A new integrable discrete generalized nonlinear Schrodinger equation and its reductions

TL;DR

This paper introduces a new integrable discrete generalized nonlinear Schrödinger equation, demonstrating its integrability, reductions to classical models, and connections to continuous counterparts, expanding the understanding of discrete integrable systems.

Contribution

The paper constructs a novel integrable discrete GNLS equation, derives its recursion operator, symmetries, conservation laws, and establishes its reductions to classical discrete and continuous NLS equations.

Findings

Discrete GNLS equation is integrable with recursion operator and symmetries.

Reductions of the discrete GNLS include the classical discrete NLS.

Continuous GNLS properties are recovered from the discrete model.

Abstract

A new integrable discrete system is constructed and studied, based on the algebraization of the difference operator. The model is named the discrete generalized nonlinear Schrodinger (GNLS) equation for which can be reduced to classical discrete nonlinear Schrodinger (NLS) equation. To show the complete integrability of the discrete GNLS equation, the recursion operator, symmetries and conservation quantities are obtained. Furthermore, all of reductions for the discrete GNLS equation are given and the discrete NLS equation is obtained by one of the reductions. At the same time, the recursion operator and symmetries of continuous GNLS equation are successfully recovered by its corresponding discrete ones.