symmetries of partial differential equations: conservation laws â€” applications â€”...

Symmetries of Partial Differential Equations Conservation Laws - Applications - Algorithms
Edited by
A.M. VINOGRADOV Moscow Stale University, U.S.S.R.
Reprinted from Acta Applicandae Mathematicae, Vol. 15, Nos. 1 & 2 and Vol. 16, Nos. 1 & 2
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging in Publication Data
Symmetries of partial differential equations conservation laws. applications. algorithms / edited by A.M. Vinogradov.
p. em. Translated from Russian. "Reprinted from Acta applicandae mathematicae. volume 15. no. 1-2
and volume 16. no. 1-2."
1. Differential equations. Partial--Numerical solutions. I. Vinogradov. A. M. (Aleksandr Mikha,lovich) II. Acta appllcandae mathematicae. QA377.S963 1990 515' .353--dc20 89-26687
ISBN-13: 978-94-010-7370-7 DOl: 10.1007/978-94-009-1948-8
e-ISBN-13: 978-94-009-1948-8
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Table of Contents
A. M. VINOGRADOV / Foreword
A. M. VINOGRADOV / Symmetries and Conservation Laws of Partial Differential Equations: Basic Notions and Results 3
V. N. GUSYATNIKOVA, A. V. SAMOKHIN, V. S. TITOV, A. M. VINOGRADOV, and V. A. YUMAGUZHIN / Symmetries and Conservation Laws of Kadomtsev-Pogutse Equations (Their computation and first applications) 23
V. N. GUSYATNIKOVA and V. A. YUMAGUZHIN / Symmetries and Conserva- tion Laws of Navier-Stokes Equations 65
N. O. SHAROMET / Symmetries, Invariant Solutions and Conservation Laws of the Nonlinear Acoustics Equation 83
A. M. VERBOVETSKY / Local Nonintegrability of Long-Short Wave Interaction Equations 121
V. S. TITOV / On Symmetries and Conservation Laws of the Equations of Shallow Water with an Axisymmetric Profile of Bottom 137
YU. R. ROMANOVSKY / On Symmetries of the Heat Equation 149
I. S. KRASIL'SHCHIK and A. M. VINOGRADOV / Nonlocal Trends in the Geometry of Differential Equations: Symmetries, Conservation Laws, and Backlund Transformations 161
PART II (Acta Appl. Math. 16, 1-142)
A. N. LEZNOV and M. V. SA VELIEV / Exactly and Completely Integrable Nonlinear Dynamical Systems
B. G. KONOPELCHENKO / Recursion and Group Structures of Soliton Equations 75
V. O. BYTEV / Building of Mathematical Models of Continuum Media on the Basis of the Invariance Principle 117
TABLE OF CONTENTS
vi
A. V. BOCHAROV and M. L. BRONSlEIN / Efficiently Implementing Two Methods of the Geometrical Theory of Differential Equations: An Experience in Algorithm and Software Design 143
E. V. PANKRAT'EV / Computations in Differential and Difference Modules 167
V. L. TOPUNOV / Reducing Systems of Linear Differential Equations to a Passive Form 191
PAlJL H. M. KERSTEN / Software to Compute Infinitesimal Symmetries of Exterior Differential Systems, with Applications 207
P. K. H. GRAGERT / Lie Algebra Computations 231
Acta Applicandae Mathematicae 15: 1-2, 1989. © I <i89 Kluwer Academic Publishers.
Foreword
This special double issue of Acta Applicandae Mathematicae and two further issues to appear in the future, are devoted to recent developments in the theory of symmetries and conservation laws for general systems of partial differential equations.
The first part deals with the problem of how to find all higher symmetries and conservation laws for a given differential equation. It opens with an article in which the necessary theoretic results are summarized and which is the starting point for the subsequent articles. The above-stated problem is partially or completely solved for concrete equations of mathematical physics which are chosen to illustrate, from different points of view, both the techniques of computations as well as the characters of the problems which arise. Some attention is also paid to applications. These are mainly related to invariant solutions. The first steps toward the general non local theory of symmetries and conservation laws are made in the final paper.
The following two points are to be stressed. Firstly, equations with more than two independent variables are mainly investigated here and, for some of these, all higher symmetries and conservation laws are computed completely. Up to now, similar strong results were obtained only for equations with two independent variables. Secondly, the presented method of generating functions for finding conservation laws does not depend on any symmetry considerations. In parti cular, it works effectively in situations where Noether-type theorems are not applicable.
The Leznov and Saveliev and Konopel'chenko papers in the second part of this special volume deal with some constructions of 'integrable' (in a sense) systems of nonlinear differential equations. Many interesting properties of these systems arising from their nature, are found here as well as explicit formulae for some wide classes of their solutions. In other words, some relations between two conceptions of 'integrability' and 'symmetry' are investigated in these papers. In the final paper of the second part, written by O. Byter, some methods for modelling continuum media based on symmetry considerations are proposed.
It is now obvious that effective applications of symmetry methods to in vestigate concrete equation are impossible without considerable computer sup port. For this reason, the third part of these special issues concerns these topics. Some recent trends along this line are presented in it. However, this issue is far from a full account of modern activities in this field.
In this volume, the editor has tried to reflect all the main recent results and tendencies in the considered domain. Of course, such a goal cannot be reached, but I hope that the reader will find here a satisfactory approximation of it.
2 FOREWORD
The authors of these issues involve not only mathematicians, but also speci alists in (mathematical) physics and computer sciences. So here the reader will find different points of view and approaches to the considered field.
A. M. VINOGRADOV
Acta Applicandae Mathematicae 15: 3-21, 1989. © 1989 Kluwer Academic Publishers.
Symmetries and Conservation Laws of Partial Differential Equations: Basic Notions and Results
A. M. VINOORADOV Department of Mathematics, Moscow State University, 117234, Moscow, U.S.S.R.
(Received: 22 August 1988)
3
Abstract. The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed.
AMS subject classifications (1980). 35A30, 58005, 58035, 58H05.
Key words. Higher symmetries, conservation laws, partial differential equations, infinitely prolonged equations, generating functions.
o. Introduction
In this paper we present the basic notions and results from the general theory of local symmetries and conservation laws of partial differential equations. More exactly, we will focus our attention on the main conceptual points as well as on the problem of how to find all higher symmetries and conservation laws for a given system of partial differential equations. Also, some general views and perspectives will be discussed. The material of this paper is used in other papers [7]-[12] of this volume in which the general theory is applied to concrete equations of mathematical physics. This demonstrates the theory in action.
Presented here are the results on higher symmetries and conservation laws found by the author in 1975-1977 and its resume was published in a short note [1] (see also [2]). However, the author was not successful in publishing full details until 1984. Between these years, other authors developed similar ideas (Ibragi mov [5], Olver [6] etc.) in the field of symmetry theory. Tsujishita's work [17] contains some results on conservation laws which are close to ours. Higher symmetries and conservation laws for spatially one-dimensional evolution equa tions have been the subject of many works on the investigation of equations integrable by the inverse scattering transform method. We do not touch on these very interesting but very special topics in this paper.
In further exposition we will use the usual coordinate language for mathemati cal physics. However, this is not the best way to think of these substances. We omit here both motivations of basic notions (these are given in [3]) and proofs or
4 A. M. VINOGRADOV
their indications for the presented results. An interested reader will cover this gap by consulting [4,13,17]. We also recommend paper [14] in which all the main technological details necessary for finding symmetries and conservation laws are demonstrated in a concrete example.
Probably the most interesting new point of the theory presented below is that in principle, it makes it possible to find all higher conservation laws for arbitrary (nonlinear) differential equations. In particular, it works effectively well in situations when the Nother theorem, as well as other symmetry considerations, are not applicable. How it looks from the practical point of view will be clear from subsequent papers.
1. On Terminology
Below, the theory of higher local infinitesimal symmetries and conservation laws of partial differential equations is discussed. We use the adjective 'higher' to stress that the symmetries and conservation laws under consideration are de scribed by means of expressions containing arbitrary order derivatives of quan tities, entering into investigating differential equations.
The adjective 'local' is used to point out that we deal with symmetries and conservation laws which admit localizations on arbitrary domains in the space of independent variables. Foundations of nonlocal theory are considered in [15] in this volume.
The classical symmetries theory, originated by S. Lie, operates, with first-order derivatives. By speaking of 'higher symmetries', we underline the aspect which differentiates the modern theory from the classical one. Some authors use 'generalized symmetries' or 'Lie-Backlund transformations' in the same sense. The last term seems to be very misleading because the notion of 'Backlund transformation' is a concept of a quite different nature. In particular, higher symmetries are infinitesimal transformations, but Backlund transformations are finite ones.
Below, using the word 'symmetry', we have in mind 'higher local symmetry'.
2. Infinitely Prolonged Equations
Informally, infinitesimal symmetries are infinitesimal transformations of manifolds of infinitely prolonged equations which conserve their natural contact structures. For this reason, we consider these notions in more detail.
Let x = (XI, ... , xn) be independent variables and u = (u l , ... , urn) be depen dent ones. Geometrically, this means that we deal with a smooth fibre bundle 1T: E ~ M, x's are the base coordinates in it and u's are the fibre coordinates.
In some situations below, the multiindexes
u = (iI, ... ,in), i = 1, 2, ... , n,
SYMMETRIES AND CONSERVATION LAWS OF PDEs
will be written in the form
u= 1 ... 1, ~
iJl u I iJ i, +- - -+in
iJxu axi' ... iJx~'
supposing that u = (il , ... , in) and I u I = il + ... + in .
