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arXiv:0706.4459v1[math.AP]29Jun2007NONLINEARSTABILITYOFSOLITARYTRAVELLING-WAVESOLUTIONSFORTHEKAWAHARA-KDVANDMODIFIEDKAWAHARA-KDVEQUATIONS.F´ABIOM.AMORINNATALI11DepartmentofMathematics,IME-USPRuadoMat˜ao1010,CEP05508-090,S˜aoPaulo,SP,Brazil.Abstract.Inthispaperweestablishthenonlinearstabilityofsolitarytravelling-wavesolutionsfortheKawahara-KdVequationut+uux+uxxx−γ1uxxxxx=0,andthemodifiedKawahara-KdVequationut+3u2ux+uxxx−γ2uxxxxx=0,whereγi∈Risapositivenumberwheni=1,2.Themainapproachusedtodeterminethestabilityofsolitarytravelling-waveswillbethetheorydevelopedbyAlbertin[1].1.Introduction.Thisworkpresentsnewresultsaboutthestabilityofsolitarytravelling-wavesolutionsassociatedwiththeKawahara-Korteweg-deVriesandmodifiedKawahara-Korteweg-deVriesequations(KawaharaandmodifiedKawaharaequationsrespectively,henceforth),ut+uux+uxxx−γ1uxxxxx=0,(1.1)andut+3u2ux+uxxx−γ2uxxxxx=0,(1.2)whereγi0wheni=1,2andu:=u(x,t)isarealfunction.Here,weconsiderx∈Randt∈R.Theseequationsmodelthepropagationonnonlinearwater-wavesinthelong-wavelengthasinthecaseKdV’sequations.Suchamodel-scenarioisexpected.Infact,letubeasmoothsolutionof(1.1)and(1.2),thenifγi→0uniformly,i=1,2weobtainthatuisasolutionoftheKorteweg-deVriesandmodifiedKorteweg-deVriesequations,ut+uux+uxxx=0,(1.3)2000MathematicsSubjectClassification.76B25,35Q51,35Q53.Keywordsandphrases.Dispersiveequations,Kawahara-KdV-typeequation,solitarytravellingwaves,nonlinearstability.SupportedbyFAPESP/SPBrazilundergrant06/61310-3Date:June22,2007.12F.NATALIut+3u2ux+uxxx=0,(1.4)respectively.Thequestionofthestabilityoftheequations(1.3)and(1.4)hasbeenstudiedbymanyresearchersinthecaseofsolitarywaves,forexamplesee[1],[2],[3],[6],[7],[17]and[22].Now,formoregeneraldispersiveevolutionequationsofthegeneralformut+upux−Mux=0,(1.5)animportantstudyofsufficientconditionsforthestabilitywasestablishedbyAlbertin[1](seealso[2])onsolitarytravellingwavesoftheformu(x,t)=ϕ(x−ct),fortheequation(M+c)ϕ−1p+1ϕp+1=0.(1.6)In(1.5)(andconsequentlyin(1.6)),p≥1isanintegerandMisaFouriermultiplieroperatordefinedbydMg(k)=δ(k)bg(k),k∈R,(1.7)wherethesymbolδisameasurable,locallybounded,evenfunctiononRandsatisfiesthatA1|k|ν≦δ(k)≦A2(1+|k|)μforν≦μ,|k|≥k0,δ(k)bforallk∈RandAi0.In[1]sufficientconditionsweredeterminedtoobtainthatthelinear,closed,unbounded,self-adjointoperatorL:D(L)→L2(R),definedonadensesubspaceofL2(R)byLζ=(M+c)ζ−ϕpζ(1.8)whereM+cisapositiveoperator,itwillhaveexactlyonenegativeeigenvaluewhichissimpleandzeroissimplewitheigenfunctionddxϕ.ThesespecificspectralpropertiesofLwereobtainedprovidedϕisapositivesolitarywavesatisfyingthatbϕ0andcϕp∈PF(2)classdefinedbyKarlinin[19].Thiswork,wewillshowanewexplicitfamilyofstablesolitarytravelling-wavesolutionsfortheKawaharaandmodifiedKawaharaequations(1.1)and(1.2)respectively.Suchsolitarywavesaregiven,inthecaseoftheKawaharaequation,byϕω(ξ)=β1sech2(bξ)+λ1sech4(bξ),(1.9)whereω0isthewavespeedwithβ1,λ1andb0.InthecaseofthemodifiedKawahara,wehaveφc(ξ)=β2sech2(αξ)(1.10)wherec0isthewavespeedwithαandβ20.However,inthisspecificcase,wecannotobtainthenontrivialsolitarytravelling-wavesolutionassociatedwiththemodifiedKorteweg-deVriesequation(1.4)asγ2→0,namelythesolitarytravelling-wavesolutionNONLINEARSTABILITYFORTHEKAWAHARA-KDVTYPEEQUATIONS3gω(x)=√2ωsechπx2√ωassociatedwiththeequation(1.4).Notethatin(1.9),ifλ1→0then,wecouldexpectaprofilesolitarywaveassociatedwiththeKdVequation(1.3).Forbothcases,wewillusethefollowingconditionsthatimplythestability(see[6],[7],[17],and[22]):(P0)thereisanon-trivialsmoothcurveofsolutionsfor(1.6)oftheform,c∈I⊆R→ϕc∈H2(R);(P1)Lhasauniquenegativeeigenvalueλ,andwhichissimple;(P2)theeigenvalue0issimple;(P3)ddcZRϕ2c(x)dx0.(1.11)Anguloin[5]showedaresultoftheinstabilityofsolitarytravelling-wavesolutionsassociatedwiththegeneralizedfifth-orderKdVequationoftheformut+uxxxxx+buxxx=(G(u,ux,uxx))x,(1.12)whereG(q,r,s)=Fq(q,r)−rFqr(q,r)−sFrr(q,r)forsomeF(q,r)whichishomoge-neousofdegreep+1forsomep1,butthesolitarywavewasobtainedbysolvingaconstrainedminimizationprobleminH2(R)whichisbasedonresultsobtainedbyLevandosky.Theinstabilityofthisclassofsolitary-wavesolutionsisdeterminedforb6=0,anditisobtainedbymakinguseofthevariationalcharacterizationofthesolitarywavesandamodificationofthetheoriesofinstabilityestablishedbyShatah&Strauss[21],Bona&Souganidis&Strauss[8]andGon¸calvesRibeiro[16].Levandosky’smethodwasalsousedbyBridges&Derks[11]toshowaresultofthelinearinstabilityofsolitarywavesasso-ciatedwiththeequation(1.12).However,theauthorsmakeuseofageometricapproach.Werecall,fromtheresultsofAlbertin[1],thesolitarywavesolutionu(x,t)=ϕ(x−c0t)=sech4x−1235t,wherec0=1235,isastablesolutionoftheKawaharaequationoftheform,ut+uux+13420uxxx−11680uxxxxx=0.Inthisresult,theauthorusedthenontrivialpolynomialsofGegenbauertodeterminethesign(strictlynegative)ofthequantityI=(χ,ϕ)L2(R).Here,χ∈L2(R)issuchthatLχ=ϕ(seeTheorem3.1theSection3).4F.NATALIHaragusetal.in[18]studiedthespectralstabilityoftheperiodictravelling-wavesolutionswithwave-speedcoftheformu(x,t)=ϕ(x+ct),associatedwiththeequationut−uxxxxx+εuxxx−uux=0,whereε∈R.Theauthorsshowedthatthespec