Inverse scattering transform for a nonlinear lattice equation under non-vanishing boundary conditions

Under investigation in this paper is the inverse scattering transform for a nonlinear lattice equation, which can be used to study the fluctuation of nonlinear optics and dynamics of anharmonic lattices. Symmetries, analyticities and asymptotic behaviors of eigenfunctions will be obtained in the direct scattering analysis to establish a suitable Riemann-Hilbert problem. The Riemann-Hilbert problem of the scattering data with simple poles will be constructed. In particular, by using the Laurent expansion and the generalized residue condition to solve the Riemann-Hilbert problem, the determinant representation of N-soliton solution for the equation will be presented. One-dark-soliton under non-vanishing boundary conditions will be displayed through some representative reflectionless potentials.