5
Now let us introduce the variables p~, 1 ~ i ~ m, 'flu. The manifold with local coordinates x, u, p~, 1 ~ I ul ~ k, 1 ~ i ~ m, is said to be the manifolds of the kth order jets of 7T, or simply the kth jet manifold. More exactly, variables x, u, ... ,p~, ... are local coordinates on the manifold J k 7T of kth-order jets associated with the fibre bundle 7T. If k < 00, then J k 7T is a finite-dimensional manifold. However, infinite-dimensional manifolds Joo = Joo 7T will be of most interest to us. The local coordinates on Joo are x, u, p~, where 1 ~ i ~ m, I u I < 00. A smooth function on J 00 is, by definition, a smooth function depending only on a finite number of variables x, u, p~. The entity of all smooth functions on Joo will be denoted by 8fr= 8fr(7T). The part of 8fr consisting of all functions depending only on variables x, u, p~, with I ul ~ k, will be denoted by 8frk = 8frk ( 7T). Evidently
Below, we will write pu instead of p~ in the case m = 1 and will sometimes use the notation p~ instead of u i •
It will be convenient for us to trait a kth-order system of partial differential equation (PDE)
(1)
(Here, U(s) denotes the totality of all derivatives iJlului/iJxu, lui = k) as a sub manifold cy in Jk which is given by the equations
FI(X,u, ... ,p:;,., ... )=O} lul~k. FI (x, U, ... , p:;", ... ) = 0
(I')
For example, from this point of view, the wave equation looks like the hyperplane in the space of variables Xl = x, X2 = t, U, PI, P2, Pll, P12, P22 whose equation is Ptt - P22 = 0 (or P(2.0) - P(O,2) = 0 if one uses the standard notations for multi-indexes.)
The full derivative operator with respect to Xk D k : 8fr ~ 8fr is defined by the
6 A. M. VINOGRADOV
(JXk u,s (Jp"
Here ak = (iI, ... , ik + 1, ... , in), supposing that a = (iI, ... , in). If m = 1, then
Let
DO' = D\. 0··· 0 D~
if a = (it , ... , in). Evidently, D;Dj = Dp; . The submanifold Woo in Joo which is given by the infinite system of PDE,
Du(F;) = 0, Va, i (2)
is called the infinite prolongation of the system W defined by (1). Generally, Woo is an infinite-dimensional manifold.
Local coordinates x, U, p~ on Joo being restricted on Woo, are not yet independent. In fact, some corrdinates can be expressed through others by using Equation (2). In other words, these coordinates may be divided into two parts: internal and external, with respect to Woo. More exactly, the maximal functionally independent part of coordinates x, U, p~ on Woo is said to be internal on Woo. Clearly, internal coordinates on Woo may be chosen in many different ways. But if such a choice is made, the remaining coordinates are said to be external with respect to Woo. For example, coordinates XI = x, X2 = t, P2 ... 2, PI2 ... 2 (the num ber of 2's here is arbitrary) may be chosen to be internal for an infinitely prolonged wave equation. In fact, for this equation system (2) may be rewritten in the form
Pull = PO'22, Va.
Another way to choose internal coordinates in this case is XI X2, PI ... J, P21 ... 1
(the number of units in the multiindexes is arbitrary). The restriction of a function f E g; on Woo will be denoted by f. If a choice of
internal and external coordinates on Woo is made, then the analytical description of 1 is as follows.
If p~ (respectively, x) is an internal coordinate, then p~ = p~ (respectively, Xj = x). If p~ is one of the external coordinates, then p~ = <p ~( ••• p~ ... ) where <p:" is a function of the internal coordinates. For example, if the internal coordinates on the infinitely prolonged wave equation are chosen by the first of the two ways described above, then
Pl1 = P22, PI12 = P222 , P111 = P122,
Pl1l1 = P2222 , etc.
SYMMETRIES AND CONSERVATION LAWS OF PDEs 7
Finally, if f = 1(···, xi"'" p~, .. . ), then f = 1(· .. , xi"'" p~, .. . ). Operators of the form
constitute a natural class of operators on Joo, called a C-differential. Here aCT is a s x s matrix whose elements are smooth functions on Joo and it is supposed that d acts on columns I = ([I , ... , Is)', Ii E gji, by the rule
d/= I aCT (I?CTh). iCTi",k DCTls
The main property of C-diflerential operators is that they are restrictable on submanifolds of the form Woo in Joo. Namely, the restriction i5i of Di on Woo is given by the equality
- a I' . a D·=-+ }-. I" PCTj a }, uXi i.CT PCT
where I' means that the summing is taken over all internal p~ and it is supposed that all Xi are chosen to be internal coordinates. Next, 15" = 15 PI 0 i5 pz 0 ••• 0 i5 P,
if (T = (PI, P2, .. " p,) and
~ = I aCT i5CT , iCTi",k
where the elements of the matrix aCT are restrictions of the corresponding elements of a" on Woo.
The described restriction operation has an inner invariant sense and, in particular, does not depend on the choice of internal coordinates on Woo. Moreover, C-diflerential operators are completely characterized by their pro perty to be restrictable on every submanifold of the form '&" in Joo.
3. Contact Structure on Infinite Jets and Infinitesimal Transformations
Geometrically, full derivative operators may be treated as vector fields on Joo. Namely, the vector of the vector field corresponding to Di and having its origin at the point (J = ( ... , xi' ... , p~, ... ) E Joo has the numbers 8ii and P~i as its components with respect to coordinates Xi and p~, respectively. It shows that the vectors corresponding to D J , ••• , Dn are independent at each point (J E J=. Therefore, they generate an n-dimensional plane Co which is tangent to J= at (J.
The field (J~ Co of n-planes on Joo is called the Cartan distribution, or the contact structure of infinite order on J=. A n-dimensional submanifold L c Joo is said to be integral if its tangent plane at (J E L coincides with Co for every (J.
One can associate with every vector function 'P = ('P\(x), ... , 'Pm(x» (m is the
8 A. M. VINOGRADOV
number of dependent variables) the n-dimensional submanifold L<p in Joo, which is given by the following equations
L<p~l~~~~~~: p;"=--
ax" ---------
Manifold L0 L<p, one can think that 'the space' of all smooth vector functions cp(x) is geometrically realized as the totality of all integral submanifolds in Joo.
Now, let <1>: Joo-->o Joo be a smooth transformation. We will say that it conserves the contact structure on Joo if it maps Co onto C(O) for every 0 E Joo. We will call such transformations contact ones. Evidently, every contact maps integral submanifolds into integral ones. In that case, (L<p) is an integral submanifold and, therefore, has (locally) the form Lop for a suitable vector function 1/1. So, in this way generates a transformation cp"""" 1/1 in 'the space' of smooth vector functions. In the following, we will need the infinitesimal variant of the above construction.
Infinitesimal transformation Xi-->OXi + ea;, p~-->op{,.+ea{,., where ai, a{,.E8i', may be thought of as a vector field on JOO or, equivalently, as the operator
n iJ . iJ X= I ai-+ Ia{,.-j'
i~1 aXi j,,, iJp"
Then the condition that it conserves the contact structure on JOO may be formulated in the form
n
[X, D;J = I Aij~, i = 1, ... , n, j~1
where [X, DJ = X 0 Di - Di 0 X and Aij are some functions on Joo. The last equations with undetermined functions Aij may be solved explicitly. The result is that
n
where f= (fl,'" ,fm), t, J.L; E 8i' and
a 3f = I Du(h) -a j'
u,j p"
(3)
(4)
Functions t, JLi may be arbitrary in these formulas. The operator 3f is said to be the evolution differentiation corresponding to the generating function f.
Infinitesimal contact transformations of the form X = L /LiDi transform every integral submanifold in ] 00 into itself. So, via the above construction, they generate the identity transformation of the space of vector functions cp(x). In that sense, these transformations are trivial. By the reason, it is natural to identify infinitesimal contact transformations X and Y if X - Y = I /LiDi. In other words, we identify such contact fields which have the common evolution part 3f in its decomposition (3).
Infinitesimal transformation in the space of vector functions cp(x) generated by the field (3) via the above-mentioned construction is
CPi(X) >-+ CPi(X) + Egi(X), i = 1, ... , m,
where the gi(X)'S are obtained from fi (x, u, ... , p~, ... ) via substitutions
In other words, the field (3) generates a flow in the space of vector functions cP (x) which is described by the following evolution-type system:
acpi ( aluICPi(X)) - = fi x, cp(x), ... , , ... , aT axu
(5)
where T (= 'time') is a new independent variable. Commutators of evolution differentiations are also evolution differentiations.
More exactly,
{f, g} = ({f, gh, ... , {f, g}m).
It follows from (6) that evolution differentiations constitute a Lie algebra with respect to the usual commutators operation as well as its generating functions with respect to the bracket {., .}. The latter is called the higher Jacobi bracket. The correspondence f>-+ 3! identifies these two algebras.
4. Higher Symmetries of Differential Equations
Let <1>: Joo ~ Joo be a contact transformation. It is a symmetry of Equations (1) if the corresponding flow on the space of the vector functions transforms the solutions of (1) into its solutions.
10 A. M. VINOGRADOV
The problem of finding all symmetries of a given PDE system is equivalent to solving a new nonlinear PDE system which is, as a rule, much more complicated than the initial one. Therefore, in practice, one cannot use symmetry con siderations to investigate a given equation, because it is not possible to find its symmetries in an explicit form. However, affairs change considerably if one adopts the infinitesimal point of view.
Namely, let us understand infinitesimal contact transformations of Joo as vector fields of the form (3). Such a field is a higher infinitesimal symmetry of (1) if it tangents to the infinite prolongation Woo of (1). This is equivalent to the fact that the flow generated by this field on the space of the vector functions (see above) transforms solutions of (1) into themselves. Vector fields of the form I !LjDj are tangent to every submanifold of the form Woo in Joo. So vector field (3) is tangent to a given Woo if and only if it does its 'evolution part' 3",. The latter is equivalent to the fact that
(7)
(8)
Now, let us introduce the universal linearization operator IF by putting
Taking into account the coordinate expression (4) for 3f, one can obtain the following coordinate expression for IF:
IF = (Ii~--~~-:Iri-~~) L -1 D" ... L ---,;; D" " ap" " iip"
or, in an alternative form,
It shows that IF is a C-differential operator and therefore may be restricted on Woo. Denote this restriction by IF. Then the equality (8) expressing the fact that cp is generating function of an infinitesimal symmetry of (1) may be rewritten in the form
(9)
This linear equation is the basic in the symmetry theory.
Let Sym uy denote the totality of all higher infinitesimal symmetries of a PDE system uy. This is a linear space which may be identified with the solution space of Equation (9). In the following, we will identify infinitesimal symmetries with generating functions or, more exactly, with their restrictions on UYoo.
