Infinitely many stationary solutions of discrete vector nonlinear Schrödinger equation with symmetry

In this paper we study the existence of stationary solutions for the following discrete vector nonlinear Schrödinger equationi∂ϕn∂t=-Δϕn+τnϕn-Jf(n,|ϕn|)ϕn, where ϕnϕn is a sequence of 2-component vector,J=0110,Δϕn=ϕn+1+ϕn-1-2ϕnis the discrete Laplacian in one spatial dimension and sequence τnτn is assumed to be N-periodic in n  , i.e. τn+N=τnτn+N=τn. We prove the existence of infinitely many nontrivial stationary solutions for this system by variational methods. The same method can also be applied to obtain infinitely many breather solutions for single discrete nonlinear Schrödinger equation.