Inverse scattering transform for the integrable discrete nonlinear Schrödinger equation with non-vanishing boundary conditions

The inverse scattering transform for an integrable discretization of the defocusing nonlinear Schrodinger equation with nonvanishing boundary values at infinity is constructed. This problem had been previously studied, and many key results had been established. Here, a suitable transformation of the scattering problem is introduced in order to address the open issue of analyticity of eigenfunctions and scattering data. Moreover, the inverse problem is formulated as a Riemann-Hilbert problem on the unit circle, and a modification of the standard procedure is required in order to deal with the dependence of asymptotics of the eigenfunctions on the potentials. The discrete analog of Gel'fand-Levitan-Marchenko equations is also derived. Finally, soliton solutions and solutions in the small-amplitude limit are obtained and the continuum limit is discussed.