If !p, (fr are symmetries of UY, then the Jacobi bracket {cp, "'} is also a symmetry of uy. Therefore, Sym uy is a Lie algebra with respect to the Jacobi bracket operation. This simple fact may be very useful within the process of solving (9) (i.e., within the process of finding Sym UY) because it makes it possible to generate new solutions of (9).
The commutator relation
(10)
where A is a C-differential operator on UYoo, is also very useful in practice, as a consequence of (9). The indetermined operator A entering into it may be evaluated in some situations. In particular, it is equal to zero for scalar evolution equations.
Let 91 c Sym uy be a Lie subalgebra. Solutions of (1) which don't change under actions of infinitesimal transformations belonging to 121, are said to be l2I-invariant. If cp (x) is an \.JI-invariant solution, then it does not change under the action of flow (5) for every f E 21. This means that
( aiuicp ) f x,cp(x), ... ,--, ... =0. ax"
Therefore, \.JI-invariant solutions of (1) satisfy the system
F=O, h = 0, ... , fs = 0,
where {fl, ... , fs } is a basis of \'1. Evidently, the converse is also true. This system for finding 21-invariant solutions is overdetermined. It is much simpler to solve than the initial system (1). Roughly speaking, under some regularity conditions, this overdetermined system is equivalent to a PDE system with n - s independent variables, where s is the dimension of 21.
5. Classical and Higher Symmetries
A classical symmetry of (1) is a transformation x ~ x' = f(x, u), u~ u' = g(x, u) of dependent and independent variables which also maps solutions of (1) into solutions. A classical infinitesimal symmetry is a vector field on the space JO of dependent and independent variables, say
12 A. M. VINOGRADOV
Xi ~ Xi + Eai(X, U),
maps solutions of (1) into themselves. The field Xu may be canonically lifted up to a contact field X on Joo.
Generating function 'P = ('PI, ... , 'Pm) of X is given by the formula
n
'Pj=bj(x,u)- L Ptak(X,U). k~1
This shows which part of 'higher' theory is the classical one. Namely, the generating functions of higher symmetries may be arbitrary functions of variables X, u"p~, 0<10"1<00. But the generating functions of classical symmetries may depend only on variables X, u, p} and, in addition, this dependence is of a special sort.
From the classical point of view, the case m> 1 described above differs from the case m = 1 (m is the number of dependent variables). In the last case, infinitesimal transformation of the space JI with local coordinates XI, ... , Xn , u, PI, ... , Pn which conserve the 'integrability condition'
du - PI dXI - ..• - Pn dXn = 0
are to be considered. These are classical contact fields on the space J I • Every such field has the form
X f = L --+ f- L pi- -+ L -+Pi- -, 1 af a ( n af) a n (a f af) a i~1 api aXi i~1 api au i~1 aXi au api
where f = f(x, u, p) may be an arbitrary function on J I .
Vector field XI can be canonically lifted up to a contact vector field X on Joo. Moreover,
X=3f+L af D i . ap;
Therefore, in the case m = 1, the classical theory is more richer than in the case m>1.
6. Existence of Higher Symmetries
How wide is the class of equations admitting nonclassical symmetries? At a first glance the answer is evident: this class is much wider than the class admitting only classical symmetries. This is because there are many more potential can didates to be a higher symmetry for a given equation than a classical one. However, there are no 'nondegenerated' nonlinear equations with the number of independent variables greater than two which possess nonclassical symmetries,
i.e., essentially higher symmetries. This phenomenon was first observed during the course of studying concrete equations.
But what prevents differential equations from having symmetries? In a more constructive form, this question means: how does one exactly describe the 'inhomogeneosities' of diffeential equations which make it nonsymmetric? In classical differential geometry, such inhomogeneosities are nothing more than differential invariants (for example, the curvature in Riemannian geometry). For this reason, it is natural to suppose that some similar things have to be in the same situation as those we are interested in. In fact, the desirable analogy exists but it is necessary to use elements of the secondary differential calculus in order to describe it (see [16]). Of course, we have no possibility to do this here and instead, we will give some informal explanations of why differential equations with more than two independent variables generally have no higher symmetries.
Let us restrict ourselves to the case of one scalar equation of the second order. The main symbol (= the second-order part of the left-hand side of such an equation) is the simplest of its differential invariants. Supposing that this symbol is nondegenerated, one can consider it as a map assigning to each of the initial equation solution a conform metric on the space of the independent variables. 'Conform' here means 'up to an arbitrary functional multiplier'. But all two dimensional conform metrics are the same. In other words, they are 'flat'. In particular, the so-called conform curvature tensor is identically equal to zero in that case. Contrary, in dimensions greater than two, the conform curvature tensor is nontrivial and, moreover, other nontrivial conform invariants exist. Then nontriviality of such invariants for a given equation, simply reflects its nonlinearity in the second-order derivatives. So these invariants whose non triviality results from the nonlinearity of the symbol of the considered equation does not allow it to be symmetric.
However, it would be a great mistake to restrict the higher symmetries theory only to the cases of one or two independent variables, taking into account what was said before. First of all, the 'higher' point of view considerably simplifies the classical theory and, in particular, simplifies calculations during the course of finding classical symmetries. Next, a development of the higher symmetries theory naturally leads to the discovery of phenomena which make differential equations nonsymmetric. These are of great importance for the theory of differential equations. And, finally, it stimulates the search for a more general point of view on the concept of symmetry in the field of differential equations. In that direction, the transition on the nonlocal point of view looks very promising. It is discussed in the paper [15] of this volume.
7. Conservation Laws
The notion of a conservation law for a given differential equation is a concept dual in the sense of the conception of symmetry. A relation between them is
14 A. M. VINOGRADOV
established in some cases by the famous N6ther theorem. The long time and very fruitful use of this theorem has lead to the widespread opinion that every conservation law is a reflection of some symmetry. In fact, it is not so and we will explain this below.
The notion of conservation law is not so simple as it may appear at a first glance. In fact, it is of a cohomological nature. In the considered context, the latter means the following.
Let us suppose that the investigating physical substance is described by Equations (1). A 'vector' n = (WI, ... , wn ), where Wi = Wi(X, u, ... , p~, ... ) E ,Cfji
is said to be a conserved current for these equations and, consequently, for the considered physical substance if
div n == L Di(WJ = 0, by virtue of (1). (11)
This is equivalent to
or to
are some C-differential operators. The standard 'physical' interpretation of the last definition is as follows.
Suppose that one of the independent variables, say, XI, is the 'time', i.e., Xl = t. Let also au be a domain in the space of variables X2, ••. , Xn • Then the quantity
Cn = I WI dX2 ... dXn
is a function of I. The Gaussian formula being applied to (11) (or to (12» then shows that the difference Ca(tz) - Cn(td is equal to the flow of the 'vector' n' = (W2, ... , wn ) through the boundary aau of au within the time interval II:S; I:S;
t2' In particular, if that flow is trivial, then Ca(t) is a constant, i.e., Cn(t) is a conserved quantity.
There exists a very simple way to construct conserved currents. Namely, let aij E :!F, 1 :s; i, j:S; n, be arbitrary functions on Joo satisfying aij = - aji and
Wi = L Dj(aij)' j
(14)
Then div n = 0 if n = (WI, ... , wn ). Therefore, n is a conserved current for (1) as
well as for every other equation posed on dependent variables U\ ... , urn. This shows that conserved currents of the form (14) are nonspecific for Equations (I) and, therefore, for the physical object described by them. In other words, they contain no meaningful information about this object. For this reason, these conserved currents should be noted as trivial.
Bearing all this in mind, it is natural to identify those conserved currents which differ from each other by a trivial current. This idea leads to the notion of conservation law. More exactly, conserved current 0 1 and O2 are said to be equivalent if 0 1 - O2 is a trivial conserved current. An equivalent class of conserved currents for (1) is said to be a conservation law for (1).
One must pay attention to the fact that the terms 'conserved current' (and, sometimes, 'conserved density') and 'conservation law' are usually identically used. But according to the previously introduced terminology, they have quite different meanings. Also, we will use the term 'conserved density' for the time component of a conserved current in the case when the time coordinate is distinguished. Below, the term 'integral of motion' is used as a synonym to 'conservation laws'.
Not entering into the details which one can find in [4], or [17], we mention that a class of equivalent conserved currents, i.e., conservation laws, is, in fact, an (n - I}-dimensional cohomology class in the so-called horizontal de Rham com plex on Woo. This cohomological interpretation of the conception of conservation law is of fundamental importance because it makes it possible to apply powerful homological methods in finding the conservation laws and also during the course of its applications. In particular, describing below the method of generating functions is of a cohomological nature.
8. Generating Functions of Conservation Laws
Operator L * conjugated with a C-differential operator L = L" a"D" is defined as
Supposing that 0 is a conserved current for (1), we define
!/In = (Ai(1), A'W), ... , A~ (1»,
where the operators Ai are taken from the equality (13). The restriction ;jin of !/In on Woo does not change if one changes n by an equivalent conserved current. Therefore, !/In characterizes the conservation law containing 0 as a unity and we will call this function the generating function of this conservation law.
The operator A entering into (13) is not uniquely defined. For this reason, many generating functions may, in general, correspond to a given conservation law. However, if the considered system (1) is normal, then the following im portant fact holds: the generating function of a given conservation law for (1)
16 A. M. VINOGRADOV
does not depend on the choice of A satisfying (13). Therefore, in that case, the generating functions completely characterize the corresponding conservation laws. In particular, the two conservation laws are different from each other if and only if their generating functions are different.
The normality condition mentioned above means that system (1) is determined, i.e., the number of essentially different equations entering into it is equal to the number of dependent variables, and its main symbol is nondegenerated. In this connection, we stress that PDE systems of practical interest are usually normal ones (see [3,4] for more details). Therefore, the problem of finding all the conservation laws for a given normal system is reduced to the problem of finding the corresponding generating functions.
The main result allowing one to solve the last problem completely in many cases is that the generating functions of the conservation laws satisfy the following equation:
n~=o. (15)
Here It is the matrix C-differential operator conjugated with IF. This means that It = II V' kill, where V' kl = Ll ik supposing that IF = II Llijll (the conjugation of scalar operators was defined at the beginning of this section). More exactly,
It = (i(~:;~;~;~~~;L{;~~-:~:-~~) u apu u apu
Comparing (9) and (15), one can see that the problems of finding higher symmetries and conservation laws after being reformulated in terms of generating functions, are reduced to solving conjugate equations. This demonstrates a duality of these notions which was mentioned earlier.
