Existence And Multiplicity Of Nontrivial Solitons For Discrete Nonlinear Schr(o|¨)dinger Equations

In this paper,we firstly look for the discrete solitons of the periodic discrete nonlinear Schrodinger equation iψn=-△ψn+εnψn-γχnfn(ψn), n∈Z, where△ψn=ψn+1+ψn-1-2ψn is the discrete Laplacian in one spatial dimension,γ=±1,fn(s)=|s|2s or fn(s)=(|s|2s)/(1+cns2) with s∈R,the gien sequences {εn},{cn} and {χn} are assumed to be N-periodic(N is a positive integer),that is,εn+N=ε,cn+N=cn and χn-N=χn for n∈Z.Due to the definition of the soliton,ψn has the form ψn=une-iwt,where {un} is a real valued sequence and ω∈R is the temporal frrequency,then we arrive at the equation-△un+εnun-ωun=γχn∫n(un), n∈Z.Actually,we shall consider a more general periodic discrete nonlinear Schrodinger equation Lun-ωun=γχngn(un), n∈Z,(DNLS1)where gn is a sequence of functions and L is a Jacobi operator[61] given by Lun=anun+1+an-1un-1+bnun. Here,{an} and {bn} are real valued N-periodic sequences, that is, an|N=an and bn+N=bn for n∈Z.If an=-1and bn=2+εn, then equation (DNLS1) reduces to-Δun+εnun-ωun=γχngn(un), n∈Z.(DNLS2)For equation (DNLS2), if the discrete potential V={εn} is not periodic and γ=1, we will study the non-periodic discrete nonlinear Schrodinger equation-Δun+εnun-ωun=gn(un), n∈Z,(DNLS3) where gn is not periodic on n∈Z.In our paper, we assume that ω belongs to a finite spectral gap of the linear operator L, or it is a spectral endpoint of the operator. If gn is asymptotically or super linear, the existence and multiplicity of nontrivial solitons of the dis-crete nonlinear Schrodinger equations is obtained by different methods, such as variant generalized weak linking theorem, a direct and simple reduction of the indefinite variational problem to a definite one and variant fountain theorem. In particular, if ω belongs to a finite spectral gap of the linear operator L, an open problem proposed by Pankov (2006Nonlinearity1927-40) is solved and a necessary and sufficient condition is obtained for the existence of gap solitons of the periodic discrete nonlinear Schrodinger equation (DNLS1). For the periodic discrete nonlinear Schrodinger equation (DNLS1) with superlinear or saturable nonlinearities, if gn(s) is odd in s∈R, we obtain infinitely many geometrically distinct solutions. For the non-periodic discrete nonlinear Schrodinger equation (DNLS3), if gn(s) is odd in s∈R, we also obtain the existence of infinitely many nontrivial solitons for this equation with gn(s) being super linear as|s|→∞with s∈R by a variant fountain theorem dues to Zou.In general, the dissertation is organized as follows.In Chapter1, we shall introduce the background and the latest research developments of the discrete nonlinear Schrodinger equations.In Chapter2, we assume that ω belongs to a finite spectral gap of the linear operator L, or it is a spectral endpoint of the operator, we shall study the ex- istence of nontrivial solitons of (DNLS1) with gnbeing super or asymptoticallylinear by a variant generalized weak linking theorem for strongly indefnite prob-lem developed by Schechter and Zou. In particular, an open problem proposedby Pankov (2006Nonlinearity1927-40) is solved and a necessary and sufcientcondition is obtained for the existence of gap solitons of the periodic discretenonlinear Schr(o|¨)dinger equation (DNLS1).In Chapter3, we assume that ω belongs to a fnite spectral gap of the linearoperator L, if gn(s) is odd in s∈R, we obtain infnitely many geometricallydistinct solutions for equation (DNLS1) with superlinear or saturable nonlinear-ities. Our method is based on a direct and simple reduction of the indefnitevariational problem to a defnite one.In Chapter4, we shall study the existence of nontrivial gap solitons of(DNLS2) with gnbeing super or asymptotically linear by a variant fountaintheorem dues to Jeanjean.In the fnal Chapter5, we obtain the existence of infnitely many nontrivialsolitons for the non-periodic discrete nonlinear Schr(o|¨)dinger equation (DNLS3)with gn(s) being super linear as|s|→∞by a variant fountain theorem dues toZou.