4347) is singu-lar However in this case the compatibility condition is replaced byβl+1 minus βl = 0 ie βl = β0 = constant It is easy to check that theobtained value of qnl through formula (4345) is consistent with thesubstitution of δ = 0 in (4312) This means that in the case δ = 0 thevalue of qnl is given by

qnl =γlen(fn +β0)

cn +αl (4348)

where the functions cn and fn are defined implicitly and can befound by discrete integration from (4340b) and (4346) respectivelyAs in the general case the value of pnl now can be recovered bysubstituting (4340b43454346) into (4314a) and applying Tminus1l Thisyields the general solution of the tHε3 equation (191c) if δ = 0

C O N C L U S I O N S

In this Thesis we gave a comprehensive account of recent findingsabout the equations possessing the Consistency Around the Cube Inparticular our main objective was the study of the trapezoidal H4

equation (191) and of the H6 equations (193) Apart from their in-troduction and classification in [3 20ndash22] before the studies of theauthor and his collaborators little was known about the integrabilityproperties of these equation These integrability properties were stud-ied from different points of view since as it was claimed in Section(11) does not exists a universal definition of integrability Thereforein each Chapter of this thesis we gave a self-contained introductionto the methods we used namely the Consistency Around the Cube inChapter 1 the Algebraic Entropy in Chapter 2 Generalized Symmetriesin Chapter 3 and the Darboux integrability in Chapter 4 Thereforewe can say that the primary object of this Thesis is to consider thetrapezoidal H4 equation (191) and of the H6 equations (193) fromthe standpoint of different forms of integrability as each form shedsa different light on the nature of the equations dealt with

Our main result was stated in Chapter 2 and it is the heuristicproof of the fact that according to Algebraic Entropy the trapezoidalH4 equation (191) and of the H6 equations (193) are actually lin-earizable equations This shows how different tests of integrabilitycan provide different outcomes and they must be considered as com-plementary to each other To support this heuristic statement we gavesome examples of explicit linearization

In Chapter 3 we showed that the three-point Generalized Sym-metries of the trapezoidal H4 equation (191) and of the H6 equations(193) are particular instances of the non-autonomous YdKN equation(399) This result completed the identification of the three-point Gen-eralized Symmetries belonging to the ABS and Bollrsquos classificationstarted in [102] and in [166] respectively Furthermore based on theobservation that the most general case of the non-autonomous YdKN(399) was not covered by such symmetries we conjectured the exis-tence of a new integrable equations which encloses all the equationscoming from the Bollrsquos classification We then found a candidate equa-tion namely (3150) which is a non-autonomous generalization of theQV equation (3137) introduced in [156] This new non-autonomousQV equation (3150) passes the Algebraic Entropy test but unfortu-nately we were not able due to the computational complexity toprove that its three-point symmetries are given by the general caseof the non-autonomous YdKN equation (399) The main significanceof this new equation is that that all the equations from Bollrsquos classifi-

191

192 conclusions

cation [20ndash22] can be seen as particular cases of a single equation Acomplete understanding of this equation will result in better compre-hension of all its particular cases

In Chapter 4 we started by showing the Darboux integrability ofthe equations possessing the Consistency Around the Cube but notthe tetrahedron property found by J Hietarinta in [82] These equa-tions were known to be linearizable [138] and this observation alongwith the peculiar symmetry structure of the tHε1 equation (191a) ledus to conjecture that the trapezoidal H4 equations (191) and the H6

equations (193) are Darboux integrable equations The Darboux inte-grability of the trapezoidal H4 equations (191) and the H6 equations(193) was then established using a modification of the method pro-posed in [55] In the final part of Chapter 4 we showed how fromDarboux integrability we can obtain linearization and the general so-lutions of the trapezoidal H4 equations (191) and of the H6 equa-tions (193) The construction of the general solutions for the trape-zoidal H4 equations and for the H6 equations is a consequence of theDarboux integrability property and it is something that cannot be in-ferred from the method used in classify them These general solutionswere obtained in three different ways but the common feature is thatthey can be found through some linear or linearizable (discrete Riccati)equations This is the great advantage of the first integral approachwith respect to the direct one which was pursued in [67] The Dar-boux integrability therefore yields extra information that it is usefulto get the final result ie the general solutions Moreover linearizationarises very naturally from first integrals also in the most complicatedcases whereas in the direct approach can be quite tricky see eg theexamples in [67] The linearization of the first integrals is anotherproof of the intimate linear nature of the H4 and H6 equations Thisresult is even more stronger than Darboux integrability alone since apriori the first integrals do not need to define linearizable equationsWe also note that our procedure of construction of the general solu-tion based on the ideas from [56] is likely to be the discrete versionof the procedure of linearization and solutions for continuous Dar-boux integrable equations presented in [178] The preeminent rocircle ofthe discrete Riccati equation in the solutions is reminiscent of the im-portance of the usual Riccati equation in the continuous case Recalleg that the first integrals of the Liouville equation (43) are Riccatiequations (44)

Now although our knowledge of the trapezoidal H4 equations(191) and of the H6 equations (193) is more deep than before wewould like to end this Thesis mentioning some problems which arestill open

bull Prove (or disprove) that the J1 equation ie the first of the Hi-etarintarsquos (128) is Darboux integrable Since it linearizable andpossess the Consistency Around the Cube in Chapter 4 we con-

conclusions 193

jectured that it should be Darboux integrable but it is possibleto prove that its first integrals (47) must be of order higher thantwo

bull Prove the validity of the connection formulaelig (3154) and (F4)and directly compute the three-point Generalized Symmetriesof the non-autonomous QV equation

bull Find and classify all the non-autonomous equations satisfyingthe ldquostrictrdquo Klein symmetries (3144) according to their integra-bility properties We believe that in studying such equationsboth the Algebraic Entropy approach and the Generalized Sym-metries approach can be very fruitful However we remark thatas showed with an example in Chapter 3 the Generalized Sym-metries of these equations can depend on more than on threepoints

bull As mentioned in Chapter 4 Darboux integrable equations ad-mit Generalized Symmetries depending on arbitrary functions[5] However for the trapezoidal H4 equations (191) and for theH6 equations (193) the explicit form of the symmetries depend-ing on arbitrary functions is known only for the tHε1 equation(191a) This poses the challenging problem of finding the ex-plicit form of such generalized symmetries These symmetrieswill be highly non-trivial especially in the case of the tHε2 equa-tion (191b) and of the tHε3 equation (191c) were the order ofthe first integrals is particularly high

As a final remark we note that the majority of the open problems orig-inates directly from the original researches reported in this Thesis

AF I N A L PA RT O F T H E P R O O F O F T H EC O M M U TAT I V E D I A G R A M 1 9

In this Appendix we conclude the proof of fact that the diagram 19is commutative by checking all the fundamental Moumlbius transforma-tions To this end we have written a python module mobgenpy whichcontains python functions in order to generate all possible Mob4 and

Mob4

transformations and compare between the results

generate_mob4 Generate an element of Mob4 with prescribed trans-formations type acting on the four vertices Types are classifiedas translations T dilation D and inversions I

generate_mobnm Generate an element of Mob4

with prescribed trans-formations type acting on the four vertices As abobe types areclassified as translations T dilation D and inversions I

tilde_eq_der Generate the equation on the lattice (148) startingfrom an abstract equation on the geometrical quad graph (129)

equality_check Checks if the result of the application of an ele-

ment of Mob4 and of an element of Mob4

is the same Theoutput is a boolean value

generate_type Generate all possible combinations (with repetition)of four Moumlbius transformations

The content of mobgenpy is the following

1 from sympy import 2

3 def generate_type()

4 Generate al l possible combinations (with repetition ) of fourMoebius transformations

5 TT= [ rsquoTrsquo rsquoDrsquo rsquo I rsquo]6 TF = []

7 for t1 in TT

8 for t2 in TT

9 for t3 in TT

10 for t4 in TT

11 TE = [t1t2t3t4]

12 TF = TF + [TE]

13 return TF

14

15 def generate_mob4(TTu)

16 Generate an element of Mob4 with prescribed transformationstype acting on the four vertices

195

196 final part of the proof of the commutative diagram 1 9

17 var( rsquoa rsquo)18 Mob4 =

19 Ind = [01212]

20 for i in range(len(TT))

21 if TT[i] == rsquoTrsquo22 MobT = u(Ind[i])+a(Ind[i])

23 elif TT[i] == rsquo I rsquo24 MobT = 1u(Ind[i])

25 elif TT[i] == rsquoDrsquo26 MobT = a(Ind[i])u(Ind[i])

27 else

28 raise ValueError( rsquoNot a fundamental type Moebiustransformation rsquo)

29 Mob4[u(Ind[i])] = MobT

30 return Mob4

31

32 def generate_mobnm(TTEMu)

33 Generate an element of Mobnm with prescribed transformationstype acting on the four vertices

34 var( rsquoa rsquo)35 var( rsquon mrsquo integer=True)

36 f0 = (1+EM[0](-1)n+EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

37 f1 = (1-EM[0](-1)n+EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

38 f2 = (1+EM[0](-1)n-EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

39 f12 = (1-EM[0](-1)n-EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

40 F = [f0f1f2f12]

41 Mobnm =

42 Ind = [01212]

43 Mob0=0


45 if TT[i] == rsquoTrsquo46 MobT = u(0)+a(Ind[i])

47 elif TT[i] == rsquo I rsquo48 MobT = 1u(0)

49 elif TT[i] == rsquoDrsquo50 MobT = a(Ind[i])u(0)

51 else


53 Mob0 += F[i]MobT

54 Mob0=(Mob0expand())simplify()

55 Mobnm[u(0)]=Mob0

56 Mob1 = Mob0subs(n n+1u(0) u(1)simultaneous=True)

57 Mobnm[u(1)]=Mob1

58 Mob2 = Mob0subs(m m+1u(0) u(2)simultaneous=True)

59 Mobnm[u(2)]=Mob2

60 Mob12 = Mob0subs(n n+1m m+1 u(0) u(12)simultaneous=

True)

final part of the proof of the commutative diagram 1 9 197

61 Mobnm[u(12)]=Mob12

62 return Mobnm

63

64 def tilde_eq_der(QEMu)

65 bQsub = u(0) u(1) u(1) u(0) u(2) u(12) u(12) u(2)

66 Qbsub = u(0) u(2) u(1) u(12) u(2) u(0) u(12) u(1)

67 bQbsub = u(0) u(12) u(1) u(2) u(2) u(1) u(12) u(0)

68

69 f0 = (1+EM[0](-1)n+EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

70 f1 = (1-EM[0](-1)n+EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

71 f2 = (1+EM[0](-1)n-EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

72 f12 = (1-EM[0](-1)n-EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

73

74 Qt = ((f0Q+f1Qsubs(bQsubsimultaneous=True)

75 +f2Qsubs(Qbsubsimultaneous=True)

76 +f12Qsubs(bQbsubsimultaneous=True))expand())simplify()

77

78 return Qt

79

80 def equality_check(QTTEu)

81 Mob4sub=generate_mob4(TTu)

82 Mobnmsub=generate_mobnm(TTEu)

83

84 Qhat=Qsubs(Mob4sub)

85 Qhattilde=tilde_eq_der(QhatEu)

86

87 Qtilde=tilde_eq_der(QEu)

88 Qtildehat=Qtildesubs(Mobnmsub)

89

90 DQ = ((Qhattilde-Qtildehat)expand())simplify()

91 if DQ == 0

92 return True

93 else

94 return False

All the functions described above are used in following programwhich generates the most general quad equation as shown in (186)and checks if the action of all the possible elements of Mob4 and

Mob4

yields the same equation

1 from sympy import 2 import time

3 from mobgen import 4

5 def seconds_to_hours(s)

6 if s gt= 3600

7 si sd = divmod(s 1)

8 si = int(si)

9 h = si3600


10 m = (si60) 60

11 ss = si 60

12 return str(h)+ rsquoh rsquo+str(m)+ rsquomrsquo+str(ss+sd)[5]+ rsquo s rsquo13 elif s lt 3600 and s gt= 60


15 si = int(si)

16 m = (si60) 60

17 ss = si 60

18 return str(m)+ rsquomrsquo+str(ss+sd)[5]+ rsquo s rsquo19 elif s lt 60 and s gt10

20 return str(s)[5]+ rsquo s rsquo21 else

22 return str(s)[4]+ rsquo s rsquo23

24 def main()

25 generate all possible Moebius transformations

26 MTot = generate_type()

27 generic quad-equation

28 var( rsquoA B C D K ursquo)29 constant part

30 Qconst = K

31 linear part

32 Qlin = D(0)u(0)+D(1)u(1)+D(2)u(2)+D(12)u(12)

33 quadratic part

34 Qquadr = C(01)u(0)u(1)+C(02)u(0)u(2)+C(012)u(0)u(12)

35 +C(12)u(1)u(2)+C(112)u(1)u(12)+A(212)u(2)u(12)

36 cubic part

37 Qcub = B(012)u(0)u(1)u(2)+B(0112)u(0)u(1)u(12)+B

(1212)u(1)u(2)u(12)

38 quartic part

39 Qquart = A(01212)u(0)u(1)u(2)u(12)

40 the equation

41 Q = Qconst + Qlin + Qquadr + Qcub + Qquart

42 print Q

43 various possible embeddings

44 R = [-11]

45 EM = []

46 for r in R

47 for s in R

48 EP = [rs]

49 EM += [EP]

50 for E in EM

51 print E

52 taking trace of the transformation

53 N=0

54 taking trace of eventual lsquolsquoerrorsrsquorsquo

55 Nerr = []

56 dt =float(0)

57 fo= open( rsquoquad_equation_check_ rsquo+str(E[0])+ rsquo_ rsquo+str(E[1])+ rsquo out rsquo rsquowrsquo)


58 fowrite( rsquoWe check the equationnrsquo+latex(Q mode= rsquoequation rsquo)+ rsquon rsquo)

59 for TT in MTot

60 N += 1

61 t0 = timetime()

62 test = equality_check(QTTEu)

63 dt0 = timetime()-t0

64 print N test dt0

65 dt += dt0

66 fowrite(str(N)+ rsquo rsquo+str(TT)[1-1]+ rsquo is rsquo+str(test)+ rsquon rsquo)

67 if test == False

68 Nerr += [N]

69

70 fowrite( rsquoOperations take rsquo+seconds_to_hours(dt)+ rsquo withaverage time rsquo+seconds_to_hours(dt81)+ rsquo n rsquo)

71

72 if Nerr = []

73 fowrite( rsquoThere were errors at rsquo+str(Nerr)[1-1]+ rsquo Check carefully what happened rsquo)

74

75 if __name__ == rsquo__main__ rsquo76 main()

Running the above program has resulted in a complete verificationof the equivalence of the two actions

BPYTHON P R O G R A M S F O R C O M P U T I N G A L G E B R A I CE N T R O P Y

In this Appendix we discuss some python programs which can com-pute the iterates of various kind of discrete an semi-discrete equationsIn particular in Section B1 we present the python module ae2dpy

which can be used to compute the degrees of iterates of systems ofquad equations this was presented in [64] In Section B2 we presentthe python module ae_differential_differencepy which can beused to compute the degrees of the iterates of systems of differenceequations andor differential-difference of arbitrary order Finally inSection B3 we present some python utilities for the analysis of theobtained sequences All the proposed modules extensively uses theclasses defined by the pure python Computer Algebra System sympy

[150] so to be used they require a basic knowledge of itFollowing the development guidelines of sympy [150] all the mod-

ule are written using the python3 standards but they maintain thepython2 compatibility To this end is used the module __future__ toredefine the division function and the printing system All the mod-ules were tested on both python versions

b1 algebraic entropy of systems of quad equations with

ae2dpy

In this Section we give a brief description of the functions containedin the python module ae2dpy The scope of the functions in ae2dpy

is to calculate the degree of the iterates of a system of quad equa-tions We underline that ae2dpy uses the multiprocessing modulewhich permit to take advantage of modern multi-core architecturesto evaluate simultaneously the degrees along the diagonals

b11 Description of the content of ae2dpy

The ae2dpy module contains the following functions1

up_or_anti_diagonal Creates a diagonal or anti-diagonal matrix Thisis an auxiliary function used in constructing the evolution matri-ces (237)

evolution_matrix Construct an evolution matrix starting from thedegrees of the iterates This representation relies on the matrix

1 The names of the functions follow the naming convention adopted by sympy devel-opers

201

202 python programs for computing algebraic entropy

class defined by sympy This is an auxiliary function automat-ically used by the evol function in case there is the need ofdisplaying the results in the form of an evolution matrix (237)

analyze_evolution_matrix This function analyze an evolution ma-trix (237) searching for repeating patterns This is a user-levelfunction and it accepts a single argument M which must be asympy matrix The user has no need to specify which is the direc-tion of the evolution The function detect the kind of evolutionmatrix from the fact that an evolution matrix (or it transposedmatrix) is a Hessenberg matrix [80] Then the matrix is trans-formed into a list of lists eliminating the zero entries and or-dering the list from the longest to the shortest Then a pairwisecomparison of the lists is carried out by comparing appropriateslice of the two lists Equal (slice of) lists are then removed Theresult of this procedure are the unequal sequences of degrees

gen_ics This function generates the set of the initial conditions fora system of quad equations The main advantage of ae2dpy

is that this function can also handle arbitrary constants andarbitrary functions of the lattice variables including shifts Fol-lowing the remarks given in Section 22 it generate a staircaseof initial values linearly parametrized (244) Assuming that wehave a system of M quad equations and that we have chosento perform N iterations we will have the following number ofinitial conditions needed

Nini = 4M (N+ 1) (B1)

We assign to those values prime numbers

Next the function has a list F of the arbitrary functions possiblywith their shifts Then we operate in the following way

1 Taken an element f isin F check on how many variables it de-pends Then we proceed in isolating all the other elementsg isin F which has the same symbol ie

fl+1 rarr fgl+1 rarr g (B2)

2 Now consider the case when f isin F depends only upon asingle variable At the end of the preceding step we haveformed a list of the form

Ff =fq+j1 fq+jkf

(B3)

Then we define

jm = mink=1kf

jk jM = mink=1kf

jk (B4)

B1 algebraic entropy of systems of quad equations with ae2dpy 203

We then form a list Fs of all the triples

(f jm jM) (B5)

So finally we have that to evaluate the functions of a singlevariable along the evolution we then need to evaluate themon the following number of points

Ns =

|Fs|sumk=1

(N+ 1+ jMiminus jmi

) (B6)

We choose as values for those functionsNs prime numbers

3 Now we consider the case in which g isin F depends on twovariables Again from 1 we have a list of the form

Fg =gl+j1m+z1 gl+jkg m+zwg

(B7)

Now we define

im = mink=1kfw=1wg

jk zw (B8a)

iM = maxk=1kfw=1wg

jk zw (B8b)

We then form a list Fd of all the triples

(g im iM) (B9)

So finally we have that to evaluate the functions of a twovariables along the evolution we then need to evaluatethem on the following number of points

Nd =

|Fd|sumk=1

[N+ 1+ 2 (iMkminus imk

)] [N+ 2+ 2 (iMkminus imk

)]

2

+

|Fd|sumk=1

[N+ 2 (iMkminus jmk

)]

(B10)

We choose as values for those functions Nd prime num-bers

Remark B11 The proposed procedure for the evaluation of thefunctions of two variables may result in considering more pointsthan needed However a more accurate procedure of evaluationwill be much more complicated and above all this procedure isquite cheap in term of resource usage

We needed the following amount of prime numbers

NTOT = Nini +Nac +Ns +Nd (B11)


where Nac is the number of arbitrary constants The prime num-bers are generated with the sympy function sieve and shuffledbefore being assigned so that running the program on the sameequation more than once will result in different initial condi-tions and different values of the arbitrary functions and arbi-trary constants

We underline that the gen_ics function is not intended for auser-level usage but is used automatically by the evol function

evol evol is the principal user-level function in ae2dpy Its scopeis to output a degree sequence or an evolution matrix Sincethe whole program is supposed to work at the affine level itreturns the degree sequence of each dependent variable Nowwe describe carefully its input its operations and the possibleform of its output

The function evol needs two mandatory arguments

EQ The system of quad equations under considerations It canbe entered as a python list or as a python dictionary In thecase of a single scalar equation it can be entered plainly

Y The dependent variables They can be entered as a python listor as a python dictionary In the case of a single dependentvariable it can be entered plainly The entries of the listmust be sympy Functions of two lattice variables eg

Y = [u(n m) v(n m)] (B12)

Remark B12 The function evol can handle only well-determinedsystems of quad equations so it will check that the number ofequations is equal to the number of dependent variables In neg-ative case it will raise an error To reduce a system of quad equa-tion to a well-determined one is the up to the user

Moreover the function evol support four optional parameters

NI Defines the number of iterations It must be a positive inte-ger and error is returned otherwise The default value isNI=8

direction Direction defines the principal diagonal of evolutionas shown in Figure 25 The possible choices are

direction=mp perform the evolution from the ∆minus+ prin-cipal diagonal this is the default argument

direction=pm perform the evolution from the ∆+minus prin-cipal diagonal

direction=mm perform the evolution from the ∆minusminus prin-cipal diagonal

direction=pp perform the evolution from the ∆++ prin-cipal diagonal


direction=all performs the evolution in all directions Ifthe evolution in one of the four direction is not possi-ble [64] then the direction is skipped

If an argument different from the listed ones is used thenthe program falls back to the default one

matrix_form This is a boolean parameter By default it is falseand it is automatically activated by the program if it findsdifferent degrees along the diagonals Setting it to true abinitio will cause the result to be outputted in matrix formregardless of the need of such representation

verbosity_level This parameter sets how much output will bedisplayed on the stout by the program It has three possi-ble values

verbosity_level=0 The function displays nothing on the stout

verbosity_level=1 The function displays on the stout onlythe progression of the iterates in the form iNI wherei is the i-th iterate this is the default value

verbosity_level=2 The function displays nothing the stout

If the value of verbosity_level is not an integer then itfalls back to the default value

The function then accepts keyword arguments Keyword argu-ments in this context can be used to set some parameters to afixed (not arbitrary in the sense of the function gen_ics) valueKeyword arguments are immediately substituted into EQ

The function then starts by checking that the given equations arerational quad equations and search for arbitrary constants andfunctions Anything not-appearing in the list Y is consideredhere arbitrary Then the system is solved in the chosen directionand the function gen_ics is recalled in order to generate theinitial conditions and the values of the arbitrary functions andconstantsRemark B13 A continuous variable t is defined as in (244)This implies that it is not possible to call one of the two discretevariable t

Now the actual evolution begins The function value_deg ofstep evaluation is defined This function encloses all the rele-vant step-wise substitutions and degree calculations The mainpoint is the fact that after the substition a factorization in thefinite field KP is carried out The prime number P is chosen tobe the first prime after p2MAX where pMAX is the biggest primegenerated by gen_ics The number of points on the diagonalto be evaluated is then divided by the number of CPU at dis-posal The function evol then exectute the value_deg function


in chuncks each one of lenght equal to the number of CPU Thisprocedure is repeated until the NI-th iteration

At this point the evolution is ended It is checked if there is theneed of a matrix form If not the result is returned as a list oftuples whose first element is the name of the dependent variable(eg u(lm) is returned as u) and whose second member is thelist of the degrees or an evolution matrix In the case of scalarequation since there is no ambiguity just the list of degrees oran evolution matrix is returned

b12 Code

The content of the file ae2dpy is the following

1 from __future__ import print_function division

2

3 from sympycoresymbol import Symbol Wild

4 from sympycorerelational import Equality

5 from sympycoreadd import Add

6 from sympycoremul import Mul

7 from sympysimplify import simplify numer denom

8 from sympysolvers import solve

9 from sympypolyspolytools import degree factor

10 from sympyfunctionselementarymiscellaneous import Max

11 from sympyntheorygenerate import sievenextprime

12 from sympymatrices import Matrix

13 from sympyfunctionselementaryintegers import ceiling

14 from array import array

15 from random import shuffle

16 from warnings import warn

17 from itertools import chain

18 from multiprocessing import Queue Processcpu_count

19

20 class WellDefinitenessError(Exception)

21 pass

22

23 def up_or_anti_diagonal(dimnVAnti=False)

24 Create a matrix which is different from 0 only on an upper

25 or lower diagonal

26 if len(V) =dim-abs(n)

27 raise ValueError(Vector must be of the appropriatelenght s str(dim-abs(n)))

28 if Anti

29 k1 = 0

30 k2 = 1

31 else

32 k1 = 1

33 k2 = 0

34 def f(ij)

35 if i == k1(j-n) + k2(dim - 1 - n -j)

36 if ngt0


37 return V[i]

38 else

39 return V[k1j+k2(j+n)]

40 else

41 return 0

42 return Matrix(dimdimf)

43

44 def evolution_matrix(VVDIR)

45 Generate an evolution matrix on

46 in all the the possible directions

47 if type(VV) = list

48 raise TypeError(Argument s must be a l i s t str(VV))

49 NI = len(VV)

50 for i in range(NI)

51 if type(VV[i]) = list

52 raise TypeError(Elements of the argument must bel i s t s s is not str(VV[i]))

53 if DIR[0]DIR[1] == 1

54 adiag = True

55 else

56 adiag = False

57 M = up_or_anti_diagonal(NI+10[1](NI+1)adiag)+

up_or_anti_diagonal(NI+1-DIR[1][1]NIadiag)

58 for i in range(len(VV))

59 M += up_or_anti_diagonal(NI+1DIR[1](i+1)VV[i]adiag)

60 return M

61

62 def analyze_evolution_matrix(M)

63 if not type(M) == MutableDenseMatrix or type(M) ==

ImmutableMatrix

64 return TypeError(s expected to be a matrix str(M))

65

66 if not Mis_square

67 raise ValueError(Matrix expected to be square )68

69 W = []

70 K = Mcols

71 for i in range(K)

72 Wappend(list(M[ i]))

73 if Mis_upper_hessenberg


75 W[i]reverse()

76 W[i] = W[i][K-1-i]

77 elif Mis_lower_hessenberg


79 W[i] = W[i][i]

80 else

81 M = Matrix([list(M[ K-1-i]) for i in range(K)])

transpose()



84 W[i]reverse()


85 W[i] = W[i][i]



88 W[i] = W[i][K-1-i]

89 else

90 raise ValueError(Matrix is not an evolution matrix )

91

92 T = []


94 Tappend((len(W[i])W[i]))

95 S = [sorted(T key = lambda couple couple[0] reverse=True)[

i][1] for i in range(K-1)]

96 patterns = S[]

97 for i in range(len(S))

98 for j in range(i+1len(S))

99 if S[i][K-j] == S[j]

100 if S[j] in patterns

101 patternsremove(S[j])

102

103

104 return patterns

105

106 def gen_ics(NIDIRsVARSyCA=[]CAF=[])

107 Generate the necessary initial conditions for the

calculation of

108 the algebraic entropy If necessary substitute arbitrary

constants

109 and functions with prime numbers

110 l = VARS[0]

111 m = VARS[1]

112 NN = 2NI

113 NEQ = len(y)

114 Fl = []

115 Fm = []

116 Flm = []

117 q = Wild(q exclude=(l))

118 p = Wild(p exclude=(m))

119 if CAF = []

120 CAFC = CAF[]

121 for C in CAF

122 if not C in CAFC

123 continue

124 CL = [K for K in CAF if Kfunc == Cfunc]

125 CLargs = [Kargs for K in CL]

126 CAFC = [ K for K in CAFC if Kfunc = Cfunc]

127 if len(set([len(Kargs) for K in CL])) gt 1

128 raise ValueError(Arbitray function s canrsquo t befunction of different number of arguments

str(Cfunc))

129 if len(CLargs[0]) == 1


130 CLatom = list(set([list(Ka[0]atoms(Symbol))[0]

for Ka in CLargs]))

131 if len(CLatom) gt 1

132 raise ValueError(Arbitrary function s canrsquot be function of different arguents

str(Cfunc))

133 if CLatom == [l]

134 qv = []

135 for arg in CLargs

136 qs = arg[0]match(l+q)

137 qvappend(qs[q])

138 Flappend((Cfuncmin(qv)max(qv)))

139 elif CLatom == [m]

140 pv = []


142 ps = arg[0]match(m+p)

143 pvappend(ps[p])

144 Fmappend((Cfuncmin(pv)max(pv)))

145 else

146 raise ValueError(Unknown variable s infunction s (str(CLatom[0]) str(C

func)))



for Ka in CLargs]))+

149 list(set([list(Ka[1]atoms(Symbol))[0] for Ka

in CLargs]))

150 if CLatom == [lm]

151 qpv = []



154 qpvappend(qs[q])


156 qpvappend(ps[p])

157 qpvm = min(qpv)

158 qpvM = max(qpv)

159 if qpvm == qpvM

160 if qpvm lt 0

161 Flmappend((Cfuncqpvm0))

162 else

163 Flmappend((Cfunc0qpvM))

164 else

165 Flmappend((CfuncqpvmqpvM))

166 else

167 raise ValueError(Unknown variables s infunction s (str(CLatom)[1-1] str(C

func)))

168 if len(CLargs[0]) gt 2

169 raise ValueError(Wrong number of arguments s in arbitrary function s

170 (str(len(CLargs[0])) str(Cfunc)))

171


172 NFl = sum(NI+1+fl[2]-fl[1] for fl in Fl)

173 NFm = sum(NI+1+fm[2]-fm[1] for fm in Fm)

174 NFlm = sum((NI+1+2(flm[2]-flm[1]))(NI+2+2(flm[2]-flm

[1]))2+NI+2(flm[2]-flm[1]) for flm in Flm)

175 NTOT = int(NEQ(2NN+4)+len(CA)+NFl+NFm+NFlm)

176 if len(sieve_list) lt NTOT

177 sieveextend_to_no(NTOT)

178 rp = sieve_list[NTOT]

179 P = nextprime(max(rp)2)

180 print( rsquoa big prime rsquoP)181 else

182 NTOT = NEQ(2NN+4)+len(CA)



185 rp=list(sieve_list[])[NTOT]


187 print( rsquoa big prime rsquoP)188 shuffle(rp)

189 al = []

190 be = []

191 for i in range(NEQ)

192 alappend(rppop())

193 beappend(rppop())

194 s1 = DIR[0]

195 s2 = DIR[1]

196 repl =


198 for j in range(NN+1)

199 ll = int((2j - (-1)j + 1)4)

200 mm = int((2j + (-1)j - 1)4)

201 ci = (rppop()+rppop()s)(al[i]+be[i]s)

202 repl[y[i](s1lls2mm)]=ci

203

204 CAs =

205 for i in range(len(CA))

206 CAs[CA[i]] = rppop()

207

208 replfunc =

209

210 for fl in Fl

211 for i in range(NI+1+fl[2]-fl[1])

212 replfunc[fl[0](s1(i+fl[1]))] = rppop()

213 for fm in Fm

214 for i in range(NI+1+fm[2]-fm[1])

215 replfunc[fm[0](s2(i+fm[1]))] = rppop()

216 for flm in Flm

217 for i in range(2(NI+2(flm[2]-flm[1]))+1)

218 ll = int((2i - (-1)i + 1)4)

219 mm = int((2i + (-1)i - 1)4)

220 if s1 == 1

221 l_s = s1(ll+flm[1])

222 else


223 l_s = s1(ll-flm[2])

224 if s2 == 1

225 m_s = s2(mm-(flm[2]-2flm[1]))

226 else

227 m_s = s2(mm-(2flm[2]-flm[1]))

228 replfunc[flm[0](l_sm_s)] = rppop()

229 for i in range(NI+2(flm[2]-flm[1])+1)

230 for j in range(NI+2(flm[2]-flm[1])-i)

231 if s1 == 1

232 l_s = s1(j+flm[1])

233 else

234 l_s = s1(j-flm[2])

235 if s2 == 1

236 m_s = s2(j+i+1-(flm[2]-2flm[1]))

237 else

238 m_s = s2(j+i+1-(2flm[2]-flm[1]))


240 return replCAsreplfuncP

241

242 def evol(EQ Y NI=8 direction=mp matrix_form = False

verbosity_level=1modfact=True kwargs)

243 if not type(NI) == int

244 NI = 8

245 if type(EQ) == set or type(EQ) == tuple

246 EQ = list(EQ)

247 if type(Y) == set or type(Y) == tuple

248 Y = list(Y)

249 if not type(EQ) == list

250 EQ = [EQ]

251 if not type(Y) == list

252 Y = [Y]

253 if not type(verbosity_level) == int

254 verbosity_level = 1

255

256 NEQ = len(EQ)

257 if NEQ = len(Y)

258 if NEQ gt len(Y)

259 raise ValueError(Overdetermined system of equations )

260 else

261 raise ValueError(Underdetermined system of equations )


263 if isinstance(EQ[i]Equality)

264 EQ[i] = EQ[i]lhs - EQ[i]rhs

265 if kwargs

266 EQ[i] = EQ[i]subs(kwargs)

267

268 y = []

269

270 if not Y[0]is_Function

271 raise TypeError(s is expected to be function Y[0])


272 if len(Y[0]args) = 2

273 raise ValueError(The dependent variable s must befunction of two discrete variables str(Y[0]func)

)

274 l = Y[0]args[0]

275 m = Y[0]args[1]

276 yappend(Y[0]func)

277

278 for i in range(1NEQ)

279 if not Y[i]is_Function

280 raise TypeError(s is expected to be function

str(Y[i]))

281 if len(Y[i]args) = 2

282 raise ValueError(The dependent variable s must befunction of two discrete variables str(Y[i]

func))

283 if Y[i]args[0] = l or Y[i]args[1] = m

284 raise ValueError(The dependent variable s must befunction of (ss ) (str(Y[i]) str(l) str(m

)))

285 yappend(Y[i]func)

286

287 N_core = cpu_count()

288

289 CA = set([])

290 CAF = set([])

291

292 Ymp = [yf(l+1m+1) for yf in y]

293 Ypp = [yf(lm+1) for yf in y]

294 Ypm = [yf(lm) for yf in y]

295 Ymm = [yf(l+1m) for yf in y]

296 Ymon = Ymp+Ypp+Ypm+Ymm

297

298 for e in EQ

299 if not eis_rational_function(str(Ymon)[1-1])

300 raise TypeError(The equation s must be a rationalfunction of the dependent variables str(e))

301 e = numer(eexpand())expand()

302 for g in Addmake_args(e)

303 for h in Mulmake_args(g)

304 h = has_base_exp()

305 print(h)

306 for w in [ w for w in h if abs(w) = 1 ]

307 if wis_Function

308 if wfunc in y

309 if not w in Ymon

310 raise ValueError(The functionmust be defined on the square got instead h str(w))

311 else

312 CAF = CAFunion(set([w]))

313 else


314 CA = CAunion(watoms(Symbol))

315 if l in CA

316 CAremove(l)

317 if m in CA

318 CAremove(m)

319 CA = list(CA)

320 CAF = list(CAF)

321

322 if direction == mp323 s1 = -1

324 s2 = 1

325 print( rsquorank rsquo(Matrix(EQ)jacobian(Ymp))rank())326 print( rsquon eqs rsquo NEQ)

327 if (Matrix(EQ)jacobian(Ymp))rank() lt NEQ

328 raise WellDefinitenessError(The system has not welldefined evolution in the ss direction (str

(s1) str(s2)))

329 ySol = solve(EQYmpdict=True)

330 if not len(ySol) == 1


(s1) str(s2)))

332 ySol = ySol[0]

333 ySub = [ySol[s] for s in Ymp]

334 lmSub = lambda ij l s1(j+1) m s2(j+i)

335 elif direction == pp336 s1 = 1

337 s2 = 1

338 print( rsquorank rsquo(Matrix(EQ)jacobian(Ypp))rank())339 if (Matrix(EQ)jacobian(Ypp))rank() lt NEQ


(s1) str(s2)))

341 ySol = solve(EQYppdict=True)



(s1)str(s2)))

344 ySol = ySol[0]

345 ySub = [ySol[s] for s in Ypp]

346 lmSub = lambda ij l s1j m s2(j+i)

347 elif direction == pm348 s1 = 1

349 s2 = -1

350 if (Matrix(EQ)jacobian(Ypm))rank() lt NEQ


(s1) str(s2)))

352 ySol = solve(EQYpmdict=True)




(s1) str(s2)))

355 ySol = ySol[0]

356 ySub = [ySol[s] for s in Ypm]

357 lmSub = lambda ij l s1j m s2(j+i+1)

358 elif direction == mm359 s1 = -1

360 s2 = -1

361 if (Matrix(EQ)jacobian(Ymm))rank() lt NEQ


(s1) str(s2)))

363 ySol = solve(EQYmmdict=True)



(s1) str(s2)))

366 ySol = ySol[0]

367 ySub = [ySol[s] for s in Ymm]

368 lmSub = lambda ij l s1(j+1) m s2(j+i+1)

369 elif direction == a l l 370 alldirs = [mp pp pm mm]371 ALL = []

372 for dire in alldirs

373 try

374 print(Direction +dire)375 ALLappend((direevol(EQ Y NI dire

matrix_form verbosity_level kwargs)))

376 except WellDefinitenessError

377 print(Evolution is not well defined in the +dire+ direction continuing to the following )

378 continue

379 if len(ALL) == 4

380 if matrix_form

381 return tuple([seq[1] for seq in ALL])

382 for E in ALL

383 for F in ALL

384 if E[1] = F[1]


386 return ALL[0][1]

387 return tuple(ALL)

388 else

389 warn(Direction not detected switching to default )390 return evol(EQ Y NI mp matrix_form kwargs)

391

392 t = Symbol( t real=True)

393 bug if a discrete variable is called t

394 CIm = dict()

395 CICAsubCAFsubP = gen_ics(NI[s1s2]t(lm)yCACAF)

396 EQ = [esubs(CAsub) for e in EQ]


397 ySub = [yssubs(CAsub) for ys in ySub]

398 if verbosity_level gt 1

399 print(CICAsubCAFsub)

400

401 DS = [[] for i in range(NEQ)]

402

403 Internal evaluation function

404 def value_deg(ijkCImCIoutEQ)

405 step = (s1js2(j+i+1))

406 ev = factor((((ySub[k]subs(lmSub(ij)))subs(CIm

simultaneous=True))subs(CIsimultaneous=True))subs(

CAFsubsimultaneous=True)modulus=P)

407 d1 = degree(numer(ev)t)

408 d2 = degree(denom(ev)t)

409 d = Max(d1d2)


411 print(ev)

412 print(d)

413 outEQput((kstepevd))

414 return True

415

416

417 Actual evolution algorithm


419 outd = Queue()

420 out = Queue()


422 print(str(i+1)++str(NI))423 processes = [ Process(target=value_deg args=(ijkCIm

CIout)) for k in range(NEQ) for j in range(NI-i)]

424 N_proc = (NI-i)NEQ

425 Cal = ceiling(len(processes)N_core)

426 for h in range(Cal+1)

427 for i in range(hN_coremin((h+1)N_corelen(

processes)))

428 processes[i]start()


processes)))

430 processes[i]join(1)

431 OutC = sorted(sorted([outget() for p in processes] key=

lambda op op[1][1]) key=lambda op op[0])

432 CIm = dict(CI)

433 CIclear()

434 for k in range(len(OutC))

435 CI[y[OutC[k][0]](OutC[k][1][0]OutC[k][1][1])] = OutC

[k][2]

436

437 for k in range(NEQ)

438 DK = [OutC[q][3] for q in range(len(OutC)) if OutC[q

][0] == k]