Not every solution", of (15) is, in fact, a generating function of a conservation law for (1). In order that it would be so, the following additional condition has to be satisfied.
An equivalent form of Equation (15) is
I}", = A(F),
where A is a matrix C-differential operator. A solution ~ of (15) is a generating function for a conservation law of (1) if and only if the operator IF +.4:* may be represented in the following form
(16)
where B is a C-differential operator. So the following procedure for finding the generating functions of conservation
laws may be devised firstly, to solve equations (15) and, secondly, to test the obtained solutions by (16).
Equations (15) and (16) are, in fact, corollaries of a theorem describing the structure of C-spectral sequences for normal equations. The theory of C-spectral sequences allows us to construct efficient procedures for finding the conservation laws for equations which are not normal. However, we will not discuss this here. Instead see [4J for more details.
It is worth paying attention to a fact which is essential for physicists who usually describe an investigated physical object by different systems of differential equations. Here we do not have in mind descriptions which differ by degrees of exactitude but those which can, in a sense, be reduced to each other. For example, such are Eulerian and Lagrangian approaches to continuum media mechanics as well as alternative descriptions of the electromagnetic field by means of its stress tensor or by its potentials. The notion of conservation law introduced above is associated with equations describing the physical object but not with this object in itself. For this reason, it may occur that different equations describing the same physical situation may have different groups of conservation laws. For example, Euler and Lagrange approaches to the same continuum media may lead to different sets of conservation laws. So it is not correct to speak about conservation laws of a physical object alongside a reference to describing its equations. This is not very satisfactory, but adopting a nonlocal point of view can overcome this phenomenon.
Concluding, we remark that there are very interesting and important global aspects in the conservation laws theory. See, for example, [4J and [17].
9. Symmetries and Conservation Laws
Now we will discuss the main interrelations between symmetries and conservation laws.
First of all, symmetries may be used to obtain new conservation laws from some of those already known. More exactly, there exists a natural action of symmetries on conservation laws which, in terms of generating functions, is described as follows.
Let cp be the generating function of a symmetry of Equations (1) and 1/1 be the generating function of its conservation law.
Because 3",( F) = () on UJJ=, 3",( F) = B( F), where B is a C-differential opera tor. Let
Then the restriction of 3",{ I/I} on UJJ= is a generating function of a conservation law for (1). Therefore, Iii t--'> 3", { I/I} is an action of the Lie algebra Sym UJJ on the space of all conservation laws of UJJ.
We now discuss the famous Nother theorems (both direct and inverse) from the point of view of generating functions theory. The main result in this
18 A. M. VINOGRADOV
(17)
if F = 0 is a Euler-Lagrange system, i.e., a system obtained from a functional by means of classical variational procedure. In other words, operator IF is self adjoint for Euler-Lagrange equations. Comparing Equations (9) and (IS) and taking into account (17), one can see that every generating function of a conservation law for a Euler-Lagrange equation is, at the same time, the generating function of its symmetry. So there exists the canonical map
{conservation laws} --------? {symmetries}
for Euler-Lagrange equations. It is not difficult to see that this map coincides with that which is given by the inverse N6ther theorem.
To a point, it is natural to illustrate the advantage of the adopted approach to symmetries and conservation laws by generalizing the inverse N6ther theorem outside the framework of variation calculus. We will call the equation F = () conformly selfadjoint if
(18)
where A is an operator. Evidently, every solution of Equations (15) is also a solution of (9) for conformly self-adjoint equations. Therefore, as above, there exists a natural map from conservation laws into symmetries for such equations. If A 11, then the conformly self-adjoint equation is not, as a rule, of the Euler-Lagrange type, i.e., cannot be put into the framework of the calculus of variations. The simplest example of this kind is the equation Ux = u, for which A=-l.
The above consideration is not invertible, i.e., does not lead to the map
{symmetries}------? {conservation laws}.
This is because not every solution of (15) is a generating function of the conservation laws (conditions (16) are also to be satisfied). However, if OY is the Euler-Lagrange equation corresponding to a Lagrangian 5£, then every sym metry of 5£ automatically satisfies (16). It follows then from (9), (15) and (17), that in this case the generating functions of the symmetries of 5£ are also generating functions for the conservation laws of the corresponding Euler Lagrange equation. In other words, we have a natural map
Iconservation laws {symmetries of a Lagrangian}-------? of corresponding Euler
Lagrange equations
This map is identical to the one that is given by the direct N6ther theorem. Hamiltonian equations form another class for which a natural correspondence
between symmetries and conservation laws exists. Not entering into details of
Hamiltonian theory for partial differential equations (see [18], where it is presen ted in the necessary general form), we recall only the fact that a Hamiltonian structure is given by a C-differential operator, say r. Now suppose that the equation F = 0 is Hamiltonian with respect to the Hamiltonian structure r. Then the' operator r maps the solution of (15) into solutions of (9). Therefore, in this case, a mapping of conservation laws into symmetries also exists.
10. Complete Integrability Problem for Partial Differential Equations
The notion of a conservation law in the form presented here is an exact analog of the notion of an integral for ordinary differential equations. It is well known that every ordinary differential equation possesses a complete integral locally in a neighbourhood of everyone of its regular points. Therefore, it is natural to ask if partial differential equations also have complete integrals locally supposing necessary regularity conditions hold? More exactly, we are interested in such a set of conservation laws (or, more physically, 'integrals of motion') which completely determine the solutions of a given equation 'in small'.
Now having to hand the above-described effective tool for finding out all conservation laws of a given PDE system, we are able to answer this question. It is negative. This is because partial differential equations, as a rule, have only a finite number of conservation laws (see the subsequent papers of this volume) but its solutions spaces are infinite-dimensional. However, the situation does not look hopeless. The following two reasons are sources of an optimism.
First of all, it seems to be true that normal linear differential equations are completely integrable 'in the small' via linear conservation laws (see [4]). Secondly, some fact forces us to suppose that 'regular' differential equations possess infinity of nonloeal conservation laws. By nonlocal, we mean such conservation laws whose generating functions depend on nonloeal variables. The exact sense of the last notion is given in [15].
11. Some Problems
]. The reader will see from the subsequent papers that calculations which are neccssary in order to find all symmetries or conservation laws of a given equation, are usually very cumbersome. So one of the more important problems now is to make the existing methods essentially more effective. Evidently, one of the ways in that direction is to develop computer methods which will allow one to carry out enormous routine symbolic calculations which arise at the very begin ning in solving Equations (9) and (15). Computers were used in some of the subsequent papers from which one may obtain somc feelings on the subject. The state of arts in computer applications to the above problem are discussed in the 'computer part' of this special volume.
Another line along which important progress may be expected is the
20 A. M. VINOGRADOV
development of differential invariants theory for differential equations. As was mentioned, such invariants provide obstructions to the symmetricity of differen tial equations.
Generally, further development of symmetries and conservation laws theory will depend on progress in the geometry of differential equations and especially in the secondary differential calculus (see [13, 19]).
2. The most interesting question for applications is what information about the equation under consideration may be taken from what is known of its symmetries and conservation laws. This problem has principle and technological aspects.
Fir:;t of all, we must state that a considerable vagueness exists over how symmetries and conservation laws may be used in the analysis of concrete equations. For example, the following are waiting to be investigated in that direction: symmetry analysis of discontinuities and singularities of solutions; the relationship between asymptotic methods and symmetry analysis; applications of conservation laws in stability theory and in the theory of 'intermediate states', etc.
The character of technological problems arising in the considered domain may be illustrated by the problem of finding invariant solutions. First of all, it is necessary to find all k-dimensional subalgebras, k < n, in the Lie algebra Sym qy. The next step is to obtain reduced systems of partial equations, i.e., systems with less than n number of independent variables whose solutions are invariant solutions of the initial equations. Finally, these reduced systems are to be solved. It is clear that all this can be carried out in detail only with aid of the appropriate computer machinery (see [7] for an illustration).
3. It is natural to suppose that both symmetries and conservation laws theories may be built for more general equations than differential ones. For example, integro-differential equations are surely such ones. One can imagine that sym metry transformations for such equations will be operators which are more general than differential. If this is the fact, more general symmetry trans formations than described before exist for differential equations. This is because the latter may be understood, for example, as integro-differential ones. This leads us to assume the existence of more general frames for symmetry theory.
These considerations, as well as some others, force us to generalize the local theory presented above. Essentially, this problem is to find an exact definition for the conceptions of 'nonlocal' symmetry and conservation laws. An approach to it is made in the paper [15] of this volume. It bases on the notion of covering in the category of differential equations. We point out that only the first steps are made in that direction and the appearing theory looks attractive because of its nonstandard features.
References
1. Vinogradov, A. M.: A spectral sequence associated with a nonlinear differential equation and algebra-geometric foundations of Lagrangian field theory with constraints, Dokl. Acad. Nauk. SSSR 238 (1978), 1028-1031 (English trapslation in Soviet Math. Dokl. 19 (1978), 144-148).
2. Vinogradov, A. M.: The theory of higher infinitesimal symmetries of nonlinear partial differential equations, Dokl. Acad. Nauk. SSSR 248 (1979), 274-278 (English translation in Soviet Math. Dok!. 20 (1979), 985-990).
3. Vinogradov, A. M. Local symmetries and conservation laws, Acta Appl. Math. 2 (1984), 21-78. 4. Vinogradov, A. M.: The C-spectral sequence, Lagrangian formalism and conservation laws, J.
Math. Anal. Appl. 100 (1984), 2-129. 5. Ibragimov, N. H.: Transformation Groups Applied to Mathematical Physics, D. Reidel, Dor
drecht, 1985. 6. Oliver, P. J.: Application of Lie Groups to Differential Equations, Springer, New York, 1986. 7. Gusyatnikova, V. N., Samokhin, A. V., Titov, V. S., Vinogradov, A. M., and Yumaguzhin, V.
A.: Symmetries and conservation laws Kadomtsev-Pogutse equations, Acta App/. Math. 15 (1989), 23-64.
