PhysicsLeuersA 163 (1992) 286—292 PHYSICSLETTERS ANorth-Holland

Bifurcationsof invariantmanifolds in the Gelfand—Dikii system

IvanDimitrovDifferential Equations,DepartmentofMathematics,SofiaUniversity,5 AntonIvanov,Sofia 1126,Bulgaria

ReceivedI May 1991;revisedmanuscriptreceived13 November1991; acceptedfor publication 16 January1992Communicatedby A.P.Fordy

The Gelfand—DikiiHamiltonianH= —q~—q~+ 3q2q~— q1 p~—

2P1P2 is studiedfroma topologicalpointofview. Thetopologyofthereallevel setsfor all valuesof theconstantsof motionis describedusingthecomplexalgebraicstructureoftheproblem.

1. Introduction

The Hamiltoniansystemof GelfandandDikii (seeref. [1J) is definedvia the Hamiltonian

~ (1.1)

It is completelyintegrable[2] asit hasa secondintegralof motion

~ (1.2)

Theaim of this paperis to classifyall real level sets.According to the classicalLiouville—Arnold theorem,we mayexpectthat theyconsistof tori, cylindersandplanes.Unfortunately,this theoremis notimmediatelyapplicableto the presentproblembecauseit is not a priori clearwhetherthe Hamiltonianflows aredefinedfor all valuesof the time variableThe Fomenkoclassification [31of all genericbifurcationsof the compactLiouville tori cannotbe appliedeitherbecausethe invariantmanifoldsof theGelfand—Dikii systemare nevercompact.Gavrilov [4] classifiedall genericbifurcationsof invariantmanifoldsin thegeneralizedHénon—Heilessystemusinga methodsimilar to ours.

In orderto studythetopologyof theproblemwe haveto studyits algebraicstructure.Thisis donein section2 usingthe techniquedevelopedin ref. [51. We provethat the genericcomplexinvariant manifoldscanbecompletedinto Abelian varieties(andthe systemis algebraicallycompletelyintegrable),eachof them beingisomorphicto theJacobivarietyof a genus-twohyperellipticcurve F (theorem1). In section3 wedeterminethetopological type of the genericreal invariantmanifolds.Non-genericinvariantmanifoldsare studiedin asimilar way. Later in the samesectionthe bifurcationsof Liouville cylindersandplanesare describedby bi-furcationsof curveson the correspondingJacobivarieties.

2. Algebraicstructure

Consideringthe Gelfand—DikiiHamiltonian (1.1),we obtain the correspondingHamiltoniansystem:

q1=—

2P2, j51_—4q~—6q1q2+p~,~2=—2p3.—2q1p2, j~2=2q2—3qf. (2.1)

Denoteby A~the affine algebraicvariety

Ac_{H~b,F=f}cC4. (2.2)

286 0375-9601/92/$ 05.00© 1992 ElsevierSciencePublishersB.V. All rights reserved.

Volume163,number4 PHYSICSLETrERSA 23 March1992

FollowingErcolaniandSiggia [6], weobtainthattheHamilton—Jacobiequationcorrespondingtothesystem(2.1) separatesin coordinatesu, v definedby the formulas

q1—u+v, q2=uv. (2.3)

Thenthe restrictionsof p~andP2 on A~canbe expressedin termsof coordinatesu, v in the following way,

u~~-v~/N~ __

Pi , P~ , (2.4)u—v u—v

whereP(z) is the polynomial

P(z)=z5+hz+f. (2.5)

Finally, using

d(u+v) . d(uv)dt =—2P2, q

2= dt =—2p1—2q1p2

andformulas (2.4) we concludethat u andv satisfy the systemof differentialequations

du dv udu vdv+ =0 + =2d1. (2.6)~ ~‘N~5

Letus supposethat thegenus-twohyperellipticcurveF: { w2= P(z)} is non-degenerate,i.e.,thediscriminant

of P(z), disc[P(z) J, is non-zero.Thus eqs.(2.6) imply that the solutionsof system(2.1), lying on A~,canbe expressedin termsof genus-twohyperelliptic thetafunctionsassociatedwith F.

If d(1, d(2 form a canonicalbasisof thespaceof holomorphicdifferentialson F (with respectto someca-

nonicalhomologybasison F), then thereexist a1, a2, b1, b2 suchthat

a1z+b1 a2z+b2dCi(z)= ~1~—~dz, d~2(z)= ~ (2.7)

Denoteby F2 thetwo-fold symmetricproductof F andlet ~be the Abel—Jacobimap:

~: F2-~J(F), ~i + ~2 -, (~fd~1+ d~1. d(2 + ~{dc~). (2.8)

whereP0 is a fixed basepointandJ(F) is the Jacobivarietyof F.Integratingsystem(2.6), we obtain

~(u+v)=2at+t°eJ(F), (2.9)

wherea= (a1, a2) and t°=(tv, t~)eJ(F)is the vectorof initial dataon J(F). Usingstandardformulas(seerefs. [7,8], expressingu+v anduv on J(F), we find

q1(t)= ~jjlnO(2at+l0+K)+di,

d O(2a1+t°—C(P5)+K)O(2a1+t°—C(P6)+K)

q2(l)= 2 02(2at+t°+K) , (2.10)

whered1 andd2 are suitableconstantsdependingon F only,

K=c(P1+P2), Ps=(O,~,/heF, P6=(0,—,JY)�F.

