Inverse scattering transform for the discrete focusing nonlinear schroedinger equation with nonvanishing boundary conditions

In this article we develop the direct and inverse scattering theory of the Ablowitz-Ladik system with potentials having limits of equal positive modulus at infinity. In particular, we introduce fundamental eigensolutions, Jost solutions, and scattering coefficients, and study their properties. We also discuss the discrete eigenvalues and the corresponding norming constants. We then go on to derive the left Marchenko equations whose solutions solve the inverse scattering problem. We specify the time evolution of the scattering data to solve the initial-value problem of the corresponding integrable discrete nonlinear Schroedinger equation. The one-soliton solution is ¨ also discussed.