8. Gusyatnikova, V. N. and Yumaguzhin, V. A.: Symmetries and conservation laws of Navier Stokes equations, Acta App!. Math. 15 (1989), 65-81.
9. Titov, V. S.: On symmetries and conservation laws of the equations of shallow water with an axisymmetric profile of bottom, Acta App/. Math. 15 (1989),137-147.
10. Verbovetsky, A. M.: Local nonintegrability of the long-short wave interaction equations, Acta Appl. Math. 15 (1989), 121-136.
11. Sharomet, N. 0.: Symmetries, invariant solutions and conservation laws of the nonlinear acoustics equation, Acta App/. Math. 15 (1989), 83-120.
12. Romanovsky, Yu. R.: On symmetries of the heat equation, Acta App/. Math. 15 (1989), 149-160.
13. Krasil'shchik, I. S" Lychagin, V. V., and Vinogradov, A. M.: Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986.
14. Senashov, S. I. and Vinogradov, A. M.: Symmetries and conservation laws of 2-dimensional ideal plasticity. Proc. Edinburgh Math. Soc. 31 (1988), 415-439.
15. Krasil'shchik, I. S. and Vinogradov, A. M.: Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Backlund transformations, Acta App/. Math. 15 (1989),161-209.
16. Gusyatnikova, V. N., Vinogradov, A. M., and Yumaguzhin, W. A.: Secondary differential operators, J. Geom. Phys. 2 (1985), 23-66.
17. Tsujishita, T.: On variation biocomplexes associated to differential equations, Osaka Math. J. 19 (1982), 311-363.
18. Astashov, A. M. and Vinogradov, A. M.: On the structure of Hamiltonian operators in field theory, J. Geom. Phys. 3 (1986), 264-2~n.
19. Vinogradov, A. M.: The category of differential equations and its significance for physics, Proc. Conf. Differential Geometry and its Applications, Nove Mesto na Morave (VSSR), 5-9 September 1983, Part 2, pp. 289-301.
Acta Applicandae Mathematicae 15: 23-64, 1989. © 1989 Kluwer Academic Publishers.
Symmetries and Conservation Laws of Kadomtsev-Pogutse Equations
(Their computation and first applications)
V. N. GUSYATNIKOVA
Program Systems Institute of the U.S.S.R. Academy of Sciences, 152140, Pereslavl-Zalessky, U.S.S.R.
A. V. SAMOKHIN Moscow Institute of Civil Aviation Engineers, Moscow, U.S.S.R.
V. S. TITOV Program Systems Institute of the U.S.S.R. Academy of Sciences, 152140, Pereslavl-Zalessky, U.S.S.R.
A. M. VINOGRADOV Moscow State University, Department of Mechanics and Mathematics, 117234, Moscow, U.S.S. R.
and
V. A. YUMAGUZHIN Program Systems Institute of the U.S.S.R. Academy of Sciences, 152140, Pereslavl-Zalessky, U.S.S.R.
(Received: 11 April 1988)
Abstract. Kadomtsev-Pogutse equations are of great interest from the viewpoint of the theory of symmetries and conservation laws and, in particular, enable us to demonstrate their potentials 'in action'. This paper presents, firstly, the results of computations of symmetries and conservation laws for these equations and the methods of obtaining these results. Apparently, all the local symmetries and conservation laws admitted by the considered equations are exhausted by those enumerated in this paper. Secondly, we point out some reductions of Kadomtsev-Pogutse equations to more simpler forms which have less independent variables and which, in some cases, allow us to construct exact solutions. Finally, the technique of solution deformation by symmetries and their physical inter pretation are demonstrated.
AMS subject classifications (1980). 35Q20, 58F07, 35G20.
Key words. Magnetohydrodynamic equations, symmetry, conservation laws, invariant solution.
o. Introduction
In 1973, B. B. Kadomtsev and O. P. Pogutse [1] suggested the now well-known simplification of the general system or magnetohydrodynamic (MHD) equations, discarding some details, unimportant from the viewpoint of the problem of holding a high temperature plasma in 'tokamak'-type facilities. They proceeded from the ideal MHD equations, since the characteristic times of most important
24 V. N. GUSYATNIKOVA ET AL.
physical processes are much less than the distinctive time of dissipation stipulated by plasma electric resistance and viscosity. Besides, it had been taken into account that for the plasma stability
(1) it is necessary that plasma pressure and the cross-component of magnetic field pressure are both much less than the pressure created by the longitudal component of a magnetic field;
(2) it is expedient that the lesser radius of a tokamak torus should be much less than the greater one. As a result, the system of the two scolar equations was obtained. After proper norming, they have the form
(1)
Here the functions cp and '" are the potentials of the velocity and the cross-component of the magnetic field (they may be also interpreted as the potential of the electric field and of the vector potential z-component of the magnetic field). The coordinate system (x, y, z) is such that the z-axis coincides with the 'tokamak' axis. Besides,
and [u, v Jz = uxvy - uyvx is a z-component of the vector product [u, v J of vectors u, v.
System (1) will be referred to as the Kadomtsev-Pogutse equations. This system has become popular in the West since the publications of White et at. (1974) [2J and Strauss (1976) [3J and often figures under the name 'reduced MHD equations' or 'Strauss equations'.
Simplicity and cleanness of Kadomtsev-Pogutse equations (in comparison with the full MHD equations) have made it possible to construct, on their base, a theory of the kink and tearing instabilities (see [4J), and a theory of the first mode reconnection which has been given experimental support. Their usage has also made it rather easier to investigate the instability by approximate methods [2].
Kadomtsev-Pogutse equations are of great interest from the viewpoint of the theory of symmetries and conservation laws and, in particular, they enable us to demonstrate their potentials 'in action'. This paper presents the results of computations of symmetries and conservation laws for Equations (1) and the methods of obtaining these results. Apparently, all the local symmetries and conservation laws admitted by Equations (1) are exhausted by those enumerated in this paper. We also point out some reductions of the Kadomtsev-Pogutse equations to more simpler forms than those which have less independent vari-
LAWS OF KADOMTSEV-POGUTSE EQUATIONS 25
abIes and which, in some cases, allow us to construct exact solutions. Finally, the technique of solution deformation by symmetries and its physical interpretation are demonstrated.
Investigation of Equations (1) from this point of view was initiated by A. V. Samokhin [6].
This should point out the Marsden-Morrison work [7], where it was shown that the system (1) may be interpreted as a Hamiltonian one in some weakened sense. It may become useful in future investigations of the Kadomtsev-Pogutse equa tions by methods of geometric theory of differential equations in the spirit of the book [8].
1. Symmetries
In this section, all the symmetries of Kadomtsev-Pogutse equations whose generating functions depend on derivatives up to the first order are described. A number of considerations indicate that hardly any symmetries were obtained which depended on derivatives of a higher order.
Even in the stated assumptions, a search for symmetries is connected with calculations of great volume, though of a routine character. In this connection, we have used the computer system 'SCoLAr' (see [9]) and, as a result, it is shown that generating functions of the symmetries of Kadomtsev-Pogutse equations linearly depend on the first-order derivatives. (We do not dwell upon our interaction with the computer. This will be explained in some detail in the last section). Remember, that the computations of symmetries in the paper [6] have been carried out assuming the linear dependence of the generating function on the first-order derivatives. A description of this computation is presented below. We emphasize that all symmetries of Kadomtsev-Pogutse equations obtained this way prove to be classical.
When computing symmetries of low orders, it is more convenient to use the following notations for coordinate functions on jet manifolds. Firstly, let us denote Xl = X, Xz = y, X3 = Z, X4 = t, Ul = 'P, Uz = 1/1. Secondly, instead of the symbols P ~, P~, we shall write 'P x'yizk,l and 1/1 x'yizk,l correspondingly where (I = xiyizkt', I(II = i + j + k + I. By these notations, Kadomtsev-Pogutse equa tions come to a form C&.
(1.1)
Fz = 'P x2, + 'P y2, + 'Px'P x2y + 'Px'P y3 - 'Py'P x3 - 'Py'P xy2 -
- I/Ix2z - I/I y2z - I/Ixl/lXy2 - I/Ixl/ly3 + I/Iyl/lx3 + I/Iyl/lxy2 = o. (1.2)
As the external coordinates on infinitely prolonged equation C&OO, we take variables of the form I/IxiyiZk,1 for I;;. 1 and 'Pxiyizk,. for j;;. "2, I;;. 1. Indeed, the equations Fl = 0 may be written in the form
0/, = C{!z - C{!xo/y + C{!yo/x .
Noting that o/,u = D( 0/,) = C{!zu + Du( - C{!xo/y + C{!yo/x), by induction in the order of entry of variable t into multi-index a, one can prove that variables o/,u on Woo are expressed in terms of o/p, C{!T' P not containing t. Similarly, by rewriting the equation F2 = 0 to the form
C{! yZ, = - C{! xZ, - C{!xC{! xZy - C{!xC{! y' + C{!yC{! x3 + C{!yC{! xyZ + + o/x2 z + 0/ y2z + o/xo/ xZy + o/xo/y3 - o/yo/x3 - o/yo/ xyZ,
one can verify that functions C{! xiyizk", j ~ 2, I ~ 1 may be expressed in terms of the internal coordinates on Woo which are functions x, y, z, t, C{!, 0/, C{!T> o/u, where a = xiyi Zk for any i, j, k ~ 0 and T = xiyi Zktl for any i, k ~ 0 and 0,,; j,,; 1 +(1/21).
Since the Kadomtsev-Pogutse system contains two dependent variables, C{! and 0/, the generating function of its symmetries are two-component. We denote it = (;::). In view of what is stated above, here we seek the generating function of a special kind
8' = S + AC{!x + BC{!y + CC{!z + EC{!, .
:Y = T + Ao/x + Bo/y + Co/z + Eo/,.
Here A, B, C, E, T, S are functions on ,0, i.e., functions of variables x, y, z, t, C{!, 0/. Passing to the internal coordinates, we get
g = S + AC{!x + BC{!y + CC{!z + EC{!, .
;!j = T + Ao/x + Bo/y + Co/z + E( c{!z - C{!xo/y + C{!yo/x).