439 print( rsquokd_k rsquokDK)440 if [s1s2] = [-11]


441 DKreverse()

442 if matrix_form == False

443 for i in range(len(DK))

444 for j in range(i+1len(DK))

445 if DK[i] = DK[j]

446 matrix_form = True

447 break

448 if matrix_form

449 break

450 DS[k]append(DK)

451

452 if matrix_form

453 if NEQ == 1

454 return evolution_matrix(DS[0][s1s2])

455 return [(y[k]evolution_matrix(DS[k][s1s2])) for k in

range(NEQ)]

456

457 if NEQ == 1

458 return [1]+[d[0] for d in DS[0]]

459

460 return [(y[k][1]+[d[0] for d in DS[k]]) for k in range(NEQ)]

b2 algebraic entropy for systems of ordinary differ-ence equations and differential difference equa-tions of arbitrary order with ae_differential_differencepy

In this Section we give a brief description of the functions containedin the python module ae_differential_differencepy This modulecontains only the function evol (not to be confused with the functionof the same name in ae2dpy) that can be used to compute the se-quence of the degrees for systems of ordinary difference equations orsystems of differential-difference equations of arbitrary order

Remark B21 The function evol cannot automatically treat systems ofmixed ordinary difference equations and differential-difference equa-tions If this is the case under consideration we suggest to the user toconsider the purely difference variables as implicitly depending onldquotimerdquo un rarr un (t) For this reason from now on we will address tothe systems under study as one-dimensional discrete systems


f The system of one-dimensional discrete equations under consider-ations It can be entered as a python list or as a python dictio-nary In the case of a single scalar equation it can be enteredplainly

y The dependent variables They can be entered as a python list oras a python dictionary In the case of a single dependent variable

B2 algebraic entropy for 1d systems with ae_differential_differencepy 217

it can be entered plainly The entries of the list must be sympy

Functions of one lattice variables eg

y = [u(n) v(n)] (B13)

or of one lattice variable and one continuous variable

y = [u(n t) v(n t)] (B14)

The continuous variable is by convention the second one

Remark B22 The function evol can handle only well-determined sys-tems of one-dimensional discrete equations so it will check that thenumber of equations is equal to the number of dependent variablesIn negative case it will raise an error To reduce a system of one-dimensional discrete equations to a well-determined one is the up tothe user

The function evol also accepts an optional argument NI NI definesthe number of iteration to be performed Its default value is NI=8

The function then accepts keyword arguments Keyword argumentsin this context can be used to set some parameters to a fixed valueKeyword arguments are immediately substituted into f

First of all the function evol analyze the equation f in order todetermine the order with respect to every variable Furthermore anysymbol (not function) different from those of the dependent variablescontained in y is considered an arbitrary constant Arbitrary functionsare not supported by evol

At this point the system if consistent is solved with respect to thehighest shifts otherwise an error is raised Then the initial conditionsin the form (244) are generated If we denote by parte the discrete orderas defined in Section (21) of an equation e in f we have that we needif M is the number of equations

Nini = 2

Msumk=1

(partek + 1) (B15)

prime numbers to evaluate them Then if Nac is the number of arbi-trary constants we need

NTOT = Nini +Nac (B16)

prime numbers The prime numbers are generated with the sympy

function sieve and shuffled before being assigned so that runningthe program multiple times will yield different initial conditions

Remark B23 In principle evol cannot perform the evolution in thedirection of the smallest shifts However this can be simply obtainedby the user performing the change nrarr minusn into f


Now the actual evolution begins and the iterates are computedThe main point is the fact that after the substition a factorizationin the finite field KP is carried out The prime number P is chosento be the next prime after p2MAX where pMAX is the biggest primenumber generated to evaluate the initial conditions and the arbitraryconstants

The output of the function evol is then a dictionary whose key-words are the elements of y and whose values are the sequences ofdegrees

The content of the file aedifferential_difference_systemspy isthe following


2

3 from sympy import 4 from random import shuffle

5 from collections import defaultdict

6

7 def evol(fyNI=8 kwargs)


9 NI = 8

10 if type(f) == set or type(f) == tuple

11 f = list(f)

12 if type(y) == set or type(y) == tuple

13 y = list(y)

14 if not type(f) == list

15 f = [f]

16 if not type(y) == list

17 y = [y]

18

19 NEQ = len(f)

20 if NEQ = len(y)

21 if NEQ gt len(y)


23 else


25

26 Convert equalities to single side equations

27 and substitute keywords arguments into the equations


29 if isinstance(f[i]Equality)

30 f[i] = f[i]lhs - f[i]rhs

31 if kwargs

32 f[i] = f[i]subs(kwargs)

33

34 if not y[0]is_Function


36 if len(y[0]args) == 1

37 n = y[0]args[0]

38 s = var( rsquo s rsquo)


39 elif len(y[0]args) == 2

40 n s = y[0]args

41 else

42 raise ValueError(s must be function of maximum twovariables y[0])

43 Y = [y[0]func]

44


46 if not y[i]is_Function


str(y[i]))

48 if y[i]args = y[0]args

49 raise ValueError(The dependent variable s must befunction of

50 (ss ) (str(y[i]) str(n) str(s)))

51 Yappend(y[i]func)

52

53 k = Wild( rsquok rsquo exclude=(n))

54

55 f = [ numer(simplify(fsratio=oo))expand() for fs in f]

56 pprint(f)

57

58 CA = set([])

59 kv = yf set([]) for yf in Y

60

61 for fs in f

62 for g in Addmake_args(fs)

63 coeff = SOne

64 kspec = None

65 for q in Mulmake_args(g)

66 q = qas_base_exp()

67 for h in q

68 if his_Derivative

69 h = hargs[0]

70 if his_Function

71 if hfunc in Y

72 result = hargs[0]match(n + k)

73 if result is not None

74 kspec = int(result[k])

75 kv[hfunc]=kv[hfunc] = kv[hfunc

]union(set([kspec]))

76 else

77 raise ValueError(

78 rsquos(s+k) rsquo expected got rsquosrsquo (y n h))

79 else


81 rsquos rsquo expected got rsquos rsquo (Y h

func))

82 else

83 coeff = h

84 CA = CAunion(coeffatoms(Symbol))


85

86 if n in CA

87 CAremove(n)

88 if s in CA

89 CAremove(s)

90 if CA

91 CA = list(CA)

92

93 N = yf max(kv[yf]) for yf in Y

94 M = yf min(kv[yf]) for yf in Y

95 ORD = yf N[yf]-M[yf] for yf in Y

96

97 svars = [y[i]subs(nn+N[Y[i]]) for i in range(NEQ)]

98

99 if (Matrix(f)jacobian(svars))rank() lt NEQ


101 rsquos rsquo must be of maximal rank with respect102 to rsquos rsquo (f svars))

103 K = 2sum(ORD[yf] for yf in Y)+2NEQ

104 NTOT = K + len(CA)




108 shuffle(rp)

109

110 rp0 = rp[02NEQ]

111 rp = rp[2NEQ]

112

113 CI =


115 den = rp0pop()s+rp0pop()

116 for k in range(M[Y[i]]N[Y[i]])

117 CI[y[i]subs(nk)] = (rppop()s+rppop())den

118 CAs =



121

122 Evolution in the right direction

123 if CAs

124 f = [f[al]subs(CAs) for al in range(len(f))]

125 ySol = solve(fsvarsdict=True)



128 rsquos rsquo must be singleminusvalued (f))

129 ySol = ySol[0]

130 yN = [ySol[yks] for yks in svars]

131

132

133 ds = y[i] [1] for i in range(NEQ)

134


136 print(str(i)+ rsquorsquo+str(NI))

B3 analysis of the degree sequences 221

137 val = [factor((yN[k]subs(ni)subs(CI))doit()modulus=P

) for k in range(NEQ)]


139 CI[y[k]subs(ni+N[Y[k]])] = val[k]

140 del CI[y[k]subs(ni+M[Y[k]])]

141 vq = numer(val[k])

142 d1 = degree(vq s)

143 vq = denom(val[k])

144 d2 = degree(vqs)

145 d = max(d1d2)

146 ds[y[k]] += [d]

147

148 return ds

b3 analysis of the degree sequences

Once obtained a finite sequence of degrees it must be analyzed us-ing eg the methods presented in Section 22 To this end we com-plement the programs for calculating the sequences of degrees withsome python functions which can compute Padeacute approximants andthe inverse Z-transform This need comes from the fact that the an-alytic evaluation of the Padeacute approximants and of the inverse Z-transform is not yet implemented in sympy We present then threemodules the padepy module which defines the calculation of thePadeacute approximants the pade_rat_gfuncpy module which computesthe generating function of a sequence using Padeacute approximants andthe z_transformpy which defines the Z-transform for rational func-tions and functions to find the coefficients and the algebraic entropyof a given generating function

b31 The padepy module

The padepy contains a single function pade_expansion This func-tion computes the Padeacute approximant of order [L M] using the for-mula (252) It has four arguments all of them mandatory

vect This argument must be a list of numbers or a function If theargument is a function its Taylor expansion of order L+ M+ 1

with respect to z is computed

L This argument must be a positive integer and it is the degree of thenumerator of the Padeacute approximant (250)

M This argument must be a positive integer and it is the degree ofthe denominator of the Padeacute approximant (250)

z This argument is the independent variable


vect can be a list and not a polynomial since it is obvious that to anylist of numbers [a0 a1 aN] we can associate the polynomial

p(z) = a0 + a1z+ aNzN (B17)

The content of the file padepy is the following


2

3 from sympycoreexpr import Expr

4 from sympypolyspolytools import factor


6

7

8 def pade_expansion(vectLMz)

9 Calculate the Pade expasion [LM] of a given function or

vector

10 in the variable z

11 Check if the first argument is a list of values If not it

is assumed

12 to be a symbolic expression

13 if not type(vect) is list

14 taylp=vectseries(z0L+M+1)removeO()

15 if M==0

16 return taylp

17 vect = [taylpcoeff(zi) for i in range(L+M+1)]

18 Check that we have enough terms in the initial vector

19 to carry out the computations needed for the LM

20 Pade approximant

21 else

22 if len(vect) = L+M+1

23 raise ValueError(Pade expansion cannon rsquo t be computed length of vector and of degree donrsquo t match )

24 if L == 0 and M ==0

25 return vect[0]

26

27 def c(i)

28 if ilt0 or i gt= len(vect)

29 return 0

30 else

31 return vect[i]

32

33 if M == 0

34 return Poly([c(i) for i in range(0L+M+1)]z)simplify()

35 def cfP(i)

36 if M-igtL

37 return 0

38 else

39 S = c(0)

40 for k in range(1max(L-M+i+11))

41 S += c(k)zk

42 return (Sz(M-i))simplify()

43 vzP = Matrix([[cfP(i) for i in range(M+1)]])


44 vzQ = Matrix([[z(M-i) for i in range(M+1)]])

45 A = Matrix([[c(L-M+i+j) for i in range(M+1)] for j in range

(1M+1)])

46 QMat = Acol_join(vzQ)

47 PMat = Acol_join(vzP)

48 QLM = ((QMat)det())simplify()

49 PLM = ((PMat)det())simplify()

50 PA = (PLMQLM)simplify()

51

52 return factor(PA)

b32 The pade_rat_gfuncpy module

The pade_rat_gfuncpy contains a single function find_rat_gen_functionThis function has two arguments both of them mandatory

V A list of numbers

s The independent variable of the generating function

The algorithm runs as follows it start with a slice V prime of the list V

with three elements and compute the Padeacute approximant g = [1 1]

through the association (B17) Then computes the Taylor expansionof g up to the length of V If its coefficients agrees with V then g isreturned If not we take a slice with two more elements V prime and com-pute the Padeacute approximant g = [2 2] through the association (B17)If its coefficients agrees with V then g is returned This is repeateduntil one of the two following condition is reached

1 The coefficients of the Taylor expansion of the Padeacute approxi-mant g agree with V

2 We cannot form a slice with two elements more than the previ-ous

In the case 1 g is returned since it is the desired generating functionIn the case 2 it is not possible to find a generating function so theboolean value False is return

The content of the file pade_rat_gfuncpy is the following


2

3 from sympyseries import series

4 from sympycoresymbol import Symbol

5 from sympyfunctionselementaryintegers import floor

6 from pade import pade_expansion

7 import warnings

8

9 def find_rat_gen_func(Vs)

10

11 Find a rational generating function by using the Pade

approximant method


12 taking a slice of the input vector V finds a Pade approximant

using

13 pade_expansion then check if it reproduces the exact

behaviour of the following

14 elements of V If not it takes a bigger slice until the

vector V is finished

15

16 if not type(V) is list

17 raise TypeError(Argument s expected to be l i s t str(

V))

18 if not sis_Symbol

19 raise TypeError(Argument s expected to be symbol

str(s))

20 M = len(V)

21 N = floor((M-1)2)

22 j = 1

23 while(jltN+1)

24 R = pade_expansion(V[2j+1]jjs)

25 ret = True

26 Rv = [Rseries(s0M+1)removeO()coeff(si) for i in

range(M)]

27 if Rv == V

28 return R

29 else

30 j = j + 1

31 return False

b33 The z_transformpy module

The z_transformpy module contains the following functions

inverse_z_transform This function computes the inverse Z-transformof a rational function It needs three mandatory arguments

f A sympy rational function If another kind of function is giventhe program raise an error

z The independent variable of the function z

n The discrete target variable

The Z-transform of f is then computed using equation (256)Remark B31 Due to some limitations in sympy this programmay misbehave in case of high order polynomials in the de-nominator

find_coefficients This function compute the asymptotic form of thecoefficients of a given ration function It needs three mandatoryarguments





Using formula (258) and the function z_transform the asymp-totic form of the coefficients are reconstructed

find_entropy This function compute the Algebraic Entropy definedby a ration generating function It needs two mandatory argu-ments



Using formula (260) the value of the Algebraic Entropy is com-puted

The content of the file z_transformpy is the following


2


4 from sympysimplify import simplify


6 from sympyfunctionselementarycomplexes import Abs

7 from sympyfunctionselementaryexponential import log

8 from sympypolyspolytools import div degree


10 from sympyseries import residue

11

12 def inverse_z_transform(fzn)

13 if not fis_rational_function(z)

14 raise TypeError(Function s expected to be rational

str(f))

15 if not zis_Symbol

16 raise TypeError(Input variable s must be of type symbol str(z))

17 if not nis_Symbol

18 raise TypeError(Output variable s must be of typesymbol str(n))

19 if not nis_integer

20 n = Symbol(str(n)integer=True)

21 f = fas_numer_denom()

22 if degree(f[0]z) gt= degree(f[1]z)

23 quotrem = div(f[0]f[1])

24 f = (rem)+f[1]

25 sing = solve(f[1]z)

26 transf = 0

27 for z0 in sing

28 s=residue((f[0]f[1]z(n-1))simplify()zz0)

29 transf = transf + s

30 return transfsimplify()

31

32 def find_coefficients(fzn)




str(f))







41 f = fsubs(z1z)simplify()

42 dn = inverse_z_transform(fzn)

43 return dn

44

45 def find_entropy(fz)



str(f))





52 m = 0

53 for z0 in sing

54 m = Max(1Abs(z0)m)

55 return log(m)simplify()

CE V O L U T I O N M AT R I C E S

In this Appendix we give the explicit form of the evolution matricesfor the trapezoidal H4 equations (191) and H6 equations (193) Us-ing the program ae2dpy contained in Appendix B we performed 32

iterations in each of the four possible direction of growth

227

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C1

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(1

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22

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c1 trapezoidal H4 equations (1 91)

c11 tHε1 equation (191a)

direction minus+ 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17

1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44 46 47

0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16

0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44

0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15

0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41

0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14

0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38

0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13

0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35

0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12

0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32

0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C1)

23

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direction ++

17 17 16 16 15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1

47 46 44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1

16 16 15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0

44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0

15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0

41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0

14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0

38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0

13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0

35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0

12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0

32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0

11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0

29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C2)

C1

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(1

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23

1

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0

12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0

12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0

13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0

13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0

14 37 13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0

14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0

15 40 14 37 13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0

15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0

16 43 15 40 14 37 13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0

16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0

17 46 16 43 15 40 14 37 13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1

17 47 16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1

(C3)

23

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direction minusminus

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4 10 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4 10 5 13 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4 10 5 13 6 16 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4 10 5 13 6 16 7 19 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9 25 10

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9 25 10 28 11

0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11

0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9 25 10 28 11 31 12

0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12

0 0 0 0 0 0 0 0 1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9 25 10 28 11 31 12 34 13

0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13

0 0 0 0 0 0 1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9 25 10 28 11 31 12 34 13 37 14

0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14

0 0 0 0 1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9 25 10 28 11 31 12 34 13 37 14 40 15

0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15

0 0 1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9 25 10 28 11 31 12 34 13 37 14 40 15 43 16

0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16

1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9 25 10 28 11 31 12 34 13 37 14 40 15 43 16 46 17

1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16 47 17

(C4)

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(1

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3

c12 tHε2 equation (191b)

direction minus+

1 2 4 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107 111 115 119 123

1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

0 1 1 2 4 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107 111 115

0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58

0 0 0 1 1 2 4 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107

0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

0 0 0 0 0 1 1 2 4 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99

0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

0 0 0 0 0 0 0 1 1 2 4 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91

0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

0 0 0 0 0 0 0 0 0 1 1 2 4 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83

0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 7 11 15 19 23 27 31 35 39 43 47 51 55 59

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 7 11 15 19 23 27 31 35 39 43 47 51

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 7 11 15 19 23 27 31 35 39 43

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 7 11 15 19 23 27 31 35

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 7 11 15 19 23 27

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 7 11 15 19

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 7 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C5)

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direction ++

123 119 115 111 107 103 99 95 91 87 83 79 75 71 67 63 59 55 51 47 43 39 35 31 27 23 19 15 11 7 4 2 1

62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1

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(C6)

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eq

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ns

(1

91)

23

5

direction +minus

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(C7)

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(C8)

C1

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(1

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23

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c13 tHε3 equation (191c)

direction minus+

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(C9)

23

8e

vo

lu

tio

nm

at

ric

es

direction ++

123 119 115 111 107 103 99 95 91 87 83 79 75 71 67 63 59 55 51 47 43 39 35 31 27 23 19 15 11 7 4 2 1

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(C10)

C1

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ns

(1

91)

23

9

direction +minus

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(C11)

24

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0 0 0 0 0 0 1 1 2 4 7 8 15 12 23 16 31 20 39 24 47 28 55 32 63 36 71 40 79 44 87 48 95

0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99

0 0 0 0 1 1 2 4 7 8 15 12 23 16 31 20 39 24 47 28 55 32 63 36 71 40 79 44 87 48 95 52 103

0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99 54 107

0 0 1 1 2 4 7 8 15 12 23 16 31 20 39 24 47 28 55 32 63 36 71 40 79 44 87 48 95 52 103 56 111

0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99 54 107 58 115

1 1 2 4 7 8 15 12 23 16 31 20 39 24 47 28 55 32 63 36 71 40 79 44 87 48 95 52 103 56 111 60 119

1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99 54 107 58 115 62 123

(C12)

C2H6

eq

ua

tio

ns

(1

93)

24

1

c2 H6 equations (1 93)

c21 1D2 equation (193b)

direction minus+ 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16 47 17

1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35 25 38 27 41 29 44 31 47

0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16

0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35 25 38 27 41 29 44

0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15

0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35 25 38 27 41

0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14

0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35 25 38

0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13

0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35

0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12

0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32

0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C13)

24

2e

vo

lu

tio

nm

at

ric

es

direction ++

17 47 16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1

47 31 44 29 41 27 38 25 35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1

16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0

44 29 41 27 38 25 35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0

15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0

41 27 38 25 35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0

14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0

38 25 35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0

13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0

35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0

12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0

32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0

11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0

29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C14)

C2H6

eq

ua

tio

ns

(1

93)

24

3

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0

32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0

12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0

35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0

13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0

38 25 35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0

14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0

41 27 38 25 35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0

15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0

44 29 41 27 38 25 35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0

16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0

47 31 44 29 41 27 38 25 35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1

17 47 16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1

(C15)

24

4e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29

0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11

0 0 0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32

0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12

0 0 0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35

0 0 0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13

0 0 0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35 25 38

0 0 0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14

0 0 0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35 25 38 27 41

0 0 0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15

0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35 25 38 27 41 29 44

0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16

1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35 25 38 27 41 29 44 31 47

1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16 47 17

(C16)

C2H6

eq

ua

tio

ns

(1

93)

24

5

c22 2D2 equation (193c)

direction minus+

1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27 42 29 45 31 48

1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39 15 42 16 45 17

0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27 42 29 45

0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39 15 42 16

0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27 42

0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39 15

0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39

0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14

0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36

0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13

0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33

0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12

0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C17)

24

6e

vo

lu

tio

nm

at

ric

es

direction ++

48 31 45 29 42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1

17 45 16 42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1

45 29 42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0

16 42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0

42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0

15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0

39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0

14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0

36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0

13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0

33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0

12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0

30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0

11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C18)

C2H6

eq

ua

tio

ns

(1

93)

24

7

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0

21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0

33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0

23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0

36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0

25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0

39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0

27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0

42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0

29 42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0

45 16 42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0

31 45 29 42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1

48 17 45 16 42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1

(C19)

24

8e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27

0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19

0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30

0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21

0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33

0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23

0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36

0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25

0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39

0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27

0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39 15 42

0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27 42 29

0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39 15 42 16 45

1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27 42 29 45 31

1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39 15 42 16 45 17 48

(C20)

C2H6

eq

ua

tio

ns

(1

93)

24

9

c23 3D2 equation (193d)

direction minus+

1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27 27 29 29 31 31 33

1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27 2 29 2 31 2

0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27 27 29 29 31

0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27 2 29 2

0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27 27 29

0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27 2

0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27

0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2

0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25

0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2

0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23

0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2

0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C21)

25

0e

vo

lu

tio

nm

at

ric

es

direction ++

33 31 31 29 29 27 27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1

2 31 2 29 2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1

31 29 29 27 27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0

2 29 2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0

29 27 27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0

2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0

27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0

2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0

25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0

2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0

23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0

2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0

21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0

2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C22)

C2H6

eq

ua

tio

ns

(1

93)

25

1

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0

21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0

23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0

23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0

25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0

25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0

27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0

27 27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0

29 2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0

29 29 27 27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0

31 2 29 2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0

31 31 29 29 27 27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1

33 2 31 2 29 2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1

(C23)

25

2e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19

0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19

0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21

0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21

0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23

0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23

0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25

0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25

0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27

0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27 27

0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27 2 29

0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27 27 29 29

0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27 2 29 2 31

1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27 27 29 29 31 31

1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27 2 29 2 31 2 33

(C24)

C2H6

eq

ua

tio

ns

(1

93)

25

3

c24 D3 equation (193e)

direction minus+

1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60 38 67 42 74 46 81 50 88 54 95 58 102 62 109

1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

0 1 1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60 38 67 42 74 46 81 50 88 54 95 58 102

0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58

0 0 0 1 1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60 38 67 42 74 46 81 50 88 54 95

0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

0 0 0 0 0 1 1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60 38 67 42 74 46 81 50 88

0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

0 0 0 0 0 0 0 1 1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60 38 67 42 74 46 81

0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60 38 67 42 74

0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60 38 67

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 18 14 25 18 32 22 39 26 46

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 18 14 25 18 32 22 39

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 18 14 25 18 32

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 18 14 25

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 18

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C25)

25

4e

vo

lu

tio

nm

at

ric

es

direction ++

109 62 102 58 95 54 88 50 81 46 74 42 67 38 60 34 53 30 46 26 39 22 32 18 25 14 18 10 11 6 4 2 1

62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1

102 58 95 54 88 50 81 46 74 42 67 38 60 34 53 30 46 26 39 22 32 18 25 14 18 10 11 6 4 2 1 1 0

58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0

95 54 88 50 81 46 74 42 67 38 60 34 53 30 46 26 39 22 32 18 25 14 18 10 11 6 4 2 1 1 0 0 0

54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0

88 50 81 46 74 42 67 38 60 34 53 30 46 26 39 22 32 18 25 14 18 10 11 6 4 2 1 1 0 0 0 0 0

50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0

81 46 74 42 67 38 60 34 53 30 46 26 39 22 32 18 25 14 18 10 11 6 4 2 1 1 0 0 0 0 0 0 0

46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0

74 42 67 38 60 34 53 30 46 26 39 22 32 18 25 14 18 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0

42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0

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60 34 53 30 46 26 39 22 32 18 25 14 18 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

32 18 25 14 18 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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25 14 18 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C26)

C2H6

eq

ua

tio

ns

(1

93)

25

5

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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25 14 18 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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102 58 95 54 88 50 81 46 74 42 67 38 60 34 53 30 46 26 39 22 32 18 25 14 18 10 11 6 4 2 1 1 0

62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1

109 62 102 58 95 54 88 50 81 46 74 42 67 38 60 34 53 30 46 26 39 22 32 18 25 14 18 10 11 6 4 2 1

(C27)

25

6e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 18 14 25

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0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

0 0 0 1 1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60 38 67 42 74 46 81 50 88 54 95

0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58

0 1 1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60 38 67 42 74 46 81 50 88 54 95 58 102

1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60 38 67 42 74 46 81 50 88 54 95 58 102 62 109

(C28)

C2H6

eq

ua

tio

ns

(1

93)

25

7

c25 1D4 equation (193f)

direction minus+

1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76 41 82 44 88 47 94

1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44 46 47

0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76 41 82 44 88

0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44

0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76 41 82

0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41

0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76

0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38

0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70

0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35

0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64

0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32

0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C29)

25

8e

vo

lu

tio

nm

at

ric

es

direction ++

94 47 88 44 82 41 76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1

47 46 44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1

88 44 82 41 76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0

44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0

82 41 76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0

41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0

76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0

38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0

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35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0

64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0

32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0

58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0

29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C30)

C2H6

eq

ua

tio

ns

(1

93)

25

9

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

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35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0

70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0

38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0

76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0

41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0

82 41 76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0

44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0

88 44 82 41 76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0

47 46 44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1

94 47 88 44 82 41 76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1

(C31)

26

0e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26

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0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29

0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58

0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32

0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64

0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35

0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70

0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38

0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76

0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41

0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76 41 82

0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44

0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76 41 82 44 88

1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44 46 47

1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76 41 82 44 88 47 94

(C32)

C2H6

eq

ua

tio

ns

(1

93)

26

1

c26 2D4 equation (193g)

direction minus+

1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99 54 107 58 115 62 123

1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41 44 45 48 49 52 53 56 57 60 61

0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99 54 107 58 115

0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41 44 45 48 49 52 53 56 57

0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99 54 107

0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41 44 45 48 49 52 53

0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99

0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41 44 45 48 49

0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91

0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41 44 45

0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83

0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41

0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C33)

26

2e

vo

lu

tio

nm

at

ric

es

direction ++

123 62 115 58 107 54 99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1

61 60 57 56 53 52 49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1

115 58 107 54 99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0

57 56 53 52 49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0

107 54 99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0

53 52 49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0

99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0

49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0

91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0

45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0

83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0

41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0

75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0

37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C34)

C2H6

eq

ua

tio

ns

(1

93)

26

3

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0

42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0

83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0

46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0

91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0

50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0

99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0

54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0

107 53 99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0

58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0

115 57 107 53 99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0

62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1

123 61 115 57 107 53 99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1

(C35)

26

4e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75

0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83

0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91

0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99

0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99 53 107

0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58

0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99 53 107 57 115

1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99 53 107 57 115 61 123

(C36)

DE X A M P L E S O F C A L C U L AT I O N S O F G E N E R A L I Z E DS Y M M E T R I E S F O R E Q U AT I O N S W I T HT W O - P E R I O D I C C O E F F I C I E N T S

In this Appendix we give two examples of explicit calculations of thethree-points generalized symmetries in the case of equation with twoperiodic coefficients as explained in Section 32 The two exampleswe will give are the D3 equation (193e) which is quite simple andthe tHε1 equation (191a) in the direction m which is instead a moredifficult one These detailed calculations were presented in [63]

d1 three-points generalized symmetries of the D3 equa-tion

Consider the D3 equation as given by formula (193e) We wish tocompute its three-points generalized symmetries in the direction nTo do so we can apply the method explained in Subsection 32 Wethen first find the system for the function z from (340) Consideringthe case in which n = 2k and m = 2l we find from (340) two equa-tions for z(++)

un+1m + unmminus1

unm minus u2nmminus1

partz(++)

partun+1m+partz(++)

partunm

+unminus1m + unmminus1


partz(++)

partunminus1m=2unmminus1unminus1m + u2nmminus1 + unm

(unm minus u2nmminus1)(unm minus u2nminus1m)

(D1a)

un+1m + unm+1

u2nm+1 minus unm

partz(++)

partun+1mminuspartz(++)

partunm

+unminus1m + unm+1

u2nm+1 minus unm

partz(++)

partunminus1m=u2nm+1 + 2unminus1munm+1 + unm

(unm minus u2nminus1m)(unm minus u2nm+1)

(D1b)

Taking the coefficients with respect to unmplusmn1 and solving we obtain

z(++) = log

[C(++)1

(unm minus u2nminus1m

(un+1m minus unminus1m)2

)] (D2)

In an analogous way we obtain the solutions for z(+minus) z(minus+) andz(minusminus)

z(minus+) = log

[C(minus+)1

(u2nm minus unminus1m


)] (D3a)

265

266 examples of calculations of generalized symmetries

z(+minus) = log[C(+minus)1

(unm + unminus1m


)] (D3b)

z(minusminus) = log[C(minusminus)1

(unm + unminus1m


)] (D3c)

Inserting the resulting value for z (384a) obtained from (D3) intothe conservation law (329) we derive the following relation betweenthe constants

C(+minus)1 = minusC

(++)1 C

(minusminus)1 = C

(minus+)1 (D4)

Proceeding in the same way we can derive the values of the func-tions v(plusmnplusmn) from (342)

v(++) = log

[C(minus+)2

(unm minus u2n+1m


)] (D5a)

v(minus+) = log

[C(minus+)2

(u2nm minus un+1m


)] (D5b)

v(+minus) = log[C(+minus)2

(un+1m + un+1m


)] (D5c)

v(minusminus) = log[C(minusminus)2

(un+1m + unm


)] (D5d)

Inserting the resulting value for v (384b) obtained from (D5) intothe conservation law (334) we derive the following relation betweenthe constants

C(+minus)2 = minusC

(++)2 C

(minusminus)2 = C

(minus+)2 (D6)

We can now insert z and v into the compatibility conditions in orderto find g From these compatibility conditions we obtain the followingrelations between the remaining constants

C(++)2 = minusC

(++)1 C

(minus+)2 = minusC

(minus+)1 (D7)

Integrating the equation for g (344) we get

g = F(+)n F

(+)m

[C(++)1

(u2nminus1m minus unm

un+1m minus unminus1m+ unminus1m

)+Φ(++)

]

+ F(minus)n F

(+)m

[C(minus+)1

unminus1m minus u2nm

un+1m minus unminus1m+Φ(minus+)

]

+ F(+)n F

(minus)m

[C(++)1

unm + unminus1m

un+1m minus unminus1m+Φ(+minus)

]+ F

(minus)n F

(minus)m

[C(minus+)1

unm + unminus1m

unminus1m minus un+1m+Φ(minusminus)

]

(D8)

D2 three-points generalized symmetries of the tHε1 in direction m 267

Inserting it in the determining equations (326) and applying the op-erator (347) we obtain a system of four equations which have to beidentically satisfied The result is C(minus+)

1 = minusC(++)1 and

Φ(++) = K1unm Φ(minus+) =1

2K1unm minus

1

2C(++)1

Φ(+minus) =1

2K1unm +

1

2C(++)1 Φ(minusminus) =

1

2K1unm +

1

2C(++)1

(D9)

where K1 is another arbitrary constant Note that the symmetry gen-erated by K1 is a point symmetry so the equation (193e) in the di-rection n possess only the genuine three-points symmetry given by

g =F(+)n F


1

2

(F(minus)m minus F

(minus)n F

(+)m



+F(minus)n F

(+)m u2nm +

(F(minus)m minus F

(+)n F

(+)m

)unm


(D10)

As a sub-product of these computation we obtain then also thepoint symmetry given by the coefficients of K1

gP =[F(+)n

(2F

(+)m + F

(minus)m

)+ F

(minus)n

]unm (D11)

Note that since (193e) is invariant under the exchange n harr m

the symmetry in the m direction is given simply by performing suchexchange in (D10) which then gives

g =F(+)n F


1

2

(F(minus)n minus F

(+)n F

(minus)m



+F(+)n F

(minus)m u2nm +

(F(minus)n minus F

(+)n F

(+)m

)unm


(D12)

Clearly these results match with those presented without all thedetails in Section 33

d2 three-points generalized symmetries of the tHε1 in

direction m

Now we consider another interesting example the tHε1 equation Wewill show how to compute the three-points generalized symmetriesof the tHε1 equation as given by (191a) in the m direction To safely


apply the methods discussed in Section 32 we have to perform theexchange of the discrete indices n harr m and then compute the sym-metries in the n direction Applying such exchange to tHε1 equation(191a) we obtain

(unm minus unm+1)(un+1m minus un+1m+1) minusα2

minus ε2α2(F(+)n un+1m+1un+1m + F

(minus)n unmunm+1)

(D13)

Since there are only two two-periodic functions in (D13) a functionw decompose as

w = F(+)n w(+) + F

(minus)n w(minus) (D14)

instead of the most general decomposition (385) Applying the methodexplained in Subsection 32 we can find the system for the function zdecomposed as in (D14) from (340) Considering the case n = 2k wefind from (340) the system for z(+)

minusα2

(1+ ε2u2n+1m

)(unm minus unmminus1)

2 + ε2α22

partz(+)

partun+1m+partz(+)

partunm

+α2

(1+ ε2u2nminus1m


2 + ε2α22

partz(+)

partunminus1m=

2ε2un+1m

(unm minus unmminus1)2 + ε2α22

(D15a)

minusα2

(1+ ε2u2n+1m

)(unm minus unm+1)

2 + ε2α22

partz(+)

partun+1m+partz(+)

partunm

+α2

(1+ ε2u2nminus1m

)(unm minus unm+1)

2 + ε2α22

partz(+)

partunminus1m=

2ε2un+1m

(unm minus unm+1)2 + ε2α22

(D15b)

Taking the coefficients with respect to unmplusmn1 we can solve this sys-tem This time the system is not overdetermined but it is underdeter-mined

z(+) = minus log(1+ ε2u2n+1m

)+ logZ(+)

nm

(ε(un+1m minus unminus1m)

1+ ε2un+1munminus1m) (D16)

The same holds true for z(minus)

z(minus) = log(1+ ε2u2nm

)+ logZ(minus)

nm(minusun+1m + unminus1m)(D17)

Using (341) we just add equations which are identically satisfiedso all the differential conditions on z are satisfied Inserting into the


conservation law (329) the only condition we get is Z(plusmn)nm = Z

(plusmn)n ie

the two arbitrary functions can depend only on one lattice variableFor v we can proceed in the same way and we obtain

v(+) = log

[(1+ ε2un+1munminus1m)2

1+ ε2u2n+1m

]

+ logV(+)nm


1+ ε2un+1munminus1m

)

(D18a)

v(minus) = log[(1+ ε2u2nm)V

(minus)nm(minusun+1m + unminus1m)

] (D18b)

Using (343) we just add equations which are identically satisfied sowe have satisfied all the differential conditions on v Inserting v intothe conservation law (334) the only condition we get is V(plusmn)

nm = V(plusmn)n

ie the two arbitrary functions depend only on one lattice variableInserting z and v into the compatibility conditions (345) we can

reduce the number of independent functions from four to two sincewe find the following relations

Z(+)n (ξ) = C

(1)n minus

(1+ ξ2

)V(+)n (ξ) (D19a)

V(minus)n (ξ) = C

(2)n minusZ(minus)(ξ) (D19b)

Then defining

G(+)n (ξ) =

intV(+)n (ξ)dξ G

(minus)n (ξ) =

intZ(minus)n (ξ)dξ (D20)

we can write the solution for g from (344)

g =minusF

(+)n

εGn


1+ ε2un+1munminus1m]

)+ F

(+)n

[C(1)n

εarctan(εun+1m) +Φ(+)

]minus F

(minus)n (1+ ε2u2nm)G

(minus)n (minusun+1m + unminus1m)

+ F(minus)n

[(1+ ε2u2nm)C

(2)n unminus1m +Φ(minus)

]

(D21)

with Φ(plusmn)nm = Φ

(plusmn)nm(unm) functions to be determined

Inserting this form of g into the determining equations we find thefollowing restrictions

C(1)n = C

(2)n = 0 Φ(+) = C

(3)n Φ(minus) = C

(4)n (1+ ε2u2nm) (D22)

where C(3)n and C(4)

n are arbitrary functions of the lattice variable nThis means that (D13) the three-points generalized symmetries aregenerated by the following function

g = F(+)n

[minus1

εG

(+)n



)+C

(3)n

]+ F

(minus)n

[minus(1+ ε2u2nm)G

(minus)n (minusun+1m + unminus1m) + (1+ ε2u2nm)C

(4)n

]


(D23)

which depends on arbitrary functions Note that we can separate thetrue generalized symmetry part from the point part noting that thecoefficients of C(3)

n and C(4)n only depend on unm and not from its

shifts Therefore we have the three-points generalized symmetry part

gGS = F(+)n

[minus1

εG

(+)n

(ε(un+1m minus unminus1m

(1+ ε2un+1munminus1m])

)]+ F

(minus)n

[minus(1+ ε2u2nm)G


] (D24)

and in the purely point part

gP = F(+)n C

(3)n + F

(minus)n (1+ ε2u2nm)C

(4)n (D25)

Reversing the transformation n harr m we obtain the result displayedin Section 33

We underline that the property of possessing generalized symme-tries depending on arbitrary functions is linked with the fact that(191a) is Darboux integrable [5] as it was proved in [69] and is dis-cussed in Chapter 4

ET H R E E - P O I N T S G E N E R A L I Z E D S Y M M E T R I E S O FT H E R H O M B I C H 4 E Q U AT I O N S

e1 the rhombic H4 equations

In Chapter 1 we presented the single-cell form of the rhombic H4

equations in formula (132) Applying the prescription of [3 21 67] wehave (with the identification α1 = α and α2 = β) that the equations(132) once written on the Z2(nm) lattice have the form

rHε1 (unm minus un+1m+1) (un+1m minus unm+1) minus (α minus β)

+ ε(αminusβ)(F(+)n+m un+1munm+1 + F


)= 0

(E1a)

rHε2 (unm minus un+1m+1)(un+1m minus unm+1)+

+ (βminusα)(unm + un+1m + unm+1 + un+1m+1) minusα2 +β2

minus ε (βminusα)3 minus ε (βminusα)(2F

(minus)n+munm + 2F

(+)n+mun+1m +α+β

)middot

middot(2F

(minus)n+mun+1m+1 + 2F

(+)n+munm+1 +α+β

)= 0

(E1b)

rHε3 α(unmun+1m + unm+1un+1m+1)

minusβ(unmunm+1 + un+1mun+1m+1) + (α2 minusβ2)δ

minusε(α2 minusβ2)

