Modulational and Parametric Instabilities of the Discrete Nonlinear Schrodinger Equation

We examine the modulational and parametric instabilities arising in a non-autonomous, discrete nonlinear Schr{\"o}dinger equation setting. The principal motivation for our study stems from the dynamics of Bose-Einstein condensates trapped in a deep optical

ModulationalandParametricInstabilitiesoftheDiscreteNonlinearSchr¨odinger

Equation

DepartmentofMathematicsandStatistics,UniversityofMassachusetts,Amherst,MA01003-4515,USA

IstitutoNazionalediFisicaperlaMateriaBEC-CRSandDipartimentodiFisica,Universita’diTrento,I-38050Povo,Italy3

TheoreticalDivisionandCenterforNonlinearStudies,LosAlamosNationalLaboratory,LosAlamos,NM87545,USA

(February6,2008)

Weexaminethemodulationalandparametricinstabilitiesarisinginanon-autonomous,dis-cretenonlinearSchr¨odingerequationsetting.TheprincipalmotivationforourstudystemsfromthedynamicsofBose-Einsteincondensatestrappedinadeepopticallattice.We ndthatunderperiodicvariationsoftheheightsoftheinterwellbarriers(orequivalentlyofthescatteringlength),additionallytothemodulationalinstability,awindowofparametricinstabilitybecomesavailabletothesystem.Weexplorethisinstabilitythroughmultiple-scaleanalysisandidentifyitnumerically.Itsprincipaldynamicalcharacteristicisthat,typically,itdevelopsovermuchlargertimesthanthemodulationalinstability,afeaturethatisqualitativelyjusti edbycomparisonofthecorrespondinginstabilitygrowthrates.

1

Z.Rapti1,P.G.Kevrekidis1,A.Smerzi2,3andA.R.Bishop3

2

arXiv:cond-mat/0404600v1 [cond-mat.soft] 25 Apr 2004

I.INTRODUCTION

Themodulationalinstability(MI)isageneralfeatureofdiscreteaswellascontinuumnonlinearwaveequations.Forthisinstability,aspeci crangeofwavenumbersofplanewavepro lesoftheformu(x,t)～exp(i(kx ωt))becomesunstabletomodulations,leadingtoanexponentialgrowthoftheunstablemodesandeventuallytodelocalization(uponexcitationofsuchwavenumbers)inmomentumspace.Thatisequivalenttolocalizationinpositionspace,andhencetheformationoflocalized,coherentsolitarywavestructures[1].

Therealizationsofthisinstabilityspanadiversesetofdisciplinesrangingfrom uiddynamics[2](whereitisusuallyreferredtoastheBenjamin-Feirinstability)andnonlinearoptics[3]toplasmaphysics[4].Oneoftheearliestcontextsinwhichitssigni cancewasappreciatedwasthelinearstabilityanalysisofdeepwaterwaves.ItwasmuchlaterrecognizedthattheconditionsforMIwouldbesigni cantlymodi edfordiscretesettingsrelevantto,forinstance,thelocaldenaturationofDNA[5]orcoupledarraysofopticalwaveguides[6,7].Inthelattercase,arelevantmodelisthediscretenonlinearSchr¨odingerequation(DNLS),anditsMIconditionswerediscussedin[8].Mostrecently,theMIhasbeenrecognizedasresponsiblefordephasingandlocalizationphenomenainthecontextofBose-Einsteincondensates(BEC)inthepresenceofanopticallatticei.e.,asinusoidalexternalpotential[9–12].

InthecontextofBECswhichareamongtheprincipalmotivationsofthiswork,anotherinterestingpossibilityarises.Fora“deep”opticallattice(i.e.,ifthewellsofthespatiallyperiodicpotentialarewell-separatedandsu cientlyhigh),ithasbeenshownthattherelevantmean eldmodelthatdescribesthebehaviorofthecondensate,atT=0,isthediscretenonlinearSchr¨odinger(DNLS)equation[10,13–15].Theopticallatticecanbecreatedbytwocounterpropagatinglaserbeamesformingastandingwaveinterferencepattern.

Ourinterestinthepresentworkisinintroducinganexplicittemporallyperiodicmodulationinthecoe cientsoftheDNLSandexaminingtheinstabilitiesthatmayarise(foruniformsolutions).IntheBECsetting,thereisanumberofpotentialrealizationsofsuchanon-autonomousDNLSequation.Forinstance,theheightsoftheinterwellbarriersoftheopticallatticeareproportionaltotheintensityofthelasers,andcanbeeasilyperiodicallymodulatedintime.Thisinducesanoscillatingtunnelingamplitudeofthecondensatesbetweenadjacentwells,aswellasanoscillatinginteractionenergyofthecondensatestrappedineachwell.AnalternativepossibilityinvolvestheperiodicmodulationofthescatteringlengthoftheinteractionbetweentheatomsviaaFeshbachresonance,i.e.,anexternalmagnetic eld;seee.g.,[16].Thepossibilityofthis,so-called,Feshbachresonancemanagement(FRM)oftheinteractionhasgeneratedalargeinterestrecentlydueitsrobuste ectonpatterns,coherentstructuresanditspotentialforavoidingcollapse,seee.g.,[22,17–20].

Inthisshortcommunication,werevisitthemodulationalinstabilitycriteriaintheDNLSequation(whichwereoriginallyderivedin[8]),butinthepresenceoftheperiodicmodulationoftheDNLStunnelingandinteractionparameters,asmotivatedabove.Wechoosethesimplestpossibleperiodicmodulation(asinusoidalvariationoftheatomicscatteringlength)andderivethemodulationalstabilityequation,whichinthiscasebecomesamodi edMathieuequation.Intheabsenceoftheperiodicperturbationwerecovertheresultsof[8].Inthepresenceofsuchaterm,anadditional,parametricinstabilitybecomespossible.Thesenewdomainsofinstabilityappearduetoparametricresonance,whenevertheparametersofasystemvaryperiodicallywithtime.Incontrasttoordinary

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