For Kadomtsev-Pogutse equations, the operator of universal linearization IF, F = (~~) has the form
I - ( - Dz + o/yDX - o/xDy D, + C{!xDy - C{!yDx ) F - D,(D; + D;) + R(C{!) - Dz(D; + D;) - R(o/)
and its restriction on Woo is
r = ( - Dz + o/yDx - o/xDy D, + C{!xDy - C{!yDX ) F D,(D~ + D;) + R(C{!) - Dz(D~ + D;) - R(o/) .
In these formulas
for (= C{! or 0/. The operators Dx, Dy, D., D" in their turn, are given by the following formulas
- iJ L iJ L- iJ --+ - -+ -Dx - .> C{!XT .> o/xu .>.1. ' uX T U'P'T U U'¥U
- 0 I 0 I- 0 D --+ - -+ -y - " 'PYT " I/Iw 0,1, ' uy T U'PT U 'i'u
- a I 0 I 0 D --+ - -+ - z - " 'PZT " I/Izu 0,1, '
uZ 'T u'P" (T 'Yu
- a I 0 I- 0 D --+ - -+ -, - " 'P'T 0 I/I,u 0,1, ' ut 1'" 'P'T (T 'Yu
where T and u are multi-indexes of the chosen internal coordinates on C[/Joo.
Since [pel> = (/p<l» an equation [pel> = 0 for the generating functions of sym metries in that case, may be written in the form
(-Dz + l/IyDX -l/IxDy)[f+(D, + 'PxDy - 'PyDx)fJ= 0 on C[/Joo, (1.3)
(D,(D~+D;)+R('P)W+(-DAD~+D;)-R(I/I))fJ=O on C[/Joo. (1.4)
As [f and fJ depend on the first-order derivatives, the left-hand side of Equation (1.3) linearly depends on the variables 'Pu, I/Iu, lui = 2, while the left-hand side of 0.4) linearly depends on 'Pu, I/Iu, lui = 4. This follows im mediately from the description of the total derivation operators Dx, Dy, Dz , D,. For example,
~[f=~S+~~A+~~B+~~C+
+ 'P,DzE + A'Pxz + B'Pyz + C'Pzz + E'P,z . (1.5)
After restricting the left-hand sides of Equations (1.3) and (1.4) on C[/Joo,
variables 'Pu, I/Iu correspondingly vanish for lui = 2 and lui = 4. This follows from the next simple consideration. Variables 'Pu, I/Iu, lui = 2 are the highest-order derivatives entering the left-hand side of (1.3) and 'Pu, I/Iu, lui = 4 are the highest-order derivatives in (1.4). But the highest-order derivatives in the expression
may only occur in summands of the form
which coincide with
That coincidence follows from the general assertion
[ '" aF'; .] = L... --j P!rx, = DsF. <T,j ap <T
But D<TF; = 0, i = 1,2, on 6JJ=. It implies that summands (1.6) turn into zero on 6JJ=, i.e., that the highest derivatives in (1.3) and (1.4) vanish after restriction on 6JJ=. We also stress that the same argument shows that functions A, B, C, E in (1.3) and (1.4) vanish on 6JJ= and only their derivatives remain.
Thus, Equation (1.3), after restricting on 6JJ=, contains only the first-order derivatives, entering polynomially. A power of homogeneous monoms varies from zero to four. Indeed, the total derivatives of functions S, T, A, B, C, Eon '1P linearly depend on variables 'P<T, o/<T, lal = 1, as follows by the structure of the total derivation operators. Therefore, Equation (1.3), after restricting on 6JJ=, contains monoms in variables i{!<T, ,fr<T, lal = 1, of a power of no more than three. For instance, the second summand in (1.3), o/yDxY' contains monom o/y'Pxo/xC",. While turning to the internal coordinates, the homogeneous power may rise according to the relation ,fr, = cpz - 'Pxo/y + 'Pyo/x . It follows from computation that no monoms of a power more than four arise and after reduction of similar terms, only monoms of the power 0, 1,2,3 remain. There are 42 such monoms and their coefficients must be identically zero. This brings to us 14 independent conditions on the functions S, T, A, B, C, E
Tz = Sz;
T",=E,;
- Sx = B,; C, = S'" ; Cy = A", ;
A", = E", ; C'" = E", ; B", = - Cx ; B", = - Ex;
E, + T", = Cz + S"'; Ax + T", = - By + Cz .
(1.7)
Further on, the variables 'P<T, r/Io., 1 ~ lal ~ 3 enter into Equation 0.4} poly nomially, and for lal = 3, enter linearly. Indeed, summands of the form (1.6) vanish on 6JJ=, as it was pointed above, so the variables 'P<T, o/<T' I al = 3 enter only the terms D<TS, D<TT, lal = 3;
and
DpK' D<T'PT' DpK' D<To/T' for Ipl = ITI = 1, lal = 2, K=A, B, C, E;
[( rp x2y + rp y3 }Dx - (rp xy2 + rp x3 }Dy]Y',
[(o/x2y + o/y3}Dx - (o/x 3 + o/xy2)Dy];J.
(1.8)
Variables rp<T, o/<T, lal = 3 enter all the enumerated terms linearly. Since the
third-order derivatives in the equation F2 = 0 are also linearly related, then, when turning to the internal coordinates in (1.4), we have the linearity in these variables being preserved.
Now the coefficients by 'P,,, I/Iu, lui = 3 in (1.4) must be identically zero. We point out that these coefficients themselves are polynomials of 'Pu, I/Iu, 10'1 = 1, as it follows from the expressions (1.8). For instance, the coefficient by 'Pxy2 has the form (after reduction of similar terms)
This polynomial of the first derivatives must be identically zero which, in turn, means that all the coefficients by 'Pu, I/Iu, 10'1 = 1 and the free terms must be identically zero.
Thus, only the consideration of variables 'Pu, I/Iu, lui = 3 in Equation (1.4) results in a very large number of conditions. Discovering these is the most toilful part of the whole work of symmetry computation. By adding the conditions obtained in this way the relations (1.7), and after some simplification, we compile the following list
A",=A",=O;
Sy=A,;
T",=S",;
(1.9)
System (1.9) is strongly overdetermined. Thanks to that, it is not hard to get the general solution
S = (a' + {3'- C1)'P + (a' - (3')I/I + !F,(x2 + y2) - xH, + yO, + R,;
T = (a' - {3')'P + (a' - (3' + C1)I/I +!FAx2 + l) - xHz + yOz + Rz;
A = !C2x + yF+ 0; B = !C2y - xF+ H; (1.10)
C = (a + (3) + ( C2 - C1) t + C3 ; E = (a - (3) + ( C2 - C1) t + C4 ;
Here a = a(z + t), {3 = (3(z - t), H(z, t), F(z, t) and R(z, t), O(z, t) are arbitrary functions, Ci - arbitrary constants, and
'(Y) = da{C} a ~ de'
Sufficiently rigid necessary conditions (1.10) make it possible to avoid further monotonous treatment of polynomials in the variables 'Pu, I/Iu, 10'1 < 3. All the remaining calculations reduce to the substitution of (1.10) into (1.3), 0.4). This results in only one additional condition
F.,= Fzz .
Whem;e, it follows F(z, t) = y(z + t) + 8(z - t), with y and 8 being arbitrary functions. Thus, the following result is stated
THEOREM. All the symmetries of Kadomtsev-Pogutse equations whose generat ing functions depend on the derivatives of the order ~ 1 are classical. The algebra Sym 0Jj of the classical symmetries is generated as a linear space over IR by symmetries with generating functions of the form
(1) s'l = (a'(cp + 1/1) + a(cpz + CPt)) " a'(cp + 1/1) + a(l/Iz + I/It) ,
(1.11)
(2) 00 = (/3'( cP - 1/1) + /3( cpz - CPt)) f3 /3'(cp-I/I)+/3(l/Iz-l/It) '
(1.12)
(3) ~ = C'(x2 + y2) + 2y(ycpx - xcpy)) "Y y'(x2 + y2) + 2 Y(YI/Ix - xl/ly) ,
(1.13)
(4) g; _ C 8'(x2 + l) + 28(ycpx - xcpy)) 8 - 8'(x2 + y2) + 28(yl/lx - xl/ly) ,
(1.14)
(1.17)
(1.18)
(10) 2= (CP + Z'Pz + t'Pt) 1/1+ zl/lz + tl/lt '
(1.20)
(11) At = CCPx + Y'Py + 2z'Pz + 2t'Pt) xl/lx + yl/ly + 2zl/lz + 2tl/l, '
(1.21)
Here a = a(z + t), /3 = /3(z - t), y = y(z + t), 8 = 8(z - t), G = G(z, t), H =
H(z, t), K = K(z, t) are arbitrary functions and
'W=daW a d( , /3'( '11) = d~~ '11) , 'w = dy«()
Y d( ,
Gt = aG(z, t)
'11 d'11' az ,
Now we describe the symmetries themselves, that is, the vector fields on J O
which correspond to the above listed generating functions. Let
a a a a a a X = al-+ a2 -+ a3-+ a4 -+ {31-+ {32-'
ax ay az at acp at/!
(The functions aj, (3j depend on x, y, z, t, cp, t/!). The correspondence in question is realized by the formulas f! = X ~ UI , ff = X ~ U2 , where
UI = dcp - CPx dx - cpy dy - cpz dz - cp, dt, U2 = dt/!- t/!x dx - l/Iy dy - t/!z dz - t/!, dt
is a system of differential forms defining the Cartan distribution on J O• We have
f!= {31 - alCPx - a2CPy - a3CPz - a4CP" ff = {32 - a1 t/!x - a2t/!y - a3t/!z - a4t/!"
so that {31 = S, (32 = T, al = -A, a2 = -B, a3 = -C, a4 = -E. Let us denote by X,,> a vector field corresponding to the generating function <1>. Then it follows that
(1) xcA.=a'(cp-t/!)C: -aat/!)-a(aaz +~),
(2) X(;g~ = (3'(cp - t/!) (a: - a:) - a (a: -~),
(3) XCf,;, = y'(x2 + y) C: + a:) - 2y (y a: - x aay) ,
(4) Xc", =8'(X2+l)(~-~)-28(Y~-X~), • at/! acp ax ay
(5) x -~ '!J> - az'
'fio 'acp z at/! ax'
a a a (8) X= =-xH--xH --H
~H , acp Z at/! ay'
a a (9) X= =K -+K -
.flK 'acp zat/!'
