On Properties of Hamiltonian Structures for a Clas

arXiv:0711.2599v2[nlin.SI]18Nov2007OnPropertiesofHamiltonianStructuresforaClassofEvolutionaryPDEsSi-QiLiu∗Chao-ZhongWu†YoujinZhang‡DepartmentofMathematicalSciences,TsinghuaUniversity,Beijing,P.R.ChinaAbstractIn[17]itisprovedthatforcertainclassofperturbationsofthehyperbolicequationut=f(u)ux,thereexistchangesofcoordinate,calledquasi-Miuratransformations,thatreducetheperturbedequationstotheunperturbedone.WeproveinthepresentpaperthatifinadditiontheperturbedequationspossessHamiltonianstructuresofcertaintype,thesamequasi-MiuratransformationsalsoreducetheHamiltonianstructurestotheirleadingterms.Byapplyingthisresult,weobtainacriterionoftheexistenceofHamiltonianstructuresforaclassofscalarevolutionaryPDEsandanalgorithmtofindouttheHamiltonianstructures.Keywords:Hamiltonianstructure,quasi-Miuratransformation,quasi-triviality1IntroductionWeconsiderinthispaperthefollowingclassofgeneralizedscalarevolutionaryPDEsoftheunknownfunctionu=u(x,t)ut=f(u)ux+∞∑k=1εkFk(u;ux,···,u(k+1)),f′(u)6=0,(1.1)whereut=∂tu,ux=∂xu,u(ℓ)=∂ℓxu.ThefunctionsFkareassumedtobepoly-nomialsofux,...,u(k+1)withcoefficientsdependingsmoothlyonu,andtheyarehomogeneousofdegreek+1underthefollowingassignmentofdegrees:degu(ℓ)=ℓ,ℓ≥1;degh(u)=0forasmoothfunctionh(u).(1.2)∗[e-mail]†[e-mail]‡[e-mail]dsideof(1.1)truncates,theequationisausualevolutionaryPDE.Ingenerally,weallowthetherighthandsideof(1.1)tobenon-truncated.Suchequationsarise,forexample,whenweconsideranevolutionaryPDEoftheformut=K(u;ux,···,u(N))(1.3)withafunctionKbeinganalyticatux=···=u(N)=0andK(u;0,···,0)=0.Wecanmaketherescalingx7→εx,t7→εt,andexpandthetherighthandsideoftheequation(1.3)intoapowerseriesofε,theresultingequationisoftheform(1.1).Equationsoftheform(1.1)alsoappearwhenweconsiderPDEsofthefollowingtypeut−uxxt=K(u;ux,···,u(N)),(1.4)wherethefunctionKisgivenasabove.Animportantexampleofsuchclassofequa-tionsistheCamassa-Holmequation(seeExample3.6below)whichisanintegrablenonlinearPDEdescribingshallowwaterwaves[1,2,13].Wecanperformthesamerescalingandrewritetheequation(1.4)asfollows:ut=(1−ε2∂2x)−1K(u;εux,...)ε=∞∑k=0ε2k∂2kxK(u;εux,...)ε,therighthandsideoftheaboveequationcanalsobeexpandedintotheform(1.1).HeretheTaylorexpansionconvergesintheformalpowerseriestopology.InthispaperweconsiderpropertiesoftheHamiltonianstructuresforequationsoftheform(1.1).Notethatequation(1.1)isaperturbationofthehyperbolicequationvt=f(v)vx,f′(v)6=0.(1.5)ThisequationpossessesinfinitelymanyHamiltonianstructures.Infact,foranysmoothfunctiong(v),thereisaPoissonbracketonthespaceoffunctionalsofv(x)whichisdefinedby{v(x),v(y)}g=g(v)δ′(x−y)+12g′(v)vxδ(x−y),(1.6)herev(x),v(y)areregardedasdistributions.Thenequation(1.5)canbeexpressedasaHamiltoniansystemvt={v(x),Hg}g,wherethefunctionalHgisdefinedbyHg=Zh(v)dx,withhsatisfyinggh′′+12g′h′=f.(1.7)2Wecalltheequation(1.1)aHamiltoniansystemoraHamiltonianperturbationoftheequation(1.5)ifitpossessesaHamiltonianstructureut={u(x),H},wherethePoissonbracketandtheHamiltonianhavetheforms{u(x),u(y)}=g(u)δ′(x−y)+12g′(u)uxδ(x−y)+∞∑k=1εkk+1∑l=0Ak,l(u;ux,···,u(l))δ(k+1−l)(x−y),(1.8)H=Zh(u)+∞∑l=1εlBl(u;ux,···,u(l))!dx.(1.9)HereAk,l,Blarehomogenousdifferentialpolynomialsofdegreel.StudiesofHamiltonianperturbationsofgeneralhyperbolicsystemswereiniti-atedbyDubrovinandtheauthorsofthepresentpaperin[8].Itwasprovedtherethatanybihamiltonianperturbationofahyperbolicsystemwhichpossessesasemisim-plebihamiltonianstructureisquasi-trivial,thismeansthattheperturbationtermscanbeeliminatedbyachangeofthedependentvariables,calledaquasi-Miuratrans-formation,ofthesystem.In[7],DubrovinstudiedtheHamiltoniansperturbationof(1.5),andprovedcertainuniversalityofbehaviorofsolutionsoftheperturbedequa-tion(1.1)nearthepointofgradientcatastrophe.Healsoprovedthatthepropertyofquasi-trivialitystillholdstrueattheapproximationuptoε4forHamiltonianpertur-bationsof(1.5),anditplaysanimportantroleinobtainingthemainresultof[7].Afurtherstudyofthequasi-trivialitypropertyofequationsoftheform(1.1)wasgivenin[17],wherethefollowingtheoremisgivenonthevalidityofthispropertywithouttheassumptionofexistenceofHamiltonianstructures.Theorem1.1(Quasi-TrivialityTheorem)ForanyevolutionaryPDEoftheform(1.1),thereexistsachangeofthedependentvariablecalledaquasi-Miuratransfor-mationv=u+∞∑k=1εk1uLkxMk∑m=0Yk,m(u;ux,···,u(Nk))(logux)m(1.10)whichreducestheequation(1.1)toitsleadingterm(1.5).HereLk,Mk,Nkarecertainintegersthatonlydependonk,andYk,marehomogenousdifferentialpolynomialsofdegreeLk+k.Thequasi-Miuratransformationoftheabovetheoremisalsocalledthereducingtransformationfor(1.1).In[17]thistheoremisalsoappliedtoobtainacriterionofintegrabilityfortheclassofequationsoftheform(1.1).3NowgivenaHamiltonianperturbation(1.1),weconsiderinthispaperthere-lationshipbetweentheHamiltonianstructureanditsreducingtransformation.Themainresultisthefollowing:Theorem1.2(MainTheorem)ForanyHamiltonianperturbation(1.1)oftheequa-tion(1.5)withaHamiltonianstructureoftheform(1.8),(1.9),th