Stationary modulated-amplitude waves in the 1-D co

arXiv:nlin/0208001v1[nlin.PS]31Jul2002Stationarymodulated-amplitudewavesinthe1-DcomplexGinzburg-LandauequationYuehengLan,NicolasGarnierandPredragCvitanovi´cCenterforNonlinearScience,SchoolofPhysics,GeorgiaInstituteofTechnology837StateStreet,Atlanta,GA30332-0430,USAAbstractWereformulatetheone-dimensionalcomplexGinzburg-Landauequationasafourthorderordinarydifferentialequationinordertofindstationaryspatially-periodicsolutions.Usingthisformalism,weprovetheexistenceandstabilityofstationarymodulated-amplitudewavesolutions.Approximateanalyticexpressionsandacomparisonwithnumericsaregiven.Keywords:complexGinzburg-Landauequation,coherentstructuresPACS:05.45.-a,47.54.+r,05.45.JnIntroductionThecubiccomplexGinzburg-Landauequation(CGLe)isagenericamplitudeequationdescribingHopfbifurcationinspatiallyextendedsystems,i.e.,Iosystems[1],withreflectionsymmetry[6,3,4].Itisofgreatinterestduetoitsgenericityandapplicationstoonsetofwavepattern-forminginstabilities[1]invariousphysicalsystemssuchasfluiddynamics,optics,chemistryandbiology.Itexhibitsrichdynamicsandhasbecomeaparadigmforthetransitiontospatio-temporalchaos.Weconsidertheone-dimensionalCGLeforthecomplexamplitudefieldA(x,t):At=μA+(1+iα)Axx−(1+iβ)|A|2A(1)whereA(x,t):R27→C,andμ,α,β∈R,x∈D.Disthespatialdomainonwhichtheequationisdefined.InterestingdomainsforusareeitherthewholerealaxisorafiniteboxoflengthLwithperiodicboundaryconditions.μisthePreprintsubmittedtoElsevierPreprintJuly29th2002controlparameter.Onlyμ0isconsideredbecausewestudythesupercrit-icalGinzburg-Landauequation;onecouldsetμ=1byappropriaterescalingofthetime,spaceandamplitude,butwekeepitasaparameterforcloserconnectionwithexperimentalresultsandpreviousliterature.Coefficientsαandβparametrizethelinearandnonlineardispersion.Ifbothαandβaresetto0,werecovertherealGinzburg-Landauequation(RGLe)inwhichonlythediffusiontermandthestabilizingcubictermcom-petewitheachotherandthelinearterm.ALyapunovfunctionalexistsinthatcase[1]andtheRGLebehaveslikeagradientsystem.Anotherlimit—thenonlinearSchr¨odingerequation—resultsfromsettingα,β→∞;wethenhaveanintegrablenonlinearPDE.Forparametervaluesintheintermediaterange,long-timebehavioroftheCGLecanvaryfromstationarytoperiodicandtospatiotemporalchaos[5].Inthispaper,weconcentrateonthestation-arysolutionsoftheCGLeinafiniteboxoflengthLwithperiodicboundaryconditions,andthecaseα6=β.Stationarysolutionsarethesimplestnon-trivialsolutions,relatedtopropagatingsolutionsbyanappropriatechangeofframeofreference(x,t)7→(x−vt,t)withfixedv∈R.Searchingforcoherentstructuresallowsonetoreduceapartialdifferentialequationintoanordinaryone,andsuchsolutionsoftheCGLearebelievedtobeextremelyimportantinmanyregimes,includingthespatiotemporalchaos[9].Recently,numericalintegrationsoftheCGLehavefocusedonaclassofsolutionscalledmodulated-amplitudewaves(MAWs)andtheirroleinthenonlinearevolutionoftheEckhausinstabilityofinitiallyhomogeneousplanewaves[12,13].MAWscanbifurcatefromthetrivialsolutionA=0(caseI)orplanewavesolutionsofzerowavenumber(caseII).Analyticalaspectsofmodulatedso-lutionsoftheCGLehavebeenaddressedbyNewtonandSirovichwhohaveappliedaperturbationanalysistostudythebifurcationincaseII[14],anddiscussedthesecondarybifurcationofthoseMAWs[15].Tak´aˇc[16]provedtheexistenceofMAWsolutionsusingastandardbifurcationanalysisintheinfinite-dimensionalphasespaceoftheCGLe,inbothcasesIandII,togetherwithastabilityanalysisincaseIbymeansofthecentermanifoldtheorem.InthisarticlewereformulatetheCGLeequationassumingacoherentstruc-tureformforthesolutions,andobtainafourth-orderordinarydifferentialequation(ODE)withaconsistencycondition.Thisformisalgebraicallycon-venient,becausethededucedsystemoffourfirst-orderODEscontainsonlyquadraticnon-linearity.IntheBenjamin-Feir-Newellregime,whereplanewavessolutionsarealwaysunstable,wegiveaproofofexistenceofMAWsinbothcaseIandIIusingourODE.ForweakperturbationsincaseIorII,wewriteapproximateanalyticsolutionsintheODEphasespace.ComingbacktothefullCGLe,wethenprovethestabilityofthoseMAWsinafiniteboxincase2II,andprovethatthebifurcationissupercritical,assuggestedbyrecentnu-mericalwork[12].Inthenextsection,wediscusssymmetriesandsolutionsoftheCGLe.Insec-tion2wetransformthesteadyCGLeforMAWsintoanequivalentODE,andgivethesufficientconditiontoidentifythesolutionsofthesetwoequations.Insection3thisODEisusedtoconstructa4-DdynamicalsystemandprovetheexistenceofsymmetricstationarysolutionsoftheCGLeinthetwocasesIandII.Insection4theapproximateanalyticformofthesolutionsisgivenandcomparedtonumericalcalculations,andthestabilityofMAWsincaseIIisproved.SeveraltheoremsneededintheproofsarereproducedinappendixB.1BasicpropertiesoftheCGLe1.1SymmetriesTheequation(1)isinvariantundertemporalandspatialtranslations.More-over,itisinvariantunderaglobalgaugetransformationA→Aexp(iφ),whereφ∈R,andunderx→−xreflection.Asaconsequence,itpreservesparityofA,i.e.,ifA(−x,0)=±A(x,0),thenA(−x,t)=±A(x,t)foranylatertimet0.IfA(x,t)hasnoparity,thenA(−x,t)givesanothersolution.1.2StokessolutionsandtheirstabilityTheglobalphaseinvarianceimpliesthattheCGLehasnonlinearplanewavesolutionsofformA(x,t