αβ

(F(+)n+m un+1munm+1 + F


)= 0

(E1c)

where F(plusmn)k is still given by formula (190) Obviously this formulaagrees with that presented in [166]

e2 three-points generalized symmetries of the rhom-bic H4 equations

The three-points generalized symmetries of the rhombicH4 equationsas given by (E1) can be computed with the methods presented inChapter 3 and were first presented in [166] Their expression is thefollowing

XrHε1n =

1minus ε(F(+)n+mun+1munminus1m + F

(minus)n+mu

2nm

)un+1m minus unminus1m

partunm (E2a)

271

272 three-points generalized symmetries of the rhombic h4 equations

XrHε1m =

1minus ε(F(+)n+munm+1unmminus1 + F

(minus)n+mu

2nm


partunm

(E2b)

XrHε2n =

(1minus 4εαF

(minus)n+m

)(un+1m + unminus1m) minus 4εF

(+)n+mun+1munminus1m


+2αminus 4εα2 minus 4εF

(minus)n+mu

2nm +

(1minus 4εαF

(minus)n+m

)unm


partunm

(E2c)

XrHε2m =

(1minus 4εβF

(minus)n+m

)(unm+1 + unmminus1) minus 4εF

(+)n+munm+1unmminus1


+2βminus 4εβ2 minus 4εF

(minus)n+mu

2nm +

(1minus 4εβF

(minus)n+m

)unm

unm+1 + unmminus1

partunm

(E2d)

XrHε3n =

[1

2

unm (un+1m + unminus1m) + 2δα


minusε

α

(F(+)n+mun+1munminus1m + F

(minus)n+mu

2nm


partunm

(E2e)

XrHε3m =

[1

2

unm (unm+1 + unmminus1) + 2δβ

unm+1 + unmminus1minus

minusε

β

(F(+)n+munm+1unmminus1 + F

(minus)n+mu

2nm

)unm+1 + unmminus1

partunm

(E2f)

As stated in [166] the fluxes of the symmetries (E2) are readily iden-tified with the corresponding cases of the non autonomous YdKNequation (399) The explicit coefficients of the corresponding non-autonomous YdKN equation (399) are displayed in Table 31

FC O N N E C T I O N F O R M U L AElig B E T W E E N T H EN O N - A U T O N O M O U S QV A N D T H EN O N - A U T O N O M O U S Y D K N I N T H E m D I R E C T I O N

For the sake of completeness let us write down the non autonomousYdKN in the m direction

dumdt

=Am(um)um+1umminus1 +Bm(um)(um+1 + umminus1) +Cm(um)

um+1 minus umminus1

(F1)

Here the m-dependent coefficients are given by

Am(um) = au2m + 2bmum + cm (F2a)

Bm(um) = bm+1u2m + dum + em+1 (F2b)

Cm(um) = cm+1u2m + 2emum + f (F2c)

where bm cm and em are 2-periodic functions ie

bm = b+β(minus1)m cm = c+ γ(minus1)m

em = e+ ε(minus1)m(F3)

The correlation formulaelig read

a = a1a50 minus a220 minus a

221 + a

222 + a

223

+ (minus1)n(2a20a21 minus 2a22a23 + a1a51)

(F4a)

b =1


+1

2(a1a60 minus a22a32 minus a23a43 + a21a51)

+1

2(minus1)na21(a50 + a30 + a40)

+1

2(minus1)n(a23a32 + a1a61 + a20a51 + a22a43)

(F4b)

β =1


minus1


+1

2(minus1)na23(a30 minus a40 minus a50)

+1

2(minus1)n(a1a63 + a21a32 minus a20a43 minus a22a51)

(F4c)

c = a20a60 minus a40a30 + a21a61 minus a23a63 minus a22a62

minus (minus1)n[a22a63 + a43a32 + a23a62 minus a20a61 minus a21a60](F4d)

273

274 connection formulaelig

γ = a21a63 minus a40a32 + a20a62 minus a23a61 minus a22a60minus

minus (minus1)n[a22a61 + a43a30 + a23a60 minus a21a62 minus a20a63](F4e)

d =1

2(a250 minus a

240 minus a

230 + a1a7 + a

232 + a

243 minus a

251)

+ 2(minus1)n(a21a60 + a20a61 + a23a62 + a22a63)

(F4f)

e =1


+1

2(a20a7 minus a51a61 + a32a62 + a43a63)

minus1

2(minus1)n(a32a63 + a43a62 + a51a60 minus a21a7)

minus1

2(minus1)na61(a30 + a40 + a50)

(F4g)

ε =1


+1

2(a32a60 minus a43a61 + a51a63 + a22a7)

+1

2(minus1)n(a43a60 minus a32a61 + a51a62 + a23a7)

+1


(F4h)

f = a50a7 minus a260 + a

262 + a

263 minus a

261

minus (minus1)n(2a62a63 minus 2a60a61 + a51a7)

(F4i)

GC A L C U L AT I O N O F T H E F I R S T I N T E G R A L S O F T H EtH 1 E Q U AT I O N

In this Appendix we present a full developed example of calculationof the first integrals of an quad equation using the method discussedin Section 41 We will discuss the example of the tHε1 equation (191a)We note that the first integrals were first presented in [69] where theywere found by direct inspection

Before proceeding we note that since the H4 equations in generaland therefore the tHε1 in particular are non-autonomous only in thedirection m though the factors F(plusmn)m we can consider a simplified ver-sion of (420) in the same spirit of what was done with the General-ized Symmetries see Appendix E

Wi = F(+)m W

(+)i + F

(minus)m W

(minus)i (G1)

If we assume W1 = W1 (un+1munm) separating the even andodd terms with respect tom in (414) we find the following equations

partW(+)1

partun+1m+partW

(+)1

partunm= 0 (G2a)

(1+ ε2u2n+1m

) partW(minus)1

partun+1m+(1+ ε2u2nm

) partW(minus)1

partunm= 0 (G2b)

whose solution is

W1 = F(+)m F (un+1m minus unm)+ F

(minus)m G

(un+1m minus unm

1+ ε2unmun+1m

)(G3)

Inserting (G3) into the functional equation (47b) we obtain that Fand G must satisfy the following identity

G (ξ) = F

(α2ξ

) (G4)

This yields

W1 = F(+)m F

(α2

un+1m minus unm

)+F

(minus)m F

(un+1m minus unm

1+ ε2unmun+1m

) (G5)

For the m direction we may also suppose our first integral to betwo-point W2 = W2 (unm+1unm) It easy to see from (419) withk = 1 that this yields only the trivial solution W2 = constant There-fore we consider the case where the integral is three-point W2 =

275

276 calculation of the first integrals of the th1 equation

W2 (unm+1unmunmminus1) From (419) with k = 1 separating theeven and odd terms we obtain

α2(1+ ε2u2nm+1

) partW(+)2

partunm+1

minus[(unm minus un+1m)2 + ε2α22

] partW(+)2

partunm

+α2(1+ ε2u2nmminus1

) partW(+)2

partunmminus1= 0

(G6a)

α2(1+ ε2u2n+1m

) partW(minus)2

partunm+1

minus (unm minus un+1m)2partW

(minus)2

partunm

+α2(1+ ε2u2n+1m

) partW(minus)2

partunmminus1= 0

(G6b)

Taking the coefficients with respect to un+1m and then solving wehave

W2 = F(+)m F



)+ F

(minus)m G (unm+1 minus unmminus1)

(G7)

Inserting (G7) into (47a) we donrsquot have any further restriction on theform of the first integral So we conclude that we have two indepen-dent first integrals in the m direction as it was observed in [69]

As explained in Section 41 since the application of the method wasfruitful we obtained first integrals depending on arbitrary functionswhich is of course redundant We can then apply the simplifyingassumptions discussed in Section 41 and consider the first integralsfor the tHε1 as given by linear functions of their arguments

W1 = F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm

1+ ε2unmun+1m (G8a)

W2 = F(+)m α




(G8b)

Here α and β are two arbitrary constants coming from the fact thatin the m direction we found two independent arbitrary functions

The first integrals for the tHε1 equation (191a) were first presentedin the form (G8) in [69] In Section 43 we show these first integralsagain in this form

HL I N E A R I Z AT I O N O F T H E tH

ε1 E Q U AT I O N

T H R O U G H T H E F I R S T I N T E G R A L I N T H ED I R E C T I O N m

In this Appendix we consider the linearization of the tHε1 equation(191a) through the integral in the direction m namely W2 given byformula (429b) Note that this case is more interesting since now weare dealing with a three-point second order integral Let us assumewithout loss of generality that α = β = 1 Our starting point is therelation W2 = ρm from which we can derive two different equationsone for the even and one for the odd m component of unm This willgive a priori a coupled system However in this case it easy to see thatchoosing m = 2k and m = 2k+ 1 we obtain the two equations

1+ ε2un2k+1un2kminus1 = ρ2k (un2k+1 minus un2kminus1) (H1a)

un2k+2 minus un2k = ρ2k+1 (H1b)

Therefore the system consists of two uncoupled equationsThe first one (H1a) is a discrete Riccati equation which can be

linearized through the non-autonomous Moumlbius transformation

un2kminus1 =1

vnk+αk ρ2k =

1+ ε2αk+1αkαk+1 minusαk

(H2)

from which we obtain(1+ ε2α2k+1

)vnk+1 + ε

2αk+1 =(1+ ε2α2k

)vnk + ε

2αk (H3)

This equation is a total difference and therefore its solution is givenby

vnk =θn minus ε2αk

1+ ε2α2k (H4)

Putting αk = κ2kminus1 we obtain the solution for un2kminus1 as

un2kminus1 =1+ κ2kminus1θnθn minus ε2κ2kminus1

(H5)

The second equation is just a linear ordinary difference equationwhich can be written as a total difference performing the substitutionρ2k+1 = κ2k+2 minus κ2k

un2k = ωn + κ2k (H6)

Therefore we obtain

unm = F(+)m (ωn + κm) + F

(minus)m

1+ κmθnθn minus ε2κm

(H7)

277

278 linearization of the tHε1 equation with the m integral

(H7) depends on three arbitrary function This is because we startedfrom a second order integral which is a consequence the discreteequation This means that there must be a relation between θn andωn This relation can be retrieved by inserting (H7) into (191a) Sowe obtain the following relation

ωn minusωn+1 = α2ε2 + θnθn+1θn+1 minus θn

(H8)

which gives us the final expression for the solution of (191a) up tothe integration given by (H8) The general solutions obtained fromdifferent first integrals are the same in the sense that one of them caneasily be transformed into the other

B I B L I O G R A P H Y

[1] V E Adler ldquoBaumlcklund transformation for the Krichever-Novikovequationrdquo In Intern Math Research Notices 1 (1998) pp 1ndash4

[2] V E Adler A I Bobenko and Yu B Suris ldquoClassification ofIntegrable Equations on Quad-Graphs The Consistency Ap-proachrdquo In Comm Math Phys 233 (2003) pp 513ndash543

[3] V E Adler A I Bobenko and Yu B Suris ldquoDiscrete non-linear hyperbolic equations Classification of integrable casesrdquoIn Funct Anal Appl 43 (2009) pp 3ndash17

[4] V E Adler A B Shabat and R I Yamilov ldquoSymmetry ap-proach to the integrability problemrdquo In Theor Math Phys1253 (2000) pp 1603ndash1661

[5] V E Adler and S Ya Startsev ldquoDiscrete analogues of the Li-ouville equationrdquo In Theor Math Phys 1212 (1999) pp 1484ndash1495

[6] V E Adler and Yu B Suris ldquoQ4 integrable master equationrelated to an elliptic curverdquo In Intern Math Research Notices47 (2004) pp 2523ndash2553

[7] V E Adler S I Svinolupov and R I Yamilov ldquoMulti-componentVolterra and Toda type integrable equationsrdquo In Phys Lett A254 (1999) pp 24ndash36

[8] I M Anderson and M Juras ldquoGeneralized Laplace invariantsand the method of Darbouxrdquo In Duke Math Journal 89 (1997)pp 351ndash375

[9] I M Anderson and N Kamran The variational bicomplex for hy-perbolic second-order scalar partial differential equations in the planeCentre de Recherches Matheacutematiques Universiteacute de Montreacuteal1994

[10] I M Anderson and N Kamran ldquoThe variational bicomplexfor hyperbolic second-order scalar partial differential equa-tions in the planerdquo In Duke Math Journal 87 (1997) pp 265ndash319

[11] R L Anderson and N H Ibragimov Lie-Baumlcklund Transfor-mations in Applications Studies in Appl Math PhiladelphiaSIAM 1979

[12] V I Arnolrsquod ldquoOn a theorem of Liouville concerning integrableproblems in dynamicsrdquo In Amer Math Soc Trasl 2nd ser 61

(1967) pp 292ndash296

279

280 Bibliography

[13] V I Arnolrsquod Mathematical Methods of Classical Mechanics 2ndVol 60 Graduate Texts in Mathematics Berlin Springer-Verlag1997

[14] J Atkinson ldquontegrable lattice equations Connection to theMoumlbius group Baumlcklund transformations and solutionsrdquo PhDthesis The University of Leeds School of Mathematics 2008

[15] Hasselblatt B and J Propp ldquoDegree-growth of monomial mapsrdquoIn Ergodic Theory and Dynamical Systems (2007)

[16] A V Baumlcklund ldquoUumlber Flachentransformationenrdquo In Math Ann9 (1876) pp 297ndash320

[17] G A Baker and P R Graves-Morris Padeacute approximants Cam-bridge University Press 1996

[18] M Bellon and C-M Viallet ldquoAlgebraic entropyrdquo In CommMath Phys 204 (1999) pp 425ndash437

[19] A I Bobenko P Schroumlder J M Sullivan and G M ZieglerDiscrete differential geometry Birkhaumluser Verlag AG 2008

[20] R Boll ldquoClassification of 3D consistent quad-equationsrdquo In JNonlinear Math Phys 183 (2011) pp 337ndash365

[21] R Boll ldquoClassification and Lagrangian structure of 3D consis-tent quad-equationsrdquo PhD thesis TU Berlin 2012

[22] R Boll ldquoCorrigendum Classification of 3D consistent quad-equationsrdquo In J Nonlinear Math Phys 194 (2012) p 1292001

[23] T Bridgman W Hereman G R W Quispel and P H vander Kamp ldquoSymbolic Computation of Lax Pairs of Partial Dif-ference Equations using Consistency Around the Cuberdquo InFound Comput Math 134 (2013) pp 517ndash544

[24] M Bruschi Ragnisco O P M Santini and G-Z Tu ldquoInte-grable symplectic mapsrdquo In Physica D 493 (1991) pp 273ndash294

[25] F Calogero ldquoWhy are certain nonlinear PDEs both widely ap-plicable and integrablerdquo In What is integrability Ed by V EZakharov Berlin-Heidelberg Springer 1991

[26] F Calogero and A Degasperis Spectral Transform and SolitonsI Amsterdam North Holland Publishing Co 1982

[27] F Calogero and M C Nucci ldquoLax pairs galorerdquo In J MathPhys 321 (1991) pp 72ndash74

[28] A L Cauchy Oevre complegravets Ser 1 Acadeacutemie de Science 1890

[29] G Darboux ldquoSur les eacutequations aux deacuteriveacutees partielles du sec-ond ordrerdquo In Ann Sci Eacutecole Norm Sup 7 (1870) pp 163ndash173

[30] G Darboux Leccedilons sur la theacuteorie geacuteneacutele de surfaces Vol II ParisHermann 1915

Bibliography 281

[31] A De Moivre Miscellanea analytica de seriebus et quadraturis JTonson amp J Watts 1730

[32] D K Demskoy and C-M Viallet ldquoAlgebraic entropy for semi-discrete equationsrdquo In J Phys A Math Theor 45 (2012) 352001

(10 pp)

[33] J Diller ldquoDynamics of birational maps of P2rdquo In Indiana UnivMath J (1996) pp 721ndash772

[34] J Diller and C Favre ldquoDynamics of bimeromorphic maps ofsurfacesrdquo In Amer J Math 1236 (2001) pp 1135ndash1169

[35] A Doliwa and P M Santini ldquoMultidimensional quadrilaterallattices are integrablerdquo In Phys Lett A 233 (1997) pp 365ndash372

[36] B A Dubrovin A T Fomenko and F T Novikov ModernGeometry - Methods and Applications Part I The Geometry of Sur-faces Trasformations Groups and Fields II New York Springer-Verlag 1992

[37] S Elaydi An introduction to Difference Equations 3rd Springer2005

[38] L Euler Institutionum calculi integralis Vol III Imp Acad ImpSaegravent 1768

[39] G Falqui and C-M Viallet ldquoSingularity complexity and quasi-integrability of rational mappingsrdquo In Comm Math Phys 154

(1993) pp 111ndash125

[40] H Flaschka ldquoThe Toda lattice II Existentence of integralsrdquoIn Phys Rev B 94 (1974) pp 1924ndash1925

[41] A S Fokas ldquoA symmetry approach to exactly solvable evolu-tion equationsrdquo In J Math Phys 216 (1980) pp 1318ndash1325

[42] A S Fokas ldquoSymmetries and integrabilityrdquo In Stud ApplMath 77 (1987) pp 253ndash299

[43] A S Fokas and B Fuchssteiner ldquoThe hierarchy of the Benjamin-Ono equationrdquo In Phys Lett A 86 (1981) pp 341ndash345

[44] J Friš V Mandrosov Ya A Smorondinski M Uhliacuter and PWinternitz ldquoOn higher symmetries in Quantum MechanicsrdquoIn Phys Lett 133 (1965)

[45] G Frobenius ldquoUumlber Relationen zwischen den Naumlherungsbruumlchenvon Potenzreihenrdquo In J Reine Angew Math 90 (1881) pp 1ndash17

[46] B Fuchssteiner ldquoMastersymmetries and higher order time-dependent symmetries and conserved densities of nonlinearevolution equationsrdquo In Prog Theor Phys 70 (1983) pp 1508ndash1522

282 Bibliography

[47] B Fuchssteiner ldquoOn the hierarchy of the Landau-Lifshitz equa-tionrdquo In Physica D 13 (1984) pp 387ndash394

[48] V I Fushchich and A G Nikitin ldquoNew and old symmetriesof the Maxwell and Dirac equationsrdquo In Sov J Part Nucl 14

(1983) pp 1ndash22

[49] G Gaeta ldquoThe Poincareacute-Lyapounov-Nekhoroshev TheoremrdquoIn Ann Phys 2971 (2002) pp 157ndash173

[50] G Gaeta ldquoThe Poincareacute-Nekhoroshev Maprdquo In J NonlinearMath Phys 101 (2003) pp 51ndash64

[51] C S Gardner J M Greene M D Kruskal and R M MiuraldquoKorteweg-devries equation and generalizations VI methodsfor exact solutionrdquo In Comm Pure Appl Math 271 (1974)pp 97ndash133

[52] R N Garifullin E V Gudkova and I T Habibullin ldquoMethodfor searching higher symmetries for quad-graph equationsrdquoIn J Phys A Math Theor 44 (2011) 325202 (16pp)

[53] R N Garifullin I T Habibullin and R I Yamilov ldquoPeculiarsymmetry structure of some known discrete nonautonomousequationsrdquo In J Phys A Math Theor 48 (2015) 235201 (27pp)

[54] R N Garifullin A V Mikhailov and R I Yamilov ldquoDiscreteequation on a square lattice with a nonstandard structure ofgeneralized symmetriesrdquo In Theor Math Phys 1801 (2014)pp 765ndash780

[55] R N Garifullin and R I Yamilov ldquoGeneralized symmetryclassification of discrete equations of a class depending ontwelve parametersrdquo In J Phys A Math Theor 45 (2012) 345205

(23pp)

[56] R N Garifullin and R I Yamilov ldquoIntegrable discrete nonau-tonomous quad-equations as Baumlcklund auto-transformationsfor known Volterra and Toda type semidiscrete equationsrdquo InJ Phys Conf Ser 621 (2015) 012005 (18pp)

[57] R N Garifullin R I Yamilov and D Levi Classification offive-point differential-difference equations

[58] E Gau ldquoSur lrsquointeacutegration des eacutequations aux deacuteriveacutees partiellesdu second ordre par la meacutethode de M Darbouxrdquo In J MathPures Appl 7 (1911) pp 123ndash240

[59] E Goursat Leccedilons sur lrsquointeacutegration des eacutequations aux deacuteriveacutees par-tielles du second orde agrave deux variables indeacutependendantes Vol I-IIParis Hermann 1896

[60] B Grammaticos R G Halburd A Ramani and C-M VialletldquoHow to detect the integrability of discrete systemsrdquo In J PhysA Math Theor 42 (2009) Newton Institute Preprint NI09060-DIS 454002 (41 pp)

Bibliography 283

[61] B Grammaticos A Ramani and V Papageorgiou ldquoDo inte-grable mappings have the Painleveacute propertyrdquo In Phys RevLett 67 (1991) p 1825

[62] B Grammaticos A Ramani and C-M Viallet ldquoSolvable chaosrdquoIn Phys Lett A 3362-3 (2005) pp 152ndash158

[63] G Gubbiotti ldquoIntegrability of difference equations throughAlgebraic Entropy and Generalized Symmetriesrdquo In Symme-tries and Integrability of Difference equations ndash Lecture notes of theAbecedarian School of SIDE12 Montreacuteal 2016 Ed by D Levi RVerge-Rebelo and P Winternitz to appear on CRM LectureNotes in Mathematecal Physics Berlin Springer-Verlag 2017Chap 3

[64] G Gubbiotti and M Hay A SymPy module to calculate algebraicentropy for difference equations and quadrilateral partial differenceequations in preparation 2016

[65] G Gubbiotti C Scimiterna and D Levi A non-autonomousgeneralization of the QV equation arXiv151200395 2015

[66] G Gubbiotti C Scimiterna and D Levi The non autonomousYdKN equation and generalized symmetries of Boll equations arXiv1510071752015

[67] G Gubbiotti C Scimiterna and D Levi ldquoAlgebraic entropysymmetries and linearization of quad equations consistent onthe cuberdquo In J Nonlinear Math Phys 234 (2016) pp 507ndash543

[68] G Gubbiotti C Scimiterna and D Levi ldquoLinearizability andfake Lax pair for a consistent around the cube nonlinear non-autonomous quad-graph equationrdquo In Theor Math Phys 18891(2016) pp 1459ndash1471

[69] G Gubbiotti C Scimiterna and D Levi ldquoOn Partial Differen-tial and Difference Equations with Symmetries Depending onArbitrary Functionsrdquo In Acta Polytechnica 563 (2016) pp 193ndash201

[70] G Gubbiotti C Scimiterna and R I Yamilov Darboux inte-grability of trapezoidal H4 and H6 families of lattice equations IIGeneral solutions In preparation 2017

[71] G Gubbiotti and R I Yamilov Darboux integrability of trape-zoidal H4 and H6 families of lattice equations I First integralsarXiv160803506v2 2017

[72] I T Habibullin ldquoCharacteristic algebras of fully discrete hy-perbolic type equationsrdquo In SIGMA 1 (2005) p 23

284 Bibliography

[73] I T Habibullin Sokolov V V and R I Yamilov ldquoMulti-componentintegrable systems and nonassociative structuresrdquo In Proceed-ings of 1st Int Workshop on Nonlinear Physics Theory and Experi-ment Gallipoli Italy 29 June - 7 July 1995 Ed by E Alfinito MBoiti L Martina and F Pempinelli World Scientific Publish-ing 1996 pp 139ndash168

[74] I T Habibullin N A Zheltukhina and A Sakieva ldquoDiscretiza-tion of hyperbolic type Darboux integrable equations preserv-ing integrabilityrdquo In J Math Phys 52 (2011) p 093507

[75] M Hay ldquoA Completeness Study on discrete 2times 2 Lax PairsrdquoIn J Math Phys 50 (2009) 103516 29pp

[76] M Hay ldquoA Completeness Study on Certain 2times 2 Lax PairsIncluding Zero Termsrdquo In SIGMA 7 (2011) 089 12pp

[77] M Hay and S Butler Simple identification of fake Lax pairs arXiv13112406v12012

[78] M Hay and S Butler ldquoTwo definitions of fake Lax pairsrdquo InAIP Conf Proc 1648 (2015) p 180006

[79] M Heacutenon ldquoA two dimentional mapping with a strange attrac-torrdquo In Comm Math Phys 501 (1976) pp 69ndash77

[80] K Hessenberg Behandlung linearer Eigenwertaufgaben mit Hilfeder Hamilton-Cayleyschen Gleichung Tech rep 1 Institut fuumlrPraktische Mathematik (IPM) ndash Technische Hochschule Darm-stadt 1940

[81] J Hietarinta ldquoA new two-dimensional lattice model that islsquoconsistent around the cubersquordquo In J Phys A Math Gen 37

(2004) pp L67ndashL73

[82] J Hietarinta ldquoSearching for CAC-mapsrdquo In J Nonlinear MathPhys 12 (2005) pp 223ndash230

[83] J Hietarinta ldquoDefinitions and Predictions of Integrability forDifference Equationsrdquo In Symmetries and Integrability of Dif-ference Equations Ed by D Levi P Olver Z Thomova and PWinternitz London Mathematical Society Lecture Notes seriesCambridge Cambridge University Press 2011 pp 83ndash114

[84] J Hietarinta and C-M Viallet ldquoSearching for integrable latticemaps using factorizationrdquo In J Phys A Math Theor 40 (2007)pp 12629ndash12643

[85] J Hietarinta and C-M Viallet ldquoWeak Lax pairs for lattice equa-tionsrdquo In Nonlinearity 25 (2012) pp 1955ndash1966

[86] R Hirota K Kimura and H Yahagi ldquoHow to find the con-served quantities of nonlinear discrete equationsrdquo In J PhysA 3448 (2001) pp 10377ndash10386

[87] N H Ibragimov Transformation Groups applied to MathematicalPhysics Boston Reidel 1985

Bibliography 285

[88] C G J Jacobi ldquoUumlber die Darstellung einer Reihe gegebnerWerthe durch eine gebrochne rationale Functionrdquo In J fuumlrReine Angewandte Math 30 (1846) pp 127ndash156

[89] N Joshi ldquoSingularity analysis and integrability of discrete sys-temsrdquo In J Math Anal Appl 184 (1994) pp 573ndash584

[90] E Jury Theory and applications of the Z-transform method RobertE Krieger 1964

[91] E G Kalnins W Miller Jr and G C Williamns ldquoMatrix oper-ator symmetries of the Dirac equation and separation of vari-ablesrdquo In J Math Phys 27 (1986) pp 1893ndash1900

[92] D J Korteweg and G de Vries ldquoXLI On the change of formof long waves advancing in a rectangular canal and on a newtype of long stationary wavesrdquo In Phil Mag Ser 5 39240

(1895) pp 422ndash443

[93] Y Kosmann-Schwarzbach ldquoGeneralized symmetries of non-linear partial differential equationsrdquo In Lett Math Phys 3

(1979) pp 395ndash404

[94] Y Kosmann-Schwarzbach ldquoVector fields and generalized vec-tor fields on fibered manifoldsrdquo In Geometry and DifferentialGeometry Ed by R Artzy and I Vaisman Lecture Notes inMath New York Springer-Verlag 1980

[95] S K Lando Lectures on Generating Functions American Math-ematical Society 2003

[96] P-S Laplace ldquoRecherches sur le calcul inteacutegral aux diffeacuterencespartiellesrdquo In Meacutemoires de lrsquoAcadeacutemie royale des sciences de Paris(1773) pp 5ndash68

[97] P D Lax ldquoIntegrals of nonlinear equations of evolution andsolitary wavesrdquo In Comm Pure Appl Math 215 (1968) pp 467ndash490

[98] P D Lax ldquoPeriodic solutions of the KdV equationrdquo In CommPure Appl Math 281 (1975) pp 141ndash188

[99] D Levi ldquoNonlinear differential-difference equations as Baumlck-lund transformationsrdquo In J Phys A Math Gen 14 (1981)pp 1083ndash1098

[100] D Levi P Olver Z Thomova and P Winternitz eds Symme-tries and Integrability of Difference Equations London Mathemat-ical Society Lecture Notes Cambridge Cambridge UniversityPress 1991

[101] D Levi and M Petrera ldquoContinuous symmetries of the latticepotential KdV equationrdquo In J Phys A 40 (2007) pp 4141ndash4159

286 Bibliography

[102] D Levi M Petrera C Scimiterna and R I Yamilov ldquoOnMiura Transformations and Volterra-Type Equations Associ-ated with the AdlerndashBobenkondashSuris Equationsrdquo In SIGMA 4

(2008) 077(14pp)

[103] D Levi O Ragnisco and M Bruschi ldquoContinuous and dis-crete matrix Burgersrsquo hierarchiesrdquo In Il Nuovo Cimento B (1971-1996) 741 (1983) pp 33ndash51

[104] D Levi A Sym and Tu G Z A working algorithm to isolateintegrable surfaces in E3 preprint DF INFN 761 Roma Oct 101990

[105] D Levi and R I Yamilov ldquoConditions for the existence ofhiger symmetries of the evolutionary equations on the latticerdquoIn J Math Phys 3812 (1997) pp 6648ndash6674

[106] D Levi and R I Yamilov ldquoThe generalized symmetry methodfor discrete equationsrdquo In J Phys A Math Theor 42 (2009)454012 (18pp)

[107] D Levi and R I Yamilov ldquoGeneralized symmetry integrabil-ity test for discrete equations on the square latticerdquo In J PhysA Math Theor 44 (2011) 145207 (22pp)

[108] D Levi R I Yamilov and P Winternitz ldquoSymmetries of thecontinuous and discrete Krichever-Novikov equationrdquo In SIGMA7 (2011) 097 (16pp)

[109] Y Q Li B Li and Lou S Y Constraints for Evolution Equationswith Some Special Forms of Lax Pairs and Distinguishing Lax Pairsby Available Constraints arXiv10081375v2 2010

[110] S Lie ldquoBegruumlndung einer Invariantentheorie der Beruumlhrungstrans-formationenrdquo In Math Ann 8 (1874) pp 215ndash288

[111] S Lie Vorlesungen uumlber Differentialgleichungen mit bekannten in-finitesimalen Transformationen Leipzig Teubner 1912

[112] J Liouville ldquoNote sur lrsquointeacutegration des eacutequations diffeacuterentiellesde la Dynamique preacutesenteacutee au Bureau des Longitudes le 29

juin 1853rdquo In J Math Pures Appl 20 (1853) pp 137ndash138

[113] J Liouville ldquoSur lrsquoeacutequation aux differences partielles d2 logλdudv plusmn

λ2a2

= 0rdquo In J Math Pures Appl 18 (1853) pp 71ndash72

[114] M Marvan ldquoReducibility of zero curvature representationswith application to recursion operatorsrdquo In Acta Appl Math83 (2004) pp 39ndash68

[115] M Marvan ldquoOn the spectral parameter problemrdquo In ActaAppl Math 109 (2010) pp 239ndash255

[116] A V Mikhailov and A B Shabat ldquoSymmetries ndash test of in-tegrabilityrdquo In Important developments in soliton theory Ed byA S Fokas and V E Zakharov Springer-Verlag 1993 pp 355ndash372

Bibliography 287

[117] A V Mikhailov A B Shabat and V V Sokolov ldquoThe sym-metry approach to classification of integrable equationsrdquo InWhat is Integrability Ed by V E Zakharov Springer-Verlag1991 pp 115ndash184

[118] A V Mikhailov A B Shabat and R I Yamilov ldquoThe symme-try approach to the classification of nonlinear equations Com-plete lists of integrable systemsrdquo In Russian Math Survey 424(1987) pp 1ndash63

[119] N N Nekhoroshev ldquoThe Poincareacute-Lyapunov-Liouville-Arnolrsquodtheoremrdquo In Funct Anal Appl 282 (1994) pp 128ndash129

[120] F W Nijhoff ldquoLax pair for the Adler (lattice Krichever-Novikov)systemrdquo In Phys Lett A 297 (2002) pp 49ndash58

[121] F W Nijhoff and H Capel ldquoThe discrete Korteweg-de Vriesequationrdquo In Acta Appl Math 39 (1995) pp 133ndash158

[122] F W Nijhoff A Ramani B Grammaticos and Y Ohta ldquoOndiscrete Painlevegrave equations associated with the lattice KdV sys-tems and the Painlevegrave VI equationrdquo In Stud Appl Math 1063(2001) pp 261ndash314

[123] F W Nijohff and A J Walker ldquoThe discrete and continousPainleveacute VI hierarchy and the Garnier systemsrdquo In Glasg MathJ 43A (2001) pp 109ndash123

[124] J J C Nimmo and W K Schief ldquoSuperposition principles as-sociated with the Moutard transformation an integrable dis-cretization of a (2 + 1)-dimensional sine-Gordon systemrdquo InProc Roy Soc London Ser A 453 (1997) pp 255ndash279

[125] E Noether ldquoInvariante Variationsproblemerdquo In Nachr d KoumlnigGesellsch d Wiss zu Goumlttingen Math-phys Klasse (1918) pp 235ndash257

[126] W Oevel and B Fuchssteiner ldquoExplicit formulas for the sym-metries and conservation laws of the Kadomtsev-Petviashviliequationrdquo In Phys Lett A 88 (1982) pp 323ndash327

[127] W Oevel B Fuchssteiner H Zhang and O Ragnisco ldquoMas-tersymmetries angle variables and recursion operator of therelativistic Toda latticerdquo In J Math Phys 30 (1989) pp 2664ndash2670

[128] W Oevel H Zhang and B Fuchssteiner ldquoMastersymmetriesand multi-Hamiltonian formulations for some integrable lat-tice systemsrdquo In Progr Theor Phys 812 (1989) pp 295ndash308

[129] P J Olver ldquoEvolution equations possessing infinitely manysymmetriesrdquo In J Math Phys 186 (1977) pp 1212ndash1215

[130] P J Olver ldquoConvervation laws in elasticity II Linear homo-geneous isotropic elastostaticsrdquo In Arch Rat Mech Anal 85

(1984) pp 131ndash160

288 Bibliography

[131] P J Olver Applications of Lie Groups to Differential EquationsBerlin Springer-Verlag 1986

[132] P J Olver ldquoConvervation laws in elasticity III Planar linearanisotropic elastostaticsrdquo In Arch Rat Mech Anal 85 (1988)pp 167ndash181

[133] L V Ovsiannikov ldquoGroups and group-invariant solutions ofdifferential equationsrdquo In Doklady Akademii Nauk SSSR 118

(1958) (in Russian) pp 439ndash442

[134] L V Ovsiannikov ldquoGroup Properties of Differential EquationsrdquoIn USSR Academy of Sciences Siberian Branch Novosibirsk 118

(1962) pp 439ndash442

[135] H Padeacute ldquoSur la reacutepresentation approcheacutee drsquoune fonction pardes fractions rationellesrdquo In Ann Eacutecole Nor 39 (1892) Thesissuppement pp 1ndash93

[136] GRW Quispel JAG Roberts and CJ Thompson ldquoIntegrablemappings and soliton equations IIrdquo In Physica D 341 (1989)pp 183ndash192

[137] A Ramani B Grammaticos and J Hietarinta ldquoDiscrete ver-sion of the Painleveacute equationsrdquo In Phys Rev Lett 67 (1991)p 1829

[138] A Ramani N Joshi B Grammaticos and T Tamizhmani ldquoDe-constructing an integrable lattice equationrdquo In J Phys A MathGen 39 (2006) pp L145ndashL149

[139] O G Rasin and P E Hydon ldquoSymmetries of integrable differ-ence equations on the quad-graphrdquo In Stud Appl Math 119

(2007) pp 253ndash269

[140] S Rauch-Wojciechovski ldquoSuperintegrability of the Calogero-Moser systemrdquo In Phys Lett A 95 (1983) pp 279ndash281

[141] A Russakovskii and B Shiffman ldquoValue distribution of se-quences of rational mappings and complex dynamicsrdquo In In-diana U Math J 46 (1997) pp 897ndash932

[142] H Sakai ldquoRational surfaces associated with affine root sys-tems and geometry of the Painleveacute Equationsrdquo In Comm MathPhys 2201 (2001) pp 165ndash229

[143] S Yu Sakovich True and fake Lax pairs how to distinguish themarXivnlin SI0112027 2001

[144] S Yu Sakovich Cyclic bases of zero-curvature representations fiveillustrations to one concept arXivnlin SI0212019 2002

[145] E Schroumlder ldquoUumlber iterirtie Funktionenrdquo In Math Ann 32(1870) pp 296ndash322

[146] V V Sokolov ldquoOn the symmetries of evolution equationsrdquo InRussian Math Surveys 435 (1988) pp 165ndash204

Bibliography 289

[147] V V Sokolov and A B Shabat ldquoClassification of integrableevolution equationsrdquo In Soviet Scientific Reviews Section C Math-ematical Physics Reviews Vol 4 New York Harwood AcademicPublishers 1984 pp 221ndash280

[148] V V Sokolov and A V Zhiber ldquoOn the Darboux integrablehyperbolic equationrdquo In Phys Lett A 208 (1995) pp 303ndash308

[149] S Ya Startsev ldquoDarboux integrable discrete equations possess-ing an autonomous first-order integralrdquo In J Phys A MathTheor 47 (2014) 105204 (16pp)

[150] SymPy Development Team SymPy Python library for symbolicmathematics 2016 url httpwww20sympy20org

[151] T Takenawa ldquoDiscrete dynamical systems associated with rootsystems of indefinite typerdquo In Comm Math Phys 2243 (2001)pp 657ndash681

[152] S Tremblay B Grammaticos and A Ramani ldquoIntegrable lat-tice equations and their growth propertiesrdquo In Phys Lett A2786 (2001) pp 319ndash324

[153] A Veselov ldquoGrowth and integrability in the dynamics of map-pingsrdquo In Comm Math Phys 145 (1992) pp 181ndash193

[154] E Vessiot ldquoSur les eacutequations aux deacuteriveacutees partielles du secondordre F(xypq r s t) = 0 inteacutegrables par la meacutethode deDarbouxrdquo In J Math Pures Appl (1939) pp 1ndash61

[155] C-M Viallet Algebraic Entropy for lattice equations arXiv06090432006

[156] C-M Viallet ldquoIntegrable lattice maps Q5 a rational version ofQ4rdquo In Glasg Math J 51A (2009) pp 157ndash163

[157] C-M Viallet Algebraic entropy for differentialndashdelay equations arXiv140861612014

[158] C-M Viallet ldquoOn the algebraic structure of rational discretedynamical systemsrdquo In J Phys A 4816 (2015) 16FT01

[159] J von zur Gathen and J Gerhard Modern Computer Algebra3rd Cambridge Cambridge University Press 2013

[160] H Weyl The Theory of Groups and Quantum Mechanics DoverBooks on Mathematics Dover Publications 1950

[161] E T Whittaker A Treatise on the Analytical Dynamics of Parti-cles and Rigid Bodies Cambridge Cambridge University Press1999

[162] E T Whittaker and G N Watson A Course of Modern Analysis4th Cambridge University Press 1927

[163] G L Wiersma ldquoLattice equations hierarchies and Hamilto-nian structuresrdquo PhD thesis Universiteit van Amsterdam 1987

290 Bibliography

[164] G L Wiersma and H W Capel ldquoLattice equations hierar-chies and Hamiltonian structuresrdquo In Physica A 1421 (1987)pp 199ndash244

[165] P D Xenitidis ldquoIntegrability and symmetries of differenceequations the Adler-Bobenko-Suris caserdquo In Proceedings of the4th Workshop Group Analysis of Differential equations and ItegrableSystems 2009 pp 226ndash242