287

Volume 163, number4 PHYSICSLETTERSA 23 March1992

Now we canstatethe following

Theorem1. If (f/4)4+ (h/5)5~&0(see(2.2)) thentheaffine algebraicvarietyA~is a smoothcomplexman-ifold. Thereexists an embeddingi: A~—.CP8suchthat i(A~)= i ( A~)uD~,whereD~is a genus-twohyperel-liptic curve.I ( A~)is an Abeliansurfacewith (2, 2) polarizationwhich is biholomorphicallyequivalentto theJacobivarietyJ(F) andthe curveD~is biholomorphicallyequivalentto the curve ~(F). The Hamiltonianflows definedby H andF on A~extendbiholomorphicallyto flows on i(A~)which are straight-linemotions.

Later in this sectionweshall provethis theorem.Note that disc[P(z)]=constx[(J74)4+(h/5)5].Accordingto (2.10) thefunctionsq

1, q2,P1, P2 havenobranchpointsandexpressions(2.3) and(2.4) implythat they are single-valuedmeromorphicfunctionson F

2 andhencealso on J(F). Considerthe mapping

i: C4—+CIP8, (q1,q2,p1,p2)—~[fo,f1,f2,f3,f4,f5,f6,f7,f8J, (2.11)

where

f0=1, f1=q1, f2=q2, f3_—p~--q~+q1q2, f4=p2, f5=q1p2+p1,

f6=q~p2+p1p2—p2q2, f7_——27q~p2+7p~+28q1q2p2—20q~p1+l0q2p1,

fs=pp2+q~—q~q2+q~. (2.12)

Further,supposethat thebasepointof ~ is P~,,,i.e. ~(P~) = 0. Let D~c J(F) be the minimal divisoralongwhich at leastoneof the functionsf0 j~blows up. According to (2.10),~ = ~(F).

LemmaI. Thefunctionsf, i= 0, 1, ..., 8, consideredassingle-valuedmeromorphicfunctionsonJ(F) providea smoothembeddingof J(F) into CP

8.

Proofof lemma1. q1, q2, P1, P~are single-valuedfunctionson J(F) andhencej also havethis property.

According to the classicalLifschitz theorem[9,p.3171 we concludethat every basisof 2( 3D~)definesanembeddingof J(r) into CP

8 (becausedim .2”( 3D~)= 3 x 3= 9). As D,~,,is the minimal divisor along whichthe functionsq

1, q2, Pi, P2 blow up, it mustbe thatfc2’(kD,~,)for someke7L.We shall provethat k=3 andthatj are linearly independenton J(F). To thisend,we shallfind theasymptoticexpansionsof thesolutionsof (2.1) asfunctionsof time t. Thisprocedureis a part ofthewell-known Painlevétest.We find that theabovesystemadmitsa family of Laurentsolutionsdependingon threeparametersa, /3, y:

q1=t2+a—3a2t2+4flt3—l0a3t4+6aflt5+yt6+~~(153a5+36fl2+9a~’)t8+O(t9),

q2 =3at

2—6a2+6flt—9a3t2—1(9y+33a4)t4—l8a2flt5+~(27a5+12fl2—3ay)t6+O(t7)

P1 = —t5+2at3—fl—4a3t+15aflt2+(l1y+22a4)t3+30a2flt4+j~j-(774a5+l20$2+96ay)t5+O(t6)

p2=t~

3+3a2t—6flt2+20a3t3—l5aflt4—3yt5—~(306a5+72fl2+l8cty)17+O(t8). (2.13)

After substituting(2.13) into (2.12),we obtain

f0=l, fj=t

2+..., f2=3at

2+..., f3=9a

2t2+..., f4=t

3+...,

f5=3at

3+..., f6_—9a

2t3+..., f7_—72a

3t3+..., f8=918t

3+.... (2.14)

Thecomplexa,/3, y, for which (q1, q2,p~,p2)eA~,parametrizethe poledivisorD,~,.Substituting(2.14) into

H(q1, q2, P1,P2) = h andF(q1, q2, Pi, P2)=f andsolvingthe simultaneoussystemfor a, /1andy, we concludethata and/3 satisfy the following equation,

288

Volume 163, number4 PHYSICSLETTERSA 23 March1992

(9fl)2=(3a)5+h(3a)+f. (2.15)

(2.14) and(2.15) imply thatthefunctionsf0,...,J~arelinearly independentonJ(F) and,obviously,fa9(3D,~).

