Travelling wave solutions for the discrete sine-Go

BICSBathInstituteforComplexSystemsTravellingwavesolutionsforthediscretesine-GordonequationwithnonlinearpairinteractionCarl-FriedrichKreinerandJohannesZimmerBathInstituteForComplexSystemsPreprint3/08(2008)andJohannesZimmeryAbstractThefocusofstudyisthenonlineardiscretesine-Gordonequation,wherethenonlinearityreferstoanonlinearinteractionofneighbouringatoms.Theexistenceoftravellingheteroclinic,homoclinicandperiodicwavesisshown.Theasymptoticstatesarechosensuchthattheactionfunctionalisnite.Theproofsemployvariationalmethods,inparticularasuitableconcentration-compactnesslemmacombinedwithdirectminimisationandmountainpassarguments.1IntroductionThisarticleisconcernedwithtravellingwavesinthediscretesine-Gordonequationqk(t)=V0(qk+1(t)qk(t))V0(qk(t)qk1(t))Ksin(qk(t));k2Z;(1)withaconstantK0.Equation(1)describestheevolutionofaninnitechainofatomswithelasticnearestneighbourinteractionandanon-sitepotential,accordingtoNewton'slaw.TheargumentoftheinteractionpotentialV:R!Risthediscretestrainqk+1(t)qk(t).Inanearlierwork[7],weassumedthatVisaquadraticfunctionV():=c2022withc00;here,weconsiderananharmonicinteraction,thatis,V()6=c2022.Weareinterestedintravellingwavesolutionsto(1),thatis,solutionsoftheformqk(t)=u(kct)forallk2Z,whereu:R!Risthewaveproleandc0isthewavespeed.Forthisansatz,(1)becomesc2u00()=V0(u(+1)u())V0(u(t)u(1))Ksin(u()):(2)Inasuitablesetting,Equation(2)istheEuler-LagrangeequationoftheactionfunctionalJ(u):=ZRc22(u0())2V(u(+1)u())2+K(1+cos(u()))d:(3)Here,RRK(1+cos(u()))distheon-sitepotential.Theresultspresentedherewill,withobviousmodications,alsoholdforanynon-negative2-periodicW1;1-functionwithzerosetf(2k+1):k2Nginsteadof(1+cos(
)).MathematischesForschungsinstitutOberwolfach,77709Oberwolfach,GermanyyDepartmentofMathematicalSciences,UniversityofBath,BA27AY,UnitedKingdom1Inthisarticle,weonlyconsidersupersonicwaves,thatis,restricttheanalysistowavespeedscV00(0).UndersuitableconditionsontheinteractionpotentialV,weshowtheexistenceofthreetypesofsolutions:{heteroclinictravellingwaves:limz!1u(z)=andlimz!+1u(z)=,(4){homoclinictravellingwaves:limz!1u(z)=limz!+1u(z)=,{periodictravellingwaves:u(z)=u(z+T)forsomeT0andforeveryz2R.Therstpartgeneralisesanexistenceresultforsupersonicheteroclinicwavesin[7]tothecaseofnonlinearinteraction.TheresultsforhomoclinicwavespresentedherearerelatedtothoseofBatesandZhang[3].Theyhave,amongotherresults,showntheexistenceofsupersonictravellingwavesforc2u00()=c20(u(+1)2u()+u(1))+Ksin(u()):(5)BatesandZhang[3]considerhomoclinicwavesthathavetheirasymptoticstatesinthemaximumoftheon-sitepotential,whichcanherebetakentobeZR[Kcos(u())1]d:Employingentirelydierentmethods,westudytheanalogoussituationwithnon-linearinteractionandthusachieveacomplementaryresult.Inaddition,weprovetheexistenceofperiodicsolutions.Thisresultisrelatedtoworkonperiodicsolutionswithnonlinearinteraction,butwithouton-sitepo-tential[1,2].Theinterestinperiodicsolutionscanbeexplainedwiththedesiretoanalysethe(non-)ergodicityofasystem;seethediscussionin[1],alsoregardingthe(non-)equipartitioningofenergyoftheFermi-Pasta-Ulamexperiment(nonlinearinteractionwithouton-sitepotential).Otherchoicesofboundaryconditionsandtheirphysicalinterpretationsaredis-cussedin[7].2HeteroclinictravellingwavesInthissection,weprovetheexistenceofheteroclinicwavesfor(2)withboundaryconditions(4).Thesolutionwillbefoundasaminimiserofapenalisedvariantoftheactionfunctional(3).Thepenalisationisnecessarysincetheactionfunctionalis,unlikeinthecaseoflinearinteraction,notboundedfrombelow.Weintroducethefunction-analyticsetting.LetusdenethespaceX:=u2H1loc(R):u02L2(R) ;whenequippedwiththeinnerproducthu;viX:=u(0)v(0)+ZRu0()v0()d,itbecomesaHilbertspace.Further,wesetM;:=fu2X:u(1)=;u(1)=g:(6)Throughoutthissection,thefollowingassumptionsaremade.2Assumption2.1(i)V2C1(R),V(0)=0,andV(x)0forallx2R.(ii)Theinteractionpotentialisgrowingatinnity,limjxj!1V(x)=1:(iii)(Super-)quadraticgrowthat0:limx!0V(x)x2existsandisnite.(iv)Thewavespeedsatisesc2c21:=2supjxj6V(x)x2:Themainresultofthissectionisasfollows.Theorem2.2LetAssumption2.1besatisedandsupposethatcislargeenoughtoensureforgivenby:=4c21c2c21+cp(c2c21):(7)Thenasolutionu2C2(R)of(2)existswithboundaryconditions(4).Assumption2.1allowsforinteractionpotentialsVwhichgrowsuperquadrat-icallyatinnity,i.e.,limx!1x2
V(x)=1;forsuchpotentialstheactionfunctionalJfrom(3)isunboundedfrombelow(andfromabove).Inthenextsub-section,wegathersomegeneralpropertiesofJandintroduceapenalisedfunctionalthatagreeswithJonasuitableneighbourhoodof02XwhichincludesarelevantpartofM;.Itisthenshownthataglobalminimiserofthepenalisedfunctional,ifitexists,liesintheinteriorofthisneighbourhoodsothatitisnecessarilyalocalminimiserofJaswell.Thelastsubsectionestablishestheexistenceofsuchaglobalminimiserofthepenalisedfunctional,whichisthesolutionclaimedinTheorem2.2