Travelling waves in the Fermi Pasta Ulam lattice Gerard Iooss

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Travelling waves in the Fermi-Pasta Ulam lattice Gerard Iooss Institut Universitaire de France, INLN, UMR CNRS-UNSA 6618 1361 route des Lucioles F-06560 Valbonne e.mail: Abstract We consider travelling wave solutions on a one-dimensional lattice, corresponding to mass particles interacting nonlinearly with their nearest neighbor (Fermi-Pasta-Ulam model). A constructive method is given, for obtaining all small bounded travelling waves for generic potentials, near the first critical value of the velocity. They all are solutions of a finite di- mensional reversible ODE. In particular, near (above) the first critical ve- locity of the waves, we construct the solitary waves whose global existence was proved by Friesecke et Wattis [1], using a variational approach. In addition, we find other travelling waves like (i) superposition of a periodic oscillation with a non zero averaged stretching or compression between particules, (ii) mainly localized waves which tend to uniformly stretched or compressed lattice at infinity, (iii) heteroclinic solutions connecting a stretched pattern with a compressed one. 1 Introduction and Formulation of the problem We consider the dynamics of the classical one-dimentional lattice given by Xn = V ?(Xn+1 ? Xn) ? V ?(Xn ? Xn?1), n ? Z (1) where Xn(t˜), t˜ ? R, gives the position of the nth particle, V is the potential due to nearest-neighbor interaction.

all small bounded

ulam lattice

delay differential

e?vx ? ∫

only eigenvalue

fermi-pasta-ulam model

differential equation

fermi

dimensional reversible

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Institut universitaire de France Differential equation

TravellingwavesintheFermi-PastaUlamlatticeGe´rardIoossInstitutUniversitairedeFrance,INLN,UMRCNRS-UNSA66181361routedesLuciolesF-06560Valbonnee.mail:iooss@inln.cnrs.frAbstractWeconsidertravellingwavesolutionsonaone-dimensionallattice,correspondingtomassparticlesinteractingnonlinearlywiththeirnearestneighbor(Fermi-Pasta-Ulammodel).Aconstructivemethodisgiven,forobtainingallsmallboundedtravellingwavesforgenericpotentials,neartheﬁrstcriticalvalueofthevelocity.Theyallaresolutionsofaﬁnitedi-mensionalreversibleODE.Inparticular,near(above)theﬁrstcriticalve-locityofthewaves,weconstructthesolitarywaveswhoseglobalexistencewasprovedbyFrieseckeetWattis[1],usingavariationalapproach.Inaddition,weﬁndothertravellingwaveslike(i)superpositionofaperiodicoscillationwithanonzeroaveragedstretchingorcompressionbetweenparticules,(ii)mainlylocalizedwaveswhichtendtouniformlystretchedorcompressedlatticeatinﬁnity,(iii)heteroclinicsolutionsconnectingastretchedpatternwithacompressedone.1IntroductionandFormulationoftheproblemWeconsiderthedynamicsoftheclassicalone-dimentionallatticegivenby00X¨n=V(Xn+1−Xn)−V(Xn−Xn−1),n∈Z(1)whereXn(te),te∈R,givesthepositionofthenthparticle,Visthepotentialduetonearest-neighborinteraction.Weareinterestedintravellingwavessolutionsof(1).Thesystem(1)hasaspecialphysicalimportance,mainlyduetoitsapparentsimplicity,andtothediscoveryfromFermi,Pasta,Ulam[1]aboutthe(numericallyfound)timebehaviorofsolutionswithsinusoidalinitialcondition,havingrecurrenceproperties,meaningthatitdoesnotmixdiﬀerentmodes,de-spiteofitsaprorinonintegrability.ImportantresultsonitslocalizedsolutionsaregivenbyFriesecke&Wattisin[2],usingavariationalapproach.Wereferthereadertoreferencesin[2]fornumericalresultsonlocalizedwavesinonedimensionallattices.Inthepresentwork,wefollowthelinesofthemethodinitiatedbyIoossandKirchga¨ssner[4]onasimilarsystem.Withtheansatz1

Xn(te)=xe(nτ−te),afterscalingthetimeaste=τt,anddenotingx(t)=xe(τt),system(1)istransformedtox¨(t)=τ2(V0[x(t+1)−x(t)]−V0[x(t)−x(t−1)])(2)whichisascalar”neutral”or”advance-delay”diﬀerentialequation.Werefertothebasicpaper[4]forgeneralreferencesonadvance-delaydiﬀerentialequations,relatedtoourtypeofproblem(secondorderdiﬀerential).However,wegiveaconstructiveoriginalmethodforobtainingthe”small”solutionsof(2)forvelocitiesofthewavesclosetotheﬁrstcriticalvalue(calledthe”soundvelocity”in[2]).Weshowthattheyallbelongtoaﬁnitedimensionalcentermanifold,andaregivenbythesmallboundedsolutionsofanordinarydiﬀerentialequation.Thisreductionresultfollowsfromthework[4]onasimilarproblem,whichwasinspiredbytheanalogousreductionavailableforellipticsystemsinstrips(seetheseminalwork[6]ofKirchga¨ssner).Wedonotreproduceherethecompleteproofofthisreductionprocess,referingthereaderto[4]fordetails.Theadditionaldiﬃcultyhereistheinvariancepropertyofoursystemunderadditionofaconstant,whichleadstoasystematicdoublezeroeigenvalueforthelinearizedoperator,forallvaluesoftheparameters(velocityofthewaves,curvatureat0ofthepotential).Wesolvethisdiﬃcultyandconcentrateontheextensivestudyofalltypesofsmallboundedsolutionsof(2).Ourresultsaresummedupin3theorems,validforgenericpotentials.Theﬁrstonegivesthelocalizedsolutions,andistheresultanalogoustotheoneoftheorem1in[2].Thesecondtheoremgives”mainlylocalized”solutions,whichareasymptotictoastretchedoracompressedlattice(”stretched”meansthatXn+1−Xnisenlarged,”compressed”meanstheopposite).Dependingonthelocalpropertiesofthepotential,wealsoshowtheexistenceofan”heteroclinic”solutionconnectingastretchedlatticewithacompressedone.Thelasttheoremgiveslargefamiliesofperiodicsolutions(x˙(t)istimeperiodic),wherethepatternhastimeperiodicoscillationsaroundastretchedoracompresseduniformstate(thisaveragedstatemaybethebasicuniformstate(x=0).Thereversibilitysymetrygivenbyx(t)7−→−x(−t)playsacapitalroleinwhatfollows.Animportantremarkisthatthissystemisinvariantunderthetransformationx7−→x+qforanyq∈R.Moreoverwehavetokeepinmindthatthereisafamilyof”trivial”solutionsgivenbyx(t)=at+bforanyaandbinR.Theycorrespondtouniformlystretchedorcompressedpatternsin(1).Inwhatfollows,weneedtospecifythebehaviorofV0near0:V0(x)=αx+βx2+γx3+δx4+...,theessentialassumptionbeingthatV00(0)=α>0.Forobtainingnontrivialresults,wealsoneedthat,atleastoneofthehigherordercoeﬃcientsoftheTaylorexpansionofV0attheoriginisnonzero.Weshallconcentrateouranalysistothecaseswhenβorγarenonzero.2

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