The integrable semi-discrete nonlinear Schr\"odinger equations with nonzero backgrounds: Bilinearization-reduction approach

TL;DR

This paper develops a bilinearization-reduction method to solve classical and nonlocal semi-discrete nonlinear Schrödinger equations with nonzero backgrounds, providing explicit solutions including rogue waves and classification schemes.

Contribution

It introduces a novel bilinearization-reduction approach for semi-discrete NLS equations with nonzero backgrounds, yielding explicit solutions and classifications.

Findings

Explicit solutions for classical and nonlocal sdNLS equations with backgrounds.

Formulas for rogue wave solutions of the focusing sdNLS.

Classification of solutions based on spectral matrix forms.

Abstract

In this paper the classical and nonlocal semi-discrete nonlinear Schr\"{o}dinger (sdNLS) equations with nonzero backgrounds are solved by means of the bilinearization-reduction approach. In the first step of this approach, the unreduced sdNLS system with a nonzero background is bilinearized and its solutions are presented in terms of quasi double Casoratians. Then, reduction techniques are implemented to deal with complex and nonlocal reductions, which yields solutions for the four classical and nonlocal sdNLS equations with a plane wave background or a hyperbolic function background. These solutions are expressed with explicit formulae and allow classifications according to canonical forms of certain spectral matrix. In particular, we present explicit formulae for general rogue waves for the classical focusing sdNLS equation. Some obtained solutions are analyzed and illustrated.