Xx = cp-+ t/!-- z-- t-, acp at/! az at
<. ;>
a a a a (11) X.A,t=-x--y--2z--2t-.
ax ay az at
In conclusion of this section, we give in Table I the commutators for the above-listed generators of the algebra Sym. (i.e., we write down their pairwise Jacobi brackets, see [8]).
2. Three-Dimensional Subalgebras of Classical Symmetries Algebra
As is known from the general theory (see [10]) the task of finding the solutions of a system qy of differential equations which are invariant with respect to a five-dimensional symmetry subalgebra of qy, reduces to solving a system with n-s independent variables (n being the number of independent variables in qy).
If, following this remark, we want to select those solutions of the Kadomtsev Pogutse equations whose findings are reduced to integrating ordinary differential equations, then we have to discover all three-dimensional subalgebras in the symmetries algebra of these equations.
A description of all three-dimensional symmetries subalgebras of the Kadom tsev-Pogutse equations is a rather cumbersome task. It compels us to restrict this task in the following way. Namely, we take a symmetries subalgebra d for the Kadomtsev-Pogutse equations which is generated by d"" OOtl, CfJ'Y' qj;s, 'l:, :¥, CfiG , 'lJeH , X K , .'£, At while choosing all functional parameters a, (3, y, 8, G, H, K to be identically unit. Thus,
d=(C, E, F, G, H, L, M),
where
L= . ( u + Zllx + tu,) v + zVx + tv, '
M= (-2n + XUx + YUy). -2v + xVx + yUy
The symbol (P, ... , Q) is used here to denote the Lie algebra generated by P, ... , Q. We shall describe every three-dimensional subalgebra .'£ of the algebra d. This task divides itself naturally in~o four cases according to the dimension of the intersection of the required subalgebra .'£ with the commutant d 1 of the algebra d. We recall that the commutant d 1 of the Lie algebra d is its ideal generated by all elements of the form [a, b], a, bEd.
Using the table of commutators for the generators of d
C E F 0 H L M
C 0 0 0 -2H 20 0 0 E 0 0 0 0 0 -E 0 F 0 0 0 0 0 -F 0 0 2H 0 0 0 0 0 -0 H -20 0 0 0 0 0 -H L 0 E F 0 0 0 0 M 0 0 0 0 H 0 0
one can deduce that its commutant d 1 is generated by E, F, 0, H, i.e.,
d 1 = (E, F, 0, H),
besides, d l is Abelian.
Case 1: dim:;en d l = 3 ~:;ec d l
Since commutant d l in this case is Abelian, it follows that any three-dimensional subspace of d is also a subalgebra. Hence, any three-dimensional subalgebra is generated by three generators
for
ei, Ii , qi, hi E IR, i = 1,2, 3,
They have to satisfy the only condition
It is possible to reduce this matrix by elementary line transformations to one of the following four forms
[~ 0 0
h'] [~ 0 gl
H 1 0 h2 ; 1 g2 0 1 h3 0 0
[~ II 0
0 0 0 0
Accordingly, the set of all three-dimensional subalgebras divides itself into four nonintersecting classes
2:'1 = (E + hlH; F+ h2H; G+ h3H), hI, h2, h3 E IR;
2:'~ = (E + gl G; F + g2 G; H), gl, g2 E IR;
2:'~ = (E + /IF; G; H), II E IR;
2:'1 = (F; G; H).
Case 2: dim 2:' n .sill = 2 In this case the following generators of a subspace 2:' may be chosen
VI = eC+ IL+ mM+ elE+ IIF+ glG+ hlH;
V2 = e2E+ hF+ g2G+ h2H;
V3 = e3E+ hF+ g3G+ h3 H ,
where e, I, m, ei, J;, gi, hi E IR, i = 1,2,3, vector (e, I, m) is nonzero, and
rank (e2 h g2 h2) = 2. e3 h g3 h3
(2.1)
Since {Vi, Vj} E .sill and .sill is Abelian ideal, it follows that VI, V2, V3 generate a subalgebra if and only if
{VI,V2}=r'V2+p'V3, r,pEIR,
{VI, V3} = q . V2 + s . V3, q, S E IR,
Taking (2.1) into account, we get the following system of equations
le2 = re2 + pe3 ;
Ih= rh+ ph;
le3 = qez + se3 ;
Ih = qh+ sh;
By elementary line transformations, the matrix
( e2 h g2 h2) e3 h g3 h3
may be reduced to one of next six forms
( 00 1 0 o 1
(1 h g2 0). o 0 0 1 '
( 0 0 1 0) o 0 0 1 .
(2.2)
Accordingly, the solving of system (2.2) divides into six subcases. We shall
consider in detail the first one (that is, for e2 = 13 = 1). (All the other subcases are studied quite analogously and, for that reason, we do not consider them and give the final result right away: see the list of all three-dimensional subalgebras of .st1., below. We note here that there are no subalgebras in the fourth subcase.)
In the first subcase, one may obviously suppose that e1 = It = O. Then system (2.1) takes the form
1= r; 0 = q;
0= p; 1= s;
or
This system implies that if the determinant
Im-I 2c I 2c m-I
= (m-l)2+4c2 =f 0,
then g2 = g3 = h2 = h3 = O. But this determinant is nonzero either if c =f 0 or if c = 0 and I =f m. Accordingly, we get subalgebras of the following two types
21.1 = (C+ IL+ mM+ glG+ hIH; E; F), I, m, gl, hI E R;
21.2 = (IL+ mM+ gIG+ hlH; E; F), I =f m, gl, hi E R.
If
1m -I 2c 1=0 2c m -I '
then c = 0, m = 1=0 and we get a subalgebra of the type
21.3 = (L+ M+ glG+ hlH; E+ gzG+ h2H; F+ g3G+ h3H), g;, II; E R, j = 1,2,3.
Case 3: dim 2n.st1.1 = 1 In this case, one may choose as the generators of a subspace I£ the following symmetries
VI = clC+ IIL+ mIM+eIE+ I1F+ glG+ hlH;
Vz = czC+ IzL+ m2M + ezE+ hF+ g2G+ h2H;
V3 = e3E+ hF+ g3G+ h3H,
where
and vector (e3, h, g3, h3) is nonzero. Since {Vi, Vj} E d l and .:£ n d l = (V3), it follows that elements VI, VZ, V3
generate a subalgebra if and only if
{VI,vZ}=r·v3, rER,
{vz, V3} = q . V3, q E R.
Matrix
(2.3)
may be reduced by elementary line transformations to one of the following three forms
(1 II 0). o 0 1 '
(0 1 0) o 0 1 .
Vector (e3, h, g3, h3) may be chosen in one of the following four forms
Accordingly, solving (2.3) divides into 12 subcases. Each of them is studied by analogy with subcases of the case .:£ n d l = 2. So we formulate the final result right away (see the 'list of three-dimensional sub algebras of d', below).
Case 4: dim.:£ n d l = 0 In this case, one may choose as the generators of subspace .:£, the following symmetries
VI = C+ elE+ [IF+ gl G+ hlH;
Vz = L + ezE + fzF + gz G + hzH;
V3 = M+ e3E+ hF+ g3G+ h3H,
where el, [I, gl, hi E R, i = 1,2,3. Since {Vi, Vj} E d l and .:£ n d l = (0) it follows that element VI, VZ, V3 generate a subalgebra if and only if
Solving this system, we get subalgebras on a unique type
.:£0 = (C+ glG+ hlH; L+ ezE+ fzF; M-!hIG+!gIH), ez, fz, gl , hi E R.
38 V. N. GUSYATNIKOVAET AL.
LIST OF ALL THREE-DIMENSIONAL SUBALGEBRAS OF ALGEBRA JIl
dim.:e n Jill = 3
.:e~ = (E + hlH; F+ hzH; G+ h3 H), h; E R, i = 1, 2,3;
.:e~ = (E+ giG; F+ gzG; H), g; ER, i = 1,2;
.:e~ = (E + /JF; G; H), II E R,
.:e~ = (F; G; H).
dim.:e n Jill = 2
.:ei.l = (C+ lL+ mM + gl G+ hlH; E; F), " m, gl, hi E R;
.:ei.z = (lL+ mM + gl G+ hlH; E; F), l = m, gl, hi;
.:ei.3 = (L + M+ gl G+ hlH; E + g2G+ hzH; F+ g3G+ h3 H), g;,h;ER, i=1,2,3;
.:e~.1 = (L+ M+ IIF+ hlH; E+ fzF+ hzH; G± H), \ fl,f2,h1,h2ER;
.:eL = (L+ mM+ IIF+ hlH; E+ fzF; G± H), m =f 1,/I,fz, hi ER;
.:e~.3 = (C+(m'F2)L+ mM+ IIF+ hlH; E+ fzF+ h2H; G± H), m,fl'fz, hi, hzER;
.:e~.4 = (C+ lL+ mM + /JF+ hlH; E+ fzF; G± H), m -l =f ±2,
" m,/J,fz, hi ER;
.:e~.l = (L + M + /JF+ gl G; E+ fzF+ gzG; H), II'fz, gl, gzER;
.:e~.z = (lL+ mM+ /JF+ giG; E+ fzF; H), l =f m, " m, /1, Iz, gl E R;
.:e~.l = (L+ M + elE + gl G; F+ gzG; H), el, gl, gz E R;
.:e~.z = (lL + mM + elE + gl G; F; H), l =f m, " m, el, gl E R;
.:e~.1 = (cC + lL + mM + elE + /IF; G; H), c, " m, el, /1 E R.
dim.:e n Jilt = 1
t ( mZ(mlgl - 2hl) .:e1.1 = C+ mlM+ glG+ hlH; L+ mzM+ fzF+ z G+
ml +4
ml, mz, fz, /3, gl, hi E R;
mz(2g1 + mlhl) H ) + mi+4 ;F ,
ml, m2, e2, gl, hi E R;
LAWS OF KADOMTSEV-POGUTSE EQUATIONS
2?L.I = (C+ IIL+ IIF+2hzG-2gzH; M+ gzG+ hzH; E+ /JF), 11'/1,/J,gz,hz ER,1110;
2?L.z = (c+ IIF+2hzG-2gzH; M+ fzF+ gzG+ hzH, E+ /JF), /I,fz,/J,gz,hzER, fz10;
2?tl.3 = (c+ IIF+2h2 G-2g2H; M+ g2G+ h2H; E+ /JF), II,/J, g2, hzER;
2?i.2.1 = (C+ IlL + e1E + 2hzG- 2g2H; M + g2G+ h2H; F), II, el , g2, h2 E R, II 1 0;
2?i.2.2 = (C+ elE +2h2G- 2gzH; M + ezE + g2G+ hzH; F), el, ez, gz, hz E R, ez1 0;
2?i.Z.3 = (C+ elE+ 2hzG- 2g2H; M + gzG+ hzH; F), el, g2, hz ER;
2?b = (L + IIF; M + gzG+ hzH; E + /JF), 11,/J,g2,hz ER;
2?t2 = (L + elE; M + gzG+ hzH; F), el, gz, h2 E R;
2?t3 = (L+ elE+ /IF; M+ hzH; G+ h3H), el, /I, hz, h3 E R;
2?j.4 = (L + elE + IIF; M + gzG; H) el, II, gz E IR.
dim 2? n s1.1 = 0
2?0 = (C+ glG+ hlH; L+ e2E+ fzF; M-1hIG+!gIH), ez, fz, gz, h1 E IR.
