Stability and instability of solitary waves of the

Stabilityandinstabilityofsolitarywavesofthefth-orderKdVequation:anumericalframeworkThomasJ.Bridges,GianneDerksandGeorgGottwaldDepartmentofMathematicsandStatistics,UniversityofSurrey,Guildford,Surrey,GU27XH,UKJuly24,2002AbstractThespectralproblemassociatedwiththelinearizationaboutsolitarywavesofthegeneralizedfth-orderKdVequationisformulatedintermsoftheEvansfunction,acomplexanalyticfunctionwhosezeroscorrespondtoeigenvalues.Anumericalframework,basedonafastro-bustshootingalgorithmonexterioralgebraspacesisintroduced.Thecompletealgorithmhasseveralnewfeatures,includingarigorousnumericalalgorithmforchoosingstartingvalues,anewmethodfornumericalanalyticcontinuationofstartingvectors,theroleoftheGrassman-nianG2(C5)inchoosingthenumericalintegrator,andtheroleoftheHodgestaroperatorforrelatingV2(C5)andV3(C5)anddeducingarangeofnumericallycomputableformsfortheEvansfunction.Thealgorithmisillustratedbycomputingthestabilityandinstabilityofsolitarywavesofthefth-orderKdVequationwithpolynomialnonlinearity.TableofContents1.Introduction..............................................................................22.LinearstabilityequationsandtheEvansfunction.........................................53.Inducedsystems,HodgedualityandtheEvansfunction...................................74.AshootingalgorithmonV2(C5)andV3(C5)..........................................115.InitialconditionsatL1fortheshootingalgorithm....................................115.1.AnalyticpathsofinitialconditionsatL1.....................................125.2.AnalyticEvansfunctionwithnon-analyticeigenvectors..............................156.Intermezzo:theGrassmanianisaninvariantmanifold....................................166.1.CantheGrassmannianbemoreattractive?.........................................177.Detailsofthesystematinnityforlinearized5th-orderKdV.............................188.Numericalresultsforaclassofsolitarywaves............................................219.Concludingremarks.....................................................................25Acknowledgements.......................................................................26References...............................................................................2711IntroductionThefth-orderKdVequationisamodelequationforplasmawaves,capillary-gravitywaterwaves,andotherdispersivephenomenawhenthecubicKdV-typedispersionisweak.Suchequationscanbewritteninthegeneralform@u@t+@3u@x3+@5u@x5=@@xf(u;ux;uxx);(1.1)forthescalar-valuedfunctionu(x;t),whereandarerealparameterswith6=0andf(u;ux;uxx)issomesmoothfunction.Theformof(1.1)whichoccursmostofteninapplicationsiswithf(u;ux;uxx)=Kup+1whereKisanonzeroconstantandp1generallyaninteger.ThisequationrstappearsintheliteratureintheworkofHasimotoandKawaharawithp=1wheregeneralizedsolitarywavesarecomputednumerically[37].Motivatedbywaterwaves,modelequationswithalargerclassofnonlinearitiesarederivedbyCraig&Groves[22].Otherformsfor(1.1)withfurthergeneralizationoffappearin[30,39,40].Thesolutionsof(1.1)ofgreatestinterestinapplicationsaretravellingsolitarywaves.Suchstates,travellingatspeedcandoftheformu(x;t)=^u(xct),satisfythefourth-ordernonlineardierentialequation^uxxxx+^uxx2c^uf(^u;^ux;^uxx)=A;(1.2)whereAisaconstantofintegration.Thissystemisnotintegrableingeneral,andcanhaveanextraordinaryrangeofsolitarywaves.AreviewoftheknownclassesisgivenbyChampneys[18].However,thereisverylittleintheliteratureaboutthestabilityofthesesolitarywaves.WhenthePDEisHamiltonian,forexamplewhenf(u;ux;uxx)isagradientfunction,onecanappealtoenergy-momentumargumentsfornonlinearstability(e.g.Ill'ichev&Semenov[33],Karpman[35],Dey,Khare&Kumar[23],Dias&Kuznetsov[24],Levandosky[40]),andthesymplecticEvansmatrixforarangeofanalyticaltechniquesforlinearinstability(Bridges&Derks[13,14]).However,theenergymomentummethodrequiresthesecondvariationofthemainfunctionaltohaveapreciseeigenvaluestructurewhichisoftenviolated.ThesymplecticEvansmatrixprovidesageometrictheoryforanalyticpredictionofinstabilityofsolitarywavesof(1.1)[13],butthesemethodsdonotapplywhenfin(1.1)isnon-gradient.Ontheotherhand,itwouldbeusefultohaveanumericalframeworkfor(1.1)evenintheHamiltoniancase.Inthenon-Hamiltoniancase,theonlyknowngeneralapproachistheEvansfunctionframe-work.Thisfunctioncanbeconstructedforthelinearizationaboutasolitarywaveofthe5th-orderKdVequation(aslongasthesolitarywaveexists),buttherearenoresultsintheliteratureontheconstructionoranalysisoftheEvansfunctionfor(1.1),exceptintheHamiltoniancase[14].Thespectralproblemforthelinearizationaboutasolitarywavecanalsobeformulatednumer-icallywithoutconsiderationoftheEvansfunction.Forexample,Beyn&Lorenz[10]consideralinearizedcomplexGinzburg-Landauequation,Barashenkov,Pelinovsky&Zemlyanaya[7]andBarashenkov&Zemlyanaya[8]consideralinearizednonlinearSchrodingersystem,andLiefvendahl&Kreiss[41]studythestabilityofviscousshockproles.Inallthreecases,theyapproachtheproblembydiscretizingthespectralproblemonthetruncateddomainx2[L1;L1]usingnitedierences,collocationoraspectralmethod,r