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

Inverse Scattering Transform for the Discrete Focusing Nonlinear Schrödinger Equation with Nonvanishing Boundary Conditions