Transverse instability and its long-term developme

arXiv:nlin/0208046v2[nlin.CD]30Aug2002Transverseinstabilityanditslong-termdevelopmentforsolitarywavesofthe(2+1)-BoussinesqequationK.B.Blyuss,T.J.Bridges,andG.DerksDepartmentofMathematics&Statistics,UniversityofSurrey,GuildfordGU27XH,UK(Dated:February8,2008)AbstractThestabilitypropertiesoflinesolitarywavesolutionsofthe(2+1)-dimensionalBoussinesqequa-tionwithrespecttotransverseperturbationsandtheirconsequencesareconsidered.Ageometricconditionarisingfromamulti-symplecticformulationofthisequationgivesanexplicitrelationbetweentheparametersfortransverseinstabilitywhenthetransversewavenumberissmall.TheEvansfunctionisthencomputedexplicitly,givingtheeigenvaluesfortransverseinstabilityforalltransversewavenumbers.Todeterminethenonlinearandlongtimeimplicationsoftransverseinstability,numericalsimulationsareperformedusingpseudospectraldiscretization.Thenumericsconfirmtheanalyticresults,andinallcasesstudied,transverseinstabilityleadstocollapse.PACSnumbers:05.45.Yv,47.35.+i1I.INTRODUCTIONOneofthefundamentalwaysthatasolitarywavetravelinginonespacedimensiongeneratesatwospacedimensionalpatternisthroughtransverseinstability.Atransverseinstabilityofalinesolitarywaveisassociatedwithaclassofperturbationstravelinginadirectiontransversetothebasicdirectionofpropagation.Inadditiontoestablishingtheexistenceoftransverseinstability,amajorquestioniswhatimplicationsthisinstabilitymeansforthelong-termbehaviourofthesystem:doesitsettleintoanewtwo-space-dimensionalpattern,orcollapse?InthispaperwestudythissequenceofquestionsforthecanonicalBoussinesqequationintwospacedimensionsutt=(f(u)+εuxx)xx+σuyy,(1)whereε=±1andσ=±1.Ingeneral,f(u)canbeanysmoothfunction,butthecanonicalformoftheBoussinesqequationhastheformf(u)=D(u2−u)withD=±1.WhenD=−1,ε=1andσ=1thisequationwasderivedbyJohnson[18]todescribethepropagationofgravitywavesonthesurfaceofwater,inparticularthehead-oncollisionofobliquewaves,anditwasderivedbyBreizmanandMalkin[8]inthecontextofLangmuirwaves.Intheabsenceofthetransversevariation(i.e,uy=0)andforε=−1,D=−1thisequationreducestotheso-called”good”Boussinesqequation,whichiswell-posed,andforwhichsech2-solutionsexistforanycwith|c|1.Thesewavesarestablewhen12|c|1[9].Forthecase|c|12itwasshownbycomputer-assistedsimulationoftheleadingtermintheTaylorexpansionoftheEvansfunctionthatthereisanunstableeigenvalue[3].Thisresultwasgeneralizedtoincludesolitarywaveswithnonzerotails,andrigorouslyprovedusingthesymplecticEvansmatrixin[13].TransverseinstabilityofsolitarywaveshasbeenwidelystudiedsincetheseminalworkofZakharov[25]onthenonlinearSchr¨odingerequationandtheworkofKadomtsev&Petviashvili[19]ontransverseinstabilityoftheKorteweg-deVriessoliton.Sincethen,transverseinstabilityofsolitarywaveshasbeeninvestigatedforawiderangeofmodels;examplesincludethenonlinearSchr¨odinger(NLS)equationandrelatedequations[21,22,224],Kadomtsev-Petviashviliequation[2,5,17,20],theZakharov-Kuznetsovequation[4,10,22],andwaterwaves[11].AreviewoftransverseinstabilityforNLSandotherrelatedmodelscanbefoundinKivshar&Pelinovsky[20].Inthispaper,wewillfirstuseageometricconditionasderivedin[10]togetanexplicitcri-terionforsmalltransversewavenumberinstability.Forthisweusethemulti-symplecticfor-mulationof(1)inanessentialway.Togetdetailedinformationforalltransversewavenum-berswecomputeexplicitlytheEvansfunctionforthe(2+1)-dimensionalBoussinesqmodellinearizedaboutalargerfamilyoflinesolitarywaves(allowingthestateatinfinitytobenonzero).Plotsofthedependenceofthegrowthrateonthetransversewavenumberarepresented.Thepost-instabilitybehaviourofthenonlinearproblemisstudiedusingdirectnumericalsimulation.Thenumericalevidenceconfirmstheanalyticresultsandsuggeststhatthepost-instabilityinthenonlinearsystemleadstocollapseinallcases.Amulti-symplecticpseu-dospectraldiscretization[15]isusedasabasisforthenumericalsimulations.Thenumericalschemeisappliedtothefulltwo-dimensionalPDEandweobservetransversemodulationandfurtherdevelopmentofthelongitudinalandtransverseinstabilities,resultinginthecollapseoftheinitiallinesolitarywaves.Intheparameterregionwheretheanalyticcrite-rionindicatesthatthesolitarywavestateislongitudinallystablebuttransverselyunstable,simulationssupporttheanalyticresultsandprovideinsightintothelong-termdevelopmentofthisinstability.II.MULTI-SYMPLECTIFYINGTHEEQUATIONSTheBoussinesqsystemhasarangeofgeometricstructures.Firstly,werecordtheLa-grangianandHamiltonianstructures.Letu=φxx,thenthesystemisLagrangianwithL=Z−12φ2xt+F(φxx)+12εφ2xxx+12σφ2xydxdydt,whereF(·)isanyfunctionsatisfyingF′(·)=f(·).TheBoussinesqequationcanberepresentedasaHamiltoniansysteminanumberofways(e.g.[23]).Forexample,letH=ZF(u)−12εu2x+12Φ2x+12σw2y+γ(u−wx)dxdy,3whereγisaLagrangemultiplierassociatedwiththeconstraintu=wx.WithHamiltonianvariables(Φ,u,w,γ)thegoverningequationstaketheform−ut=δHδΦ=−ΦxxΦt=δHδu=f(u)+εuxx+γ,0=δHδw=γx−σwyy,0=δHδγ=u−wx.(2)However,themostinterestingformof(1)forthepresentpurposesisthemulti-symplecticformulationwhichcanberepresentedinthecanonicalform[14]MZt+KZx+LZy=∇S(Z)Z∈R6,(3)whereZ=q1q2q3p1p2p3,M=