A KAM Theorem for Hamiltonian Networks with Long R

AKAMTHEOREMFORHAMILTONIANNETWORKSWITHLONGRANGEDCOUPLINGSJIANSHENGGENGANDYINGFEIYIAbstract.WeconsiderHamiltoniannetworksoflongrangedandweaklycoupledoscillatorswithvariablefrequencies.ByderivinganabstractinfinitedimensionalKAMtypeoftheorem,weshowthatforanygivenpositiveintegerNandafixed,positivemeasuresetOofNvariablefrequencies,thereisasubsetO∗⊂Oofpositivemeasuresuchthateachω∈O∗correspondstoasmallamplitude,quasi-periodicbreather(i.e.,asolutionwhichisquasi-periodicintimeandexponentiallylocalizedinspace)oftheHamiltoniannetworkwithN-frequencieswhichareslightlydeformedfromω.1.IntroductionandMainResultAssociatedwiththesymplecticstructurePnpn∧qn,weconsiderHamiltoniannetworksdefinedbyrealanalyticHamiltoniansoftheform(1.1)H=Xn∈Z(p2n2+Vn(qn))+W({qn}),whereVn’saretheon-sitepotentialssatisfyingVn(0)=V0n(0)=0andV00n(0)≡β2n2,βn0,andWisacouplingpotential.Hamiltoniannetworkshavebeenusedinsolidstatephysicsindescribingthevibrationofparticles(atoms)inalattice(see[10,11,20])andalsousedtomodelDNAchains(see[10,14,34]).TheyalsoarisenaturallyasspatialdiscretizationofHamiltonianPDEssuchasnonlinearwaveequations.AmongthesolutionsofaHamiltoniannetwork,ofparticularphysicalinterestsaretheso-calledbreathersorquasi-periodicbreathers,whichareself-localized,timeperiodicorquasi-periodic,solutionswhoseamplitudesdecayatleastexponentiallyas|n|→∞.Breathersorquasi-periodicbreathersareoftenreferredtoasdynamicalsolitonsorintrinsiclocalizedmodesinphysicsandtheyhavebeenlargelyfoundvianumericsinmanyphysicalmodels(see[10,29]andreferencestherein).TheexistenceofbreathersinHamiltoniannetworksassociatedwithHamiltonians(1.1)wasrigorouslyanalyzedwhenβn≡βbyAubry[1,2],Mackay–Aubry[25]fortheinter-particle,nearestneighborcouplingpotentialW({qn})=Xn(qn+1−qn)2andbyBambusi[3]forthelong–rangecouplingpotentialW({qn})=Xn6=m1|n−m|α(qn−qm)2,α1.Likein[25],breathersinthenearestneighborcouplingcasecanbestudiednearafixedperiodicor-bitoftheuncoupledHamiltonianbycertaincontinuationorperturbationarguments,providedthatthecouplingsare“weak”,and,nosmalldivisorsneedtobeconsideredinsuchperturbationprob-lems.Theseperturbationtechniquesarealsoapplicableinfindingquasi-periodicbreatherswith1991MathematicsSubjectClassification.Primary37K60,37K55.Keywordsandphrases.Coupledoscillators,Hamiltoniannetworks,longrangedcoupling,KAMtheory,quasi-periodicbreathers.ThesecondauthorisPartiallysupportedbyNSFgrantDMS0204119.12JIANSHENGGENGANDYINGFEIYItwoorthreefrequenciesforcertainmodelswithsymmetries(seeBambusi–Vella[4],Johansson–Aubry[19]).UsingamodifiedKAMtechnique,theexistenceofquasi-periodicbreatherswithanyfinitenumberoffrequencieswasrecentlyshownbyYuan[33]forthehigherorder,nearest-neighborcouplingpotential(1.2)W({qn})=Xn(qn+1−qn)3.Almostperiodicbreatherswithinfinitelymanyfrequencieshavealsobeeninvestigated.Associatedwiththepotential(1.2),Fr¨ohlich-Spencer-Wayne[15]consideredthecasewhenthefrequenciesarenon-negativerandomvariableswithsmoothdistributionoffastdecayatinfinityandshowedthatthereisasetΩ⊂R∞+withpositiveprobabilitymeasuresuchthateachω∈Ωcorrespondstoanalmostperiodicbreatherwithinfinitemanyfrequencies(seealsoP¨oschel[28]formoregeneralspatialstructures).Inthispaper,wewillstudytheexistenceofquasi-periodicbreathersfortheHamiltonian(1.1)withthefollowinghigherorder,long-rangedcouplingpotential(1.3)W({qn})=13Xn6=me−|n−m|α(qn−qm)3,α≥1,orequivalentlytheHamiltoniannetwork(1.4)d2qndt2+V0n(qn)=−Xm∈Ze−|n−m|α(qn−qm)2.ForagivenintegerN1,wespecifyNintegers{i1,···,iN}andletZ1=Z\{i1,···,iN}.Wetreatω=(βi1,···,βiN)asparametersinaboundedclosedregionOinRN+andassumethefollowingspectralgapcondition:SG)Thereexist1≤d∞andγ0suchthat{βn}n∈Z1=∪∞l=1ΛlwhereΛl,l=1,2,···,aresetssatisfying#(Λl)≤d,foralll,and|βn−βm|≥γ,forallβn∈Λl,βm∈Λj,l6=j.Wewillshowthefollowingresult.TheoremA.AssumeSG)withγsufficientlysmall.ThenthereexistsaCantorsetOγ⊂O,withmeas(O\Oγ)=O(γ),suchthatforanyω∈Oγ,theHamiltoniannetwork(1.4)associatedwithωadmitsasmallamplitude,linearlystable,quasi-periodicbreatherq(t)=({qn(t)})ofN-frequencyω∗whichisclosetoω,andmoreover,|qn|∼e−|n|.TheconditionSG)clearlyholdswhenβn=|n|,n∈Z1.Comparingwiththecasesofnonlinearwaveequations[12,17,26,27]inwhichβn∼|n|,n∈Z,thevalidityofTheoremAcruciallydependsonthecouplingpotentialorperturbation(1.3)whichadmitsaweakerregularitybutahigherorderperturbation.Forthecaseofnearest-neighboringcoupledHamiltoniannetworks,breatherswereshowntobesuper-exponentiallylocalizedinspace([33]).Thisisduetotheatmostlineargrowthofthenormalcomponentsinthenormalformassociatedwiththeshort-rangedcouplingpotential(1.2).Ourresultonlyassertstheexponentiallocalizationofquasi-periodicbreathersduetotheexponentialgrowthofthenormalcomponentsinthenormalformassociatedwiththeexponentiallyweighted,long-rangedcouplingpotential(1.3).Ifthelong-rangedcouplingpotentialW({qn})=13Xn6=m1|n−m|α(qn−qm)3,α1AKAMTHEOREMFORHAMILTONIANNETWORKSWITHLONGRANGEDCOUPLINGS3isconsideredinstead,thenthenormalcomponentsintheassociatednormalformwillhaveasuper-exponentialgrowth,andourmethodwillequallyapplicabletoyieldquasi-periodicbreatherswhicharelocalizedlike1|n