Discrete solitons of the Ablowitz-Ladik equation with nonzero boundary conditions via inverse scattering

Soliton solutions of the focusing Ablowitz-Ladik (AL) equation with nonzero boundary conditions at infinity are derived within the framework of the inverse scattering transform (IST). After reviewing the relevant aspects of the direct and inverse problems, explicit soliton solutions are obtained which are the discrete analog of the Tajiri-Watanabe and Kutznetsov-Ma solutions to the focusing NLS equation on a finite background. Then, by performing suitable limits of the above solutions, discrete analog of the celebrated Akhmediev and Peregrine solutions are also presented. The latter, which can be thought of as a discrete “rogue” wave, is expressed as a family of rational functions of the discrete spatial variable n ∈ Z and time t ∈ R, parametrically depending on the amplitudeQo of the background. These solutions, which had been recently derived by direct methods, are obtained for the first time within the framework of the IST, thus also providing a spectral characterization of the solutions and a description of the singular limit process.