[166] P D Xenitidis and V G Papageorgiou ldquoSymmetries and in-tegrability of discrete equations defined on a blackndashwhite lat-ticerdquo In J Phys A Math Theor 4235 (2009) p 454025

[167] R I Yamilov ldquoClassification of discrete evolution equationsrdquoIn Usp Mat Nauk 38 (1983) in Russian pp 155ndash156

[168] R I Yamilov ldquoClassification of discrete integrable equationsrdquoin Russian PhD thesis Ufa 1984

[169] R I Yamilov ldquoSymmetries as integrability criteria for differen-tial difference equationsrdquo In J Phys A 39 (2006) R541ndashR623

[170] Cherdantsev I Yu and R I Yamilov ldquoMaster symmetries fordifferential-difference equations of the Volterra typerdquo In Phys-ica D 87 (1995) pp 140ndash144

[171] Cherdantsev I Yu and R I Yamilov ldquoLocal master symme-tries of differential-difference equationsrdquo In Proceedings of 1stInt Workshop on Symmetries and Integrability of Difference Equa-tions SIDE-1 Montreal Canada 22-29 May 1994 Centre de RecherchesMathematiques Vol 9 CRM Proceedings and Lecture Notes1996 pp 51ndash61

[172] V E Zakharov ed What Is Integrability Springer Series inNonlinear Dynamics Berlin Heidelberg Springer-Verlag 1991

[173] H Zhang G Tu W Oevel and B Fuchssteiner ldquoSymmetriesconserved quantities and hierarchies for some lattice systemswith soliton structurerdquo In J Math Phys 327 (1991) pp 1908ndash1918

[174] A V Zhiber N Kh Ibragimov and A B Shabat ldquoEquationsof Liouville typerdquo In Soviet Math Dokl 20 (1979) pp 607ndash609

[175] A V Zhiber and A B Shabat ldquoKlein-Gordon equations with anontrivial grouprdquo In Soviet Phys Dokl 24 (1979) pp 607ndash609

[176] A V Zhiber and A B Shabat ldquoSystems of equations ux =

p(u v) vy = q(u v) possessing symmetriesrdquo In Soviet MathDokl 30 (1984) pp 23ndash26

[177] A V Zhiber and V V Sokolov ldquoNew example of a nonlinearhyperbolic equation possessing integralsrdquo In Theor and MathPhys 120 (1999) pp 834ndash839

Bibliography 291

[178] A V Zhiber and V V Sokolov ldquoExactly integrable hyperbolicequations of Liouville typerdquo In Russian Math Surveys 561(2011) pp 61ndash101

[179] A V Zhiber V V Sokolov and S Ya Startsev ldquoOn nonlinearDarboux integrable hyperbolic equationsrdquo In Doklady Math52 (1995) pp 128ndash130

colophon

This document was typeset using the typographical look-and-feelclassicthesis developed by Andreacute Miede The style was inspiredby Robert Bringhurstrsquos seminal book on typography ldquoThe Elements ofTypographic Stylerdquo classicthesis is available for both LATEX and LYX

httpsbitbucketorgamiedeclassicthesis

Final Version as of April 9 2017 (classicthesis)

Dedication

Abstract

Sommario

Acknowledgments

Ringraziamenti

Contents

List of Figures

List of Tables

Introduction

1 Integrability and the Consistency Around the Cube

11 The meaning of integrability

12 The Consistency Around the Cube

13 The Consistency Around the Cube as a search method


132 Bolls classification non-autonomous lattices


14 Construction of the 2D3D lattice in the general case

15 Classification tools on the 2D lattice

16 Independent equations on the 2D-lattice

2 Algebraic Entropy and direct linearization

21 Definition and basic properties




22 Computational tools

23 Algebraic Entropy test for (191 193)

24 Examples of direct linearization

241 The tH1 equation


3 Generalized symmetries of the trapezoidal H4 and H6 equations and the non-autonomous YdKN equation

31 Generalized Symmetries and integrability

32 Three-point Generalized Symmetries for quad equation

33 Three-point generalized symmetries of the trapezoidal H4 and H6 equations


332 H6 equations

34 The non-autonomous YdKN equation

35 Integrability properties of the non autonomous YdKN equation and its sub-cases

36 A non-autonomous generalization of the QV equation

4 Darboux integrability and general solutions

41 Darboux integrability

42 Darboux integrability and Consistency Around the Cube

43 First integrals for the trapezoidal H4 and H6 equations


4311 1 equation

4312 2 equation

4313 3 equation

432 H6 equations

4321 1D2 equation

4322 2D2 equation

4323 3D2 equation

4324 D3 equation

4325 1D4 equation

4326 2D4 equation

44 General solutions through the first integrals

441 The iD2 equations i=123

4411 1D2 equation

4412 2D2 equation

4413 3D2 equation

442 The D3 and the iD4 equations i=12

4421 D3 equation

4422 1D4 equation

4423 2D4 equation


4431 The 1 equation

4432 The 2 equation

4433 The 3 equation

Conclusions

A Final part of the proof of the commutative diagram 19

B python programs for computing Algebraic Entropy

B1 Algebraic Entropy of systems of quad equations with ae2dpy

B11 Description of the content of ae2dpy

B12 Code

B2 Algebraic Entropy for systems of ordinary difference equations and differential difference equations of arbitrary order with ae_differential_differencepy

B3 Analysis of the degree sequences

B31 The padepy module

B32 The pade_rat_gfuncpy module

B33 The z_transformpy module

C Evolution Matrices

C1 Trapezoidal H4 equations (191)

C11 tH1 equation (191a)

C12 tH2 equation (191b)

C13 tH3 equation (191c)

C2 H6 equations (193)

C21 1D2 equation (193b)

C22 2D2 equation (193c)

C23 3D2 equation (193d)

C24 D3 equation (193e)

C25 1D4 equation (193f)

C26 2D4 equation (193g)

D Examples of calculations of Generalized Symmetries for equations with two-periodic coefficients

D1 Three-points generalized symmetries of the D3 equation

D2 Three-points generalized symmetries of the tH1 in direction m

E Three-points generalized symmetries of the rhombic H4 equations

E1 The rhombic H4 equations

E2 Three-points generalized symmetries of the rhombic H4 equations

F Connection formulaelig between the non-autonomous QV and the non-autonomous YdKN in the m direction

G Calculation of the first integrals of the tH1 equation

H Linearization of the tH1 equation through the first integral in the direction m

Bibliography

Colophon

Giorgio Gubbiotti Integrability properties of quad equations consistent onthe cube copy AA 2016ndash2017


A B S T R A C T



S O M M A R I O


v





vii



viii





ix


A tutti voi grazie

x

C O N T E N T S

introduction 1


cube 5






















equation 79





332 H6 equations 99








xi

xii contents

























conclusions 190





b12 Code 206













contents xiii

















bibliography 279











32










xiv








xv









1

2 introduction




introduction 3

Mob4

















HiH = 0 (12)

5



HiHj= 0 i 6= j = 1 N (13)


rankpart(H1 Hn)















xn+1m+1

xnm+1+ ynm =

xn+1m+1

xn+1m (17a)

xn+1m

xnm+ yn+1m =

xnm+1

xnm (17b)


Lnm =

(xn+1mxnm λ

λ 0

) (18a)

Mnm =

(xnm+1xnm λ

λ ynm

) (18b)


(xn+1m+1

xnm+1+ ynm minus

xn+1m+1

xn+1m

)minusλ

(xn+1m

xnm+ yn+1m minus

xnm+1

xnm

)0

=

(0 0

0 0

)

(19)







un+1 = 4un (1minus un) (110)


un = sin2(c02

nminus1)

(111)


dundc0

= 2n sin(c02

nminus1)

cos(c02

nminus1)

(112)








unm

un+1m

unm+1

un+1m+1

p

p

qq




unmp un+1mp

unm+1p

unmp+1

un+1m+1p

un+1mp+1

unm+1p+1 un+1m+1p+1

A

A

B B

C

C




)= 0

(114a)


)= 0

(114b)


)= 0

(114c)


)= 0

(114d)



)= 0

(114e)


)= 0

(114f)





unmp+1 =fnm

gnm un+1mp+1 =

fn+1m

gn+1m

unm+1p+1 =fnm+1

gnm+1

(115)



l21unmp+1 + l22 (116a)


m21unmp+1 +m22 (116b)


fn+1m



fnm+1





Φnm =

(fnm

gnm

) (118)


Lnm = γnm

(l11 l12

l21 l22

) Mnm = γ primenm

(m11 m12

m21 m22

) (119)





τnm =γnm+1γ

primenm

γnmγprimen+1m

(121)















(122)




(123a)





(123c)






(124a)









4αβ= 0

(124c)







+ k5(unm + un+1m + unm+1 + un+1m+1) + k6 = 0

(124d)

with


k3 =αβ(α+β)


g24)

k4 =αβ(α+β)


g24)

k5 =g32k0 +

g24k1 k6 =

g2216k0 + g3k1

(125)

where







J1 unm + e2unm + e1

un+1m+1 + o2un+1m+1 + o1

=un+1m + e2un+1m + o1


(127)



J2 unm + un+1m + unm+1 + un+1m+1 = 0 (128a)

J3 unmun+1m+1 + un+1munm+1 = 0 (128b)



+ unm + un+1m + unm+1 + un+1m+1 = 0

(128c)




Q (x x1 x2 x12α1α2) = 0 (129)




x

x1

x2

x12

α1

α1

α2α2



A (x x1 x2 x12α1α2) = 0 (130a)

A (x3 x13 x23 x123α1α2) = 0 (130b)

B (x x2 x3 x23α3α2) = 0 (130c)

B (x1 x12 x13 x123α3α2) = 0 (130d)

C (x x1 x3 x13α1α3) = 0 (130e)

C (x2 x12 x23 x123α1α3) = 0 (130f)


x x1

x2

x3

x12

x13

x23 x123

A

A

B B

C

C









+ ε(α2 minusα1)(2x1 +α1 +α2)(2x2 +α1 +α2)


22

(132b)


+ (α21 minusα22)

(δ+

εx1x2α1α2

) (132c)




hij =partQ

partxk

partQ

partxlminusQ

part2Q

partxk partxl (133)






2 Here x0 = x



Q(x x1 x2 x12α1α2) = Q(x1 x x12 x2α1α2) (134)











tHε1 (ε2 ε2 0 0

) Eq B of 3101



tHε2 (1 + 4εx 1 + 4εx2 1 1) Eq B of 3102


+εα22

(2x3 + 2α3 +α2) (2x23 + 2α3 +α2)

+ (α2 +α3)2 minusα23 +

εα322

= 0

(135b)


2ε2 x23 x223)

Eq B of 3103



)(δ2 +

ε2x3x23e4α3+2α2

)= 0

(135c)


D1 (0 0 0 0) Eq A of 3121 and 3131

x+ x1 + x2 + x12 = 0 (136a)


1D2 (δ21 (δ1δ2 + δ1 minus 1)

2 1 0)

Eq A of 3122


D3 (4x 1 1 1) Eq A of 3123

x+ x1x2 + x1x12 + x2x12 = 0 (136c)


1D4 (x2 + 4δ1δ2δ3 x21 x212 x22

) Eq A of 3124

xx12 + x1x2 + δ1x1x12 + δ2x2x12 + δ3 = 0 (136d)


2D2 (δ21 0 1 (δ1δ2 + δ1 minus 1)

2)

Eq C of 3132


3D2 (δ21 0 (δ1δ2 + δ1 minus 1)

2 1)

Eq C of 3133


(136f)


2D4 (x2 + 4δ1δ2δ3 x21 x22 x212

) Eq A of 3135

xx1 + δ2x1x2 + δ1x1x12 + x2x12 + δ3 = 0 (136g)






case


Q = Q(x x1 x2 x12α1α2) = 0 (137a)

|Q = Q(x1 x x12 x2α1α2) = 0 (137b)

Q = Q(x2 x12 x x1α1α2) = 0 (137c)

|Q = Q(x12 x2 x1 xα1α2) = 0 (137d)



Q = |Q = Q = |Q (138)


x x1 x

x2x12

x2

x x1 x

Q |Q

Q |Q




Q = |Q Q = |Q (139)


x x1 x

x2x12

x2

x x1 x

Q |Q

|Q Q



Q = |Q Q = |Q (140)



x x1 x

x2x12

x2

x x1 x

Q Q

Q Q



Q [u] =





(141)



rHε1 =






|n|+ |m| = 2k+ 1 k isin Z

(142)







xx1

x2

x3

x12

x13

x23x123

x

x

x2

xx1

x3

x3

x23

x3x13

A |A

A |A

A |A

A |A

B B B

B B B

C

C

C

|C

|C

|C



B(x x2 x3 x23) = B(x2 x x23 x3) (143a)

B(x1 x12 x13 x123) = B(x12 x1 x123 x13) (143b)

|C(x x1 x3 x13) = C(x1 x x13 x3) (143c)

|C(x2 x12 x23 x123) = C(x12 x2 x123 x23) (143d)




˜A [u] =





B [u] =





(144b)

˜B [u] =





C [u] =





˜C [u] =






A A A ˜A B B B ˜B C C C ˜C (145)












A =






|n|+ |m| = 2k+ 1 k isin Z

(147a)

A =






|n|+ |m| = 2k+ 1 k isin Z

(147b)

B =






|n|+ |m| = 2k+ 1 k isin Z

(147c)

B =






|n|+ |m| = 2k+ 1 k isin Z

(147d)

C =






|n|+ |m| = 2k+ 1 k isin Z

(147e)

C =






|n|+ |m| = 2k+ 1 k isin Z

(147f)











(148)







whose solution is


m + c3 (minus1)n+m (150)



f2k2k = 1 f2k+12k = f2k2k+1 = f2k+12k+1 = 0 (151)


which yield


4 (152a)



4 (152b)


4 (152c)


4 (152d)




Using (152)


(minus)n+m (154)

where

F(plusmn)k =

1plusmn (minus1)k

2 k isin Z (155)

we obtain

Q = F(+)n+mQ+ F

(minus)n+m|Q (156)





(minus)m (158)

we obtain

Q = F(+)m Q+ F

(minus)m Q (159)




mula (154) and get




)= 0

(160)





+ fnmA+ |fnm|A = 0(161b)



+ fnmB+ |fnm|B = 0(161d)



+ fnmC+ |fnm|C = 0(161f)





B = TnB C = TmC (162)




middot[1+ ε2



)]= 0


middot[1+ ε2



)]= 0

(163b)


middot[1+ ε2



)]= 0

(163c)


middot[1+ ε2



)]= 0

(163d)


Lnm =


1 minusun+1m

)

+ (α1 minusα3)ε2

(F(+)n+mun+1m 0


) (164a)

Mnm =


1 minusunm+1

)


(F(+)n+munm+1 0


)

(164b)


Lnm+1 =


1 minusun+1m+1

)

+ (α1 minusα3)ε2


0 minusF(+)n+munm+1

)(164c)

Mn+1m =


1 minusun+1m+1

)



0 minusF(+)n+mun+1m

)

(164d)


τ =1+ ε2

(F(minus)n+munm + F

(+)n+mun+1m

)1+ ε2

(F(minus)n+munm + F

(+)n+munm+1

) (165)



G =

(0 1

minus1 0

) (166)


G =

(0 (minus1)n+m

minus(minus1)n+m 0

) (167)


fσ1σ2nm =



4 (168a)

|fσ1σ2nm =



4 (168b)

fσ1σ2nm =



4 (168c)

|fσ1σ2nm =



4 (168d)




F(plusmnσ)k =

1plusmn σ(minus1)k

2 (169)








(171)



(172)



dminus+ = 1 2 4 9 21 50 120 289 (173)



s3 + s2 minus 3s+ 1(174)



(175)






a1x1 + b1c1x1 + d1

a2x2 + b2c2x2 + d2

a12x12 + b12c12x12 + d12

)

(176)




The group Mob4



Q

Q M(Q)


4

M isin (Mob)4

11



mations Mob4








(177)






Φ

ax+ b


T

7minusrarrfnm

aunm + b

cunm + d+ |fnm




(178a)


Φminus1

fnmα(0)unm +β(0)


α(1)unm +β(1)

γ(1)unm + δ(1)



α(3)unm +β(3)

γ(3)unm + δ(3)

7minusrarr

α(0)x+β(0)

γ(0)x+ δ(0)

α(1)x1 +β(1)

γ(1)x1 + δ(1)

α(2)x2 +β(2)

γ(2)x2 + δ(2)

α(3)x12 +β(3)

γ(3)x12 + δ(3)

T

(178b)





Mnm (unm) =


)unm


(



(179)





cnmunm + dnm (180)


W1nm

(W0nm (unm)

)=

(a0nma1nm + b1nmc

0nm)unm + a1nmb

0nm + b1nmd

0nm

(a0nmc1nm + c0nmd

1nm)unm + b0nmc

1nm + d0nmd

1nm

(181)



Wminus1nm(unm) =

dnmunm minus bnm




nm isin Mob4

with parameters(a


M2nm

(M1nm (unm)

)=

(a(2)0 fnm + a

(2)1 |fnm + a

(2)2 fnm + a

(2)12 |fnm

)unm

+ b(2)0 fnm + b

(2)1 |fnm + b

(2)2 fnm + b

(2)12 |fnm

(c(2)0 fnm + c

(2)1 |fnm + c

(2)2 fnm + c

(2)12 |fnm

)unm

+ d(2)0 fnm + d

(2)1 |fnm + d

(2)2 fnm + d

(2)12 |fnm

(183)


a(2)0 = a

(0)0 a

(1)0 + b

(1)0 c

(0)0 a

(2)1 = a

(0)1 a

(1)1 + b

(1)1 c

(0)1

a(2)2 = a

(0)2 a

(1)2 + b

(1)2 c

(0)2 a

(2)12 = a

(0)12 a

(1)12 + b

(1)12 c

(0)12

b(2)0 = a

(1)0 b

(0)0 + b

(1)0 d

(0)0 b

(2)1 = a

(1)1 b

(0)1 + b

(1)1 d

(0)1

b(2)2 = a

(1)2 b

(0)2 + b

(1)2 d

(0)2 b

(2)12 = a

(1)12 b

(0)12 + b

(1)12 d

(0)12

c(2)0 = a

(0)0 c

(1)0 + c

(0)0 d

(1)0 c

(2)1 = a

(0)1 c

(1)1 + c

(0)1 d

(1)1

c(2)2 = a

(0)2 c

(1)2 + c

(0)2 d

(1)2 c

(2)12 = a

(0)12 c

(1)12 + c

(0)12 d

(1)12

d(0)0 = b

(0)0 c

(1)0 + d

(0)0 d

(1)0 d

(0)1 = b

(0)1 c

(1)1 + d

(0)1 d

(1)1

d(0)2 = b

(0)2 c

(1)2 + d

(0)2 d

(1)2 d

(0)12 = b

(0)12 c

(1)12 + d

(0)12 d

(1)12

(184)


Mminus1nm(unm) =





(185)






fnm fnm 0 0 0

|fnm 0 |fnm 0 0

fnm 0 0 fnm 0

|fnm 0 0 0 |fnm



Qgen (x x1 x2 x12) = A01212xx1x2x12

+B012xx1x2 +B0112xx1x12

+B0212xx2x12 +B1212x1x2x12

+C01xx1 +C02xx2 +C012xx12

+C12x1x2 +C112x1x12 +C212x2x12

+D0x+D1x1 +D2x2 +D12x12 +K

(186)




m(z) =az+ b



Ta(z) = z+ a (188a)

dilatation

Da(z) = az (188b)

and inversion

I(z) =1

z (188c)



ie


)(z) (189)




group Mob4


F(plusmn)k =

1plusmn (minus1)k

2 (190)




(minus)m unmun+1m

)minusα2 = 0

(191a)



+α2 (unm + un+1m + unm+1 + un+1m+1)

+εα22

(2F

(+)m unm+1 + 2α3 +α2

)(2F

(+)m un+1m+1 + 2α3 +α2

)+εα22

(2F


)(2F


)+ (α3 +α2)


(191b)



)δ2+


α3α2

(F(+)m unm+1un+1m+1 + F

(minus)m unmun+1m

)= 0

(191c)




D1 unm + un+1m + unm+1 + un+1m+1 = 0 (193a)


(+)n F

(minus)m + δ2F

(+)n F

(+)m

)unm


(minus)n F

(minus)m + δ2F

(minus)n F

(+)m

)un+1m+


(+)n F

(+)m + δ2F

(+)n F

(minus)m

)unm+1


(minus)n F

(+)m + δ2F

(minus)n F

(minus)m

)un+1m+1+

+ δ1


(+)m unm+1un+1m+1

)+ F



(193b)


(+)n F

(minus)m + δ2F

(+)n F

(+)m minus δ1λF

(minus)n F

(+)m

)unm


(minus)n F

(minus)m + δ2F

(minus)n F

(+)m minus δ1λF

(+)n F

(+)m

)un+1m

+(F(+)m minus δ1F

(+)n F

(+)m + δ2F

(+)n F


(minus)n F

(minus)m

)unm+1

+(F(+)m minus δ1F

(minus)n F

(+)m + δ2F

(minus)n F


(+)n F

(minus)m

)un+1m+1

+ δ1


(+)n+mun+1munm+1

)+ F

(+)m unmun+1m + F


(193c)



(minus)n F

(minus)m + δ2F

(+)n F

(+)m minus δ1λF

(minus)n F

(+)m

)unm


(+)n F

(minus)m + δ2F

(minus)n F

(+)m minus δ1λF

(+)n F

(+)m

)un+1m

+(F(+)m minus δ1F

(minus)n F

(+)m + δ2F

(+)n F


(minus)n F

(minus)m

)unm+1

+(F(+)m minus δ1F

(+)n F

(+)m + δ2F

(minus)n F


(+)n F

(minus)m

)un+1m+1

+ δ1


(+)n un+1mun+1m+1

)+ F



(193d)

D3 F(+)n F

(+)m unm + F

(minus)n F

(+)m un+1m + F

(+)n F

(minus)m unm+1

+ F(minus)n F


(minus)m unmun+1m




(+)n un+1mun+1m+1

+ F(+)m unm+1un+1m+1 = 0

(193e)

1D4 δ1


(+)n un+1mun+1m+1

)+

+ δ2


(+)m unm+1un+1m+1

)+

+ unmun+1m+1 + un+1munm+1 + δ3 = 0(193f)

2D4 δ1


(+)n un+1mun+1m+1

)+

+ δ2


(+)n+mun+1munm+1

)+

+ unmun+1m + unm+1un+1m+1 + δ3 = 0(193g)



fnm = F(+)n F

(+)m |fnm = F

(minus)n F

(+)m

fnm = F(+)n F

(minus)m |fnm = F

(minus)n F

(minus)m

(194)











41







(r3 + r2

)u20 + r

2u0 (22b)


(6r7 minus 2r6

)u60 + lot (22c)












u(p)n = fn


) (24)


partfn

partun+l

partfn






) n isin Z (27)








un+1 =aun + b

cun + d (29)


un =x

y un+1 =

X

y (210)


X = yax+ by

cx+ dy (211)




(Xy) =(yax+ by















(j)l prime+1(t) u

(j)lminus2(t)u

(j)lminus1(t)

jisinN0

(217)




d1d2d3d4d5 dk (218)








η 6 degϕ (221)


(ϕk)6









implying










yn+1 = βxn (225b)




1 1 2 4 8 16 32 64 128 256 512 (227)




minus04

minus02

0

02

04

x

y








1 1 2 3 5 8 13 21 34 55 89 144 233 377 (229)




dk sim

(1+radic5

2

)k(231)


η = log

(1+radic5

2

) (232)







(234)

ie



1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 (236)



(1)

(2) (3)(4)












∆(3)[11]

∆(2)[minus13] ∆

(4)[minus3minus2]

∆(3)[2minus1













∆minusminus

∆+minus ∆++

∆minus+



1 d(Nminus1)1 d

(Nminus2)2 d

(2)Nminus1 d

(1)N

1 1 d(Nminus2)1 d

(3)2 d

(1)Nminus1

1 1 d(3)1 d

(2)2

1 1 d(2)1 d

(1)2

1 1 d(1)1

1 1

(237)
























Σ

Γ

Π Π prime

Γ prime

Σ prime









ui isin IN (244)



R =α1t+β1α0t+β0

P(t)

Q(t) (245)




R = λP(t)

Q(t) or R =

1

λ

P(t)

Q(t) (247)





d0d1 dN (248)




f (s) =

infinsumk=0

dksk (249)





[L M] (s) =P[LM] (s)


L

1+ b1s+ bMsM (250)


f (s) = [L M] (s) + O(sL+M+1

) (251)




Q[LM] (s) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣





sM sMminus1 s 1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(252a)

P[LM] (z) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣





dksM+k

LminusM+1sumk=0

dksM+kminus1

Lsumk=0

dksk

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(252b)



[L M] (s) =P[LM] (s)

Q[LM] (s) (253)




f(τ) =

infinsumk=0

dkτminusk (254)


Zminus1 [f(τ)]k =1

2πi

intC



Zminus1 [f(τ)] =

Psumj=1


(256)



g(s) =

infinsumk=0

dksk (257)



]k

(258)


)is







1k (259)



log |s|minus1 (260)



[1 1](s) =1minus s

1minus 2s (261)


[1 1](s) = 1+ s+ 2s2 + 4s3 + 8s4 + 16s5 + 32s5 + O(s6)

(262)





[2 2] (s) =1



dk =5+ 3

radic5

10

(1+radic5

2

)k+5minus 3

radic5

10

(1minusradic5

2

)k (264)



radic5

2(265)



[3 3](s) =s3 + s2 minus s+ 1

(1minus s)2(266)




1 2 4 7 11 16 22 29 37 (267)









1D2 L9L10 L9L10 L9L10 L9L10

2D2 L14L15 L14L15 L15L14 L15L14


1D4 L13L8 L13L8 L13L8 L13L8

2D4 L18L8 L18L8 L19L20 L19L20



g =s2 minus s+ 1

(1minus s)3 (268)


dk =k(k+ 1)

2+ 1 (269)










g(s) =P(s)

(sminus 1)2(s+ 1)2 (270)


g(s) = P0(s) +P1(s)

(sminus 1)2(s+ 1)2 (271)


g(s) = P0(s) +1

4


(s+ 1)2+α+ γ+β+ δ

(sminus 1)2

+2α+βminus δ


s+ 1

]

(272)


g(s) = P0(s) +

infinsumk=0

[A0 +A1(minus1)


(273)








23

al

ge

br

aic

en

tr

op

yt

es

tf

or

(1

91

19

3)6

3



dk g(s) dk η

L0 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1minus s1 0

L1 1 2 2 5 3 8 4 11 5 14 6 17 7 s3 + 2s+ 1


4(minus2k+ 1) + k+

3

40

L2 1 2 4 3 7 4 10 5 13 6 16 7 19 minuss3 + 2s2 + 2s+ 1


4(2kminus 1) + k+

5

40

L3 1 2 2 3 3 4 4 5 5 6 6 7 7 minuss2 + s+ 1


4+k

2+5

40

L4 1 2 4 5 7 8 10 11 13 14 16 17 19 s2 + s+ 1


(minus1)k

4+3k

2+3

40

L5 1 2 4 6 11 10 19 14 27 18 35 22 43 s6 + 4s4 + 2s3 + 2s2 + 2s+ 1


(kminus

5

2

)+ 3kminus

5

20

L6 1 2 4 7 8 15 12 23 16 31 20 39 24 3s5 + s4 + 3s3 + 2s2 + 2s+ 1


(minusk+

5

2

)+ 3kminus

5

20

L7 1 2 4 7 11 15 19 23 27 31 35 39 43 s4 + s3 + s2 + 1



64

al

ge

br

aic

en

tr

op

ya

nd

dir

ec

tl

in

ea

riz

at

io

n



dk g(s) dk η

L8 1 2 4 6 8 10 12 14 16 18 20 22 24 s2 + 1

(sminus 1)22k 0

L9 1 2 2 5 3 8 4 11 5 14 6 17 7 s3 + 2s+ 1


4(minus2k+ 1) + k+

3

40

L10 1 2 3 5 5 8 7 11 9 14 11 17 13 s3 + s2 + 2s+ 1


4(minusk+ 1) +

5k

4+3

40

L11 1 2 4 5 10 8 16 11 22 14 28 17 34 3s4 + s3 + 2s2 + 2s+ 1


4(3kminus 5) +

9k

4minus3

40

L12 1 2 4 5 7 8 10 11 13 14 16 17 19 s2 + s+ 1

(sminus 1)2 (s+ 1)

(minus1)k

4+3k

2+3

40

L13 1 2 4 6 11 10 18 14 25 18 32 22 39 4s4 + 2s3 + 2s2 + 2s+ 1


4(kminus 2) +

11k

4minus3

20

L14 1 2 3 3 6 4 9 5 12 6 15 7 18 s4 minus s3 + s2 + 2s+ 1


4(2kminus 3) + k+

3

40

L15 1 1 3 3 6 5 9 7 12 9 15 11 18 s4 + s3 + s2 + s+ 1


4

((minus1)k + 5

)0


23

al

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or

(1

91

19

3)6

5



dk g(s) dk η

L16 1 2 3 2 5 2 7 2 9 2 11 2 13 minus2s3 + s2 + 2s+ 1


2(kminus 1) +

k

2+3

20

L17 1 1 3 3 5 5 7 7 9 9 11 11 13 s2 + 1

(sminus 1)2 (s+ 1)

(minus1)k

2+ k+

1

20

L18 1 2 4 5 11 9 19 13 27 17 35 21 43 s6 + s5 + 4s4 + s3 + 2s2 + 2s+ 1


L19 1 2 4 6 11 10 19 14 27 18 35 22 43 s6 + 4s4 + 2s3 + 2s2 + 2s+ 1


(kminus

5

2

)+ 3kminus

5

20

L20 1 1 3 2 6 3 9 4 12 5 15 6 18 s4 + s2 + s+ 1


4(2kminus 1) + k+

1

40








(274a)


)= 0 (274b)


)= 0 (274c)



)= 0 (274e)


)= 0 (274f)





(275)




)minusα1

(upn minus upn+1



(+)m

(276a)



(minus)m unmun+1m

)

(276b)

C =(upm minus up+1m


)minusα1


(minus)m upmup+1m

)

(276c)



Lnm =


1 minusunm

)

minus ε2α1

(minusF

(minus)m unm 0

0 F(+)m unm+1

)

(277a)


(1 0

0 1

)

+α2


1 minusunm

)


(0 1

0 0

)

(277b)


τnm = α21+ ε2

(F(minus)m u2nm + F

(+)m u2nm+1

)(unm minus unm+1)

2 + ε2α22F(+)m

(278)


γnm =1


(+)m unm+1

) (279a)

γ primenm =1


(279b)






ε1





(282a)




(282b)





fk = plusmn iε

(284)






zn+1k minus znk =1

κn (286)


znk =

jk +

nminus1suml=n0

1

κl n gt n0 + 1

jk minus

n0minus1suml=n

1

κl n 6 n0 minus 1

(287)



znk =iε

ynk minus 1

ynk + 1(288)


to




ynk =

jk

nminus1prodl=n0


n gt n0 + 1

jk

n0minus1prodl=n


n 6 n0 minus 1

(290)







1+wnm (292)


wnm+1 +wnm = 0 (293)


unm =

km +

radicα2

l=nminus1suml=n0


n gt n0 + 1

km minusradicα2

l=n0minus1suml=n


n 6 n0 minus 1

(294)













Θnk =1

i minus εznk



)Ξnk (298)



(1)nkΞnk (299)

where

L(1)nk = minusα1

(1 wnk+1 minuswnk

0 1

) (2100a)

M(1)nk =



(1 0

0 1

)

+α2



1 minuswnk

)



(0 1

0 0

)

(2100b)







(1 wnk

0 1

)(2102)


L(1)nk =

(1 0

0 1

) (2103a)

M(1)nk =

wnk minuswn+1k


(1 wnk minuswn+1k

α2α1 1

)

+α2



1α1 0

)

(2103b)







+ λ [x+ x3 minus δ1 (x2 + x23)] + δ1λ

(2104b)



(2104c)








(+)p F

(minus)n + δ2F

(+)p F

(+)n

)upn


(minus)p F

(minus)n + δ2F

(minus)p F

(+)n

)up+1n


(+)p F

(+)n + δ2F

(+)p F

(minus)n

)upn+1


(minus)p F

(+)n + δ2F

(minus)p F

(minus)n

)up+1n+1

+ δ1


(+)n upn+1up+1n+1

)+ F



(2105)

B = λ[



(minus)p F

(minus)n

(unm + unm+1)

+ λ[



(minus)p F

(+)n

(un+1m + un+1m+1)



(+)n un+1mun+1m+1


+ δ1λ2F

(+)p + 2 (δ1 minus 1) δ2λF

(minus)p

(2106)

C = λ[


]F(+)n + F

(minus)p F

(minus)n

(upm + upm+1

)+ λ[


]F(+)n + F

(+)p F

(minus)n

(up+1m + up+1m+1

)+(upm minus upm+1


)+ 2δ2λF


2F(+)n

(2107)




vst = u2s2t+1 zst = u2s+12t+1(2108)




+ (δ1vs+1t +ws+1t) zst = 0(2109b)


+ (δ1vst +wst+1) zst = 0(2109c)



+ (δ1vs+1t +ws+1t+1) zst = 0(2109d)


















(2112a)











(2112b)


If

δ1(1+ δ2) = 1 (2113)



(2114)









(2116)






vst =

js +sumtminus1k=t0

(hs + lk+1) ik t gt t0 + 1


(2118)








θst =1







θst =


lprodj=t0

pminus1j

tminus1prodk=t0

pk t gt t0 + 1

us

t0minus1prodk=t

pminus1k +

t0minus1suml=t

lprodj=t


(2123)













(2126b)






vst = hs + kt (2128)









ft primeprime+1



+ (minus1)t zs0



(2130)








znm (2131)

ie


α2 (2132)


znm+1 + znm = 2iπκ κ = 0 1 (2133)





79












ut = f(uu1 uk) (31)


form

uτ = g(uu1u2 um) (32)


part2u




D =part

partx+sumigt0

ui+1part

partui (34a)

Dt =part

partt+sumigt0

Difpart

partui (34b)

Dτ =part

partτ+sumigt0

Digpart

partui (34c)




ut = 6uu1 + u3 (35)


Lt = [LM] (36)





Ltj =[LMj

] (38)


L M]= V (39)





V = LV + ux (311)

where

LV = minus1

4Vxx + uV minus

1

2ux

intinfinx

V(y)dy (312)


ut = F (L)ux (313)







unminus1munm

un+1m













(315)


gpartQ

partunm+ [Tng]

partQ

partun+1m

+ [Tmg]partQ

partunm+1+ [TnTmg]

partQ

partun+1m+1= 0

(316)









part2TnTmg


part2g







(319)



(320)





dunm




Q(unun+1 un un+1) = 0 (323)








un+1m+1 = f (un+1munmunm+1) (325)


TnTmg = Tngpartf

partun+1m+ Tmg

partf

partunm+1+ g

partf

partunm(326)


TnTmpartg

partun+1mTn

partf

partun+1m=

partf

partun+1mTn

partg

partun+1m (327)




Define

z = logpartg

partun+1m(328)



)log

partf

partun+1m(329)



(partf

partunm

partf

partunm+1

)Tminus1n g

+ Tminus1n

(1

partf

partunm+1

)Tmg

minus Tminus1n

(partf

partun+1m

partf

partunm+1

)g

(330)


Tminus1n Tmpartg

partunminus1m

partunminus2m+1


(partf

partunm

partf

partunm+1

)Tminus1n

partg

partunminus1m

(331)


partunminus2m+1


partunminus1m+1


(partf

partunm

partf

partunm+1

) (332)

So introducing

v = logpartg

partunminus1m(333)



partf

partunm

partf

partunm+1

) (334)


Yminus1 =part

partunmminus

partfpartunm

partfpartunm+1

part

partun+1m

minus Tminus1n

(partfpartun+1m

partfpartunm

)part

partunminus1m

(335)




Y1 =part


partf

partunm+1

part

partun+1m


(partf

partunm+1

)minus1part

partunminus1m

(336)



)log

partf

partun+1m (337)



)log

partf

partun+1m(338)


Y1z = Y1Tminus1m

(Tminus1n minus Id

)log

partf

partun+1m (339)


Y1z = Y1Tminus1m

(Tminus1n minus Id

)log

partf

partun+1m (340a)


)log

partf

partun+1m (340b)

partz

partunm+1=

partz






)log

partf

partun+1m (341)



(partf

partunm

partf

partunm+1

) (342a)


partf

partunm

partf

partunm+1

) (342b)

partv

partunm+1=

partv







)log(

partf

partunm

partf

partunm+1

)

(343)


partg

partun+1m= ez

partg



partez

partunminus1m=

partev

partun+1m (345)


g =




S =part

partunmminus

partfpartunm

partfpartun+1m

part

partun+1m (347)







partz

partunm




(348)


in (348)

u2n+1mpartz


partz

partunm

+ u2nminus1mpartz


(349a)

un+1mpartz


partz


partz

partunminus1m+

partz

partun+1m= 0 (349c)



] (350)




] (351)


g =C1



unm+1dg(1)


dg(1)

dunm(unm)


dun+1m(un+1m)

+ un+1mdg(1)


(353)


du2nm =



g =C1






g =C1




dukdt

=1



symmetry reduction



+(un+1m + 1)(unm+1 minus 1) = 0(357)



u2nminus1m minus 1

2unm

partz

partunminus1m+

partz

partunm

+2un+1m

u2nm minus 1

partz


unm(u2nm minus 1)

(358a)

2unminus1m

u2nm minus 1

partz

partunminus1mminus

partz


2unm

partz

partun+1m


unm(1minus u2nm)

(358b)


partz



partz

partun+1m= minus

2(unm + 1)


partz

partunm=

1

unm+

1

unm + 1+

1

unm minus 1



(359c)





logminusC1nm+1

C1nm= 0 (361)







u2nminus1m minus 1

2unm

partv

partunminus1m+

partv


unm(u2nm minus 1)

(363a)

2unminus1m

u2nm minus 1

partv

partunminus1mminus

partv

partunm= minus







logminusC3nm+1

C3nm= 0 (365)








(366)


2unm(u2n+1m minus 1

) (u2nm+1 minus 1


(367)


d2g(1)

du2nmminus

1

unm

dg(1)

dunm+g(1)

u2nm=


) (C2n minusC4n+1


(368)


d2g(1)

du2nmminus

1

unm

dg(1)

dunm+g(1)

u2nm= 0 C2n = C4n+1 (369)



4n+1

)(minus1)m = 0


4n+2

)(minus1)m = 0

(370)






(372)




dunm







C1






(1)m +

1



U(0)m+1 +U

(0)m +

C1

2= 0 (378a)(

U(0)m+1 +U

(0)m

)2minus(U

(1)m+1 minusU

(1)m

)2+

(C1)2

2+C1U

(0)m +C1U


(378b)


U(0)m = K1(minus1)

m minus1

4(1+ (minus1)m)C1 (379)



U(1)m+1 minusU

(1)m = plusmn1

2

radicminus4α1 + 4α2 + (C1)2 (380)


U(1)m = K2 plusmn

m

2

radicminus4α1 + 4α2 + (C1)2 (381)



C1

4

)+K2


C1

4+C1

4n

(382)



F(+)n F

(+)m Q(++) + F

(+)n F

(minus)m Q(+minus)

