Stability-of-periodic-travelling-shallow-water-wav

arXiv:0707.4365v1[nlin.SI]30Jul2007Stabilityofperiodictravellingshallow-waterwavesdeterminedbyNewton’sequationSevdzhanHakkaev1,IliyaD.Iliev2andKirilKirchev21FacultyofMathematicsandInformatics,ShumenUniversity,9712Shumen,Bulgaria2InstituteofMathematicsandInformatics,BulgarianAcademyofSciences,1113Sofia,BulgariaE-mail:[e-mail],[e-mail],[e-mail]abilityofperiodictravelling-wavesolutionsforgeneralizedBenjamin-Bona-MahonyandCamassa-Holmequations.Toproveorbitalsta-bility,weusetheabstractresultsofGrillakis-Shatah-StraussandtheFloquettheoryforperiodiceigenvalueproblems.MathematicsSubjectClassification:35B10,35Q35,35Q53,35B25,34C08,34L401.Introduction.Considerthefollowingequationut+(a(u))x−uxxt=b′(u)u2x2+b(u)uxx!x(1.1)wherea,b:R→Raresmoothfunctionsanda(0)=0.Inthispaperwestudytheproblemsoftheexistenceandstabilityofperiodictravelling-wavesolutionsu=ϕ(x−vt)for(1.1).Itiseasytoseethatwhatevera,bbe,theequationforϕhasnodissipativeterms.Hence,anytravelling-wavesolutionof(1.1)isdeterminedfromNewton’sequationwhichwewillwritebelowintheformϕ′2=U(ϕ).Thereforebyusingthewell-knownpropertiesofthephaseportraitofNewton’sequationinthe(ϕ,ϕ′)-plane,onecanestablishthatunderfairlybroadconditions,(1.1)hasatleastonethree-parameterfamilyofperiodicsolutionsϕ(y)=ϕ(v,c1;ϕ0;y)wherec1isaconstantofintegrationandϕ0=minϕ(seeProposition1).Theparametersv,c1determinethephaseportraitwhilstϕ0servestofixtheperiodicorbitwithin.Moreover,ifT=T(v,c1,ϕ0)istheminimal(sometimescalledfundamental)periodofϕ,thenϕhasexactlyonelocalminimumandonelocalmaximumin[0,T).Thereforeϕ′hasjusttwozeroesineachsemi-openintervaloflengthT.ByFloquettheory,thismeansthatϕ′iseitherthesecondorthethirdeigenfunctionoftheperiodiceigenvalueproblemobtainedfromthesecondvariationalongϕofanappropriateconservativefunctionalM(u).Ifthefirstcaseoccurs,thenonecanusetheabstractresultofGrillakis-Shatah-Strauss([25])toproveorbitalstabilitywhenever¨d(v)=(d2/dv2)M(ϕ)ispositive.Intheperiodiccasewedealwith,itisnotalwayssoeasytodeterminethesignof¨d(v).Toovercomethisproblem,wefirstestablishageneralresult(seeProposition16)expressing¨d(v)throughsomespeciallineintegralsalongtheenergylevelorbit{H=h}oftheNewtonianfunctionH(X,Y)=Y2−U(X)whichcorrespondstoϕ.Whena,barepolynomials,thesearecompleteAbelianintegralsandonecanapplymethodsfromalgebraicgeometry(Picard-Fuchsequations,etc.)todeterminethepossiblevaluesofv,c1andϕ0where¨d(v)changessign.Letusmentionthatevenforvandc1fixed,thesignof¨d(v)mightdependontheamplitudeofϕ(ruledbyϕ0)asshowninProposition8.Inthisconnection,wecalculateexplicitlythemaintermof¨d(v)inthecaseofarbitrarysmall-amplitudeperiodicsolutionsϕof(1.1),seeformula(7.7).Itisshownthatthemaintermdependsonthefirsttwoisochronousconstantsrelatedtothecenter(X0,0)intowhichtheorbit(ϕ,ϕ′)shrinkswhenε=maxϕ−minϕ→0,andonX0itselfaswell.Weapplyourresultstoproveorbitalstabilityforseveralparticularexamples.TheoremI.(ThemodifiedBBMequation).Leta(u)=2ωu+βu3,b(u)=0,β0andu=ϕ(x−vt)wherev0,ϕ(y)=ϕ(v,0;ϕ0;y)beaperiodictravelling-wavesolutionof(1.1)whichdoesnotoscillatearoundzero.Thenϕisorbitallystableinanyofthecases:(i)3v2−8ω2≥0;(ii)3v2−8ω20,2v2−2ωv−ω20andtheperiogofϕissufficientlylarge.TheoremII.(Theperturbedsingle-powerBBMequation).Leta(u)=βu2,b(u)=γβu,β0,andletu=ϕ(x−vt)wherev0,ϕ(y)=ϕ(v,0;ϕ0;y)beaperiodictravelling-wavesolutionof(1.1).Thenϕisorbitallystableforsmall|γ|.TheoremIII.(Theperturbedsingle-powermBBMequation).Leta(u)=βu3,b(u)=γβu2,β0,andletu=ϕ(x−vt)wherev0,ϕ(y)=ϕ(v,0;ϕ0;y)beaperiodictravelling-wavesolutionof(1.1)whichdoesnotoscillatearoundzero.Thenϕisorbitallystableforsmall|γ|.TheoremIV.(Small-amplitudewavesoftheperturbedBBMequation.)Leta(u)=2ωu+32u2,b(u)=γg(u)andu=ϕ(x−vt)wherev0,ϕ(y)=ϕ(v,c1;ϕ0;y)beaperiodictravelling-wavesolutionof(1.1)havingasmallamplitude.Thenϕisorbitallystableforsmall|γ|and(ω/v,c1/v2)takeninappropriatedomainΩ⊂R2.TheoremV.(Small-amplitudewavesoftheperturbedmBBMequation).Leta(u)=2ωu+βu3,b(u)=γg(u),β0andu=ϕ(x−vt)wherev0,ϕ(y)=ϕ(v,0;ϕ0;y)beaperiodictravelling-wavesolutionof(1.1)whichhasasmallamplitudeanddoesnotoscillatearoundzero.Thenϕisorbitallystablefor3v2−8ω20andsmall|γ|.Wepointoutthat,unliketheothercases,inTheoremIVtheconstantofintegrationc1isnotfixed,thereforeweconsiderthewholefamilyofsmall-amplitudewaves.TheexplicitexpressionofΩisgivenintheproof.Letusmentionthatfora(u)=2ku+32u2andb(u)=u,equation(1.1)becomes2thewell-knownCamassa-Holmequationut+2kux+3uux−uxxt=2uxuxx+uuxxx.(1.2)Equation(1.2)wasderivedasabi-HamiltoniangeneralizationoftheKorteweg-deVriesequation[23]andlaterCamassaandHolm[10]recovereditasawater-wavemodel.TheCamassa-Holmequationislocallywell-posedinHsfors32.Moreover,whilesomesolutionsofequation(1.2)areglobal,othersblowupinfinitetime(inboththeperiodicandnon-periodiccases)[11,12,13,14,15,38,42].ThesolitarywavesofCamassa-Holmequationaresmoothinthecasek0andpeakedfork=0.Theirstabilityisconsideredin[16,17,18,26,27,36,37].Fora(u)=2ku+32u2andb(u)=γu,equation(1.1)servesasamodelequationformechanicalvibrationsinacompressibleelasticrod[21,