The deformations of Whitham systems and Lagrangian

arXiv:nlin/0601050v3[nlin.SI]30Mar2006ThedeformationsofWhithamsystemsandLagrangianformalism.A.Ya.MaltsevL.D.LandauInstituteforTheoreticalPhysics,119334ul.Kosygina2,Moscow,[e-mail]agrangianformalismofthedeformationsofWhithamsystemshavingDubrovin-Zhangform.Asanexamplethecaseofmodulatedone-phasesolutionsofthenon-linear”V-Gordon”equationisconsidered.1Introduction.Thispaperisacontinuationofthepaper[72]connectedwiththedeformationsofthehyperbolicWhithamsystems.ThemethodofdeformationsofWhithamsystemssuggestedin[72]isconnectedwiththeslowmodulationsofm-phasequasiperiodicsolutionsϕi(x,t)=Φi(k(U)x+ω(U)t+θ0,U),i=1,...,n(1.1)ofsomesystemFi(ϕ,ϕt,ϕx,...)=0,i=1,...,n(1.2)HerethefunctionsΦi(θ,U)are2π-periodicfunctionsw.r.t.eachθα,α=1,...,m,thevaluesθ0=(θ10,...,θ1m)arearbitraryinitialphaseshiftsandthevariablesU=(U1,...,UN)playtheroleoftheparametersofm-phasesolutions.Thefunctionsω(U)=(ω1(U),...,ωm(U))andk(U)=(k1(U),...,km(U))arethe”frequencies”andthe”wavenumbers”ofthesolution(1.1)suchthatthefunctionsΦi(θ,U)satisfythesystemFiΦ,ωα(U)Φθα,kβ(U)Φθβ,...
=0(1.3)foreveryθandU.InWhithammethodthesmallparameterǫisintroducedsuchthatX=ǫxandT=ǫtplaytheroleofthe”slow”coordinatesin(x,t)-space.Thecorrespondingslowlymodulatedsolutionsof(1.1)arerepresentedintheasymptoticform1φi(θ,x,t)=Xk≥0Ψi(k)S(X,T)ǫ+θ0,X,Tǫk(1.4)whereallΨ(k)(θ,X,T)are2π-periodicw.r.t.eachθαfunctions.ThefunctionS(X,T)=(S1(X,T),...,Sm(X,T))isthe”modulatedphase”ofthesolution(1.4).ThefunctionΨ(0)(θ,X,T)satisfiesthesystem(1.3)andbelongstothefamilyΛ={Φ(θ+θ0,U)}ateveryfixedXandT.WehavethenΨ(0)(θ,X,T)=Φ(θ+θ0(X,T),U(X,T))(1.5)andSαT=ωα(U(X,T)),SαX=kα(U(X,T))asfollowsfromthesubstitutionof(1.4)inthesystem(1.2).ThefunctionsΨ(k)(θ,X,T)aredefinedfromthelinearsystemsˆLij[U,θ0](X,T)Ψj(k)(θ,X,T)=fi(k)(θ,X,T)(1.6)whereˆLij[U,θ0](X,T)isalinearoperatorgivenbythelinearizingofthesystem(1.3)onthesolution(1.5).Theresolvabilityconditionsofthesystem(1.6)canbewrittenastheorthogonalityconditionsofthefunctionsf(k)(θ,X,T)toallthe”lefteigenvectors”(theeigenvectorsofadjointoperator)κ(q)[U(X,T)](θ+θ0(X,T))oftheoperatorˆLij[U,θ0](X,T)correspondingtozeroeigen-values.Theresolvabilityconditionsof(1.6)fork=1togetherwithkαT=ωαXgivetheWhithamsystemform-phasesolutionsof(1.2)playingthecentralroleintheslowmodulationsapproach.Likein[72]wewillassumeherethattheparameters(k,ω)canbeconsideredastheindependentparametersonthefamilyΛsuchthatthefullsetofindependentparametersU(exceptinitialphasesθα0)canberepresentedintheform(k,ω,n)wherekandωarethewavenumbersandthefrequenciesofthesolutionandn=(n1,...ns)aresomeadditionalparameters(iftheyexist).EasytoseethatthefunctionsΦθα(θ+θ0,k,ω,n),α=1,...,mandΦnl(θ+θ0,k,ω,n),l=1,...,sgivetheeigen-vectorsoftheoperatorˆLij[θ0,k,ω,n]correspondingtozeroeigen-values.2Letusgivealsothedefinitionofthefullregularfamilyofm-phasesolutionsof(1.2).1Definition1.1.WecallthefamilyΛthefullregularfamilyofm-phasesolutionsof(1.2)if1)ThefunctionsΦθα(θ,k,ω,n),Φnl(θ,k,ω,n)arelinearlyindependentandgive(forgenerickandω)thefullbasisinthekerneloftheoperatorˆLij[θ0,k,ω,n];2)TheoperatorˆLij[θ0,k,ω,n]has(forgenerickandω)exactlym+slinearlyindependent”lefteigenvectors”κ(q)[U](θ+θ0)=κ(q)[k,ω,n](θ+θ0)dependingontheparametersUinasmoothwayandcorrespondingtozeroeigen-values.ItcanbeshownthatforfullregularfamilyΛthecorrespondingWhithamsystemputstherestrictionsonlyonthefunctionsU(X,T)=(k(X,T),ω(X,T),n(X,T))havingtheformkαT=ωαXC(q)ν(U)UνT−D(q)ν(U)UνX=0(1.7)(q=1,...,m+s,ν=1,...,N=2m+s)anddoesnotincludetheinitialphaseshiftsθα0(X,T).Definition1.2.LetuscalltheWhithamsystem(1.7)non-degeneratehyperbolicWhithamsystemif:1)Thesystem(1.7)isresolvablewithrespecttothetimederivativesofparametersUνandcanbewrittenintheformUνT=Vνμ(U)UμX,ν,μ=1,...,N(1.8)2)ThematrixVνμ(U)hasNlinearlyindependentrealeigen-vectorswithrealeigen-values.Providedthatthesystem(1.7)issatisfiedwecanfindthefirstcorrectionΨ(1)(θ,X,T)inthesolution(1.4)modulothelinearcombinationofthefunctionsΦθα(θ+θ0,k,ω,n),Φnl(θ+θ0,k,ω,n).Ingeneralschemewetrytofindrecursivelythehigherordercor-rectionsΨ(k)(θ,X,T)fromthelinearsystems(1.6).Thefunctionsθα0(X,T)andthefreedominthedeterminationofthefunctionsΨ(k)(θ,X,T)areusedtosatisfythecom-patibilityconditionsofthesystems(1.6)inhigherordersofǫ,sowegettherecursive1Thisdefinitioncorrespondstothe”weak”definitionoffullregularfamilyofm-phasesolutionsgivenin[72].3restrictionsonthecorrespondingparameters.2ThesolutionoftheWhithamsystem(1.7)(or(1.8))isconsideredusuallyasthecentralpointoftheprocedurewhichdefinestheglobalpropertiesofthemodulatedsolution.LetusalsomentionthewellknownfactthattheWhithamsystemscorrespondingtotheintegrablesystems(1.2)possessalsotheintegrabilityproperties.ThefirstconsiderationofdispersivecorrectionstoWhithamsystemsweremadein[5](seealso[6]-[7])wherethemulti-phaseWhithamwasalsofirstdiscussed.Aswaspointedoutin[5]thedispersivecorrectionscannaturallyariseintheWhithammethodbothinone-phaseandmulti-phasesituations.HereweconsiderthedeformationsofWhithamsystems(1.8)havingtheformofDubrovin-ZhangdeformationsofFrobeniusmanifolds([63,