+F(minus)n F

(+)m Q(++) + F

(minus)n F


(383)




z = F(+)n F

(+)m z(++) + F

(+)n F

(minus)m z(+minus)

+ F(minus)n F

(+)m z(++) + F

(minus)n F


(384a)

v = F(+)n F

(+)m v(++) + F

(+)n F

(minus)m v(+minus)

+ F(minus)n F

(+)m v(++) + F

(minus)n F


(384b)





g = F(+)n F

(+)m g(++) + F

(+)n F

(minus)m g(+minus)

+ F(minus)n F

(+)m g(++) + F

(minus)n F


(385)



+ F(+)n F

(+)m ϕ(++) + F

(+)n F

(minus)m ϕ(+minus)

+ F(minus)n F

(+)m ϕ(minus+) + F

(minus)n F


(386)










XtHε2n =

[(unm + εα22F





(+)m α32 + 8εF

(+)m α22α3 + ε

2F(+)m α42


]partunm

(387a)

XtHε2m =

[1


(+)m


(+)m unm+1unmminus1


minusεF

(minus)m u2nm minus


(minus)m

]unm +α3 + ε (α2 +α3)

2


partunm

(387b)

XtHε3n =



minusα22u


2F(+)m


]partunm

(387c)

XtHε3m =


(+)m unm+1unmminus1


minusε2F


2


]partunm

(387d)




XtHε1n1 =

F(+)m


middot


)Bn

(α2

un+1m minus unm

)


)Bnminus1

(α2

unm minus unminus1m

)]

+ F(minus)m


2unminus1m)unm




middot[Bn

(un+1m minus unm

1+ ε2un+1munm

)minusBn


1+ ε2unmunminus1m

)]partunm

(388a)

XtHε1n2 =

F(+)m

[unm minus



]

minusF(minus)m





+(4minus 4un+1mε


)]partunm

(388b)


XtHε1n = Xt

Hε1n1 +αXt

Hε1n2 (389)


XtHε1m =

[F(+)m Bm



)

+ F(minus)m

(1+ ε2u2nm


]partunm

(390)



YtHε1 =

[F(+)m κm + F

(minus)m λm

(1+ ε2u2nm

)]partunm (391)



332 H6 equations



X1D2n =

(F(+)n F

(minus)m minus δ1F

(+)n F

(+)m



+

(F(+)n F

(minus)m minus δ1F

(+)n F

(+)m minus δ1δ2

)F(minus)n F

(+)m unminus1m


+(F

(+)n+m minus δ1F

(+)m minus δ1δ2F

(+)n F

(+)m )unm + δ2F

(minus)m


]partunm

(392a)

X1D2m =

[δ1F

(minus)n F




+δ1F

(+)m unm+1 + δ1δ2F

(minus)n F


(minus)n F

(minus)m u2nm


+

[F(+)n+m + δ1

(F(+)n F

(minus)m minus F

(minus)n F

(minus)m

)+ δ1δ2F

(minus)n F

(minus)m

]unm



(minus)n


]partunm

(392b)

X2D2n =

(F(minus)n F

(minus)m δ1 + F

(minus)n F


(minus)n F

(minus)m

)un+1m


+

(F(+)n F

(+)m δ1 minus F

(+)n F

(minus)m

)unminus1m


+

(δ1F

(minus)n+m minus F

(minus)m + δ1δ2F

(+)n F

(minus)m


(+)m


partunm

(392c)


X2D2m

F(minus)n F


(δ1δ2F

(minus)n F

(minus)m + F

(+)n F

(minus)m

)unm+1


++(δ1F

(+)n F

(+)m + F

(minus)n F

(minus)m minus δ1F

(minus)n F

(minus)m

)unmminus1


+δ1F

(minus)n F

(+)m u2nm +

[F(+)n F


(minus)n F

(+)m + F

(+)m

]unm


+δ2(1minus δ2)F


(+)n


]partunm

(392d)

X3D2n =

(δ1F

(+)n F

(minus)m + δ1δ2F

(+)n F

(minus)m minus F

(+)n F

(minus)m

)un+1m


+

(F(+)n F

(+)m δ1 minus F

(minus)n F

(minus)m

)unminus1m


+

(δ1F

(minus)n F

(minus)m δ2 + F


(minus)m


(+)m


partunm

(392e)

X3D2m =


(+)n F

(minus)m unm+1


+

(F(minus)n F

(minus)m minus F

(+)n F

(+)m δ1

)unmminus1 + δ2F

(minus)n


+

(F(+)m minus δ1F

(+)n minus δ1δ2F

(+)n F

(+)m

)unm



(+)n


]partunm

(392f)


Y1D21 =(F(+)n F

(+)m + F

(+)n F

(minus)m + F

(minus)n F

(+)m

)unmpartunm (393a)

Y1D22 =[δ1F

(+)n F

(+)m + [1minus δ1(1+ δ2)]F

(minus)n F

(+)m + F

(+)n F

(minus)m

]partunm

(393b)

Y2D21 =[(F(+)n F

(+)m + F

(+)n F

(minus)m + F

(minus)n F

(+)m

)unmminus

minusλF(+)n F


(minus)n F

(minus)m

]partunm

(393c)

Y2D22 =[δ1F

(+)n F

(+)m + F

(+)n F


(minus)n F

(minus)m

]partunm

(393d)


Y3D21 =[(F(+)n F

(+)m + F

(+)n F

(minus)m + F

(minus)n F

(+)m

)unmminus

minusλF(minus)n F


(minus)n F

(minus)m

]partunm

(393e)

Y3D22 =[δ1F

(+)n F

(+)m + [1minus δ1(1+ δ2)]F

(+)n F

(minus)m minus F

(minus)n F

(minus)m

]partunm

(393f)


XD3n =

F(+)n F


1

2

(F(minus)m minus F

(minus)n F

(+)m



+F(minus)n F

(+)m u2nm +

(F(minus)m minus F

(+)n F

(+)m

)unm


partunm

(394a)

XD3m =

F(+)n F


1

2

(F(minus)n minus F

(+)n F

(minus)m



+F(+)n F

(minus)m u2nm +

(F(minus)n minus F

(+)n F

(+)m

)unm


partunm

(394b)


YD3 =[F(+)n

(2F

(+)m + F

(minus)m

)+ F

(minus)n

]unmpartunm (395)



X1D4n =


1



+minusδ1F


(+)m


]partunm

(396a)


X1D4m =


1



+δ2F


(+)n


]partunm

(396b)

X2D4n =

minusδ1δ2F(+)n F


1



+minusδ1δ2F

(minus)n F

(+)m u2nm + δ3


]partunm

(396c)

X2D4m =


1



+δ2F

(minus)n+mu

2nm minus δ1δ3F

(+)n


]partunm

(396d)




dqkdt




where

A(qk) = aq2k + 2bqk + c (398a)

B(qk) = bq2k + dqk + e (398b)

C(qk) = cq2k + 2eqk + f (398c)



dqkdt





Bk(qk) = bk+1q2k + dqk + ek+1 (3100b)

Ck(qk) = ck+1q2k + 2ekqk + f (3100c)








Bn (ξ) = minus1

ξ α = 0 (3101)


XtHε1n =




Bm(ξ) =1

ξ Cm(ξ) =

1

ξ (3103)


XtHε1m =

F(+)m


)+ F

(minus)m

(1+ ε2u2nm


partunm

(3104)




iD22 j = nm i = 1 2 3 (3105)



2

K1D22j = K2D22j = K3D22j = 0(3106)




10

6g

en

er

al

iz

ed

sy

mm

et

rie

sa

nd

th

en

ay

dk

ne

qu

at

io

n


Eq k a bk ck d ek f

rHε1



rHε2





rHε3

n 0 0 minusεF

(+)n+m

α

1

20 δα

m 0 0 minusεF

(+)n+m

β

1

20 δβ

tHε1


m 0 0 ε2F(+)m 0 0 2

tHε2


22

(4α2 + 8α3 + εα

22

)F(+)m


1



2


34

th

en

on

-a

ut

on

om

ou

sy

dk

ne

qu

at

io

n1

07


Eq k a bk ck d ek f

tHε3

n 0 0 minusα221

2α2(1+α

22) 0 minusε2δ2F

(+)m (1minusα22)

2

m 0 0 minusε2F(+)m

1

2α3 0 minusα23δ

2

1D2n 0 0 0 0

1

2[δ1(1+ δ2) minus 1]F

(+)n F

(+)m

+1

2F(minus)n F

(+)m minus

1

2F(minus)n F

(minus)m

minusδ2F(minus)m


(+)m δ1 0

1


(minus)n F

(minus)m

minus1

2F(+)n F

(+)m minus

1

2δ1F

(minus)n F

(+)m


2D2n 0 0 0 0

1


(+)n F

(minus)m

+1

2F(minus)n F

(minus)m minus

1

2δ1F

(minus)n F

(+)m

(δ1 minus 1)F(+)m


(minus)m 0

1


(minus)n F

(+)m

minus1

2F(+)n F

(+)m minus

1

2δ1F

(minus)n F

(+)m


(+)n


10

8g

en

er

al

iz

ed

sy

mm

et

rie

sa

nd

th

en

ay

dk

ne

qu

at

io

n


Eq k a bk ck d ek f

3D2n 0 0 0 0

1

2[δ1(1+ δ2) minus 1]F

(minus)n F

(minus)m

+1

2F(+)n F

(minus)m +

1

2δ1F

(minus)n F

(+)m

(1minus δ1)F(+)m

m 0 0 0 0

1


(+)n F

(+)m

minus1

2F(+)n F

(minus)m +

1

2δ1F

(minus)n F

(+)m


(minus)n

D3n 0 0 F

(+)n F

(+)m 0

1

2

(F(+)n F

(minus)m + F

(minus)n F

(minus)m minus F

(+)n F

(+)m

)0

m 0 0 F(+)n F

(+)m 0

1

2

(F(minus)n F

(+)m + F

(minus)n F

(minus)m minus F

(+)n F

(+)m

)0

1D4n 0 0 minusδ1

(F(+)n F

(+)m + F

(+)n F

(minus)m

)minus1

20 δ2δ3F

(+)m

m 0 0 δ2

(F(+)n F

(+)m + F

(minus)n F

(+)m

) 1

20 minusδ1δ3F

(+)n

2D4n 0 0 minusF

(+)n F

(+)m δ1δ2

1

20 δ3

m 0 0 δ2

(F(+)n F

(+)m + F

(minus)n F

(minus)m

) 1

20 minusδ1δ3F

(+)n












(2)n (3108)


Dτi =part

partτi+sumjisinZ

ϕ(i)n+j

part

partun+j (3109)






n and ϕ(2)n

[ϕ(1)n ϕ(2)


(2)n (3110)




[[ϕ(1)n ϕ(2)

n ]ϕ(3)n ] = [[ϕ

(1)n ϕ(3)

n ]ϕ(2)n ] + [ϕ

(1)n [ϕ(2)

n ϕ(3)n ]] (3111)


[ϕ(1)n ϕ(2)

n ] = minus(Dτ1ϕ

(2)n minusDτ2ϕ

(1)n

)= minus[ϕ

(2)n ϕ(1)

n ] (3112)







gn = [ϕn fn] (3115)


[[ϕn fn] fn] = 0 (3116)





ki rarr κi = κi (τ) i = 1 M (3118)




κi (0) = ki (3119b)






]qk


(3121a)


]q2k + d(τ)qk


(3121b)


]q2k


]qk + f(τ)

(3121c)


dqkdt


qk+1 minus qkminus1

(3122)


dqkdτ


qk+1 minus qkminus1

(3123)


dadτ



dbdτ


dβdτ


dcdτ


dγdτ


dddτ


dedτ


dεdτ


dfdτ




dcdτ

(τ) = minusd (τ) c (τ) (3125a)

dγdτ

(τ) = minusd (τ)γ (τ) (3125b)

dddτ

(τ) = minusc2 (τ) + γ2 (τ) (3125c)

dedτ


dεdτ

(τ) = γ (τ) e (τ) + c (τ) ε (τ) (3125e)

dfdτ



35

in

te

gr

ab

il

it

yp

ro

pe

rt

ie

s1

13


tHε1

nminus1

τ+ 10

1

τ+ 1

mε2

2

ε2

20

tHε2

nminus1

τ+ 10

1

τ+ 1

m minusε

2minusε

20

tHε3

nα22(α22 minus 1

)eminus

12τα2(α

22minus1)


01

2

α2(1minusα22

) [eminusτα2(α

22minus1) +α22

]+eminusτα2(α

22minus1) minusα22

m minusε2eminus

12α3τ

2minusε2eminus

12α3τ

2

α32


tHε1

n 0 0 minusε2α22F(+)m (τ+ 1)

m 0 0 2

tHε2

n εα22F

(+)m (τ+ 1) 0 minusα22

[minus1+α2

2F(+)m (τ+ 1)2 ε2 + 4F

(+)m (α2 + 2α3) ε

](τ+ 1)

m1

2+1


1


1


2 εminusα3 minus1

2τ

tHε3

n 0 0ε2δ2F

(+)m

(α22 minus 1

) (eminusτα2(α

22minus1) minusα2

2)


m 0 0 minusα23δ2e

12α3τ





1 1 3 7 13 21 31 43 57 73 91 111 133 157 (3126)



(1minus z)3(3127)


dl = l(lminus 1) + 1 (3128)



m = 2k 1 1 3 7 10 17 23 33 42 55 67 83 98 117 (3129a)

m = 2k+ 1 1 1 3 4 9 13 21 28 39 49 63 76 93 109 (3129b)


m = 2kg(z) =


(1minus z)3(z+ 1)

dl =3

4l2 minus lminus

5(minus1)l minus 21

8

(3130a)

m = 2k+ 1g(z) =

(z2 + z+ 1)(2z2 minus 2z+ 1)

(1minus z)3(z+ 1)

dl =3

4l2 minus

3

2lminus

5(minus1)l minus 19

8

(3130b)






dl =3

4l2 minus

5

4l+ (minus1)l

l

4+

(minus1)l + 15

8 (3131)









p1 = logpartfn

partun+1= 2 log



)(3133)

we must have

dp1dt







= (T minus 1)gn


(3134)



1 1 3 9 27 81 273 729 (3135)


g(s) =1minus 2s

1minus 3s (3136)














+a60 (unm + un+1m + unm+1 + un+1m+1)

+a7 = 0

(3137)


dunm

dt=

hn


2partun+1mhn (3138)

where


QV

minus(partunm+1

QV) (partun+1m+1

QV) (3139)


dunm

dt=

hm


2partunm+1

hm (3140)

where


minus(partun+1mQV

) (partun+1m+1

QV) (3141)



a = a30a1 minus a220 (3142a)

b =1


c = a20a60 minus a40a50 (3142c)

d =1

2[a230 minus a

240 minus a

250 + a1a7] (3142d)

e =1


f = a30a7 minus a260 (3142f)


a = a50a1 minus a220 (3143a)


b =1


c = a20a60 minus a40a30 (3143c)

d =1

2[a250 minus a

240 minus a

230 + a1a7] (3143d)

e =1


f = a50a7 minus a260 (3143f)








(3144)











(3145)




(3146)





p12 = p13 = p14 = p15 equiv a6 p16 = a7

(3147)




a6 = 1+ (minus1)n a7 = 4+ 2(minus1)n

(3148)









(3149)





]unmunm+1un+1m+1


]un+1munm+1un+1m+1


]unmun+1mun+1m+1


]unmun+1munm+1




]unmun+1m+1

+[a40 + (minus1)n+m a43

]un+1munm+1




]unm


]un+1m


]unm+1


]un+1m+1

+a7 = 0

(3150)










1 3 7 13 21 31 43 57 73 91 111 133 (3151)


g(z) =1+ z2

(1minus z)3 (3152)


dk = k(k+ 1) + 1 (3153)









221 minus a

222 + a

223


(3154a)


b =1


+1


minus1

2(minus1)ma22(a50 + a30 + a40)

+1

2(minus1)m(a23a51 + a1a62 + a20a32 + a21a43)

(3154b)

β =1

2a21(a30 minus a40 + a50)

+1


+1


minus1


(3154c)





d =1

2(a230 minus a

240 minus a

250 + a1a7 minus a

232 + a

243 + a

251)


(3154f)

e =1


+1

2(a20a7 + a51a61 minus a32a62 + a43a63)

+1


minus1

2(minus1)ma62(a30 + a40 + a50)

(3154g)

ε =1

2a61(a30 minus a40 + a50)

minus1

2(a51a60 minus a43a62 + a32a63 + a21a7)

+1


+1


(3154h)


262 + a

263 + a

261


(3154i)





Eq a30 a32 a40 a43 a50 a51 a60 a61 a62 a63 a7

rHε1 1 0 1



(β2 minusα2

)minus(αminusβ)

(2εα2 +α+ 2εβ2 +β

)rHε3 α 0 1

2

ε(β2minusα2

)αβ

12

ε(β2minusα2


(α2 minusβ2




2α2 [2+ ε(2α2 +α3)] 0 εα2α3 + 12 εα2

2 0 α2

[α2 + 2α3 + ε (α2 +α3)

2]

tHε3

12

ε2(1minusα2

2)

α3α2

12

ε2(1minusα2

2)

α3α2α2 0 minus1 0 0 0 0 0 δ2α3

(1minusα22

)1D2

12 δ1

12 δ1

12 minus 12 0 0 1



12 minus 1

4 (δ1 + δ2) 0

2D212 minus 12

12 δ1

12 δ1 0 0 1



12 minus 1


3D212 minus 12 0 0 1


4 (δ1 minus δ2 + δ1λ)14 (δ1 + δ1λ+ δ2)

12 minus 1


D312

12

12

12

12 minus 12

14


1D412 δ2

12 δ2 1 0 1

2 δ1 minus 12 δ1 0 0 0 0 δ3

2D4 1 0 12 δ2

12 δ2

12 δ1 minus 12 δ1 0 0 0 0 δ3








∣∣∣∣uxt=f

equiv 0


∣∣∣∣uxt=f

equiv 0(42)

125





uxt = eu (43)


X = uxx minus1

2u2x T = utt minus

1

2u2t (44)



unmun+1m+1

(1+

1

un+1m

)(1+

1

unm+1

)= 1 (45)






where














W1 = ξn W2 = λm (411)



W1 =

[1+

unm (1+ unminus1m)

unminus1m

] [1+

unm

un+1m (1+ unm)

]

(412a)

W1 =

[1+

unm (1+ unmminus1)

unmminus1

] [1+

unm

unm+1 (1+ unm)

]

(412b)






Yminus1 = Tmpart






Yminusk = Tkmpart






TmW1 =aW1 + b

cW1 + d (417)



Zminusk = Tknpart




which are such that



Wi = F(+)n F

(+)m W

(++)i + F

(minus)n F

(+)m W

(minus+)i

+ F(+)n F

(minus)m W

(+minus)i + F

(minus)n F

(minus)m W

(minusminus)i

(420)

with

F(plusmn)k =

1plusmn (minus1)k

2(421)





cube









W1 = (minus1)mun+1m

unm (423a)

W2 = (minus1)nunm+1

unm (423b)




1+ unmun+1m (424a)


1+ unmunm+1 (424b)





W1 = Fm

((unm + e2)(un+1m + o1)

(unm + e1)(un+1m + e2)

) (425)



In =(unm + e2)(un+1m + o1)

(unm + e1)(un+1m + e2)(426)

















4311 tHε1 equation


W1 = F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm



W2 = F(+)m α




(429b)


4312 tHε2 equation



W1 = F(+)m



minus F(minus)m



(430a)


W2 = F(+)m α



]middot[


]

+ F(minus)m β







(430b)



Wε=01 = (minus1)m



Wε=02 =






(432)



4α2 +

1

2α3 (433)


4313 tHε3 equation



W1 = F(+)m


3un+1munm

+(unm



+ F

(minus)m



(434a)



W2 = F(+)m α




δ2α23 + u2nm+1ε

2 minusα3unmunm+1

)

minus F(minus)m β

α3u





2α32unmminus2


(434b)




Wε=01 = (minus1)m



2

] (435a)

Wε=02 =



(435b)




(436)


b2 = minus1

α2 c4 =

δ2α3(1minusα22

)α2

(437)





432 H6 equations


4321 1D2 equation


W1 = F(+)n F

(+)m α



+ F(+)n F

(minus)m α


+ F(minus)n F


minus F(minus)n F

(minus)m β


δ2 + unm

(438a)

W2 = F(+)n F

(+)m α


unm + δ1unmminus1

+ F(+)n F


minus F(minus)n F

(+)m α



minus F(minus)n F

(minus)m β


δ2 + unm

(438b)


W(0δ2)1 = F

(+)n F

(+)m α


unm

minus F(+)n F


+ F(minus)n F


+ F(minus)n F

(minus)m β


δ2 + unm

(439a)

W(0δ2)2 = F

(+)n F

(+)m α


unm

+ F(+)n F


minus F(minus)n F


minus F(minus)n F

(minus)m β


δ2 + unm

(439b)






W((1+δ2)

minus1δ2)1 = F

(+)n F

(+)m α

un+1m

unminus1m

+ F(+)n F

(minus)m α

δ2 + un+1m

δ2 + unminus1m

+ F(minus)n F


minus F(minus)n F

(minus)m β


δ2 + 1

(440a)


(+)n F

(+)m α [(1+ δ2)unm + unm+1]

+ F(+)n F

(minus)m β

unm + (1+ δ2)unm+1

δ2 + 1

minus F(minus)n F

(+)m α

(δ2 + 1)unm

δ2 + unm+1

minus F(minus)n F

(minus)m β

unm+1

δ2 + unm

(440b)



minus1

4322 2D2 equation


W1 = F(+)n F

(+)m α

δ2 + un+1m

δ2 + unminus1m

+ F(+)n F

(minus)m α



+ F(minus)n F

(+)m β



minus F(minus)n F


(441a)

W2 = F(+)n F


minus F(+)n F

(minus)m β



minus F(minus)n F

(+)m α



minus F(minus)n F

(minus)m β


unm+1 + δ2

(441b)



W(0δ2)1 = F

(+)n F


minus F(+)n F


+ F(minus)n F


minus F(minus)n F


(442a)

W(0δ2)2 = F

(+)n F


+ F(+)n F

(minus)m β

unm+1

unmminus1

minus F(minus)n F


minus F(minus)n F

(minus)m β


unm+1 + δ2

(442b)






(+)n F

(+)m α

δ2 + un+1m

δ2 + unminus1m

+ F(+)n F

(minus)m α

un+1m

unminus1m

minus F(minus)n F


minus F(minus)n F

(minus)m β


1+ δ2

(443a)


(+)n F

(+)m α [(1+ δ2)unm + unm+1]

+ F(+)n F


+ F(minus)n F

(+)m α


unm + δ2

+ F(minus)n F

(minus)m β


unm+1 + δ2

(443b)





minus1

4323 3D2 equation


W1 = F(+)n F

(+)m α



+ F(+)n F

(minus)m α



+ F(minus)n F


minus F(minus)n F


(444a)

W2 = F(+)n F


minus F(+)n F

(minus)m β



+ F(minus)n F


+ F(minus)n F

(minus)m β



(444b)


W(0δ2)1 = F

(+)n F


minus F(+)n F


+ F(minus)n F


minus F(minus)n F


(445a)

W(0δ2)2 = F

(+)n F


+ F(+)n F

(minus)m β

unm+1

unmminus1

minus F(minus)n F


+ F(minus)n F

(minus)m β

δ2 + unmminus1

δ2 + unm+1

(445b)







(+)n F

(+)m α



+ F(+)n F

(minus)m α


(δ2 + 1)unm

minus F(minus)n F


minus F(minus)n F


(446a)


(+)n F


+ F(+)n F

(minus)m β


unm

minus F(minus)n F

(+)m α


δ2 + 1

+ F(minus)n F

(minus)m β



(446b)


minus1

4324 D3 equation


W1 = F(+)n F

(+)m α


u2n+1m minus unm

+ F(+)n F

(minus)m α


)unm + unminus1m

minus F(minus)n F

(+)m β


un+1m minus u2nm

+ F(minus)n F

(minus)m β


un+1m + un+2m

(447a)

W2 = F(+)n F

(+)m α


u2nm+1 minus unm

minus F(+)n F

(minus)m β


unm+1 minus u2nm

+ F(minus)n F

(+)m α


unm + unmminus1

+ F(minus)n F

(minus)m β


unm+1 + unm+2

(447b)



4325 1D4 equation


W1 = F(+)n F

(+)m α



+ F(+)n F

(minus)m α



+ F(minus)n F

(+)m β



+ F(minus)n F

(minus)m β



(448a)

W2 = F(+)n F

(+)m α



minus F(+)n F

(minus)m β



+ F(minus)n F

(+)m α



+ F(minus)n F

(minus)m β



(448b)




unm (449a)



unm (449b)





un+1m+1unm + un+1munm+1 + 1 = 0 (450)




W(000)1 = (minus1)m

un+1m

unm W

(000)2 = (minus1)n

unm+1

unm (451)


4326 2D4 equation


W1 = F(+)n F

(+)m α



](δ2u


)unminus1m

minus F(+)n F

(minus)m α


minus F(minus)n F

(+)m β



+ F(minus)n F

(minus)m β


un+1mun+2m + δ3(452a)

W2 = F(+)n F

(+)m α



minus F(+)n F

(minus)m β



+ F(minus)n F

(+)m α



+ F(minus)n F

(minus)m β



(452b)



(unmun+1m +

δ32

) W

(00δ3)2 =

(unm+1

unmminus1

)(minus1)n

(453)





unmun+1m + unm+1un+1m+1 + c4 = 0 (454)











u2k2l+1 = ykl u2k+12l+1 = zkl (455b)








unm = wn + υm (458)













4411 1D2 equation





F(+)n




(460)



δ1

minus(F(+)m ξn + F

(minus)m

)unminus12l minus F

(minus)m ξn = 0

(461)





where



vkl = αl + ak (464)








δ1

(1

bkminus

1

bkminus1

)(αl + ak) (466)


Wkl = βl +δ

δ1

αlbk

+ ck (467)



δ1

(1

bkminus

1

bkminus1

)ak (468)


ak = minusδ1δ








δ1αl (470b)



+ F(minus)m


δ2 + un2l+1= ξn

(471)



(1minus

1

δ1








zkl

bkminuszkminus1l

bkminus1=

(1minus

1

δ1

)(1

bkminus

1

bkminus1

) (473)


zkl = 1minus1

δ1+ bkγl (474)



(1

δ1+

δ

δ21bkγl



ykl = δl +δdk

δ21γlminusakδ1

(476)



bk (477)




(478)


zkl = 1minus1

δ1+ bkγl (479a)


minusckδ1γl

+1

δ


bk minus bkminus1

(479b)





δl =1

δ1


(481)







δ1αl (482b)

zkl = 1minus1

δ1minus bk


(482c)

ykl =1

δ1


+1

δ


bk minus bkminus1

+


+ckδ1


(482d)



δ1 =1

1+ δ2 (483)






wkl = akαl (485a)



δ2 + zkl

ak=δ2 + zkminus1m

akminus1 (486a)















wkl = akαl (490a)



minus δ2 (490c)

ykl = minus(1+ δ2)


αl+1 minusαl+ bk

) (490d)





un2l+ F







vkl = ak +αl (493)





(495)



+αl (496a)




(minus)n


δ2 + un2l+1= ξn

(497)













+ ck


(4102)



minus ck (4103)





minus ck

(4104b)








is


+αl (4107a)



minus bk (4107c)





minus ck

(4107d)


4412 2D2 equation





F(+)n α

δ2 + un+12l

δ2 + unminus12l

minus F(minus)n β



(4108)



and putting


δ1(4110)


F(+)nUn+12l

Unminus12lminus F

(minus)n


Un2l + δ= ξn (4111)





(1+

δ

Wkl

) (4112b)


Wkl = akαl (4113)



(1+

δ

akαl

) (4114)

If we define


(4115)





(4116a)


(4116b)



+1



+1






(4118)


δδ1ykl + zkl





(4120)


zkl

akminuszkminus1l

akminus1=

(1

akminus

1

akminus1












(4123b)



21λ+ δδ1 + δ1







+1



+1



21λ+ δδ1 + δ1

δ1






αl+1 minusαl

(4125c)


21λ+ δδ1 + δ1

δ21



(4125d)




F(+)n (δ2 + un+12l)un2l + F



(4126)


vkl(δ2 +wkl) = ξ2k (4127a)

vk+1l(δ2 +wkl) = ξ2k+1 (4127b)



(4128)



vkl = 0 (4129)

Defining

ξ2k+1 =ak+1ak

ξ2k (4130)


vkl = akαl (4131a)


(4131b)


F(+)n (un+12l+1 + un2l+1)+ F


(4132)




ξ2k (4133b)





yk+1l minus ykl =

(1minus

ak+1ak

)ξ2k (4135)

Defining


(4136)






vkl = akαl (4138a)


αl


(4138b)





4413 3D2 equation



F(+)n




(4139)







(δ2 +wkl) (4140b)


wkl = minusδ2 +1

Wkl (4141)

into


(ξ2k minus 1) (4142a)




(4143)



(4144)





δ1+

δ(bk+1 minus bk)


(4146a)







(4147)


zkl minus1

ξ2kzkminus1l =

1

δ

(1minus

1

ξ2k

)ykl (4148a)








zkl =bk minus γlδ


(δl minus

ckδ

) (4151)






γl =δ

δ1



) (4153a)

δl = minus1

δ


(4153b)





δ1+

δ(bk+1 minus bk)


zkl =bkδ

minus1

δ1



)+1

δ



+ ck

)

(4154c)

ykl = minusbk +δ

δ1



) (4154d)






F(+)n




(4155)


wkl minuswkminus1l








Wkl = ak +βl (4158)





(4160)









(4162)





)

(4163a)








)

(4165)




+ dk

)+ δl (4166)






) (4167)



minus ck+1 (4168)




[(γl + bk+1)


minus ck+1

]+ δl

(4169b)






(4171a)








minus bk (4172c)


[(bk+1 minus



minus ck+1

]+ (δ2 + 1)


+ λ

(4172d)





ξ2k (4173a)


(δ2 +wkl) (4173b)





(4175)

Defining



vkl = βl +bkαl

(4177a)



zkl minusakakminus1

zkminus1l =

(akakminus1

minus 1

)ykl (4178a)




(4179)


ykl = ck + γl (4180)





zkminus1l =



)(ck + γl)

(4181)







ckminus1

(4183)



ck (4184)


ykl = ck + γl (4185a)







βl =δ0αl

δl = δ0 (4187a)


vkl =bk + δ0αl

(4188a)


(4188b)


ykl = ck + γl (4188c)







4421 D3 equation








wkl + ykl+


wkl + ykl (4190)



wkl+1 + ykl+


wkl+1 + ykl (4191)





)middot


(4192)








2


(4194)





2




F(+)n



+ F(minus)n


un+12l+1 + un+22l+1= ξn

(4197)













Ykminus1l + ξ2k = 0 (4201)

Defining



Ykl = ak (bk +αl) (4203)



Setting


(4205)

we have




2




+ek




2








(4211)





4422 1D4 equation



yklwkl+1 + zklvkl+1


wklyk+1l + vk+1lzkl






δ3 +wklyklvkl+


δ3 +wklykl(4214)




δ3 + yklwkl+1vkl+1


2kl+1 minus δ2δ3

(yklwkl+1 + δ3

(4215)




]middot


(4216)



we obtain for wkl







2



ykl minus yklminus1

(4218)






2





F(+)n



+ F(minus)n


un2l+1 (un+22l+1δ1 + un+12l+1)= ξn

(4221)





zkl(zk+1lδ1 + yk+1l)= ξ2k+1 (4222b)






ak+1 (4224a)


bk (4224b)




ak =1

ck

1


1

ckminus1

1







(ck minus ck+1)2


+δ1δ3αlminus1


[1


minus1


]

(4228)



Vkl = γl +δ1δ3


[αlminus1


](4229)



2


(4230)



(4231)


γl =δlδ2






4423 2D4 equation












δ1wkl + yklvkl


2klδ1 minus δ3)

(δ1wkl + ykl)wkl

(4235)



δ1wkl+1 + yklvkl+1


2kl+1δ1 minus δ3)


(4236)




]middot(


)= 0

(4237)



we obtain

wkl =1

δ1









2klminus1




y2klminus1

(4239)



minus δ2ykl (4240)




2





+ F(minus)n


un+12l+1un+22l+1 + δ3= ξn

(4242)



= ξ2k (4243a)


(yk+1lzk+1l + δ3)= ξ2k+1 (4243b)





ak+1 (4245a)


bk (4245b)



If we define

ak =1

ck

1


1

ckminus1

1



(4246) is solved by




2


minusαlδ1δ3


[1

(αldk+1 +βl)c2k+1

minus1

(αldk +βl)c2k

]

(4249)

whose solution is



+δ3δ1ek


(4250)



2


(4251)



(4252)

ie

γl =δlδ2












(Tm minus Id)(F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm

1+ ε2unmun+1m

)= 0

(4256)


F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm

1+ ε2unmun+1m= λn (4257)




(4258)

which has solution

un2k = θ2k +ωn (4259)



ω0 = 0 (4260)





un2k+1 =iε

1minus vn2k+1

1+ vn2k+1 (4262)



If we define

λn =iε


(4264)


vn2k+1 = κnθ2k+1 (4265)


un2k+1 =iε


(4266)



(minus)m

iε


(4267)










+εα22

(2qnl + 2α3 +α2)(2qn+1l + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2


(4268a)



+εα22

(2qnl + 2α3 +α2)(2qn+1l + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2 minusα23


(4268b)




+εα22

(2qnl+1 + 2α3 +α2)(2qn+1l+1 + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2


(4269a)



+εα22

(2qnl + 2α3 +α2)(2qn+1l + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2 minusα23


(4269b)


pnl+1 =




]qn+1l+1

+ε (α2 + qnl+1) (α2 + qnl)qn+1l+1


]


(4270a)


pn+1l+1 =







]


(4270b)
















(nminus 1 l)

(nminus 1 l+ 1)

(n l)

(n l+ 1) (n+ 1 l+ 1)

(n+ 1 l)










qnl = an +αl (4274)


αl+1 minusαl = 0 (4275)




α2 + an minus an+1

+

α2ε

[α22 + (2α0 + an+1 + 2α3 + an)α2

+ 2 (an+1 +α0 +α3) (α3 + an +α0)


(4276)



=bn+1bn

(4277)



pn+1l

bn+1=pnl

bn+


bnbn+1

minus ε

[(α2 +α0 + an +α3)


bnbn+1

(4278)


pnl = bn (βl + cn) (4279)



bnbn+1

minus ε

[(α2 +α0 + an +α3)


bnbn+1

(4280)






(4281)






(4283)


brings (4282) into

Qn+1l +1

ξnQnl= 1 (4284)


ξn =1

an (1minus an+1) (4285)


Qnl = an +1

Znl(4286)


Zn+1l =anZnl + 1

1minus an+1 (4287)

Introducing

an =bnminus1

bn + bnminus1(4288)

we obtain



2n


(4289)



we obtain




as
















Rnl



= 0(4296)





en



(4298a)


fn



(4298b)












= 0 (4302)





= 0

(4303a)





(4304)



(1

cn+1 minus cn+

1

cn minus cnminus1

) (4305)



Tn+1l

cn+1 minus cnminus

Tnl



Pnl minus Pnminus1l



Pnl = (cn +αl)βl + γl (4308)



cn +αl (4309)





1

ε

) (4310)






cn +α0 (4311)








(δ2 + ε2

qnlqn+1l

α23α2

)= 0

(4312a)




(δ2 + ε2

qnlqn+1l

α23α2

)= 0

(4312b)








(δ2 + ε2

qnl+1qn+1l+1

α23α2

)= 0

(4313a)




(δ2 + ε2

qnlqn+1l

α23α2

)= 0

(4313b)


pnl+1 =


(ε2qnlqnl+1 + δ

2α32)




(4314a)

pn+1l+1 =






(4314b)













W1 =qn+1l

qnl (4317a)

W2 =qnl+1

qnl (4317b)


qnl = anαl (4318)




αl+1 minusαl = 0 (4320)




2anα20an+1


Defining


=bn+1bn

(4322)



pn+1l

bn+1=pnl

bn+δ2α23α

22bn minus bn+1

(δ2α23 + ε

2a2nα20

)α2 + ε

2a2nα20bn

bnanα0α2α3bn+1

(4323)


pnl = bn (βl + cn) (4324)


cn+1 = cn+δ2α23α

22bn minus bn+1

(δ2α23 + ε

2a2nα20

)α2 + ε

2a2nα20bn

bnanα0α2α3bn+1

(4325)





(434a) with k = 2l+ 1










Qn+1l +1

ξnQnl= 1 (4329)















Pnl =Rnl

Rnminus1l (4333)



1

Rnl= α2


(4334)



1

dn= α2


(4335)


zn+1zn


(4336)




Rnl = dn +1

Snl (4338)



cn+1 minus cnSnl =




Defining

dn =en

enminus1 (4340a)


(4340b)



2nminus1



Rnl =en(fn +βl)



Pnl

en(fn +βl)=

Pnminus1l


ie



qnl =γlen(fn +βl)

cn +αl (4345)




(4346)



αl+1 minusαl =ε2

α2δ2α23





qnl =γlen(fn +β0)

cn +αl (4348)







191

192 conclusions






conclusions 193








Mob4















7 for t1 in TT

8 for t2 in TT

9 for t3 in TT

10 for t4 in TT

11 TE = [t1t2t3t4]

12 TF = TF + [TE]

13 return TF

14



195



19 Ind = [01212]





27 else



30 return Mob4

31




36 f0 = (1+EM[0](-1)n+EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

37 f1 = (1-EM[0](-1)n+EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

38 f2 = (1+EM[0](-1)n-EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

39 f12 = (1-EM[0](-1)n-EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

40 F = [f0f1f2f12]

41 Mobnm =

42 Ind = [01212]

43 Mob0=0





51 else


53 Mob0 += F[i]MobT


55 Mobnm[u(0)]=Mob0


57 Mobnm[u(1)]=Mob1


59 Mobnm[u(2)]=Mob2


True)



62 return Mobnm

63


65 bQsub = u(0) u(1) u(1) u(0) u(2) u(12) u(12) u(2)

66 Qbsub = u(0) u(2) u(1) u(12) u(2) u(0) u(12) u(1)

67 bQbsub = u(0) u(12) u(1) u(2) u(2) u(1) u(12) u(0)

68

69 f0 = (1+EM[0](-1)n+EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

70 f1 = (1-EM[0](-1)n+EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

71 f2 = (1+EM[0](-1)n-EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

72 f12 = (1-EM[0](-1)n-EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

73




77

78 return Qt

79




83



86



89


91 if DQ == 0

92 return True

93 else

94 return False


Mob4





6 if s gt= 3600


8 si = int(si)

9 h = si3600


10 m = (si60) 60

11 ss = si 60



15 si = int(si)

16 m = (si60) 60

17 ss = si 60




24 def main()





30 Qconst = K

31 linear part

32 Qlin = D(0)u(0)+D(1)u(1)+D(2)u(2)+D(12)u(12)

33 quadratic part


35 +C(12)u(1)u(2)+C(112)u(1)u(12)+A(212)u(2)u(12)

36 cubic part

37 Qcub = B(012)u(0)u(1)u(2)+B(0112)u(0)u(1)u(12)+B

(1212)u(1)u(2)u(12)

