20Stability of small periodic waves for the nonlin

arXiv:math/0609026v1[math.AP]1Sep2006StabilityofsmallperiodicwavesforthenonlinearSchr¨odingerequationThierryGallayInstitutFourierUniversit´edeGrenobleIB.P.7438402Saint-Martin-d’H`eres,FranceMarianaH˘ar˘agu¸sD´epartementdeMath´ematiquesUniversit´edeFranche-Comt´e16routedeGray25030Besan¸con,FranceAbstractThenonlinearSchr¨odingerequationpossessesthreedistinctsix-parameterfamiliesofcomplex-valuedquasi-periodictravellingwaves,oneinthedefocusingcaseandtwointhefocusingcase.Allthesesolutionshavethepropertythattheirmodulusisaperiodicfunctionofx−ctforsomec∈R.Inthispaperweinvestigatethestabilityofthesmallamplitudetravellingwaves,bothinthedefocusingandthefocusingcase.OurfirstresultshowsthatthesewavesareorbitallystablewithintheclassofsolutionswhichhavethesameperiodandthesameFloquetexponentastheoriginalwave.Next,weconsidergeneralboundedperturbationsandfocusonspectralstability.Weshowthatthesmallamplitudetravellingwavesarestableinthedefocusingcase,butunstableinthefocusingcase.Theinstabilityisofside-bandtype,andthereforecannotbedetectedintheperiodicset-upusedfortheanalysisoforbitalstability.Runninghead:PeriodicwavesintheNLSequationCorrespondingauthor:MarianaH˘ar˘agu¸s,[e-mail]:nonlinearSchr¨odingerequation,periodicwaves,orbitalstability,spectralstability1IntroductionWeconsidertheone-dimensionalcubicnonlinearSchr¨odingerequation(NLS)iUt(x,t)+Uxx(x,t)±|U(x,t)|2U(x,t)=0,wherex∈R,t∈R,U(x,t)∈C,andthesigns+and−inthenonlineartermcorrespondtothefocusingandthedefocusingcase,respectively.InbothcasestheNLSequationpossessesquasi-periodicsolutionsofthegeneralformU(x,t)=ei(px−ωt)V(x−ct),x∈R,t∈R,(1.1)wherep,ω,carerealparametersandthewaveprofileVisacomplex-valuedperiodicfunctionofitsargument.Theaimofthepresentpaperistoinvestigatethestabilitypropertiesoftheseparticularsolutions,atleastwhenthewaveprofileVissmall.Itturnsoutthatthediscussionisverysimilarinbothcases,soforsimplicitywerestrictourpresentationtothedefocusingequationiUt(x,t)+Uxx(x,t)−|U(x,t)|2U(x,t)=0,(1.2)andonlydiscussthedifferenceswhichoccurinthefocusingcaseattheendofthepaper.AcrucialroleinthestabilityanalysisisplayedbythevarioussymmetriesoftheNLSequation.Themostimportantonesforourpurposesarethefourcontinuoussymmetries:•phaseinvariance:U(x,t)7→U(x,t)eiϕ,ϕ∈R;•translationinvariance:U(x,t)7→U(x+ξ,t),ξ∈R;•Galileaninvariance:U(x,t)7→e−i(v2x+v24t)U(x+vt,t),v∈R;•dilationinvariance:U(x,t)7→λU(λx,λ2t),λ0;andthetwodiscretesymmetries:•reflectionsymmetry:U(x,t)7→U(−x,t),•conjugationsymmetry:U(x,t)7→U(x,−t).Asiswell-known,theCauchyproblemforequation(1.2)isgloballywell-posedonthewholereallineintheSobolevspaceH1(R,C),seee.g.[8,12,13,19].Alternatively,onecansolvetheNLSequationonaboundedinterval[0,L]withperiodicboundaryconditions,inwhichcaseanappropriatefunctionspaceisH1per([0,L],C).Inbothsituations,wehavethefollowingconservedquantities:E1(U)=12ZI|U(x,t)|2dx,E2(U)=i2ZIU(x,t)Ux(x,t)dx,E3(U)=ZI12|Ux(x,t)|2+14|U(x,t)|4dx,2whereIdenoteseitherthewholereallineortheboundedinterval[0,L].ThequantitiesE1andE2areconservedduetothephaseinvarianceandthetranslationinvariance,respectively,whereastheconservationofE3originatesinthefactthatequation(1.2)isautonomous.Thesymmetrieslistedabovearealsousefultounderstandthestructureofthesetofallquasi-periodicsolutionsof(1.2).AssumethatU(x,t)isasolutionof(1.2)oftheform(1.1),whereV:R→Cisaboundedfunction.Since|U(x,t)|=|V(x−ct)|,thetranslationspeedc∈RisuniquelydeterminedbyU,exceptifthemodulus|V|isconstant.Inanycase,usingtheGalileaninvariance,wecantransformU(x,t)intoanothersolutionoftheform(1.1)withc=0.Oncethisisdone,thetemporalfrequencyωisinturnuniquelydeterminedbyU(x,t),exceptinthetrivialcasewhereVisidenticallyzero.Inviewofthedilationinvariance,onlythesignofωisimportant,sowecanassumewithoutlossofgeneralitythatω∈{−1;0;1}.SettingU(x,t)=e−iωtW(x),weseethatW(x)=eipxV(x)isaboundedsolutionoftheordinarydifferentialequationWxx(x)+ωW(x)−|W(x)|2W(x)=0,x∈R.(1.3)Ifω=0orω=−1,isitstraightforwardtoverifythatW≡0istheonlyboundedsolutionof(1.3),thusweassumefromnowonthatω=1.Equation(1.3)isactuallythestationaryGinzburg-Landauequationandthesetofitsboundedsolutionsiswell-known[5,9,10,11].Therearetwokindsofsolutionsof(1.3)whichleadtoquasi-periodicsolutionsoftheNLSequationoftheform(1.1):•AfamilyofperiodicsolutionswithconstantmodulusW(x)=(1−p2)1/2ei(px+ϕ),wherep∈[−1,1]andϕ∈[0,2π].Thecorrespondingsolutionsof(1.2)arecalledplanewaves.ThegeneralformofthesewavesisU(x,t)=ei(px−ωt)V,wherep∈R,ω∈R,andV∈Csatisfythedispersionrelationω=p2+|V|2.•Afamilyofquasi-periodicsolutionsoftheformW(x)=r(x)eiϕ(x),wherethemodulusr(x)andthederivativeofthephaseϕ(x)areperiodicwiththesameperiod.AnysuchsolutioncanbewrittenintheequivalentformW(x)=eipxQ(2kx),wherep∈R,k0,andQ:R→Cis2π-periodic.Inparticular,U(x,t)=e−itW(x)=ei(px−t)Q(2kx)(1.4)isaquasi-periodicsolutionof(1.2)oftheform(1.1)(withc=0andω=1).Weshallrefertosuchasolutionasaperiodicwave,becauseitsprofile|U(x,t)|isa(non-trivial)periodicfunctionofthespacevariablex.Importantquantitiesrelatedtotheperiodicwave(1.4)aretheperiodofthemodulusT=π/k,andtheFloquetm