Discrete and continuous coupled nonlinear integrable systems via the dressing method

TL;DR

This paper introduces a discrete analogue of the dressing method to derive new integrable nonlinear evolution equations, including coupled systems of nonlinear Schrödinger type, with applications to both discrete and continuous models.

Contribution

It presents a novel discrete dressing method and constructs new coupled integrable systems, extending to multi-component cases and their continuum limits.

Findings

Derived a discrete matrix nonlinear Schrödinger system with Lax pair.

Constructed coupled integrable systems of nonlinear Schrödinger type.

Extended systems to multi-component and continuum limits.

Abstract

A discrete analogue of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear Schrodinger type. First, a demonstration is given of how discrete nonlinear integrable equations can be derived starting from their linear counterparts. Then, starting from two uncoupled, discrete one-directional linear wave equations, an appropriate matrix Riemann-Hilbert problem is constructed, and a discrete matrix nonlinear Schrodinger system of equations is derived, together with its Lax pair. The corresponding compatible vector reductions admitted by these systems are also discussed, as well as their continuum limits. Finally, by increasing the size of the problem, three-component discrete and continuous integrable discrete systems are derived, as well as their generalizations to systems with an arbitrary number of components.