This completesthe proofof lemma 1.

Proofoftheorem1. Identifying J(F) andi(A~),wedenotetheimageof D,,, in T~~Jagainby D~.Accordingto theproofof lemma I we concludethatD,,. is anaffine curveparametrizedby the curve (2.15). Obviously,the curvesF and (2.15) are bothnon-degenerateif oneof themis non-degenerate.Hencei(A~)=J(F)\D,~is a smoothcomplex manifold. Notice that i is a biholomorphicmappingbetweensome neighbourhood

C4 of A~and i(VA~) c CP8. Indeed,if (q

1, q2, Pi, P2) eA~thendet[8 (Jj,f2,f4,f5 )/8 (q1, q2, Pi, P2)1= 1andhencerank(i) = 4. As i( A~)is asmoothcomplexmanifold,it is concludedthatA~isalsoa smoothcomplexmanifold.

It remainsto provethattheHamiltonianflows g},. andg~correspondingtoH andF linearizeonJ(I’). (2.9)implies that the flow g~,linearizeson J(F). Theflow g~canbe extendedholomorphicallyto a flow on J(F)(and hence it can be linearizedon J(F)) in the following standardway: If ZEDcX, we define g~(z) =

gnTg~.gj,~(z)for suitable r. Thisdefinitiondoesnotdependupon thechoiceoft asthe aboveflows commute.

This completesthe proofof theorem1.

3. Topologicalanalysis

In this sectionwe considersystem(2.1) asa systemof realdifferentionalequationsandthe flows g~and

g~will be real flows (i.e. taR).Denoteby AR the real invariantmanifold of system(2.1):

AR={H=h, F=f} =R’~.

In thissectionwe shall describethe topologicalnatureof the real flow g~,.Thismeans(in the contextof thepresentpaper) thatwe haveto describe

(i) the topologicaltypeof A~for all constantsof motion h andj(ii) how the sets AR fit togethertopologicallyas h andf varyto makeup R~(q

1, q2, Pi~P2).

As will be seenfrom the next theorem,the topologicaltypeof AR maychangeonly asthepoint (h,f) passesthroughthe set

B={(h,J)eR21(h/5)5+(f/4)4=0}.

Forthat reasonthesetB will becalledthe bifurcationsetof system(2.1). In thenext theoremthe topologicaltype of the non-degeneratelevel setsA~is determined:

Theorem2. AR consistsof(i) one plane,when (h/5)5+ (J/4)4>0,(ii) oneplaneandonecylinder, when (h/5)5+ (f/4)4<0.

Proof AR is the realpart of A~andthusit is parametrizedby formulas (2.3) and (2.4) with u andv co-ordinatessuchthat q

1, q2,Pt, P2arereal.Thentherearetwo possibilitiesfor u andv foreach(q1,q2,Pt,P2) EAR:

(i) uelR, vaR, P(u)~0, P(v)~0;

(ii) u=v,

ThepolynomialP(z) hasoneor threesimplereal roots dependingon the sign of B(h, f) = (h/5 )~+ (~f/4)4•

Namely, it hasthreereal roots if B(h,f) <0 andone if B(h,f)>0.

289

Volume163, number4 PHYSICSLETTERSA 23 March1992

First weconsiderthe caseB(h,f)>O. Denoteby z0 theuniquereal root of P(z). Thenformulas(2.3) and

(2.4) parametrizeAR wheneither

(i) uaR, vaLR, u>~z0,

or

(ii) u=v,

WeshallprovethatAR isconnected.Let (qft),q~,p~,ps])) and(q~21,q~2),p~2),p~2))betwo differentpoints

on AR which are parametrizedby (u~,vU)) and (ut2~,vt2~),respectively,and

(A) uU)_~vU), \/P(u~’~)=..~/P(v”~),ut=vt2~, FP(U(2)) =~/P(V2)

Either u~’~andu12~or u~’~andvtt~are on the sameside of the real line P. Weconnectthis pairwith a pathwhich doesnot crossP. The imageof this pathon AR definedby (2.3) and (2.4) connects(q~),~p I) pp)) and (q~2),q~2),~ p~2)).

(B) uU)=v(l), \/‘P(U(I))\/’p(v(l)) ut2~aR, v~2~eR,u~2~?~z0,v

t2~~z0

Herewe choosethe pathin the following way: at the endof the pathu~2~andvt2~coincidewith a point b on

Pand..JP(u (2)) and,.JP(vt2~)coincide.Thenwe connectb withu ~ andvu) witha pairof complexconjugatepaths.The imageof the productof thesepathson AR connects(q~I),q ,p~,p~’’)and (q~2),q~2),~

(C) UW6P, vt’~eP, u~>~zo, v

t’~>~z0, i=l, 2.