38 quartic part

39 Qquart = A(01212)u(0)u(1)u(2)u(12)

40 the equation


42 print Q


44 R = [-11]

45 EM = []

46 for r in R

47 for s in R

48 EP = [rs]

49 EM += [EP]

50 for E in EM

51 print E


53 N=0


55 Nerr = []

56 dt =float(0)




59 for TT in MTot

60 N += 1

61 t0 = timetime()



64 print N test dt0

65 dt += dt0


67 if test == False

68 Nerr += [N]

69


71

72 if Nerr = []


74









ae2dpy








201






Nini = 4M (N+ 1) (B1)






Ff =fq+j1 fq+jkf

(B3)

Then we define

jm = mink=1kf

jk jM = mink=1kf

jk (B4)



(f jm jM) (B5)


Ns =

|Fs|sumk=1

(N+ 1+ jMiminus jmi

) (B6)




(B7)

Now we define

im = mink=1kfw=1wg

jk zw (B8a)

iM = maxk=1kfw=1wg

jk zw (B8b)


(g im iM) (B9)


Nd =

|Fd|sumk=1



)]

2

+

|Fd|sumk=1

[N+ 2 (iMkminus jmk

)]

(B10)












Y = [u(n m) v(n m)] (B12)
























b12 Code



2

















19


21 pass

22






28 if Anti

29 k1 = 0

30 k2 = 1

31 else

32 k1 = 1

33 k2 = 0

34 def f(ij)

35 if i == k1(j-n) + k2(dim - 1 - n -j)

36 if ngt0


37 return V[i]

38 else


40 else

41 return 0


43






49 NI = len(VV)




53 if DIR[0]DIR[1] == 1

54 adiag = True

55 else

56 adiag = False





60 return M

61



ImmutableMatrix


65



69 W = []

70 K = Mcols





75 W[i]reverse()

76 W[i] = W[i][K-1-i]



79 W[i] = W[i][i]

80 else


transpose()



84 W[i]reverse()


85 W[i] = W[i][i]



88 W[i] = W[i][K-1-i]

89 else


91

92 T = []





96 patterns = S[]



99 if S[i][K-j] == S[j]



102

103

104 return patterns

105



calculation of


constants


110 l = VARS[0]

111 m = VARS[1]

112 NN = 2NI

113 NEQ = len(y)

114 Fl = []

115 Fm = []

116 Flm = []



119 if CAF = []

120 CAFC = CAF[]

121 for C in CAF


123 continue






str(Cfunc))




for Ka in CLargs]))



str(Cfunc))


134 qv = []



137 qvappend(qs[q])



140 pv = []



143 pvappend(ps[p])


145 else


func)))





in CLargs]))


151 qpv = []






157 qpvm = min(qpv)

158 qpvM = max(qpv)

159 if qpvm == qpvM

160 if qpvm lt 0


162 else


164 else


166 else


func)))




171


















189 al = []

190 be = []




194 s1 = DIR[0]

195 s2 = DIR[1]

196 repl =



199 ll = int((2j - (-1)j + 1)4)

200 mm = int((2j + (-1)j - 1)4)



203

204 CAs =



207

208 replfunc =

209

210 for fl in Fl



213 for fm in Fm



216 for flm in Flm


218 ll = int((2i - (-1)i + 1)4)

219 mm = int((2i + (-1)i - 1)4)

220 if s1 == 1

221 l_s = s1(ll+flm[1])

222 else


223 l_s = s1(ll-flm[2])

224 if s2 == 1

225 m_s = s2(mm-(flm[2]-2flm[1]))

226 else

227 m_s = s2(mm-(2flm[2]-flm[1]))




231 if s1 == 1

232 l_s = s1(j+flm[1])

233 else

234 l_s = s1(j-flm[2])

235 if s2 == 1

236 m_s = s2(j+i+1-(flm[2]-2flm[1]))

237 else

238 m_s = s2(j+i+1-(2flm[2]-flm[1]))



241




244 NI = 8


246 EQ = list(EQ)


248 Y = list(Y)


250 EQ = [EQ]


252 Y = [Y]



255

256 NEQ = len(EQ)

257 if NEQ = len(Y)



260 else





265 if kwargs


267

268 y = []

269






)

274 l = Y[0]args[0]

275 m = Y[0]args[1]


277




str(Y[i]))



func))



)))


286


288

289 CA = set([])

290 CAF = set([])

291






297

298 for e in EQ







305 print(h)


307 if wis_Function

308 if wfunc in y



311 else


313 else



315 if l in CA

316 CAremove(l)

317 if m in CA

318 CAremove(m)

319 CA = list(CA)

320 CAF = list(CAF)

321


324 s2 = 1




(s1) str(s2)))




(s1) str(s2)))

332 ySol = ySol[0]




337 s2 = 1



(s1) str(s2)))




(s1)str(s2)))

344 ySol = ySol[0]




349 s2 = -1



(s1) str(s2)))





(s1) str(s2)))

355 ySol = ySol[0]




360 s2 = -1



(s1) str(s2)))




(s1) str(s2)))

366 ySol = ySol[0]





373 try





378 continue


380 if matrix_form


382 for E in ALL

383 for F in ALL

384 if E[1] = F[1]




388 else


391



394 CIm = dict()







400


402



405 step = (s1js2(j+i+1))






409 d = Max(d1d2)


411 print(ev)

412 print(d)


414 return True

415

416



419 outd = Queue()

420 out = Queue()








processes)))



processes)))




432 CIm = dict(CI)

433 CIclear()



[k][2]

436



][0] == k]



441 DKreverse()






447 break

448 if matrix_form

449 break

450 DS[k]append(DK)

451

452 if matrix_form

453 if NEQ == 1



range(NEQ)]

456

457 if NEQ == 1


459











y = [u(n) v(n)] (B13)


y = [u(n t) v(n t)] (B14)







Nini = 2

Msumk=1

(partek + 1) (B15)











2



6



9 NI = 8


11 f = list(f)


13 y = list(y)


15 f = [f]


17 y = [y]

18

19 NEQ = len(f)

20 if NEQ = len(y)

21 if NEQ gt len(y)


23 else


25






31 if kwargs


33




37 n = y[0]args[0]




40 n s = y[0]args

41 else


43 Y = [y[0]func]

44




str(y[i]))





52


54


56 pprint(f)

57

58 CA = set([])


60

61 for fs in f


63 coeff = SOne

64 kspec = None



67 for h in q


69 h = hargs[0]

70 if his_Function

71 if hfunc in Y






76 else



79 else



func))

82 else

83 coeff = h



85

86 if n in CA

87 CAremove(n)

88 if s in CA

89 CAremove(s)

90 if CA

91 CA = list(CA)

92




96


98









108 shuffle(rp)

109

110 rp0 = rp[02NEQ]

111 rp = rp[2NEQ]

112

113 CI =





118 CAs =



121


123 if CAs






129 ySol = ySol[0]


131

132


134













145 d = max(d1d2)

146 ds[y[k]] += [d]

147

148 return ds












p(z) = a0 + a1z+ aNzN (B17)



2




6

7



vector



is assumed




15 if M==0

16 return taylp




20 Pade approximant

21 else



24 if L == 0 and M ==0

25 return vect[0]

26

27 def c(i)


29 return 0

30 else

31 return vect[i]

32

33 if M == 0


35 def cfP(i)

36 if M-igtL

37 return 0

38 else

39 S = c(0)


41 S += c(k)zk






(1M+1)])






51




V A list of numbers










2





7 import warnings

8


10


approximant method



using





15



V))



str(s))

20 M = len(V)

21 N = floor((M-1)2)

22 j = 1

23 while(jltN+1)


25 ret = True


range(M)]

27 if Rv == V

28 return R

29 else

30 j = j + 1

31 return False




















2









11




str(f))










24 f = (rem)+f[1]


26 transf = 0

27 for z0 in sing




31





str(f))









43 return dn

44




str(f))





52 m = 0

53 for z0 in sing






227

22

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C1

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alH4

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(1

91)

22

9



direction minus+ 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17

1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44 46 47

0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16

0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44

0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15

0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41

0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14

0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38

0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13

0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35

0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12

0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32

0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C1)

23

0e

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direction ++

17 17 16 16 15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1

47 46 44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1

16 16 15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0

44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0

15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0

41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0

14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0

38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0

13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0

35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0

12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0

32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0

11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0

29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C2)

C1

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(1

91)

23

1

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0

12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0

12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0

13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0

13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0

14 37 13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0

14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0

15 40 14 37 13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0

15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0

16 43 15 40 14 37 13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0

16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0

17 46 16 43 15 40 14 37 13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1

17 47 16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1

(C3)

23

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(C10)

C1

tr

ap

ez

oid

alH4

eq

ua

tio

ns

(1

91)

23

9

direction +minus

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(C11)

24

0e

vo

lu

tio

nm

at

ric

es


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(C12)

C2H6

eq

ua

tio

ns

(1

93)

24

1



direction minus+ 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16 47 17

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(C13)

24

2e

vo

lu

tio

nm

at

ric

es

direction ++

17 47 16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1

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(C14)

C2H6

eq

ua

tio

ns

(1

93)

24

3

direction +minus

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17 47 16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1

(C15)

24

4e

vo

lu

tio

nm

at

ric

es


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1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16 47 17

(C16)

C2H6

eq

ua

tio

ns

(1

93)

24

5


direction minus+

1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27 42 29 45 31 48

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C17)

24

6e

vo

lu

tio

nm

at

ric

es

direction ++

48 31 45 29 42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1

17 45 16 42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1

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16 42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0

42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0

15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0

39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0

14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0

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30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0

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24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C18)

C2H6

eq

ua

tio

ns

(1

93)

24

7

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0 0

29 42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0

45 16 42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0

31 45 29 42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1

48 17 45 16 42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1

(C19)

24

8e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

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0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27

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0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27 42 29

0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39 15 42 16 45

1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27 42 29 45 31

1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39 15 42 16 45 17 48

(C20)

C2H6

eq

ua

tio

ns

(1

93)

24

9


direction minus+

1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27 27 29 29 31 31 33

1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27 2 29 2 31 2

0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27 27 29 29 31

0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27 2 29 2

0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27 27 29

0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27 2

0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27

0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2

0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11 11 13 13 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2 11 2 13 2

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9 9 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2 5 2 7 2 9 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 5 5 7 7 9

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C21)

25

0e

vo

lu

tio

nm

at

ric

es

direction ++

33 31 31 29 29 27 27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1

2 31 2 29 2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1

31 29 29 27 27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0

2 29 2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0

29 27 27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0

2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0

27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0

2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0

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21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0

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2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C22)

C2H6

eq

ua

tio

ns

(1

93)

25

1

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0 0 0 0 0

23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1 0 0 0 0 0 0 0 0

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27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0 0 0

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29 2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0 0 0

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31 2 29 2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1 1 0

31 31 29 29 27 27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1 1

33 2 31 2 29 2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1

(C23)

25

2e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

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(C26)

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(C27)

25

6e

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at

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(C28)

C2H6

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93)

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7


direction minus+

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(C29)

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direction ++

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(C30)

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direction +minus

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4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0

32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0

64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0 0 0

35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0

70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0 0 0

38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0

76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0 0 0

41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0

82 41 76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0 0 0

44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0

88 44 82 41 76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1 1 0

47 46 44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1

94 47 88 44 82 41 76 38 70 35 64 32 58 29 52 26 46 23 40 20 34 17 28 14 22 11 16 8 10 5 4 2 1

(C31)

26

0e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29

0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58

0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32

0 0 0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64

0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35

0 0 0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70

0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38

0 0 0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76

0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41

0 0 0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76 41 82

0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44

0 1 1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76 41 82 44 88

1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44 46 47

1 2 4 5 10 8 16 11 22 14 28 17 34 20 40 23 46 26 52 29 58 32 64 35 70 38 76 41 82 44 88 47 94

(C32)

C2H6

eq

ua

tio

ns

(1

93)

26

1


direction minus+

1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99 54 107 58 115 62 123

1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41 44 45 48 49 52 53 56 57 60 61

0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99 54 107 58 115

0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41 44 45 48 49 52 53 56 57

0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99 54 107

0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41 44 45 48 49 52 53

0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91 50 99

0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41 44 45 48 49

0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83 46 91

0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41 44 45

0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75 42 83

0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41

0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67 38 75

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59 34 67

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51 30 59

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25 28 29

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43 26 51

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21 24 25

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35 22 43

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17 20 21

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27 18 35

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13 16 17

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19 14 27

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9 12 13

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11 10 19

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 8 9

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C33)

26

2e

vo

lu

tio

nm

at

ric

es

direction ++

123 62 115 58 107 54 99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1

61 60 57 56 53 52 49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1

115 58 107 54 99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0

57 56 53 52 49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0

107 54 99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0

53 52 49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0

99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0

49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0

91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0

45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0

83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0

41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0

75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0

37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C34)

C2H6

eq

ua

tio

ns

(1

93)

26

3

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0

42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0

83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0

46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0

91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0

50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0

99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0

54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0

107 53 99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0

58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0

115 57 107 53 99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0

62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1

123 61 115 57 107 53 99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1

(C35)

26

4e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75

0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83

0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91

0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99

0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99 53 107

0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58

0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99 53 107 57 115

1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99 53 107 57 115 61 123

(C36)





un+1m + unmminus1


partz(++)

partun+1m+partz(++)

partunm



partz(++)



(D1a)

un+1m + unm+1

u2nm+1 minus unm

partz(++)


partunm

+unminus1m + unm+1

u2nm+1 minus unm

partz(++)



(D1b)


z(++) = log

[C(++)1



)] (D2)


z(minus+) = log

[C(minus+)1



)] (D3a)

265



(unm + unminus1m


)] (D3b)


(unm + unminus1m


)] (D3c)


C(+minus)1 = minusC

(++)1 C

(minusminus)1 = C

(minus+)1 (D4)


v(++) = log

[C(minus+)2

(unm minus u2n+1m


)] (D5a)

v(minus+) = log

[C(minus+)2

(u2nm minus un+1m


)] (D5b)


(un+1m + un+1m


)] (D5c)


(un+1m + unm


)] (D5d)


C(+minus)2 = minusC

(++)2 C

(minusminus)2 = C

(minus+)2 (D6)


C(++)2 = minusC

(++)1 C

(minus+)2 = minusC

(minus+)1 (D7)


g = F(+)n F

(+)m

[C(++)1



)+Φ(++)

]

+ F(minus)n F

(+)m

[C(minus+)1



]

+ F(+)n F

(minus)m

[C(++)1

unm + unminus1m


]+ F

(minus)n F

(minus)m

[C(minus+)1

unm + unminus1m


]

(D8)



1 = minusC(++)1 and


2K1unm minus

1

2C(++)1

Φ(+minus) =1

2K1unm +

1


1

2K1unm +

1

2C(++)1

(D9)


g =F(+)n F


1

2

(F(minus)m minus F

(minus)n F

(+)m



+F(minus)n F

(+)m u2nm +

(F(minus)m minus F

(+)n F

(+)m

)unm


(D10)


gP =[F(+)n

(2F

(+)m + F

(minus)m

)+ F

(minus)n

]unm (D11)



g =F(+)n F


1

2

(F(minus)n minus F

(+)n F

(minus)m



+F(+)n F

(minus)m u2nm +

(F(minus)n minus F

(+)n F

(+)m

)unm


(D12)



direction m






(minus)n unmunm+1)

(D13)


w = F(+)n w(+) + F



minusα2

(1+ ε2u2n+1m


2 + ε2α22

partz(+)

partun+1m+partz(+)

partunm

+α2

(1+ ε2u2nminus1m


2 + ε2α22

partz(+)

partunminus1m=

2ε2un+1m


(D15a)

minusα2

(1+ ε2u2n+1m

)(unm minus unm+1)

2 + ε2α22

partz(+)

partun+1m+partz(+)

partunm

+α2

(1+ ε2u2nminus1m

)(unm minus unm+1)

2 + ε2α22

partz(+)

partunminus1m=

2ε2un+1m


(D15b)



)+ logZ(+)

nm





)+ logZ(minus)





(plusmn)n ie


v(+) = log


1+ ε2u2n+1m

]

+ logV(+)nm



)

(D18a)



] (D18b)


nm = V(plusmn)n



Z(+)n (ξ) = C

(1)n minus

(1+ ξ2

)V(+)n (ξ) (D19a)

V(minus)n (ξ) = C


Then defining

G(+)n (ξ) =

intV(+)n (ξ)dξ G

(minus)n (ξ) =



g =minusF

(+)n

εGn



)+ F

(+)n

[C(1)n


]minus F



+ F(minus)n

[(1+ ε2u2nm)C


]

(D21)




C(1)n = C

(2)n = 0 Φ(+) = C

(3)n Φ(minus) = C

(4)n (1+ ε2u2nm) (D22)



g = F(+)n

[minus1

εG

(+)n



)+C

(3)n

]+ F

(minus)n

[minus(1+ ε2u2nm)G


(4)n

]


(D23)




gGS = F(+)n

[minus1

εG

(+)n



)]+ F

(minus)n

[minus(1+ ε2u2nm)G


] (D24)


gP = F(+)n C

(3)n + F


(4)n (D25)










)= 0

(E1a)




(minus)n+munm + 2F

(+)n+mun+1m +α+β

)middot

middot(2F


(+)n+munm+1 +α+β

)= 0

(E1b)




αβ



)= 0

(E1c)




XrHε1n =


(minus)n+mu

2nm


partunm (E2a)

271


XrHε1m =


(minus)n+mu

2nm


partunm

(E2b)

XrHε2n =

(1minus 4εαF

(minus)n+m





(minus)n+mu

2nm +

(1minus 4εαF

(minus)n+m

)unm


partunm

(E2c)

XrHε2m =

(1minus 4εβF

(minus)n+m





(minus)n+mu

2nm +

(1minus 4εβF

(minus)n+m

)unm

unm+1 + unmminus1

partunm

(E2d)

XrHε3n =

[1

2



minusε

α


(minus)n+mu

2nm


partunm

(E2e)

XrHε3m =

[1

2



minusε

β


(minus)n+mu

2nm

)unm+1 + unmminus1

partunm

(E2f)




dumdt


um+1 minus umminus1

(F1)










221 + a

222 + a

223


(F4a)

b =1


+1


+1

2(minus1)na21(a50 + a30 + a40)

+1

2(minus1)n(a23a32 + a1a61 + a20a51 + a22a43)

(F4b)

β =1


minus1


+1


+1


(F4c)



273




d =1

2(a250 minus a

240 minus a

230 + a1a7 + a

232 + a

243 minus a

251)

+ 2(minus1)n(a21a60 + a20a61 + a23a62 + a22a63)

(F4f)

e =1


+1

2(a20a7 minus a51a61 + a32a62 + a43a63)

minus1


minus1

2(minus1)na61(a30 + a40 + a50)

(F4g)

ε =1


+1

2(a32a60 minus a43a61 + a51a63 + a22a7)

+1


+1


(F4h)


262 + a

263 minus a

261


(F4i)




Wi = F(+)m W

(+)i + F

(minus)m W

(minus)i (G1)


partW(+)1

partun+1m+partW

(+)1

partunm= 0 (G2a)

(1+ ε2u2n+1m

) partW(minus)1


) partW(minus)1

partunm= 0 (G2b)

whose solution is


(minus)m G

(un+1m minus unm

1+ ε2unmun+1m

)(G3)


G (ξ) = F

(α2ξ

) (G4)

This yields

W1 = F(+)m F

(α2

un+1m minus unm

)+F

(minus)m F

(un+1m minus unm

1+ ε2unmun+1m

) (G5)


275



α2(1+ ε2u2nm+1

) partW(+)2

partunm+1


] partW(+)2

partunm


) partW(+)2

partunmminus1= 0

(G6a)

α2(1+ ε2u2n+1m

) partW(minus)2

partunm+1


(minus)2

partunm

+α2(1+ ε2u2n+1m

) partW(minus)2

partunmminus1= 0

(G6b)


W2 = F(+)m F



)+ F


(G7)



W1 = F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm


W2 = F(+)m α




(G8b)




ε1 E Q U AT I O N







un2kminus1 =1

vnk+αk ρ2k =


(H2)


)vnk+1 + ε

2αk+1 =(1+ ε2α2k

)vnk + ε

2αk (H3)



1+ ε2α2k (H4)



(H5)



Therefore we obtain


(minus)m


(H7)

277




(H8)















(1967) pp 292ndash296

279

280 Bibliography



















Bibliography 281



(10 pp)








(1993) pp 111ndash125








282 Bibliography



(1983) pp 1ndash22








(23pp)






Bibliography 283













284 Bibliography

















Bibliography 285






(1895) pp 422ndash443


(1979) pp 395ndash404









286 Bibliography


(2008) 077(14pp)













λ2a2





Bibliography 287















(1984) pp 131ndash160

288 Bibliography






(1962) pp 439ndash442






(2007) pp 253ndash269








Bibliography 289


















290 Bibliography
















Bibliography 291



colophon




Dedication

Abstract

Sommario

Acknowledgments

Ringraziamenti

Contents

List of Figures

List of Tables

Introduction


























332 H6 equations









4311 1 equation

4312 2 equation

4313 3 equation

432 H6 equations

4321 1D2 equation

4322 2D2 equation

4323 3D2 equation

4324 D3 equation

4325 1D4 equation

4326 2D4 equation



4411 1D2 equation

4412 2D2 equation

4413 3D2 equation


4421 D3 equation

4422 1D4 equation

4423 2D4 equation


4431 The 1 equation

4432 The 2 equation

4433 The 3 equation

Conclusions





B12 Code



























Bibliography

Colophon


A B S T R A C T



S O M M A R I O


v





vii



viii





ix


A tutti voi grazie

x

C O N T E N T S

introduction 1


cube 5






















equation 79





332 H6 equations 99








xi

xii contents

























conclusions 190





b12 Code 206













contents xiii

















bibliography 279











32










xiv








xv









1

2 introduction




introduction 3

Mob4

















HiH = 0 (12)

5



HiHj= 0 i 6= j = 1 N (13)


rankpart(H1 Hn)















xn+1m+1

xnm+1+ ynm =

xn+1m+1

xn+1m (17a)

xn+1m

xnm+ yn+1m =

xnm+1

xnm (17b)


Lnm =

(xn+1mxnm λ

λ 0

) (18a)

Mnm =

(xnm+1xnm λ

λ ynm

) (18b)


(xn+1m+1

xnm+1+ ynm minus

xn+1m+1

xn+1m

)minusλ

(xn+1m

xnm+ yn+1m minus

xnm+1

xnm

)0

=

(0 0

0 0

)

(19)







un+1 = 4un (1minus un) (110)


un = sin2(c02

nminus1)

(111)


dundc0

= 2n sin(c02

nminus1)

cos(c02

nminus1)

(112)








unm

un+1m

unm+1

un+1m+1

p

p

qq




unmp un+1mp

unm+1p

unmp+1

un+1m+1p

un+1mp+1

unm+1p+1 un+1m+1p+1

A

A

B B

C

C




)= 0

(114a)


)= 0

(114b)


)= 0

(114c)


)= 0

(114d)



)= 0

(114e)


)= 0

(114f)





unmp+1 =fnm

gnm un+1mp+1 =

fn+1m

gn+1m

unm+1p+1 =fnm+1

gnm+1

(115)



l21unmp+1 + l22 (116a)


m21unmp+1 +m22 (116b)


fn+1m



fnm+1





Φnm =

(fnm

gnm

) (118)


Lnm = γnm

(l11 l12

l21 l22

) Mnm = γ primenm

(m11 m12

m21 m22

) (119)





τnm =γnm+1γ

primenm

γnmγprimen+1m

(121)















(122)




(123a)





(123c)






(124a)









4αβ= 0

(124c)







+ k5(unm + un+1m + unm+1 + un+1m+1) + k6 = 0

(124d)

with


k3 =αβ(α+β)


g24)

k4 =αβ(α+β)


g24)

k5 =g32k0 +

g24k1 k6 =

g2216k0 + g3k1

(125)

where







J1 unm + e2unm + e1

un+1m+1 + o2un+1m+1 + o1

=un+1m + e2un+1m + o1


(127)



J2 unm + un+1m + unm+1 + un+1m+1 = 0 (128a)

J3 unmun+1m+1 + un+1munm+1 = 0 (128b)



+ unm + un+1m + unm+1 + un+1m+1 = 0

(128c)




Q (x x1 x2 x12α1α2) = 0 (129)




x

x1

x2

x12

α1

α1

α2α2



A (x x1 x2 x12α1α2) = 0 (130a)

A (x3 x13 x23 x123α1α2) = 0 (130b)

B (x x2 x3 x23α3α2) = 0 (130c)

B (x1 x12 x13 x123α3α2) = 0 (130d)

C (x x1 x3 x13α1α3) = 0 (130e)

C (x2 x12 x23 x123α1α3) = 0 (130f)


x x1

x2

x3

x12

x13

x23 x123

A

A

B B

C

C









+ ε(α2 minusα1)(2x1 +α1 +α2)(2x2 +α1 +α2)


22

(132b)


+ (α21 minusα22)

(δ+

εx1x2α1α2

) (132c)




hij =partQ

partxk

partQ

partxlminusQ

part2Q

partxk partxl (133)






2 Here x0 = x



Q(x x1 x2 x12α1α2) = Q(x1 x x12 x2α1α2) (134)











tHε1 (ε2 ε2 0 0

) Eq B of 3101



tHε2 (1 + 4εx 1 + 4εx2 1 1) Eq B of 3102


+εα22

(2x3 + 2α3 +α2) (2x23 + 2α3 +α2)

+ (α2 +α3)2 minusα23 +

εα322

= 0

(135b)


2ε2 x23 x223)

Eq B of 3103



)(δ2 +

ε2x3x23e4α3+2α2

)= 0

(135c)


D1 (0 0 0 0) Eq A of 3121 and 3131

x+ x1 + x2 + x12 = 0 (136a)


1D2 (δ21 (δ1δ2 + δ1 minus 1)

2 1 0)

Eq A of 3122


D3 (4x 1 1 1) Eq A of 3123

x+ x1x2 + x1x12 + x2x12 = 0 (136c)


1D4 (x2 + 4δ1δ2δ3 x21 x212 x22

) Eq A of 3124

xx12 + x1x2 + δ1x1x12 + δ2x2x12 + δ3 = 0 (136d)


2D2 (δ21 0 1 (δ1δ2 + δ1 minus 1)

2)

Eq C of 3132


3D2 (δ21 0 (δ1δ2 + δ1 minus 1)

2 1)

Eq C of 3133


(136f)


2D4 (x2 + 4δ1δ2δ3 x21 x22 x212

) Eq A of 3135

xx1 + δ2x1x2 + δ1x1x12 + x2x12 + δ3 = 0 (136g)






case


Q = Q(x x1 x2 x12α1α2) = 0 (137a)

|Q = Q(x1 x x12 x2α1α2) = 0 (137b)

Q = Q(x2 x12 x x1α1α2) = 0 (137c)

|Q = Q(x12 x2 x1 xα1α2) = 0 (137d)



Q = |Q = Q = |Q (138)


x x1 x

x2x12

x2

x x1 x

Q |Q

Q |Q




Q = |Q Q = |Q (139)


x x1 x

x2x12

x2

x x1 x

Q |Q

|Q Q



Q = |Q Q = |Q (140)



x x1 x

x2x12

x2

x x1 x

Q Q

Q Q



Q [u] =





(141)



rHε1 =






|n|+ |m| = 2k+ 1 k isin Z

(142)







xx1

x2

x3

x12

x13

x23x123

x

x

x2

xx1

x3

x3

x23

x3x13

A |A

A |A

A |A

A |A

B B B

B B B

C

C

C

|C

|C

|C



B(x x2 x3 x23) = B(x2 x x23 x3) (143a)

B(x1 x12 x13 x123) = B(x12 x1 x123 x13) (143b)

|C(x x1 x3 x13) = C(x1 x x13 x3) (143c)

|C(x2 x12 x23 x123) = C(x12 x2 x123 x23) (143d)




˜A [u] =





B [u] =





(144b)

˜B [u] =





C [u] =





˜C [u] =






A A A ˜A B B B ˜B C C C ˜C (145)












A =






|n|+ |m| = 2k+ 1 k isin Z

(147a)

A =






|n|+ |m| = 2k+ 1 k isin Z

(147b)

B =






|n|+ |m| = 2k+ 1 k isin Z

(147c)

B =






|n|+ |m| = 2k+ 1 k isin Z

(147d)

C =






|n|+ |m| = 2k+ 1 k isin Z

(147e)

C =






|n|+ |m| = 2k+ 1 k isin Z

(147f)











(148)







whose solution is


m + c3 (minus1)n+m (150)



f2k2k = 1 f2k+12k = f2k2k+1 = f2k+12k+1 = 0 (151)


which yield


4 (152a)



4 (152b)


4 (152c)


4 (152d)




Using (152)


(minus)n+m (154)

where

F(plusmn)k =

1plusmn (minus1)k

2 k isin Z (155)

we obtain

Q = F(+)n+mQ+ F

(minus)n+m|Q (156)





(minus)m (158)

we obtain

Q = F(+)m Q+ F

(minus)m Q (159)




mula (154) and get




)= 0

(160)





+ fnmA+ |fnm|A = 0(161b)



+ fnmB+ |fnm|B = 0(161d)



+ fnmC+ |fnm|C = 0(161f)





B = TnB C = TmC (162)




middot[1+ ε2



)]= 0


middot[1+ ε2



)]= 0

(163b)


middot[1+ ε2



)]= 0

(163c)


middot[1+ ε2



)]= 0

(163d)


Lnm =


1 minusun+1m

)

+ (α1 minusα3)ε2

(F(+)n+mun+1m 0


) (164a)

Mnm =


1 minusunm+1

)


(F(+)n+munm+1 0


)

(164b)


Lnm+1 =


1 minusun+1m+1

)

+ (α1 minusα3)ε2


0 minusF(+)n+munm+1

)(164c)

Mn+1m =


1 minusun+1m+1

)



0 minusF(+)n+mun+1m

)

(164d)


τ =1+ ε2

(F(minus)n+munm + F

(+)n+mun+1m

)1+ ε2

(F(minus)n+munm + F

(+)n+munm+1

) (165)



G =

(0 1

minus1 0

) (166)


G =

(0 (minus1)n+m

minus(minus1)n+m 0

) (167)


fσ1σ2nm =



4 (168a)

|fσ1σ2nm =



4 (168b)

fσ1σ2nm =



4 (168c)

|fσ1σ2nm =



4 (168d)




F(plusmnσ)k =

1plusmn σ(minus1)k

2 (169)








(171)



(172)



dminus+ = 1 2 4 9 21 50 120 289 (173)



s3 + s2 minus 3s+ 1(174)



(175)






a1x1 + b1c1x1 + d1

a2x2 + b2c2x2 + d2

a12x12 + b12c12x12 + d12

)

(176)




The group Mob4



Q

Q M(Q)


4

M isin (Mob)4

11



mations Mob4








(177)






Φ

ax+ b


T

7minusrarrfnm

aunm + b

cunm + d+ |fnm




(178a)


Φminus1

fnmα(0)unm +β(0)


α(1)unm +β(1)

γ(1)unm + δ(1)



α(3)unm +β(3)

γ(3)unm + δ(3)

7minusrarr

α(0)x+β(0)

γ(0)x+ δ(0)

α(1)x1 +β(1)

γ(1)x1 + δ(1)

α(2)x2 +β(2)

γ(2)x2 + δ(2)

α(3)x12 +β(3)

γ(3)x12 + δ(3)

T

(178b)





Mnm (unm) =


)unm


(



(179)





cnmunm + dnm (180)


W1nm

(W0nm (unm)

)=

(a0nma1nm + b1nmc

0nm)unm + a1nmb

0nm + b1nmd

0nm

(a0nmc1nm + c0nmd

1nm)unm + b0nmc

1nm + d0nmd

1nm

(181)



Wminus1nm(unm) =

dnmunm minus bnm




nm isin Mob4

with parameters(a


M2nm

(M1nm (unm)

)=

(a(2)0 fnm + a

(2)1 |fnm + a

(2)2 fnm + a

(2)12 |fnm

)unm

+ b(2)0 fnm + b

(2)1 |fnm + b

(2)2 fnm + b

(2)12 |fnm

(c(2)0 fnm + c

(2)1 |fnm + c

(2)2 fnm + c

(2)12 |fnm

)unm

+ d(2)0 fnm + d

(2)1 |fnm + d

(2)2 fnm + d

(2)12 |fnm

(183)


a(2)0 = a

(0)0 a

(1)0 + b

(1)0 c

(0)0 a

(2)1 = a

(0)1 a

(1)1 + b

(1)1 c

(0)1

a(2)2 = a

(0)2 a

(1)2 + b

(1)2 c

(0)2 a

(2)12 = a

(0)12 a

(1)12 + b

(1)12 c

(0)12

b(2)0 = a

(1)0 b

(0)0 + b

(1)0 d

(0)0 b

(2)1 = a

(1)1 b

(0)1 + b

(1)1 d

(0)1

b(2)2 = a

(1)2 b

(0)2 + b

(1)2 d

(0)2 b

(2)12 = a

(1)12 b

(0)12 + b

(1)12 d

(0)12

c(2)0 = a

(0)0 c

(1)0 + c

(0)0 d

(1)0 c

(2)1 = a

(0)1 c

(1)1 + c

(0)1 d

(1)1

c(2)2 = a

(0)2 c

(1)2 + c

(0)2 d

(1)2 c

(2)12 = a

(0)12 c

(1)12 + c

(0)12 d

(1)12

d(0)0 = b

(0)0 c

(1)0 + d

(0)0 d

(1)0 d

(0)1 = b

(0)1 c

(1)1 + d

(0)1 d

(1)1

d(0)2 = b

(0)2 c

(1)2 + d

(0)2 d

(1)2 d

(0)12 = b

(0)12 c

(1)12 + d

(0)12 d

(1)12

(184)


Mminus1nm(unm) =





(185)






fnm fnm 0 0 0

|fnm 0 |fnm 0 0

fnm 0 0 fnm 0

|fnm 0 0 0 |fnm



Qgen (x x1 x2 x12) = A01212xx1x2x12

+B012xx1x2 +B0112xx1x12

+B0212xx2x12 +B1212x1x2x12

+C01xx1 +C02xx2 +C012xx12

+C12x1x2 +C112x1x12 +C212x2x12

+D0x+D1x1 +D2x2 +D12x12 +K

(186)




m(z) =az+ b



Ta(z) = z+ a (188a)

dilatation

Da(z) = az (188b)

and inversion

I(z) =1

z (188c)



ie


)(z) (189)




group Mob4


F(plusmn)k =

1plusmn (minus1)k

2 (190)




(minus)m unmun+1m

)minusα2 = 0

(191a)



+α2 (unm + un+1m + unm+1 + un+1m+1)

+εα22

(2F

(+)m unm+1 + 2α3 +α2

)(2F

(+)m un+1m+1 + 2α3 +α2

)+εα22

(2F


)(2F


)+ (α3 +α2)


(191b)



)δ2+


α3α2

(F(+)m unm+1un+1m+1 + F

(minus)m unmun+1m

)= 0

(191c)




D1 unm + un+1m + unm+1 + un+1m+1 = 0 (193a)


(+)n F

(minus)m + δ2F

(+)n F

(+)m

)unm


(minus)n F

(minus)m + δ2F

(minus)n F

(+)m

)un+1m+


(+)n F

(+)m + δ2F

(+)n F

(minus)m

)unm+1


(minus)n F

(+)m + δ2F

(minus)n F

(minus)m

)un+1m+1+

+ δ1


(+)m unm+1un+1m+1

)+ F



(193b)


(+)n F

(minus)m + δ2F

(+)n F

(+)m minus δ1λF

(minus)n F

(+)m

)unm


(minus)n F

(minus)m + δ2F

(minus)n F

(+)m minus δ1λF

(+)n F

(+)m

)un+1m

+(F(+)m minus δ1F

(+)n F

(+)m + δ2F

(+)n F


(minus)n F

(minus)m

)unm+1

+(F(+)m minus δ1F

(minus)n F

(+)m + δ2F

(minus)n F


(+)n F

(minus)m

)un+1m+1

+ δ1


(+)n+mun+1munm+1

)+ F

(+)m unmun+1m + F


(193c)



(minus)n F

(minus)m + δ2F

(+)n F

(+)m minus δ1λF

(minus)n F

(+)m

)unm


(+)n F

(minus)m + δ2F

(minus)n F

(+)m minus δ1λF

(+)n F

(+)m

)un+1m

+(F(+)m minus δ1F

(minus)n F

(+)m + δ2F

(+)n F


(minus)n F

(minus)m

)unm+1

+(F(+)m minus δ1F

(+)n F

(+)m + δ2F

(minus)n F


(+)n F

(minus)m

)un+1m+1

+ δ1


(+)n un+1mun+1m+1

)+ F



(193d)

D3 F(+)n F

(+)m unm + F

(minus)n F

(+)m un+1m + F

(+)n F

(minus)m unm+1

+ F(minus)n F


(minus)m unmun+1m




(+)n un+1mun+1m+1

+ F(+)m unm+1un+1m+1 = 0

(193e)

1D4 δ1


(+)n un+1mun+1m+1

)+

+ δ2


(+)m unm+1un+1m+1

)+

+ unmun+1m+1 + un+1munm+1 + δ3 = 0(193f)

2D4 δ1


(+)n un+1mun+1m+1

)+

+ δ2


(+)n+mun+1munm+1

)+

+ unmun+1m + unm+1un+1m+1 + δ3 = 0(193g)



fnm = F(+)n F

(+)m |fnm = F

(minus)n F

(+)m

fnm = F(+)n F

(minus)m |fnm = F

(minus)n F

(minus)m

(194)











41







(r3 + r2

)u20 + r

2u0 (22b)


(6r7 minus 2r6

)u60 + lot (22c)












u(p)n = fn


) (24)


partfn

partun+l

partfn






) n isin Z (27)








un+1 =aun + b

cun + d (29)


un =x

y un+1 =

X

y (210)


X = yax+ by

cx+ dy (211)




(Xy) =(yax+ by















(j)l prime+1(t) u

(j)lminus2(t)u

(j)lminus1(t)

jisinN0

(217)




d1d2d3d4d5 dk (218)








η 6 degϕ (221)


(ϕk)6









implying










yn+1 = βxn (225b)




1 1 2 4 8 16 32 64 128 256 512 (227)




minus04

minus02

0

02

04

x

y








1 1 2 3 5 8 13 21 34 55 89 144 233 377 (229)




dk sim

(1+radic5

2

)k(231)


η = log

(1+radic5

2

) (232)







(234)

ie



1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 (236)



(1)

(2) (3)(4)












∆(3)[11]

∆(2)[minus13] ∆

(4)[minus3minus2]

∆(3)[2minus1













∆minusminus

∆+minus ∆++

∆minus+



1 d(Nminus1)1 d

(Nminus2)2 d

(2)Nminus1 d

(1)N

1 1 d(Nminus2)1 d

(3)2 d

(1)Nminus1

1 1 d(3)1 d

(2)2

1 1 d(2)1 d

(1)2

1 1 d(1)1

1 1

(237)
























Σ

Γ

Π Π prime

Γ prime

Σ prime









ui isin IN (244)



R =α1t+β1α0t+β0

P(t)

Q(t) (245)




R = λP(t)

Q(t) or R =

1

λ

P(t)

Q(t) (247)





d0d1 dN (248)




f (s) =

infinsumk=0

dksk (249)





[L M] (s) =P[LM] (s)