3. Solution Invariant with Respect to Three-Dimensional Subalgebras
39
In this section, we describe a system of ordinary differential equations which are to be integrated to obtain invariant solutions for the three-dimensional subalge bras of Section 2. A consistent and the systematical studying of all these subalgebras may be substantially simplified, due to a particular consideration. For this simplification, we divide all subalgebras into four classes which are in vestigated separately.
Class 1 Here we use the following observation. It is evident that any solution invariant with respect to the subalgebra 2? (2?-invariant), are invariant with respect to any symmetry belonging to 2? Therefore, whenever it is possible to obtain explicit solutions invariant with respect to some symmetry S, the task of finding the solutions, invariant with respect to the algebras containing 5£ greatly simplifies.
The symmetries S = aC + hH + gG, where a, h, g are arbitrary constants, happen to be the symmetries of such a kind for the Kadomtsev-Pogutse equa tions. The three-dimensional subalgebras of Section 2 containing S are
Let us describe the procedure for finding S-invariant solutions of the Kadomtsev-Pogutse equations in more detail. To obtain them, as follows from the general theory, one has to supplement system (1.1) with the equations
2a(ycpx - xCpy) + gcpx + hcpy = 0,
2a(y"'x - X"'y) + g"'x + h",y = ° and to solve the resulting system of the four equations.
Rewriting system (3.1) to a form
(2ay + g)cpx + (-2ax + h)cpy = 0,
(2ay + g) "'x + (-2ax + h)",y = ° and adding the operator ~.i to it, we get
(2ay + g)~.iCPx + (-2ax + h)~.icpy = 0,
(2ay + g)~.iCPx + (-2ax + h)~.i"'y = 0.
(3.1)
(3.2)
(3.3)
Equations (3.2) and (3.3) mean the linear dependence of the lines in the determinants
Therefore, the nonlinear summands in the Kadomtsev-Pogutse equations vanish. Thus, the solutions of the Kadomtsev-Pogutse equations invariant with respect
to the S = aC + hH + gG satisfy the linear system of the equations
(2ay + g)cpx + (-2ax + h)cpy = 0,
(2ay + g)"'x + (-2ax + h)",y = 0, (3.4)
"', = CPz,
The first pair of these equations yields a dependence of cp and '" on x and y
cp = cp(O, Z, t), '" = ",(0, z, t),
where O=(2ay+g)2+(-2ax+h)2. Taking this into account, the last pair of equations (3.4) reduce to the form
"', = CPz,
One can easily integrate this system by changing the variables (= Z + t, 7) = Z - t. As a result, we get the following formulas for cp and '"
cp = F(O, ()+(O, 7))+(U,- U,,)ln O+(V,- V,,)
= F(O, ()+(O, 7))+(U,+ U,,) In O+(V,+ V,,),
where
(3.5)
LAWS OF KADOMTSEV-POGUTSE EQUATIONS
~ = z + t, TJ = Z - t; U = U(~, TJ), v = V(~, TJ),
F( 0, ~) and cfJ( 0, ~) are arbitrary functions
dU u" = -, and so on
dTJ
41
Naturally, the arbitrary functions U, V, F, entering into (3.5) should be specialized in a proper way for selecting the solutions, invariant with respect to one of the above-listed subalgebras.
Class 2 Here we use the following general remark. Let L = (<1>1, ... , <1>5) be a symmetries subalgebra of the system OY. It is evident that the existence of the 2-invariant solutions requires the compatibility of the system <1>1 = 0, ... , <l>s = 0 (if the latter is linear, then compatibility means the existence of a nontrivial solution). Thus, the incompatibility of the system <1>; = 0 means that all the 2-invariant solutions are exhausted by trivial ones.
Such a situation takes place for some of the subalgebras contained in the list given in Section 2, namely .'l'Ll, 2Lz, 2tLl-2t1.3, .'l'tZ.l, 2tz.z, .'l'L.3. Below, we will take the subalgebra 2 = 2tl.3 as a characteristic example.
The subalgebra 21.1.3 is generated as a linear space by elements of the form
<1>1 = c+ /tF+2hzG-2gzH,
<l>z = M + fzF+ gzG+ hzH,
<1>3 = E+ /JF.
Hence, the system <1>1 = 0, <l>z = 0, <1>3 = 0 has the form
(y + 2hz)cpx - (x + 2gz)cpy + /tcp. = 0,
(y + 2hz)"'x - (x + 2gz)"'y + 11"'. = 0,
(x + gz)cpx + (y + hz)cpy + fzcp, - 2cp = 0,
(x + gz)"'x + (y + hz)cpy + fz"', - 2", = 0,
cpz + /Jcp, = 0,
"'z + /J"', =0,
and splits into equations containing either the function cp or the function "'. We single out those containing cp:
(y + 2hz)CPx - (x + 2gz)cpx + I1CP, = 0,
(x + gz)cpx + (y + hz)cpy + Izcp, - zcp = 0,
cpz + /Jcp. = O.
It follows from the last equation that
rp = rp(X, y, 0), 0 = t- hz.
Then, rpt = rp8. Furthermore, solving the first pair of equations of (3.6) with respect to rpx and rpy and writing down the compatibility condition rpxy = rpyx , one gets the following equation
[fz(h2x - g2Y) + !I(g2X + h2y + g~ + hm x
x rp8 + 2(g2Y - h2X)rp = O.
Its general solution is
_ ( ). ([f2(h2X - g2Y) + h2(g2 X + h2y + g2 + h2)]) rp - c x, y 0 2( h) . g2Y - 2 X
For brevity we denote the power of 0 by S. Let us substitute rp = c(x, y) . OS into the first equation of (3.6)
(y + 2h2)(cxOS + c . OS In 0 . Sx) - (x + 2g2 ) x x (cyOS + cOs In OSy) + IIcOS - 1 = O.
This equation contains an unknown function c(x, y), while it must be valid for any o. But the left-hand part is a linear combination of Os, OS In 0, OS-I with coefficients depending on x and y only. Thus, the coefficients at Os, oS-I, OS In 0 must be identically zero. In particular, a coefficient at oS-I, II . C = 0, whence, it follows that c(x, y) = O. Hence, system (3.6) has only a zero solution rp == O. The I{I == 0 identity is obtained analogously.
Class 3 This case unites two sub algebras, IeL and IeI.2, from the list in Section 2. Both of them contain as generators the translations E and F by the variables z and t. Solutions invariant with respect to these subalgebras, do not depend on z and t. This makes it possible to reduce the Kadomtsev-Pogutse equations to the form
rpxl{ly - rpyl{lx = 0,
rpxA1-rpy - rpyA1-rpx = I{Ix A1-l{Iy -l{IyA1-l{Ix.
The third generator of the subalgebra Iei.l yields two more conditions
(1- 2m)rp + (y + mx + gl)rpx + (- x + my + h1)rpy = 0,
(I-2m)l{I+ mx + gl)l{Ix + (-x + my + h1)l{Iy = 0,
(3.7)
(3.8)
which are solved by the characteristic method. Together with the first of Equations (3.7), it implies I{I = krp, k = const and
rp = rp( 0),
where 0 = O(x, y) is a known function whose explicit form may be obtained from (3.8).
We note now that if the condition 0/ = k'P holds, then the first equation of system (3.7) is automatically satisfied and the second one reduces to the equation
(k 2 - l)[V"-'P, V "-d"-'P]z = o. The latter is the identity if k = ± 1 and reduces to the Liouville equation
studied in Section 5. The analogical deductions can be drawn with respect to the subalgebra ,;t'i.2'
Class 4 Here we study the remaining three subalgebras from the list in Section 2 which have not been included in the previous considerations. They are 2:I3, 2: L, and
2:~,2' The system for finding 2:~.3-invariant solutions of the Kadomtsev-Pogutse
equations is of the form
(x + gl)'Px + (y + hl)'Py,+ Z'Pz + ('P, = 0,
(x + gdo/x + (y + hl)o/y + zo/z + to/, = 0,
gzo/x + hzo/y + O/Z = 0,
gz'Px + hz'Py + 'Pz = 0,
g30/x + h30/Y + 0/, = 0,
0/, + [V"-'P, V"-o/Jz = 'Pz·
(3.9)
The first six equations of this system are linear and are solved by the characteristic method:
Let
(3.10)
By substituting (3.10) into the Kadomtsev-Pogutse equations, we get the follow ing system of two ordinary differential equations
((z + 1)(<1><1>'" - WW"') + ((z + 1)(g3( - h3)(<1>'" - W''') +
+ ((z + 1)(<1>'<1>" - W'W") + (3g3(z -2h3 ( + g3)(" - W") +
+ 2((<1><1>' - WW') = 0,
'W - <l>W' + (gz( - hz)(' - W') - gz<l> + g3 W = O.
(3.11)
44
fP = «()(x - g2Z - g3 t + gl),
'" = 'I'(C)(x - g2Z - g3t+ gl),
where
C = y + h2 z - h3 t + hi, x- g2Z-

symmetries of partial differential equations: conservation laws â€” applications â€”...

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