Travelling waves in the Fermi Pasta Ulam lattice Gerard Iooss

Gérard Iooss

Description

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

Sujets

• Institut universitaire de France
• Differential equation

Informations

• Publié par: pefav
• Nombre de lectures: 6
• Langue: English

Extrait

TravellingwavesintheFermi-PastaUlamlattice Ge´rardIooss InstitutUniversitairedeFrance, INLN,UMRCNRS-UNSA6618 1361routedesLucioles F-06560Valbonne e.mail:[e-mail] Abstract Weconsidertravellingwavesolutionsonaone-dimensionallattice, correspondingtomassparticlesinteractingnonlinearlywiththeirnearest neighbor(Fermi-Pasta-Ulammodel).Aconstructivemethodisgiven,for obtainingallsmallboundedtravellingwavesforgenericpotentials,near thefirstcriticalvalueofthevelocity.Theyallaresolutionsofa finitedi- mensionalreversibleODE .Inparticular,near(above)thefirstcriticalve- locityofthewaves,weconstructthe solitarywaves whoseglobalexistence wasprovedbyFrieseckeetWattis[1],usingavariationalapproach.In addition,wefindothertravellingwaveslike(i)superpositionofaperiodic oscillationwithanonzeroaveragedstretchingorcompressionbetween particules,(ii) mainlylocalizedwaves whichtendtouniformlystretched orcompressedlatticeatinfinity,(iii)heteroclinicsolutionsconnectinga stretchedpatternwithacompressedone. 1IntroductionandFormulationoftheproblem Weconsiderthedynamicsoftheclassicalone-dimentionallatticegivenby 00 X ¨ n =V(X n+1 −X n )−V(X n −X n−1 ),n∈ Z (1) whereX n (t e ),t e ∈ R ,givesthepositionofthenthparticle,Visthepotentialdue tonearest-neighborinteraction.Weareinterestedintravellingwavessolutions of(1).Thesystem(1)hasaspecialphysicalimportance,mainlyduetoits apparentsimplicity,andtothediscoveryfromFermi,Pasta,Ulam[1]aboutthe (numericallyfound)timebehaviorofsolutionswithsinusoidalinitialcondition, havingrecurrenceproperties,meaningthatitdoesnotmixdifferentmodes,de- spiteofitsaprorinonintegrability.Importantresultsonitslocalizedsolutions aregivenbyFriesecke&Wattisin[2],usingavariationalapproach.Werefer thereadertoreferencesin[2]fornumericalresultsonlocalizedwavesinone dimensionallattices.Inthepresentwork,wefollowthelinesofthemethod initiatedbyIoossandKirchga¨ssner[4]onasimilarsystem.Withtheansatz 1

X n (t e )=xe(nτ−t e ),afterscalingthetimeast e =τt,anddenotingx(t)=xe(τt), system(1)istransformedto x¨(t)=τ 2 (V 0 [x(t+1)−x(t)]−V 0 [x(t)−x(t−1)])(2) whichisascalar”neutral”or”advance-delay”differentialequation.Wereferto thebasicpaper[4]forgeneralreferencesonadvance-delaydifferentialequations, relatedtoourtypeofproblem(secondorderdifferential).However,wegive aconstructiveoriginalmethodforobtainingthe”small”solutionsof(2)for velocitiesofthewavesclosetothefirstcriticalvalue(calledthe”soundvelocity” in[2]).Weshowthattheyallbelongtoafinitedimensionalcentermanifold,and aregivenbythesmallboundedsolutionsofanordinarydifferentialequation. Thisreductionresultfollowsfromthework[4]onasimilarproblem,which wasinspiredbytheanalogousreductionavailableforellipticsystemsinstrips (seetheseminalwork[6]ofKirchga¨ssner).Wedonotreproduceherethe completeproofofthisreductionprocess,referingthereaderto[4]fordetails. Theadditionaldifficultyhereistheinvariancepropertyofoursystemunder additionofaconstant,whichleadstoasystematicdoublezeroeigenvaluefor thelinearizedoperator,forallvaluesoftheparameters(velocityofthewaves, curvatureat0ofthepotential).Wesolvethisdifficultyandconcentrateonthe extensivestudyofalltypesofsmallboundedsolutionsof(2).Ourresultsare summedupin3theorems,validforgenericpotentials.Thefirstonegivesthe localizedsolutions,andistheresultanalogoustotheoneoftheorem1in[2].The secondtheoremgives”mainlylocalized”solutions,whichareasymptotictoa stretchedoracompressedlattice(”stretched”meansthatX n+1 −X n isenlarged, ”compressed”meanstheopposite).Dependingonthelocalpropertiesofthe potential,wealsoshowtheexistenceofan”heteroclinic”solutionconnectinga stretchedlatticewithacompressedone.Thelasttheoremgiveslargefamilies ofperiodicsolutions(x˙(t)istimeperiodic),wherethepatternhastimeperiodic oscillationsaroundastretchedoracompresseduniformstate(thisaveraged statemaybethebasicuniformstate(x=0). Thereversibilitysymetrygivenbyx(t)7−→−x(−t)playsacapitalrolein whatfollows.Animportantremarkisthatthissystemisinvariantunderthe transformationx7−→x+qforanyq∈ R .Moreoverwehavetokeepinmind thatthereisafamilyof”trivial”solutionsgivenbyx(t)=at+bforanyaand bin R .Theycorrespondtouniformlystretchedorcompressedpatternsin(1). Inwhatfollows,weneedtospecifythebehaviorofV 0 near0: V 0 (x)=αx+βx 2 +γx 3 +δx 4 +..., theessentialassumptionbeingthat V 00 (0)=α>0. Forobtainingnontrivialresults,wealsoneedthat,atleastoneofthehigher ordercoefficientsoftheTaylorexpansionofV 0 attheoriginisnonzero.We shallconcentrateouranalysistothecaseswhenβorγarenonzero. 2