L

1+ b1s+ bMsM (250)


f (s) = [L M] (s) + O(sL+M+1

) (251)




Q[LM] (s) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣





sM sMminus1 s 1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(252a)

P[LM] (z) =

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣





dksM+k

LminusM+1sumk=0

dksM+kminus1

Lsumk=0

dksk

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(252b)



[L M] (s) =P[LM] (s)

Q[LM] (s) (253)




f(τ) =

infinsumk=0

dkτminusk (254)


Zminus1 [f(τ)]k =1

2πi

intC



Zminus1 [f(τ)] =

Psumj=1


(256)



g(s) =

infinsumk=0

dksk (257)



]k

(258)


)is







1k (259)



log |s|minus1 (260)



[1 1](s) =1minus s

1minus 2s (261)


[1 1](s) = 1+ s+ 2s2 + 4s3 + 8s4 + 16s5 + 32s5 + O(s6)

(262)





[2 2] (s) =1



dk =5+ 3

radic5

10

(1+radic5

2

)k+5minus 3

radic5

10

(1minusradic5

2

)k (264)



radic5

2(265)



[3 3](s) =s3 + s2 minus s+ 1

(1minus s)2(266)




1 2 4 7 11 16 22 29 37 (267)









1D2 L9L10 L9L10 L9L10 L9L10

2D2 L14L15 L14L15 L15L14 L15L14


1D4 L13L8 L13L8 L13L8 L13L8

2D4 L18L8 L18L8 L19L20 L19L20



g =s2 minus s+ 1

(1minus s)3 (268)


dk =k(k+ 1)

2+ 1 (269)










g(s) =P(s)

(sminus 1)2(s+ 1)2 (270)


g(s) = P0(s) +P1(s)

(sminus 1)2(s+ 1)2 (271)


g(s) = P0(s) +1

4


(s+ 1)2+α+ γ+β+ δ

(sminus 1)2

+2α+βminus δ


s+ 1

]

(272)


g(s) = P0(s) +

infinsumk=0

[A0 +A1(minus1)


(273)








23

al

ge

br

aic

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tr

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yt

es

tf

or

(1

91

19

3)6

3



dk g(s) dk η

L0 1 1 1 1 1 1 1 1 1 1 1 1 1 1

1minus s1 0

L1 1 2 2 5 3 8 4 11 5 14 6 17 7 s3 + 2s+ 1


4(minus2k+ 1) + k+

3

40

L2 1 2 4 3 7 4 10 5 13 6 16 7 19 minuss3 + 2s2 + 2s+ 1


4(2kminus 1) + k+

5

40

L3 1 2 2 3 3 4 4 5 5 6 6 7 7 minuss2 + s+ 1


4+k

2+5

40

L4 1 2 4 5 7 8 10 11 13 14 16 17 19 s2 + s+ 1


(minus1)k

4+3k

2+3

40

L5 1 2 4 6 11 10 19 14 27 18 35 22 43 s6 + 4s4 + 2s3 + 2s2 + 2s+ 1


(kminus

5

2

)+ 3kminus

5

20

L6 1 2 4 7 8 15 12 23 16 31 20 39 24 3s5 + s4 + 3s3 + 2s2 + 2s+ 1


(minusk+

5

2

)+ 3kminus

5

20

L7 1 2 4 7 11 15 19 23 27 31 35 39 43 s4 + s3 + s2 + 1



64

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ge

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tr

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nd

dir

ec

tl

in

ea

riz

at

io

n



dk g(s) dk η

L8 1 2 4 6 8 10 12 14 16 18 20 22 24 s2 + 1

(sminus 1)22k 0

L9 1 2 2 5 3 8 4 11 5 14 6 17 7 s3 + 2s+ 1


4(minus2k+ 1) + k+

3

40

L10 1 2 3 5 5 8 7 11 9 14 11 17 13 s3 + s2 + 2s+ 1


4(minusk+ 1) +

5k

4+3

40

L11 1 2 4 5 10 8 16 11 22 14 28 17 34 3s4 + s3 + 2s2 + 2s+ 1


4(3kminus 5) +

9k

4minus3

40

L12 1 2 4 5 7 8 10 11 13 14 16 17 19 s2 + s+ 1

(sminus 1)2 (s+ 1)

(minus1)k

4+3k

2+3

40

L13 1 2 4 6 11 10 18 14 25 18 32 22 39 4s4 + 2s3 + 2s2 + 2s+ 1


4(kminus 2) +

11k

4minus3

20

L14 1 2 3 3 6 4 9 5 12 6 15 7 18 s4 minus s3 + s2 + 2s+ 1


4(2kminus 3) + k+

3

40

L15 1 1 3 3 6 5 9 7 12 9 15 11 18 s4 + s3 + s2 + s+ 1


4

((minus1)k + 5

)0


23

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or

(1

91

19

3)6

5



dk g(s) dk η

L16 1 2 3 2 5 2 7 2 9 2 11 2 13 minus2s3 + s2 + 2s+ 1


2(kminus 1) +

k

2+3

20

L17 1 1 3 3 5 5 7 7 9 9 11 11 13 s2 + 1

(sminus 1)2 (s+ 1)

(minus1)k

2+ k+

1

20

L18 1 2 4 5 11 9 19 13 27 17 35 21 43 s6 + s5 + 4s4 + s3 + 2s2 + 2s+ 1


L19 1 2 4 6 11 10 19 14 27 18 35 22 43 s6 + 4s4 + 2s3 + 2s2 + 2s+ 1


(kminus

5

2

)+ 3kminus

5

20

L20 1 1 3 2 6 3 9 4 12 5 15 6 18 s4 + s2 + s+ 1


4(2kminus 1) + k+

1

40








(274a)


)= 0 (274b)


)= 0 (274c)



)= 0 (274e)


)= 0 (274f)





(275)




)minusα1

(upn minus upn+1



(+)m

(276a)



(minus)m unmun+1m

)

(276b)

C =(upm minus up+1m


)minusα1


(minus)m upmup+1m

)

(276c)



Lnm =


1 minusunm

)

minus ε2α1

(minusF

(minus)m unm 0

0 F(+)m unm+1

)

(277a)


(1 0

0 1

)

+α2


1 minusunm

)


(0 1

0 0

)

(277b)


τnm = α21+ ε2

(F(minus)m u2nm + F

(+)m u2nm+1

)(unm minus unm+1)

2 + ε2α22F(+)m

(278)


γnm =1


(+)m unm+1

) (279a)

γ primenm =1


(279b)






ε1





(282a)




(282b)





fk = plusmn iε

(284)






zn+1k minus znk =1

κn (286)


znk =

jk +

nminus1suml=n0

1

κl n gt n0 + 1

jk minus

n0minus1suml=n

1

κl n 6 n0 minus 1

(287)



znk =iε

ynk minus 1

ynk + 1(288)


to




ynk =

jk

nminus1prodl=n0


n gt n0 + 1

jk

n0minus1prodl=n


n 6 n0 minus 1

(290)







1+wnm (292)


wnm+1 +wnm = 0 (293)


unm =

km +

radicα2

l=nminus1suml=n0


n gt n0 + 1

km minusradicα2

l=n0minus1suml=n


n 6 n0 minus 1

(294)













Θnk =1

i minus εznk



)Ξnk (298)



(1)nkΞnk (299)

where

L(1)nk = minusα1

(1 wnk+1 minuswnk

0 1

) (2100a)

M(1)nk =



(1 0

0 1

)

+α2



1 minuswnk

)



(0 1

0 0

)

(2100b)







(1 wnk

0 1

)(2102)


L(1)nk =

(1 0

0 1

) (2103a)

M(1)nk =

wnk minuswn+1k


(1 wnk minuswn+1k

α2α1 1

)

+α2



1α1 0

)

(2103b)







+ λ [x+ x3 minus δ1 (x2 + x23)] + δ1λ

(2104b)



(2104c)








(+)p F

(minus)n + δ2F

(+)p F

(+)n

)upn


(minus)p F

(minus)n + δ2F

(minus)p F

(+)n

)up+1n


(+)p F

(+)n + δ2F

(+)p F

(minus)n

)upn+1


(minus)p F

(+)n + δ2F

(minus)p F

(minus)n

)up+1n+1

+ δ1


(+)n upn+1up+1n+1

)+ F



(2105)

B = λ[



(minus)p F

(minus)n

(unm + unm+1)

+ λ[



(minus)p F

(+)n

(un+1m + un+1m+1)



(+)n un+1mun+1m+1


+ δ1λ2F

(+)p + 2 (δ1 minus 1) δ2λF

(minus)p

(2106)

C = λ[


]F(+)n + F

(minus)p F

(minus)n

(upm + upm+1

)+ λ[


]F(+)n + F

(+)p F

(minus)n

(up+1m + up+1m+1

)+(upm minus upm+1


)+ 2δ2λF


2F(+)n

(2107)




vst = u2s2t+1 zst = u2s+12t+1(2108)




+ (δ1vs+1t +ws+1t) zst = 0(2109b)


+ (δ1vst +wst+1) zst = 0(2109c)



+ (δ1vs+1t +ws+1t+1) zst = 0(2109d)


















(2112a)











(2112b)


If

δ1(1+ δ2) = 1 (2113)



(2114)









(2116)






vst =

js +sumtminus1k=t0

(hs + lk+1) ik t gt t0 + 1


(2118)








θst =1







θst =


lprodj=t0

pminus1j

tminus1prodk=t0

pk t gt t0 + 1

us

t0minus1prodk=t

pminus1k +

t0minus1suml=t

lprodj=t


(2123)













(2126b)






vst = hs + kt (2128)









ft primeprime+1



+ (minus1)t zs0



(2130)








znm (2131)

ie


α2 (2132)


znm+1 + znm = 2iπκ κ = 0 1 (2133)





79












ut = f(uu1 uk) (31)


form

uτ = g(uu1u2 um) (32)


part2u




D =part

partx+sumigt0

ui+1part

partui (34a)

Dt =part

partt+sumigt0

Difpart

partui (34b)

Dτ =part

partτ+sumigt0

Digpart

partui (34c)




ut = 6uu1 + u3 (35)


Lt = [LM] (36)





Ltj =[LMj

] (38)


L M]= V (39)





V = LV + ux (311)

where

LV = minus1

4Vxx + uV minus

1

2ux

intinfinx

V(y)dy (312)


ut = F (L)ux (313)







unminus1munm

un+1m













(315)


gpartQ

partunm+ [Tng]

partQ

partun+1m

+ [Tmg]partQ

partunm+1+ [TnTmg]

partQ

partun+1m+1= 0

(316)









part2TnTmg


part2g







(319)



(320)





dunm




Q(unun+1 un un+1) = 0 (323)








un+1m+1 = f (un+1munmunm+1) (325)


TnTmg = Tngpartf

partun+1m+ Tmg

partf

partunm+1+ g

partf

partunm(326)


TnTmpartg

partun+1mTn

partf

partun+1m=

partf

partun+1mTn

partg

partun+1m (327)




Define

z = logpartg

partun+1m(328)



)log

partf

partun+1m(329)



(partf

partunm

partf

partunm+1

)Tminus1n g

+ Tminus1n

(1

partf

partunm+1

)Tmg

minus Tminus1n

(partf

partun+1m

partf

partunm+1

)g

(330)


Tminus1n Tmpartg

partunminus1m

partunminus2m+1


(partf

partunm

partf

partunm+1

)Tminus1n

partg

partunminus1m

(331)


partunminus2m+1


partunminus1m+1


(partf

partunm

partf

partunm+1

) (332)

So introducing

v = logpartg

partunminus1m(333)



partf

partunm

partf

partunm+1

) (334)


Yminus1 =part

partunmminus

partfpartunm

partfpartunm+1

part

partun+1m

minus Tminus1n

(partfpartun+1m

partfpartunm

)part

partunminus1m

(335)




Y1 =part


partf

partunm+1

part

partun+1m


(partf

partunm+1

)minus1part

partunminus1m

(336)



)log

partf

partun+1m (337)



)log

partf

partun+1m(338)


Y1z = Y1Tminus1m

(Tminus1n minus Id

)log

partf

partun+1m (339)


Y1z = Y1Tminus1m

(Tminus1n minus Id

)log

partf

partun+1m (340a)


)log

partf

partun+1m (340b)

partz

partunm+1=

partz






)log

partf

partun+1m (341)



(partf

partunm

partf

partunm+1

) (342a)


partf

partunm

partf

partunm+1

) (342b)

partv

partunm+1=

partv







)log(

partf

partunm

partf

partunm+1

)

(343)


partg

partun+1m= ez

partg



partez

partunminus1m=

partev

partun+1m (345)


g =




S =part

partunmminus

partfpartunm

partfpartun+1m

part

partun+1m (347)







partz

partunm




(348)


in (348)

u2n+1mpartz


partz

partunm

+ u2nminus1mpartz


(349a)

un+1mpartz


partz


partz

partunminus1m+

partz

partun+1m= 0 (349c)



] (350)




] (351)


g =C1



unm+1dg(1)


dg(1)

dunm(unm)


dun+1m(un+1m)

+ un+1mdg(1)


(353)


du2nm =



g =C1






g =C1




dukdt

=1



symmetry reduction



+(un+1m + 1)(unm+1 minus 1) = 0(357)



u2nminus1m minus 1

2unm

partz

partunminus1m+

partz

partunm

+2un+1m

u2nm minus 1

partz


unm(u2nm minus 1)

(358a)

2unminus1m

u2nm minus 1

partz

partunminus1mminus

partz


2unm

partz

partun+1m


unm(1minus u2nm)

(358b)


partz



partz

partun+1m= minus

2(unm + 1)


partz

partunm=

1

unm+

1

unm + 1+

1

unm minus 1



(359c)





logminusC1nm+1

C1nm= 0 (361)







u2nminus1m minus 1

2unm

partv

partunminus1m+

partv


unm(u2nm minus 1)

(363a)

2unminus1m

u2nm minus 1

partv

partunminus1mminus

partv

partunm= minus







logminusC3nm+1

C3nm= 0 (365)








(366)


2unm(u2n+1m minus 1

) (u2nm+1 minus 1


(367)


d2g(1)

du2nmminus

1

unm

dg(1)

dunm+g(1)

u2nm=


) (C2n minusC4n+1


(368)


d2g(1)

du2nmminus

1

unm

dg(1)

dunm+g(1)

u2nm= 0 C2n = C4n+1 (369)



4n+1

)(minus1)m = 0


4n+2

)(minus1)m = 0

(370)






(372)




dunm







C1






(1)m +

1



U(0)m+1 +U

(0)m +

C1

2= 0 (378a)(

U(0)m+1 +U

(0)m

)2minus(U

(1)m+1 minusU

(1)m

)2+

(C1)2

2+C1U

(0)m +C1U


(378b)


U(0)m = K1(minus1)

m minus1

4(1+ (minus1)m)C1 (379)



U(1)m+1 minusU

(1)m = plusmn1

2

radicminus4α1 + 4α2 + (C1)2 (380)


U(1)m = K2 plusmn

m

2

radicminus4α1 + 4α2 + (C1)2 (381)



C1

4

)+K2


C1

4+C1

4n

(382)



F(+)n F

(+)m Q(++) + F

(+)n F

(minus)m Q(+minus)

+F(minus)n F

(+)m Q(++) + F

(minus)n F


(383)




z = F(+)n F

(+)m z(++) + F

(+)n F

(minus)m z(+minus)

+ F(minus)n F

(+)m z(++) + F

(minus)n F


(384a)

v = F(+)n F

(+)m v(++) + F

(+)n F

(minus)m v(+minus)

+ F(minus)n F

(+)m v(++) + F

(minus)n F


(384b)





g = F(+)n F

(+)m g(++) + F

(+)n F

(minus)m g(+minus)

+ F(minus)n F

(+)m g(++) + F

(minus)n F


(385)



+ F(+)n F

(+)m ϕ(++) + F

(+)n F

(minus)m ϕ(+minus)

+ F(minus)n F

(+)m ϕ(minus+) + F

(minus)n F


(386)










XtHε2n =

[(unm + εα22F





(+)m α32 + 8εF

(+)m α22α3 + ε

2F(+)m α42


]partunm

(387a)

XtHε2m =

[1


(+)m


(+)m unm+1unmminus1


minusεF

(minus)m u2nm minus


(minus)m

]unm +α3 + ε (α2 +α3)

2


partunm

(387b)

XtHε3n =



minusα22u


2F(+)m


]partunm

(387c)

XtHε3m =


(+)m unm+1unmminus1


minusε2F


2


]partunm

(387d)




XtHε1n1 =

F(+)m


middot


)Bn

(α2

un+1m minus unm

)


)Bnminus1

(α2

unm minus unminus1m

)]

+ F(minus)m


2unminus1m)unm




middot[Bn

(un+1m minus unm

1+ ε2un+1munm

)minusBn


1+ ε2unmunminus1m

)]partunm

(388a)

XtHε1n2 =

F(+)m

[unm minus



]

minusF(minus)m





+(4minus 4un+1mε


)]partunm

(388b)


XtHε1n = Xt

Hε1n1 +αXt

Hε1n2 (389)


XtHε1m =

[F(+)m Bm



)

+ F(minus)m

(1+ ε2u2nm


]partunm

(390)



YtHε1 =

[F(+)m κm + F

(minus)m λm

(1+ ε2u2nm

)]partunm (391)



332 H6 equations



X1D2n =

(F(+)n F

(minus)m minus δ1F

(+)n F

(+)m



+

(F(+)n F

(minus)m minus δ1F

(+)n F

(+)m minus δ1δ2

)F(minus)n F

(+)m unminus1m


+(F

(+)n+m minus δ1F

(+)m minus δ1δ2F

(+)n F

(+)m )unm + δ2F

(minus)m


]partunm

(392a)

X1D2m =

[δ1F

(minus)n F




+δ1F

(+)m unm+1 + δ1δ2F

(minus)n F


(minus)n F

(minus)m u2nm


+

[F(+)n+m + δ1

(F(+)n F

(minus)m minus F

(minus)n F

(minus)m

)+ δ1δ2F

(minus)n F

(minus)m

]unm



(minus)n


]partunm

(392b)

X2D2n =

(F(minus)n F

(minus)m δ1 + F

(minus)n F


(minus)n F

(minus)m

)un+1m


+

(F(+)n F

(+)m δ1 minus F

(+)n F

(minus)m

)unminus1m


+

(δ1F

(minus)n+m minus F

(minus)m + δ1δ2F

(+)n F

(minus)m


(+)m


partunm

(392c)


X2D2m

F(minus)n F


(δ1δ2F

(minus)n F

(minus)m + F

(+)n F

(minus)m

)unm+1


++(δ1F

(+)n F

(+)m + F

(minus)n F

(minus)m minus δ1F

(minus)n F

(minus)m

)unmminus1


+δ1F

(minus)n F

(+)m u2nm +

[F(+)n F


(minus)n F

(+)m + F

(+)m

]unm


+δ2(1minus δ2)F


(+)n


]partunm

(392d)

X3D2n =

(δ1F

(+)n F

(minus)m + δ1δ2F

(+)n F

(minus)m minus F

(+)n F

(minus)m

)un+1m


+

(F(+)n F

(+)m δ1 minus F

(minus)n F

(minus)m

)unminus1m


+

(δ1F

(minus)n F

(minus)m δ2 + F


(minus)m


(+)m


partunm

(392e)

X3D2m =


(+)n F

(minus)m unm+1


+

(F(minus)n F

(minus)m minus F

(+)n F

(+)m δ1

)unmminus1 + δ2F

(minus)n


+

(F(+)m minus δ1F

(+)n minus δ1δ2F

(+)n F

(+)m

)unm



(+)n


]partunm

(392f)


Y1D21 =(F(+)n F

(+)m + F

(+)n F

(minus)m + F

(minus)n F

(+)m

)unmpartunm (393a)

Y1D22 =[δ1F

(+)n F

(+)m + [1minus δ1(1+ δ2)]F

(minus)n F

(+)m + F

(+)n F

(minus)m

]partunm

(393b)

Y2D21 =[(F(+)n F

(+)m + F

(+)n F

(minus)m + F

(minus)n F

(+)m

)unmminus

minusλF(+)n F


(minus)n F

(minus)m

]partunm

(393c)

Y2D22 =[δ1F

(+)n F

(+)m + F

(+)n F


(minus)n F

(minus)m

]partunm

(393d)


Y3D21 =[(F(+)n F

(+)m + F

(+)n F

(minus)m + F

(minus)n F

(+)m

)unmminus

minusλF(minus)n F


(minus)n F

(minus)m

]partunm

(393e)

Y3D22 =[δ1F

(+)n F

(+)m + [1minus δ1(1+ δ2)]F

(+)n F

(minus)m minus F

(minus)n F

(minus)m

]partunm

(393f)


XD3n =

F(+)n F


1

2

(F(minus)m minus F

(minus)n F

(+)m



+F(minus)n F

(+)m u2nm +

(F(minus)m minus F

(+)n F

(+)m

)unm


partunm

(394a)

XD3m =

F(+)n F


1

2

(F(minus)n minus F

(+)n F

(minus)m



+F(+)n F

(minus)m u2nm +

(F(minus)n minus F

(+)n F

(+)m

)unm


partunm

(394b)


YD3 =[F(+)n

(2F

(+)m + F

(minus)m

)+ F

(minus)n

]unmpartunm (395)



X1D4n =


1



+minusδ1F


(+)m


]partunm

(396a)


X1D4m =


1



+δ2F


(+)n


]partunm

(396b)

X2D4n =

minusδ1δ2F(+)n F


1



+minusδ1δ2F

(minus)n F

(+)m u2nm + δ3


]partunm

(396c)

X2D4m =


1



+δ2F

(minus)n+mu

2nm minus δ1δ3F

(+)n


]partunm

(396d)




dqkdt




where

A(qk) = aq2k + 2bqk + c (398a)

B(qk) = bq2k + dqk + e (398b)

C(qk) = cq2k + 2eqk + f (398c)



dqkdt





Bk(qk) = bk+1q2k + dqk + ek+1 (3100b)

Ck(qk) = ck+1q2k + 2ekqk + f (3100c)








Bn (ξ) = minus1

ξ α = 0 (3101)


XtHε1n =




Bm(ξ) =1

ξ Cm(ξ) =

1

ξ (3103)


XtHε1m =

F(+)m


)+ F

(minus)m

(1+ ε2u2nm


partunm

(3104)




iD22 j = nm i = 1 2 3 (3105)



2

K1D22j = K2D22j = K3D22j = 0(3106)




10

6g

en

er

al

iz

ed

sy

mm

et

rie

sa

nd

th

en

ay

dk

ne

qu

at

io

n


Eq k a bk ck d ek f

rHε1



rHε2





rHε3

n 0 0 minusεF

(+)n+m

α

1

20 δα

m 0 0 minusεF

(+)n+m

β

1

20 δβ

tHε1


m 0 0 ε2F(+)m 0 0 2

tHε2


22

(4α2 + 8α3 + εα

22

)F(+)m


1



2


34

th

en

on

-a

ut

on

om

ou

sy

dk

ne

qu

at

io

n1

07


Eq k a bk ck d ek f

tHε3

n 0 0 minusα221

2α2(1+α

22) 0 minusε2δ2F

(+)m (1minusα22)

2

m 0 0 minusε2F(+)m

1

2α3 0 minusα23δ

2

1D2n 0 0 0 0

1

2[δ1(1+ δ2) minus 1]F

(+)n F

(+)m

+1

2F(minus)n F

(+)m minus

1

2F(minus)n F

(minus)m

minusδ2F(minus)m


(+)m δ1 0

1


(minus)n F

(minus)m

minus1

2F(+)n F

(+)m minus

1

2δ1F

(minus)n F

(+)m


2D2n 0 0 0 0

1


(+)n F

(minus)m

+1

2F(minus)n F

(minus)m minus

1

2δ1F

(minus)n F

(+)m

(δ1 minus 1)F(+)m


(minus)m 0

1


(minus)n F

(+)m

minus1

2F(+)n F

(+)m minus

1

2δ1F

(minus)n F

(+)m


(+)n


10

8g

en

er

al

iz

ed

sy

mm

et

rie

sa

nd

th

en

ay

dk

ne

qu

at

io

n


Eq k a bk ck d ek f

3D2n 0 0 0 0

1

2[δ1(1+ δ2) minus 1]F

(minus)n F

(minus)m

+1

2F(+)n F

(minus)m +

1

2δ1F

(minus)n F

(+)m

(1minus δ1)F(+)m

m 0 0 0 0

1


(+)n F

(+)m

minus1

2F(+)n F

(minus)m +

1

2δ1F

(minus)n F

(+)m


(minus)n

D3n 0 0 F

(+)n F

(+)m 0

1

2

(F(+)n F

(minus)m + F

(minus)n F

(minus)m minus F

(+)n F

(+)m

)0

m 0 0 F(+)n F

(+)m 0

1

2

(F(minus)n F

(+)m + F

(minus)n F

(minus)m minus F

(+)n F

(+)m

)0

1D4n 0 0 minusδ1

(F(+)n F

(+)m + F

(+)n F

(minus)m

)minus1

20 δ2δ3F

(+)m

m 0 0 δ2

(F(+)n F

(+)m + F

(minus)n F

(+)m

) 1

20 minusδ1δ3F

(+)n

2D4n 0 0 minusF

(+)n F

(+)m δ1δ2

1

20 δ3

m 0 0 δ2

(F(+)n F

(+)m + F

(minus)n F

(minus)m

) 1

20 minusδ1δ3F

(+)n












(2)n (3108)


Dτi =part

partτi+sumjisinZ

ϕ(i)n+j

part

partun+j (3109)






n and ϕ(2)n

[ϕ(1)n ϕ(2)


(2)n (3110)




[[ϕ(1)n ϕ(2)

n ]ϕ(3)n ] = [[ϕ

(1)n ϕ(3)

n ]ϕ(2)n ] + [ϕ

(1)n [ϕ(2)

n ϕ(3)n ]] (3111)


[ϕ(1)n ϕ(2)

n ] = minus(Dτ1ϕ

(2)n minusDτ2ϕ

(1)n

)= minus[ϕ

(2)n ϕ(1)

n ] (3112)







gn = [ϕn fn] (3115)


[[ϕn fn] fn] = 0 (3116)





ki rarr κi = κi (τ) i = 1 M (3118)




κi (0) = ki (3119b)






]qk


(3121a)


]q2k + d(τ)qk


(3121b)


]q2k


]qk + f(τ)

(3121c)


dqkdt


qk+1 minus qkminus1

(3122)


dqkdτ


qk+1 minus qkminus1

(3123)


dadτ



dbdτ


dβdτ


dcdτ


dγdτ


dddτ


dedτ


dεdτ


dfdτ




dcdτ

(τ) = minusd (τ) c (τ) (3125a)

dγdτ

(τ) = minusd (τ)γ (τ) (3125b)

dddτ

(τ) = minusc2 (τ) + γ2 (τ) (3125c)

dedτ


dεdτ

(τ) = γ (τ) e (τ) + c (τ) ε (τ) (3125e)

dfdτ



35

in

te

gr

ab

il

it

yp

ro

pe

rt

ie

s1

13


tHε1

nminus1

τ+ 10

1

τ+ 1

mε2

2

ε2

20

tHε2

nminus1

τ+ 10

1

τ+ 1

m minusε

2minusε

20

tHε3

nα22(α22 minus 1

)eminus

12τα2(α

22minus1)


01

2

α2(1minusα22

) [eminusτα2(α

22minus1) +α22

]+eminusτα2(α

22minus1) minusα22

m minusε2eminus

12α3τ

2minusε2eminus

12α3τ

2

α32


tHε1

n 0 0 minusε2α22F(+)m (τ+ 1)

m 0 0 2

tHε2

n εα22F

(+)m (τ+ 1) 0 minusα22

[minus1+α2

2F(+)m (τ+ 1)2 ε2 + 4F

(+)m (α2 + 2α3) ε

](τ+ 1)

m1

2+1


1


1


2 εminusα3 minus1

2τ

tHε3

n 0 0ε2δ2F

(+)m

(α22 minus 1

) (eminusτα2(α

22minus1) minusα2

2)


m 0 0 minusα23δ2e

12α3τ





1 1 3 7 13 21 31 43 57 73 91 111 133 157 (3126)



(1minus z)3(3127)


dl = l(lminus 1) + 1 (3128)



m = 2k 1 1 3 7 10 17 23 33 42 55 67 83 98 117 (3129a)

m = 2k+ 1 1 1 3 4 9 13 21 28 39 49 63 76 93 109 (3129b)


m = 2kg(z) =


(1minus z)3(z+ 1)

dl =3

4l2 minus lminus

5(minus1)l minus 21

8

(3130a)

m = 2k+ 1g(z) =

(z2 + z+ 1)(2z2 minus 2z+ 1)

(1minus z)3(z+ 1)

dl =3

4l2 minus

3

2lminus

5(minus1)l minus 19

8

(3130b)






dl =3

4l2 minus

5

4l+ (minus1)l

l

4+

(minus1)l + 15

8 (3131)









p1 = logpartfn

partun+1= 2 log



)(3133)

we must have

dp1dt







= (T minus 1)gn


(3134)



1 1 3 9 27 81 273 729 (3135)


g(s) =1minus 2s

1minus 3s (3136)














+a60 (unm + un+1m + unm+1 + un+1m+1)

+a7 = 0

(3137)


dunm

dt=

hn


2partun+1mhn (3138)

where


QV

minus(partunm+1

QV) (partun+1m+1

QV) (3139)


dunm

dt=

hm


2partunm+1

hm (3140)

where


minus(partun+1mQV

) (partun+1m+1

QV) (3141)



a = a30a1 minus a220 (3142a)

b =1


c = a20a60 minus a40a50 (3142c)

d =1

2[a230 minus a

240 minus a

250 + a1a7] (3142d)

e =1


f = a30a7 minus a260 (3142f)


a = a50a1 minus a220 (3143a)


b =1


c = a20a60 minus a40a30 (3143c)

d =1

2[a250 minus a

240 minus a

230 + a1a7] (3143d)

e =1


f = a50a7 minus a260 (3143f)








(3144)











(3145)




(3146)





p12 = p13 = p14 = p15 equiv a6 p16 = a7

(3147)




a6 = 1+ (minus1)n a7 = 4+ 2(minus1)n

(3148)









(3149)





]unmunm+1un+1m+1


]un+1munm+1un+1m+1


]unmun+1mun+1m+1


]unmun+1munm+1




]unmun+1m+1

+[a40 + (minus1)n+m a43

]un+1munm+1




]unm


]un+1m


]unm+1


]un+1m+1

+a7 = 0

(3150)










1 3 7 13 21 31 43 57 73 91 111 133 (3151)


g(z) =1+ z2

(1minus z)3 (3152)


dk = k(k+ 1) + 1 (3153)









221 minus a

222 + a

223


(3154a)


b =1


+1


minus1

2(minus1)ma22(a50 + a30 + a40)

+1

2(minus1)m(a23a51 + a1a62 + a20a32 + a21a43)

(3154b)

β =1

2a21(a30 minus a40 + a50)

+1


+1


minus1


(3154c)





d =1

2(a230 minus a

240 minus a

250 + a1a7 minus a

232 + a

243 + a

251)


(3154f)

e =1


+1

2(a20a7 + a51a61 minus a32a62 + a43a63)

+1


minus1

2(minus1)ma62(a30 + a40 + a50)

(3154g)

ε =1

2a61(a30 minus a40 + a50)

minus1

2(a51a60 minus a43a62 + a32a63 + a21a7)

+1


+1


(3154h)


262 + a

263 + a

261


(3154i)





Eq a30 a32 a40 a43 a50 a51 a60 a61 a62 a63 a7

rHε1 1 0 1



(β2 minusα2

)minus(αminusβ)

(2εα2 +α+ 2εβ2 +β

)rHε3 α 0 1

2

ε(β2minusα2

)αβ

12

ε(β2minusα2


(α2 minusβ2




2α2 [2+ ε(2α2 +α3)] 0 εα2α3 + 12 εα2

2 0 α2

[α2 + 2α3 + ε (α2 +α3)

2]

tHε3

12

ε2(1minusα2

2)

α3α2

12

ε2(1minusα2

2)

α3α2α2 0 minus1 0 0 0 0 0 δ2α3

(1minusα22

)1D2

12 δ1

12 δ1

12 minus 12 0 0 1



12 minus 1

4 (δ1 + δ2) 0

2D212 minus 12

12 δ1

12 δ1 0 0 1



12 minus 1


3D212 minus 12 0 0 1


4 (δ1 minus δ2 + δ1λ)14 (δ1 + δ1λ+ δ2)

12 minus 1


D312

12

12

12

12 minus 12

14


1D412 δ2

12 δ2 1 0 1

2 δ1 minus 12 δ1 0 0 0 0 δ3

2D4 1 0 12 δ2

12 δ2

12 δ1 minus 12 δ1 0 0 0 0 δ3








∣∣∣∣uxt=f

equiv 0


∣∣∣∣uxt=f

equiv 0(42)

125





uxt = eu (43)


X = uxx minus1

2u2x T = utt minus

1

2u2t (44)



unmun+1m+1

(1+

1

un+1m

)(1+

1

unm+1

)= 1 (45)






where














W1 = ξn W2 = λm (411)



W1 =

[1+

unm (1+ unminus1m)

unminus1m

] [1+

unm

un+1m (1+ unm)

]

(412a)

W1 =

[1+

unm (1+ unmminus1)

unmminus1

] [1+

unm

unm+1 (1+ unm)

]

(412b)






Yminus1 = Tmpart






Yminusk = Tkmpart






TmW1 =aW1 + b

cW1 + d (417)



Zminusk = Tknpart




which are such that



Wi = F(+)n F

(+)m W

(++)i + F

(minus)n F

(+)m W

(minus+)i

+ F(+)n F

(minus)m W

(+minus)i + F

(minus)n F

(minus)m W

(minusminus)i

(420)

with

F(plusmn)k =

1plusmn (minus1)k

2(421)





cube









W1 = (minus1)mun+1m

unm (423a)

W2 = (minus1)nunm+1

unm (423b)




1+ unmun+1m (424a)


1+ unmunm+1 (424b)





W1 = Fm

((unm + e2)(un+1m + o1)

(unm + e1)(un+1m + e2)

) (425)



In =(unm + e2)(un+1m + o1)

(unm + e1)(un+1m + e2)(426)

















4311 tHε1 equation


W1 = F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm



W2 = F(+)m α




(429b)


4312 tHε2 equation



W1 = F(+)m



minus F(minus)m



(430a)


W2 = F(+)m α



]middot[


]

+ F(minus)m β







(430b)



Wε=01 = (minus1)m



Wε=02 =






(432)



4α2 +

1

2α3 (433)


4313 tHε3 equation



W1 = F(+)m


3un+1munm

+(unm



+ F

(minus)m



(434a)



W2 = F(+)m α




δ2α23 + u2nm+1ε

2 minusα3unmunm+1

)

minus F(minus)m β

α3u





2α32unmminus2


(434b)




Wε=01 = (minus1)m



2

] (435a)

Wε=02 =



(435b)




(436)


b2 = minus1

α2 c4 =

δ2α3(1minusα22

)α2

(437)





432 H6 equations


4321 1D2 equation


W1 = F(+)n F

(+)m α



+ F(+)n F

(minus)m α


+ F(minus)n F


minus F(minus)n F

(minus)m β


δ2 + unm

(438a)

W2 = F(+)n F

(+)m α


unm + δ1unmminus1

+ F(+)n F


minus F(minus)n F

(+)m α



minus F(minus)n F

(minus)m β


δ2 + unm

(438b)


W(0δ2)1 = F

(+)n F

(+)m α


unm

minus F(+)n F


+ F(minus)n F


+ F(minus)n F

(minus)m β


δ2 + unm

(439a)

W(0δ2)2 = F

(+)n F

(+)m α


unm

+ F(+)n F


minus F(minus)n F


minus F(minus)n F

(minus)m β


δ2 + unm

(439b)






W((1+δ2)

minus1δ2)1 = F

(+)n F

(+)m α

un+1m

unminus1m

+ F(+)n F

(minus)m α

δ2 + un+1m

δ2 + unminus1m

+ F(minus)n F


minus F(minus)n F

(minus)m β


δ2 + 1

(440a)


(+)n F

(+)m α [(1+ δ2)unm + unm+1]

+ F(+)n F

(minus)m β

unm + (1+ δ2)unm+1

δ2 + 1

minus F(minus)n F

(+)m α

(δ2 + 1)unm

δ2 + unm+1

minus F(minus)n F

(minus)m β

unm+1

δ2 + unm

(440b)



minus1

4322 2D2 equation


W1 = F(+)n F

(+)m α

δ2 + un+1m

δ2 + unminus1m

+ F(+)n F

(minus)m α



+ F(minus)n F

(+)m β



minus F(minus)n F


(441a)

W2 = F(+)n F


minus F(+)n F

(minus)m β



minus F(minus)n F

(+)m α



minus F(minus)n F

(minus)m β


unm+1 + δ2

(441b)



W(0δ2)1 = F

(+)n F


minus F(+)n F


+ F(minus)n F


minus F(minus)n F


(442a)

W(0δ2)2 = F

(+)n F


+ F(+)n F

(minus)m β

unm+1

unmminus1

minus F(minus)n F


minus F(minus)n F

(minus)m β


unm+1 + δ2

(442b)






(+)n F

(+)m α

δ2 + un+1m

δ2 + unminus1m

+ F(+)n F

(minus)m α

un+1m

unminus1m

minus F(minus)n F


minus F(minus)n F

(minus)m β


1+ δ2

(443a)


(+)n F

(+)m α [(1+ δ2)unm + unm+1]

+ F(+)n F


+ F(minus)n F

(+)m α


unm + δ2

+ F(minus)n F

(minus)m β


unm+1 + δ2

(443b)





minus1

4323 3D2 equation


W1 = F(+)n F

(+)m α



+ F(+)n F

(minus)m α



+ F(minus)n F


minus F(minus)n F


(444a)

W2 = F(+)n F


minus F(+)n F

(minus)m β



+ F(minus)n F


+ F(minus)n F

(minus)m β



(444b)


W(0δ2)1 = F

(+)n F


minus F(+)n F


+ F(minus)n F


minus F(minus)n F


(445a)

W(0δ2)2 = F

(+)n F


+ F(+)n F

(minus)m β

unm+1

unmminus1

minus F(minus)n F


+ F(minus)n F

(minus)m β

δ2 + unmminus1

δ2 + unm+1

(445b)







(+)n F

(+)m α



+ F(+)n F

(minus)m α


(δ2 + 1)unm

minus F(minus)n F


minus F(minus)n F


(446a)


(+)n F


+ F(+)n F

(minus)m β


unm

minus F(minus)n F

(+)m α


δ2 + 1

+ F(minus)n F

(minus)m β



(446b)


minus1

4324 D3 equation


W1 = F(+)n F

(+)m α


u2n+1m minus unm

+ F(+)n F

(minus)m α


)unm + unminus1m

minus F(minus)n F

(+)m β


un+1m minus u2nm

+ F(minus)n F

(minus)m β


un+1m + un+2m

(447a)

W2 = F(+)n F

(+)m α


u2nm+1 minus unm

minus F(+)n F

(minus)m β


unm+1 minus u2nm

+ F(minus)n F

(+)m α


unm + unmminus1

+ F(minus)n F

(minus)m β


unm+1 + unm+2

(447b)