Herewe choosethe pathas in the abovecases.This completesthe proof that AR is connected.Thereis noclosedpathon AR which is not homotopicto zeroand,obviously,AR is not compact.ThusAR is a planebe-cause,accordingto theoremI, Hamiltonianflows are complete.

It remainsto considerthecaseB(h,f) <0. ThenP(z) hasthreerealroots z1<z2 < z1~.Thecoordinatesu andv parametrizethe realpointswhen:

(i) ua[z1,z2], ve[z0,x);

(ii) ue[z1,z2], va[z1,z21, or ua[z0,cc), ve[z0,cc), or u=iJ~P;

v~=vT~.

It is easyto prove (asin the caseB(h,f)>0) that (ii) definesoneconnectedcomponentof AR which is aplane.Obviously, (i) definesa connectedcomponentalso.Thesetsdefinedby (i) and (ii) haveno commonpoints. Indeed,let za(z2,z0) be a fixed point, then

(z—u)(z—v)=z2—q

1z+q2<0, foruandvdefinedby(i),

(z—u)(z—v)=z2—q

1z+q2>0, foru andvdefinedby(ii).

Thustheplanez2— q

1z+ q2 = 0 dividesthem.Finally, thesetfrom (i) is a productof onecircle correspondingto [z1, z2] andone line correspondingto [z0, cc). Thus the set from (i) is topologicallyequivalentto onecylinder.

Thiscompletesthe proofof theorem2.

Furtherweshalldeterminethebifurcationsofthe systemwhenthepoint (h,f) passesthroughB. Thegraphof B consistsof threedifferentparts (fig. l):f>0,f=0 andf<zO.

Wecannotuseformulas (2.3)and(2.4) for (h,f ) aBbutit is easyto seethat theymakesenseaftera limitingprocedure.Further,we shall use (2.3) and (2.4) for all valuesof (h,f).

290

Volume 163, number4 PHYSICSLETTERSA 23 March1992

p2 p2-f ~ P2 + C

Fig. 1. Thebifurcationset. Fig. 2. Thebifurcation through(1).

p2 p2 p2 ÷ c p2 p2 p2 + c

Fig. 3. Thebifurcationthrough(2). Fig. 4. Thebifurcation through(3).

(1) Let f< 0. Thepolynomial P(z) hasa double real root z1 anda simpleroot z0>z1. Thereare two pos-

sibilitiesfor the (u, v)-coordinates:

(i) u=z1, yE [z0,co);

(ii) ua [z0,cc), va[z0, co), or u=tJ~R; ~

It is easyto concludethat (i) parametrizesone line and(ii) onecylinder (in a similarway asin theorem2).Thus we seethat the bifurcationin case(1) is as in fig. 2.

(2) Let f> 0. ThenP(z) hasa doubleroot z0 andasimpleroot z1<z0. Thebifurcationin case(2) is shownin fig. 3.

(3)f=h=0. P(z) hasroot 0 with multiplicity five. Formulas(2.3)and(2.4) presentall pointsonAR exceptthe point (0, 0, 0, 0). The bifurcationin case (3) is shown in fig. 4.

Thiscompletesthe topologicalanalysis.

4. Conclusion

Themethodusedin thispapercaneasily be appliedto the Gelfand—Dikii hierarchyin all dimensions(seeref. [6]). Moreover, thebifurcationsare easyto describe.

In principlethe topologicalanalysisin the caseconsideredin thispapercanbe doneusing (2.10).To this

291

Volume163, number4 PHYSICSLETTERSA 23 March 1992

end,wehaveto determinetheconstantsd1 andd2. But thisseemsto bedifficult to apply in higherdimensions.This showsthe advantageof our method.

Acknowledgement

I thank Emil Horozovfor helpful discussions.

References

[1] 1.M. GelfandandL.A. Dikii, Funct.Anal. AppI. 13 (1979)8.[2] V.1.Arnold, Mathematicalmethodsof classicalmechanics(Springer,Berlin, 1980).[3] V.V. TrofimovandA.T.Fomenko,Itogi Nauk.Tech.29 (1986) 3.[4] L. Gavrilov,PhysicaD 34 (1989)223.[5] M. Adler andP. vanMoerbeke,A systematicapproachtowardssolvingintegrablesystems,in: Perspectivesin malhematics(Academic

Press,New York, 1988).6] N. ErcolaniandE. Siggia,PhysicaD 34 (1989)303.7] B.A. Dubrovin,Russ.Math.Surv. 36 (1981) 11.

[8] D. Mumford, Tatalectureson thetafunctionsII (Birkhauser,Basel,1984).[9] P.Griffiths andJ. Harris,Principlesof algebraicgeometry(Wiley—Interscience,New York, 1978).

292