4325 1D4 equation


W1 = F(+)n F

(+)m α



+ F(+)n F

(minus)m α



+ F(minus)n F

(+)m β



+ F(minus)n F

(minus)m β



(448a)

W2 = F(+)n F

(+)m α



minus F(+)n F

(minus)m β



+ F(minus)n F

(+)m α



+ F(minus)n F

(minus)m β



(448b)




unm (449a)



unm (449b)





un+1m+1unm + un+1munm+1 + 1 = 0 (450)




W(000)1 = (minus1)m

un+1m

unm W

(000)2 = (minus1)n

unm+1

unm (451)


4326 2D4 equation


W1 = F(+)n F

(+)m α



](δ2u


)unminus1m

minus F(+)n F

(minus)m α


minus F(minus)n F

(+)m β



+ F(minus)n F

(minus)m β


un+1mun+2m + δ3(452a)

W2 = F(+)n F

(+)m α



minus F(+)n F

(minus)m β



+ F(minus)n F

(+)m α



+ F(minus)n F

(minus)m β



(452b)



(unmun+1m +

δ32

) W

(00δ3)2 =

(unm+1

unmminus1

)(minus1)n

(453)





unmun+1m + unm+1un+1m+1 + c4 = 0 (454)











u2k2l+1 = ykl u2k+12l+1 = zkl (455b)








unm = wn + υm (458)













4411 1D2 equation





F(+)n




(460)



δ1

minus(F(+)m ξn + F

(minus)m

)unminus12l minus F

(minus)m ξn = 0

(461)





where



vkl = αl + ak (464)








δ1

(1

bkminus

1

bkminus1

)(αl + ak) (466)


Wkl = βl +δ

δ1

αlbk

+ ck (467)



δ1

(1

bkminus

1

bkminus1

)ak (468)


ak = minusδ1δ








δ1αl (470b)



+ F(minus)m


δ2 + un2l+1= ξn

(471)



(1minus

1

δ1








zkl

bkminuszkminus1l

bkminus1=

(1minus

1

δ1

)(1

bkminus

1

bkminus1

) (473)


zkl = 1minus1

δ1+ bkγl (474)



(1

δ1+

δ

δ21bkγl



ykl = δl +δdk

δ21γlminusakδ1

(476)



bk (477)




(478)


zkl = 1minus1

δ1+ bkγl (479a)


minusckδ1γl

+1

δ


bk minus bkminus1

(479b)





δl =1

δ1


(481)







δ1αl (482b)

zkl = 1minus1

δ1minus bk


(482c)

ykl =1

δ1


+1

δ


bk minus bkminus1

+


+ckδ1


(482d)



δ1 =1

1+ δ2 (483)






wkl = akαl (485a)



δ2 + zkl

ak=δ2 + zkminus1m

akminus1 (486a)















wkl = akαl (490a)



minus δ2 (490c)

ykl = minus(1+ δ2)


αl+1 minusαl+ bk

) (490d)





un2l+ F







vkl = ak +αl (493)





(495)



+αl (496a)




(minus)n


δ2 + un2l+1= ξn

(497)













+ ck


(4102)



minus ck (4103)





minus ck

(4104b)








is


+αl (4107a)



minus bk (4107c)





minus ck

(4107d)


4412 2D2 equation





F(+)n α

δ2 + un+12l

δ2 + unminus12l

minus F(minus)n β



(4108)



and putting


δ1(4110)


F(+)nUn+12l

Unminus12lminus F

(minus)n


Un2l + δ= ξn (4111)





(1+

δ

Wkl

) (4112b)


Wkl = akαl (4113)



(1+

δ

akαl

) (4114)

If we define


(4115)





(4116a)


(4116b)



+1



+1






(4118)


δδ1ykl + zkl





(4120)


zkl

akminuszkminus1l

akminus1=

(1

akminus

1

akminus1












(4123b)



21λ+ δδ1 + δ1







+1



+1



21λ+ δδ1 + δ1

δ1






αl+1 minusαl

(4125c)


21λ+ δδ1 + δ1

δ21



(4125d)




F(+)n (δ2 + un+12l)un2l + F



(4126)


vkl(δ2 +wkl) = ξ2k (4127a)

vk+1l(δ2 +wkl) = ξ2k+1 (4127b)



(4128)



vkl = 0 (4129)

Defining

ξ2k+1 =ak+1ak

ξ2k (4130)


vkl = akαl (4131a)


(4131b)


F(+)n (un+12l+1 + un2l+1)+ F


(4132)




ξ2k (4133b)





yk+1l minus ykl =

(1minus

ak+1ak

)ξ2k (4135)

Defining


(4136)






vkl = akαl (4138a)


αl


(4138b)





4413 3D2 equation



F(+)n




(4139)







(δ2 +wkl) (4140b)


wkl = minusδ2 +1

Wkl (4141)

into


(ξ2k minus 1) (4142a)




(4143)



(4144)





δ1+

δ(bk+1 minus bk)


(4146a)







(4147)


zkl minus1

ξ2kzkminus1l =

1

δ

(1minus

1

ξ2k

)ykl (4148a)








zkl =bk minus γlδ


(δl minus

ckδ

) (4151)






γl =δ

δ1



) (4153a)

δl = minus1

δ


(4153b)





δ1+

δ(bk+1 minus bk)


zkl =bkδ

minus1

δ1



)+1

δ



+ ck

)

(4154c)

ykl = minusbk +δ

δ1



) (4154d)






F(+)n




(4155)


wkl minuswkminus1l








Wkl = ak +βl (4158)





(4160)









(4162)





)

(4163a)








)

(4165)




+ dk

)+ δl (4166)






) (4167)



minus ck+1 (4168)




[(γl + bk+1)


minus ck+1

]+ δl

(4169b)






(4171a)








minus bk (4172c)


[(bk+1 minus



minus ck+1

]+ (δ2 + 1)


+ λ

(4172d)





ξ2k (4173a)


(δ2 +wkl) (4173b)





(4175)

Defining



vkl = βl +bkαl

(4177a)



zkl minusakakminus1

zkminus1l =

(akakminus1

minus 1

)ykl (4178a)




(4179)


ykl = ck + γl (4180)





zkminus1l =



)(ck + γl)

(4181)







ckminus1

(4183)



ck (4184)


ykl = ck + γl (4185a)







βl =δ0αl

δl = δ0 (4187a)


vkl =bk + δ0αl

(4188a)


(4188b)


ykl = ck + γl (4188c)







4421 D3 equation








wkl + ykl+


wkl + ykl (4190)



wkl+1 + ykl+


wkl+1 + ykl (4191)





)middot


(4192)








2


(4194)





2




F(+)n



+ F(minus)n


un+12l+1 + un+22l+1= ξn

(4197)













Ykminus1l + ξ2k = 0 (4201)

Defining



Ykl = ak (bk +αl) (4203)



Setting


(4205)

we have




2




+ek




2








(4211)





4422 1D4 equation



yklwkl+1 + zklvkl+1


wklyk+1l + vk+1lzkl






δ3 +wklyklvkl+


δ3 +wklykl(4214)




δ3 + yklwkl+1vkl+1


2kl+1 minus δ2δ3

(yklwkl+1 + δ3

(4215)




]middot


(4216)



we obtain for wkl







2



ykl minus yklminus1

(4218)






2





F(+)n



+ F(minus)n


un2l+1 (un+22l+1δ1 + un+12l+1)= ξn

(4221)





zkl(zk+1lδ1 + yk+1l)= ξ2k+1 (4222b)






ak+1 (4224a)


bk (4224b)




ak =1

ck

1


1

ckminus1

1







(ck minus ck+1)2


+δ1δ3αlminus1


[1


minus1


]

(4228)



Vkl = γl +δ1δ3


[αlminus1


](4229)



2


(4230)



(4231)


γl =δlδ2






4423 2D4 equation












δ1wkl + yklvkl


2klδ1 minus δ3)

(δ1wkl + ykl)wkl

(4235)



δ1wkl+1 + yklvkl+1


2kl+1δ1 minus δ3)


(4236)




]middot(


)= 0

(4237)



we obtain

wkl =1

δ1









2klminus1




y2klminus1

(4239)



minus δ2ykl (4240)




2





+ F(minus)n


un+12l+1un+22l+1 + δ3= ξn

(4242)



= ξ2k (4243a)


(yk+1lzk+1l + δ3)= ξ2k+1 (4243b)





ak+1 (4245a)


bk (4245b)



If we define

ak =1

ck

1


1

ckminus1

1



(4246) is solved by




2


minusαlδ1δ3


[1

(αldk+1 +βl)c2k+1

minus1

(αldk +βl)c2k

]

(4249)

whose solution is



+δ3δ1ek


(4250)



2


(4251)



(4252)

ie

γl =δlδ2












(Tm minus Id)(F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm

1+ ε2unmun+1m

)= 0

(4256)


F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm

1+ ε2unmun+1m= λn (4257)




(4258)

which has solution

un2k = θ2k +ωn (4259)



ω0 = 0 (4260)





un2k+1 =iε

1minus vn2k+1

1+ vn2k+1 (4262)



If we define

λn =iε


(4264)


vn2k+1 = κnθ2k+1 (4265)


un2k+1 =iε


(4266)



(minus)m

iε


(4267)










+εα22

(2qnl + 2α3 +α2)(2qn+1l + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2


(4268a)



+εα22

(2qnl + 2α3 +α2)(2qn+1l + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2 minusα23


(4268b)




+εα22

(2qnl+1 + 2α3 +α2)(2qn+1l+1 + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2


(4269a)



+εα22

(2qnl + 2α3 +α2)(2qn+1l + 2α3 +α2)

+εα22

(2α3 +α2)2 + (α2 +α3)

2 minusα23


(4269b)


pnl+1 =




]qn+1l+1

+ε (α2 + qnl+1) (α2 + qnl)qn+1l+1


]


(4270a)


pn+1l+1 =







]


(4270b)
















(nminus 1 l)

(nminus 1 l+ 1)

(n l)

(n l+ 1) (n+ 1 l+ 1)

(n+ 1 l)










qnl = an +αl (4274)


αl+1 minusαl = 0 (4275)




α2 + an minus an+1

+

α2ε

[α22 + (2α0 + an+1 + 2α3 + an)α2

+ 2 (an+1 +α0 +α3) (α3 + an +α0)


(4276)



=bn+1bn

(4277)



pn+1l

bn+1=pnl

bn+


bnbn+1

minus ε

[(α2 +α0 + an +α3)


bnbn+1

(4278)


pnl = bn (βl + cn) (4279)



bnbn+1

minus ε

[(α2 +α0 + an +α3)


bnbn+1

(4280)






(4281)






(4283)


brings (4282) into

Qn+1l +1

ξnQnl= 1 (4284)


ξn =1

an (1minus an+1) (4285)


Qnl = an +1

Znl(4286)


Zn+1l =anZnl + 1

1minus an+1 (4287)

Introducing

an =bnminus1

bn + bnminus1(4288)

we obtain



2n


(4289)



we obtain




as
















Rnl



= 0(4296)





en



(4298a)


fn



(4298b)












= 0 (4302)





= 0

(4303a)





(4304)



(1

cn+1 minus cn+

1

cn minus cnminus1

) (4305)



Tn+1l

cn+1 minus cnminus

Tnl



Pnl minus Pnminus1l



Pnl = (cn +αl)βl + γl (4308)



cn +αl (4309)





1

ε

) (4310)






cn +α0 (4311)








(δ2 + ε2

qnlqn+1l

α23α2

)= 0

(4312a)




(δ2 + ε2

qnlqn+1l

α23α2

)= 0

(4312b)








(δ2 + ε2

qnl+1qn+1l+1

α23α2

)= 0

(4313a)




(δ2 + ε2

qnlqn+1l

α23α2

)= 0

(4313b)


pnl+1 =


(ε2qnlqnl+1 + δ

2α32)




(4314a)

pn+1l+1 =






(4314b)













W1 =qn+1l

qnl (4317a)

W2 =qnl+1

qnl (4317b)


qnl = anαl (4318)




αl+1 minusαl = 0 (4320)




2anα20an+1


Defining


=bn+1bn

(4322)



pn+1l

bn+1=pnl

bn+δ2α23α

22bn minus bn+1

(δ2α23 + ε

2a2nα20

)α2 + ε

2a2nα20bn

bnanα0α2α3bn+1

(4323)


pnl = bn (βl + cn) (4324)


cn+1 = cn+δ2α23α

22bn minus bn+1

(δ2α23 + ε

2a2nα20

)α2 + ε

2a2nα20bn

bnanα0α2α3bn+1

(4325)





(434a) with k = 2l+ 1










Qn+1l +1

ξnQnl= 1 (4329)















Pnl =Rnl

Rnminus1l (4333)



1

Rnl= α2


(4334)



1

dn= α2


(4335)


zn+1zn


(4336)




Rnl = dn +1

Snl (4338)



cn+1 minus cnSnl =




Defining

dn =en

enminus1 (4340a)


(4340b)



2nminus1



Rnl =en(fn +βl)



Pnl

en(fn +βl)=

Pnminus1l


ie



qnl =γlen(fn +βl)

cn +αl (4345)




(4346)



αl+1 minusαl =ε2

α2δ2α23





qnl =γlen(fn +β0)

cn +αl (4348)







191

192 conclusions






conclusions 193








Mob4















7 for t1 in TT

8 for t2 in TT

9 for t3 in TT

10 for t4 in TT

11 TE = [t1t2t3t4]

12 TF = TF + [TE]

13 return TF

14



195



19 Ind = [01212]





27 else



30 return Mob4

31




36 f0 = (1+EM[0](-1)n+EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

37 f1 = (1-EM[0](-1)n+EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

38 f2 = (1+EM[0](-1)n-EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

39 f12 = (1-EM[0](-1)n-EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

40 F = [f0f1f2f12]

41 Mobnm =

42 Ind = [01212]

43 Mob0=0





51 else


53 Mob0 += F[i]MobT


55 Mobnm[u(0)]=Mob0


57 Mobnm[u(1)]=Mob1


59 Mobnm[u(2)]=Mob2


True)



62 return Mobnm

63


65 bQsub = u(0) u(1) u(1) u(0) u(2) u(12) u(12) u(2)

66 Qbsub = u(0) u(2) u(1) u(12) u(2) u(0) u(12) u(1)

67 bQbsub = u(0) u(12) u(1) u(2) u(2) u(1) u(12) u(0)

68

69 f0 = (1+EM[0](-1)n+EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

70 f1 = (1-EM[0](-1)n+EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

71 f2 = (1+EM[0](-1)n-EM[1](-1)m-EM[0]EM[1](-1)(n+m))

4

72 f12 = (1-EM[0](-1)n-EM[1](-1)m+EM[0]EM[1](-1)(n+m))

4

73




77

78 return Qt

79




83



86



89


91 if DQ == 0

92 return True

93 else

94 return False


Mob4





6 if s gt= 3600


8 si = int(si)

9 h = si3600


10 m = (si60) 60

11 ss = si 60



15 si = int(si)

16 m = (si60) 60

17 ss = si 60




24 def main()





30 Qconst = K

31 linear part

32 Qlin = D(0)u(0)+D(1)u(1)+D(2)u(2)+D(12)u(12)

33 quadratic part


35 +C(12)u(1)u(2)+C(112)u(1)u(12)+A(212)u(2)u(12)

36 cubic part

37 Qcub = B(012)u(0)u(1)u(2)+B(0112)u(0)u(1)u(12)+B

(1212)u(1)u(2)u(12)

38 quartic part

39 Qquart = A(01212)u(0)u(1)u(2)u(12)

40 the equation


42 print Q


44 R = [-11]

45 EM = []

46 for r in R

47 for s in R

48 EP = [rs]

49 EM += [EP]

50 for E in EM

51 print E


53 N=0


55 Nerr = []

56 dt =float(0)




59 for TT in MTot

60 N += 1

61 t0 = timetime()



64 print N test dt0

65 dt += dt0


67 if test == False

68 Nerr += [N]

69


71

72 if Nerr = []


74









ae2dpy








201






Nini = 4M (N+ 1) (B1)






Ff =fq+j1 fq+jkf

(B3)

Then we define

jm = mink=1kf

jk jM = mink=1kf

jk (B4)



(f jm jM) (B5)


Ns =

|Fs|sumk=1

(N+ 1+ jMiminus jmi

) (B6)




(B7)

Now we define

im = mink=1kfw=1wg

jk zw (B8a)

iM = maxk=1kfw=1wg

jk zw (B8b)


(g im iM) (B9)


Nd =

|Fd|sumk=1



)]

2

+

|Fd|sumk=1

[N+ 2 (iMkminus jmk

)]

(B10)












Y = [u(n m) v(n m)] (B12)
























b12 Code



2

















19


21 pass

22






28 if Anti

29 k1 = 0

30 k2 = 1

31 else

32 k1 = 1

33 k2 = 0

34 def f(ij)

35 if i == k1(j-n) + k2(dim - 1 - n -j)

36 if ngt0


37 return V[i]

38 else


40 else

41 return 0


43






49 NI = len(VV)




53 if DIR[0]DIR[1] == 1

54 adiag = True

55 else

56 adiag = False





60 return M

61



ImmutableMatrix


65



69 W = []

70 K = Mcols





75 W[i]reverse()

76 W[i] = W[i][K-1-i]



79 W[i] = W[i][i]

80 else


transpose()



84 W[i]reverse()


85 W[i] = W[i][i]



88 W[i] = W[i][K-1-i]

89 else


91

92 T = []





96 patterns = S[]



99 if S[i][K-j] == S[j]



102

103

104 return patterns

105



calculation of


constants


110 l = VARS[0]

111 m = VARS[1]

112 NN = 2NI

113 NEQ = len(y)

114 Fl = []

115 Fm = []

116 Flm = []



119 if CAF = []

120 CAFC = CAF[]

121 for C in CAF


123 continue






str(Cfunc))




for Ka in CLargs]))



str(Cfunc))


134 qv = []



137 qvappend(qs[q])



140 pv = []



143 pvappend(ps[p])


145 else


func)))





in CLargs]))


151 qpv = []






157 qpvm = min(qpv)

158 qpvM = max(qpv)

159 if qpvm == qpvM

160 if qpvm lt 0


162 else


164 else


166 else


func)))




171


















189 al = []

190 be = []




194 s1 = DIR[0]

195 s2 = DIR[1]

196 repl =



199 ll = int((2j - (-1)j + 1)4)

200 mm = int((2j + (-1)j - 1)4)



203

204 CAs =



207

208 replfunc =

209

210 for fl in Fl



213 for fm in Fm



216 for flm in Flm


218 ll = int((2i - (-1)i + 1)4)

219 mm = int((2i + (-1)i - 1)4)

220 if s1 == 1

221 l_s = s1(ll+flm[1])

222 else


223 l_s = s1(ll-flm[2])

224 if s2 == 1

225 m_s = s2(mm-(flm[2]-2flm[1]))

226 else

227 m_s = s2(mm-(2flm[2]-flm[1]))




231 if s1 == 1

232 l_s = s1(j+flm[1])

233 else

234 l_s = s1(j-flm[2])

235 if s2 == 1

236 m_s = s2(j+i+1-(flm[2]-2flm[1]))

237 else

238 m_s = s2(j+i+1-(2flm[2]-flm[1]))



241




244 NI = 8


246 EQ = list(EQ)


248 Y = list(Y)


250 EQ = [EQ]


252 Y = [Y]



255

256 NEQ = len(EQ)

257 if NEQ = len(Y)



260 else





265 if kwargs


267

268 y = []

269






)

274 l = Y[0]args[0]

275 m = Y[0]args[1]


277




str(Y[i]))



func))



)))


286


288

289 CA = set([])

290 CAF = set([])

291






297

298 for e in EQ







305 print(h)


307 if wis_Function

308 if wfunc in y



311 else


313 else



315 if l in CA

316 CAremove(l)

317 if m in CA

318 CAremove(m)

319 CA = list(CA)

320 CAF = list(CAF)

321


324 s2 = 1




(s1) str(s2)))




(s1) str(s2)))

332 ySol = ySol[0]




337 s2 = 1



(s1) str(s2)))




(s1)str(s2)))

344 ySol = ySol[0]




349 s2 = -1



(s1) str(s2)))





(s1) str(s2)))

355 ySol = ySol[0]




360 s2 = -1



(s1) str(s2)))




(s1) str(s2)))

366 ySol = ySol[0]





373 try





378 continue


380 if matrix_form


382 for E in ALL

383 for F in ALL

384 if E[1] = F[1]




388 else


391



394 CIm = dict()







400


402



405 step = (s1js2(j+i+1))






409 d = Max(d1d2)


411 print(ev)

412 print(d)


414 return True

415

416



419 outd = Queue()

420 out = Queue()








processes)))



processes)))




432 CIm = dict(CI)

433 CIclear()



[k][2]

436



][0] == k]



441 DKreverse()






447 break

448 if matrix_form

449 break

450 DS[k]append(DK)

451

452 if matrix_form

453 if NEQ == 1



range(NEQ)]

456

457 if NEQ == 1


459











y = [u(n) v(n)] (B13)


y = [u(n t) v(n t)] (B14)







Nini = 2

Msumk=1

(partek + 1) (B15)











2



6



9 NI = 8


11 f = list(f)


13 y = list(y)


15 f = [f]


17 y = [y]

18

19 NEQ = len(f)

20 if NEQ = len(y)

21 if NEQ gt len(y)


23 else


25






31 if kwargs


33




37 n = y[0]args[0]




40 n s = y[0]args

41 else


43 Y = [y[0]func]

44




str(y[i]))





52


54


56 pprint(f)

57

58 CA = set([])


60

61 for fs in f


63 coeff = SOne

64 kspec = None



67 for h in q


69 h = hargs[0]

70 if his_Function

71 if hfunc in Y






76 else



79 else



func))

82 else

83 coeff = h



85

86 if n in CA

87 CAremove(n)

88 if s in CA

89 CAremove(s)

90 if CA

91 CA = list(CA)

92




96


98









108 shuffle(rp)

109

110 rp0 = rp[02NEQ]

111 rp = rp[2NEQ]

112

113 CI =





118 CAs =



121


123 if CAs






129 ySol = ySol[0]


131

132


134













145 d = max(d1d2)

146 ds[y[k]] += [d]

147

148 return ds












p(z) = a0 + a1z+ aNzN (B17)



2




6

7



vector



is assumed




15 if M==0

16 return taylp




20 Pade approximant

21 else



24 if L == 0 and M ==0

25 return vect[0]

26

27 def c(i)


29 return 0

30 else

31 return vect[i]

32

33 if M == 0


35 def cfP(i)

36 if M-igtL

37 return 0

38 else

39 S = c(0)


41 S += c(k)zk






(1M+1)])






51




V A list of numbers










2





7 import warnings

8


10


approximant method



using





15



V))



str(s))

20 M = len(V)

21 N = floor((M-1)2)

22 j = 1

23 while(jltN+1)


25 ret = True


range(M)]

27 if Rv == V

28 return R

29 else

30 j = j + 1

31 return False




















2









11




str(f))










24 f = (rem)+f[1]


26 transf = 0

27 for z0 in sing




31





str(f))









43 return dn

44




str(f))





52 m = 0

53 for z0 in sing






227

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9



direction minus+ 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17

1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44 46 47

0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16

0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41 43 44

0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15

0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41

0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14

0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38

0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13

0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35

0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12

0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32

0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3 4 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 7 8

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 3 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C1)

23

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direction ++

17 17 16 16 15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1

47 46 44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1

16 16 15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0

44 43 41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0

15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0

41 40 38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0

14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0

38 37 35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0

13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0

35 34 32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0

12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0

32 31 29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0

11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0

29 28 26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

26 25 23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

23 22 20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 19 17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 16 14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 13 11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 5 4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 10 8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 4 3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 7 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 3 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C2)

C1

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1

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

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12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0 0 0

12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0

13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0 0 0 0 0

13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0

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14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0

15 40 14 37 13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0 0 0

15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0

16 43 15 40 14 37 13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1 0 0

16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0

17 46 16 43 15 40 14 37 13 34 12 31 11 28 10 25 9 22 8 19 7 16 6 13 5 10 4 7 3 4 2 1 1

17 47 16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1

(C3)

23

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es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4 10 5 13 6

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 3 7 4 10 5 13 6 16 7

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0 0 1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9 25 10 28 11 31 12 34 13 37 14 40 15 43 16

0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16

1 1 2 4 3 7 4 10 5 13 6 16 7 19 8 22 9 25 10 28 11 31 12 34 13 37 14 40 15 43 16 46 17

1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16 47 17

(C4)

C1

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direction minus+

1 2 4 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107 111 115 119 123

1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

0 1 1 2 4 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107 111 115

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 7 11 15 19

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 7 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C5)

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direction ++

123 119 115 111 107 103 99 95 91 87 83 79 75 71 67 63 59 55 51 47 43 39 35 31 27 23 19 15 11 7 4 2 1

62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1

115 111 107 103 99 95 91 87 83 79 75 71 67 63 59 55 51 47 43 39 35 31 27 23 19 15 11 7 4 2 1 1 0

58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0

107 103 99 95 91 87 83 79 75 71 67 63 59 55 51 47 43 39 35 31 27 23 19 15 11 7 4 2 1 1 0 0 0

54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0

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34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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51 47 43 39 35 31 27 23 19 15 11 7 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19 15 11 7 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 5 3 8 4

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C13)

24

2e

vo

lu

tio

nm

at

ric

es

direction ++

17 47 16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1

47 31 44 29 41 27 38 25 35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1

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15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0

41 27 38 25 35 23 32 21 29 19 26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0

14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0

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11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0

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26 17 23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

8 20 7 17 6 14 5 11 4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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4 8 3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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3 5 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C14)

C2H6

eq

ua

tio

ns

(1

93)

24

3

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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23 15 20 13 17 11 14 9 11 7 8 5 5 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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17 47 16 44 15 41 14 38 13 35 12 32 11 29 10 26 9 23 8 20 7 17 6 14 5 11 4 8 3 5 2 2 1

(C15)

24

4e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

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0 0 1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35 25 38 27 41 29 44

0 1 1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16

1 1 2 3 5 5 8 7 11 9 14 11 17 13 20 15 23 17 26 19 29 21 32 23 35 25 38 27 41 29 44 31 47

1 2 2 5 3 8 4 11 5 14 6 17 7 20 8 23 9 26 10 29 11 32 12 35 13 38 14 41 15 44 16 47 17

(C16)

C2H6

eq

ua

tio

ns

(1

93)

24

5


direction minus+

1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27 42 29 45 31 48

1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39 15 42 16 45 17

0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27 42 29 45

0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39 15 42 16

0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39 27 42

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0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15 11 18 13 21 15 24 17 27 19 30 21 33 23 36 25 39

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6 15 7 18 8

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 6 5 9 7 12 9 15

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 3 3 6 4 9 5 12 6

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(C17)

24

6e

vo

lu

tio

nm

at

ric

es

direction ++

48 31 45 29 42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1

17 45 16 42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1

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16 42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1 1 0 0

42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0

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21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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4 6 3 3 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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(C18)

C2H6

eq

ua

tio

ns

(1

93)

24

7

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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31 45 29 42 27 39 25 36 23 33 21 30 19 27 17 24 15 21 13 18 11 15 9 12 7 9 5 6 3 3 1 1 1

48 17 45 16 42 15 39 14 36 13 33 12 30 11 27 10 24 9 21 8 18 7 15 6 12 5 9 4 6 3 3 2 1

(C19)

24

8e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

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1 2 3 3 6 4 9 5 12 6 15 7 18 8 21 9 24 10 27 11 30 12 33 13 36 14 39 15 42 16 45 17 48

(C20)

C2H6

eq

ua

tio

ns

(1

93)

24

9


direction minus+

1 1 3 3 5 5 7 7 9 9 11 11 13 13 15 15 17 17 19 19 21 21 23 23 25 25 27 27 29 29 31 31 33

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(C21)

25

0e

vo

lu

tio

nm

at

ric

es

direction ++

33 31 31 29 29 27 27 25 25 23 23 21 21 19 19 17 17 15 15 13 13 11 11 9 9 7 7 5 5 3 3 1 1

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(C22)

C2H6

eq

ua

tio

ns

(1

93)

25

1

direction +minus

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33 2 31 2 29 2 27 2 25 2 23 2 21 2 19 2 17 2 15 2 13 2 11 2 9 2 7 2 5 2 3 2 1

(C23)

25

2e

vo

lu

tio

nm

at

ric

es


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1 2 3 2 5 2 7 2 9 2 11 2 13 2 15 2 17 2 19 2 21 2 23 2 25 2 27 2 29 2 31 2 33

(C24)

C2H6

eq

ua

tio

ns

(1

93)

25

3


direction minus+

1 2 4 6 11 10 18 14 25 18 32 22 39 26 46 30 53 34 60 38 67 42 74 46 81 50 88 54 95 58 102 62 109

1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C25)

25

4e

vo

lu

tio

nm

at

ric

es

direction ++

109 62 102 58 95 54 88 50 81 46 74 42 67 38 60 34 53 30 46 26 39 22 32 18 25 14 18 10 11 6 4 2 1

62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1

102 58 95 54 88 50 81 46 74 42 67 38 60 34 53 30 46 26 39 22 32 18 25 14 18 10 11 6 4 2 1 1 0

58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0

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30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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(C26)

C2H6

eq

ua

tio

ns

(1

93)

25

5

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

(C33)

26

2e

vo

lu

tio

nm

at

ric

es

direction ++

123 62 115 58 107 54 99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1

61 60 57 56 53 52 49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1

115 58 107 54 99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0

57 56 53 52 49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0

107 54 99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0

53 52 49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0

99 50 91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0

49 48 45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0

91 46 83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0

45 44 41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0

83 42 75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0

41 40 37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0

75 38 67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0

37 36 33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

67 34 59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

33 32 29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

59 30 51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

29 28 25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

51 26 43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

25 24 21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

43 22 35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

21 20 17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

35 18 27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

17 16 13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

27 14 19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

13 12 9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19 10 11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

9 8 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

(C34)

C2H6

eq

ua

tio

ns

(1

93)

26

3

direction +minus

1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0

38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0

75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0 0 0

42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0 0 0

83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0 0 0

46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0 0 0

91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0 0 0

50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0 0 0

99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0 0 0

54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0 0 0

107 53 99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0 0 0

58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1 0 0

115 57 107 53 99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1 1 0

62 60 58 56 54 52 50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 1 1

123 61 115 57 107 53 99 49 91 45 83 41 75 37 67 33 59 29 51 25 43 21 35 17 27 13 19 9 11 5 4 2 1

(C35)

26

4e

vo

lu

tio

nm

at

ric

es


0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67

0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38

0 0 0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75

0 0 0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

0 0 0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83

0 0 0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

0 0 0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91

0 0 0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

0 0 0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99

0 0 0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

0 0 0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99 53 107

0 0 1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58

0 1 1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99 53 107 57 115

1 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62

1 2 4 5 11 9 19 13 27 17 35 21 43 25 51 29 59 33 67 37 75 41 83 45 91 49 99 53 107 57 115 61 123

(C36)





un+1m + unmminus1


partz(++)

partun+1m+partz(++)

partunm



partz(++)



(D1a)

un+1m + unm+1

u2nm+1 minus unm

partz(++)


partunm

+unminus1m + unm+1

u2nm+1 minus unm

partz(++)



(D1b)


z(++) = log

[C(++)1



)] (D2)


z(minus+) = log

[C(minus+)1



)] (D3a)

265



(unm + unminus1m


)] (D3b)


(unm + unminus1m


)] (D3c)


C(+minus)1 = minusC

(++)1 C

(minusminus)1 = C

(minus+)1 (D4)


v(++) = log

[C(minus+)2

(unm minus u2n+1m


)] (D5a)

v(minus+) = log

[C(minus+)2

(u2nm minus un+1m


)] (D5b)


(un+1m + un+1m


)] (D5c)


(un+1m + unm


)] (D5d)


C(+minus)2 = minusC

(++)2 C

(minusminus)2 = C

(minus+)2 (D6)


C(++)2 = minusC

(++)1 C

(minus+)2 = minusC

(minus+)1 (D7)


g = F(+)n F

(+)m

[C(++)1



)+Φ(++)

]

+ F(minus)n F

(+)m

[C(minus+)1



]

+ F(+)n F

(minus)m

[C(++)1

unm + unminus1m


]+ F

(minus)n F

(minus)m

[C(minus+)1

unm + unminus1m


]

(D8)



1 = minusC(++)1 and


2K1unm minus

1

2C(++)1

Φ(+minus) =1

2K1unm +

1


1

2K1unm +

1

2C(++)1

(D9)


g =F(+)n F


1

2

(F(minus)m minus F

(minus)n F

(+)m



+F(minus)n F

(+)m u2nm +

(F(minus)m minus F

(+)n F

(+)m

)unm


(D10)


gP =[F(+)n

(2F

(+)m + F

(minus)m

)+ F

(minus)n

]unm (D11)



g =F(+)n F


1

2

(F(minus)n minus F

(+)n F

(minus)m



+F(+)n F

(minus)m u2nm +

(F(minus)n minus F

(+)n F

(+)m

)unm


(D12)



direction m






(minus)n unmunm+1)

(D13)


w = F(+)n w(+) + F



minusα2

(1+ ε2u2n+1m


2 + ε2α22

partz(+)

partun+1m+partz(+)

partunm

+α2

(1+ ε2u2nminus1m


2 + ε2α22

partz(+)

partunminus1m=

2ε2un+1m


(D15a)

minusα2

(1+ ε2u2n+1m

)(unm minus unm+1)

2 + ε2α22

partz(+)

partun+1m+partz(+)

partunm

+α2

(1+ ε2u2nminus1m

)(unm minus unm+1)

2 + ε2α22

partz(+)

partunminus1m=

2ε2un+1m


(D15b)



)+ logZ(+)

nm





)+ logZ(minus)





(plusmn)n ie


v(+) = log


1+ ε2u2n+1m

]

+ logV(+)nm



)

(D18a)



] (D18b)


nm = V(plusmn)n



Z(+)n (ξ) = C

(1)n minus

(1+ ξ2

)V(+)n (ξ) (D19a)

V(minus)n (ξ) = C


Then defining

G(+)n (ξ) =

intV(+)n (ξ)dξ G

(minus)n (ξ) =



g =minusF

(+)n

εGn



)+ F

(+)n

[C(1)n


]minus F



+ F(minus)n

[(1+ ε2u2nm)C


]

(D21)




C(1)n = C

(2)n = 0 Φ(+) = C

(3)n Φ(minus) = C

(4)n (1+ ε2u2nm) (D22)



g = F(+)n

[minus1

εG

(+)n



)+C

(3)n

]+ F

(minus)n

[minus(1+ ε2u2nm)G


(4)n

]


(D23)




gGS = F(+)n

[minus1

εG

(+)n



)]+ F

(minus)n

[minus(1+ ε2u2nm)G


] (D24)


gP = F(+)n C

(3)n + F


(4)n (D25)










)= 0

(E1a)




(minus)n+munm + 2F

(+)n+mun+1m +α+β

)middot

middot(2F


(+)n+munm+1 +α+β

)= 0

(E1b)




αβ



)= 0

(E1c)




XrHε1n =


(minus)n+mu

2nm


partunm (E2a)

271


XrHε1m =


(minus)n+mu

2nm


partunm

(E2b)

XrHε2n =

(1minus 4εαF

(minus)n+m





(minus)n+mu

2nm +

(1minus 4εαF

(minus)n+m

)unm


partunm

(E2c)

XrHε2m =

(1minus 4εβF

(minus)n+m





(minus)n+mu

2nm +

(1minus 4εβF

(minus)n+m

)unm

unm+1 + unmminus1

partunm

(E2d)

XrHε3n =

[1

2



minusε

α


(minus)n+mu

2nm


partunm

(E2e)

XrHε3m =

[1

2



minusε

β


(minus)n+mu

2nm

)unm+1 + unmminus1

partunm

(E2f)




dumdt


um+1 minus umminus1

(F1)










221 + a

222 + a

223


(F4a)

b =1


+1


+1

2(minus1)na21(a50 + a30 + a40)

+1

2(minus1)n(a23a32 + a1a61 + a20a51 + a22a43)

(F4b)

β =1


minus1


+1


+1


(F4c)



273




d =1

2(a250 minus a

240 minus a

230 + a1a7 + a

232 + a

243 minus a

251)

+ 2(minus1)n(a21a60 + a20a61 + a23a62 + a22a63)

(F4f)

e =1


+1

2(a20a7 minus a51a61 + a32a62 + a43a63)

minus1


minus1

2(minus1)na61(a30 + a40 + a50)

(F4g)

ε =1


+1

2(a32a60 minus a43a61 + a51a63 + a22a7)

+1


+1


(F4h)


262 + a

263 minus a

261


(F4i)




Wi = F(+)m W

(+)i + F

(minus)m W

(minus)i (G1)


partW(+)1

partun+1m+partW

(+)1

partunm= 0 (G2a)

(1+ ε2u2n+1m

) partW(minus)1


) partW(minus)1

partunm= 0 (G2b)

whose solution is


(minus)m G

(un+1m minus unm

1+ ε2unmun+1m

)(G3)


G (ξ) = F

(α2ξ

) (G4)

This yields

W1 = F(+)m F

(α2

un+1m minus unm

)+F

(minus)m F

(un+1m minus unm

1+ ε2unmun+1m

) (G5)


275



α2(1+ ε2u2nm+1

) partW(+)2

partunm+1


] partW(+)2

partunm


) partW(+)2

partunmminus1= 0

(G6a)

α2(1+ ε2u2n+1m

) partW(minus)2

partunm+1


(minus)2

partunm

+α2(1+ ε2u2n+1m

) partW(minus)2

partunmminus1= 0

(G6b)


W2 = F(+)m F



)+ F


(G7)



W1 = F(+)m

α2un+1m minus unm

+ F(minus)m

un+1m minus unm


W2 = F(+)m α




(G8b)




ε1 E Q U AT I O N







un2kminus1 =1

vnk+αk ρ2k =


(H2)


)vnk+1 + ε

2αk+1 =(1+ ε2α2k

)vnk + ε

2αk (H3)



1+ ε2α2k (H4)



(H5)



Therefore we obtain


(minus)m


(H7)

277




(H8)















(1967) pp 292ndash296

279

280 Bibliography



















Bibliography 281



(10 pp)








(1993) pp 111ndash125








282 Bibliography



(1983) pp 1ndash22








(23pp)






Bibliography 283













284 Bibliography

















Bibliography 285






(1895) pp 422ndash443


(1979) pp 395ndash404









286 Bibliography


(2008) 077(14pp)













λ2a2





Bibliography 287















(1984) pp 131ndash160

288 Bibliography






(1962) pp 439ndash442






(2007) pp 253ndash269








Bibliography 289


















290 Bibliography
















Bibliography 291



colophon




Dedication

Abstract

Sommario

Acknowledgments

Ringraziamenti

Contents

List of Figures

List of Tables

Introduction


























332 H6 equations









4311 1 equation

4312 2 equation

4313 3 equation

432 H6 equations

4321 1D2 equation

4322 2D2 equation

4323 3D2 equation

4324 D3 equation

4325 1D4 equation

4326 2D4 equation



4411 1D2 equation

4412 2D2 equation

4413 3D2 equation


4421 D3 equation

4422 1D4 equation

4423 2D4 equation


4431 The 1 equation

4432 The 2 equation

4433 The 3 equation

Conclusions





B12 Code



























Bibliography

Colophon

integrability properties of quad equations consistent on the cube · versity in october 2013 i knew...

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