Discrete Darboux system with self-consistent sources and its symmetric reduction

TL;DR

This paper constructs a discrete non-commutative Darboux system with self-consistent sources, introduces a symmetric reduction, and simplifies the equations using a tau/sigma form, connecting to classical systems in the continuous limit.

Contribution

It develops a novel discrete Darboux system with sources, presents a symmetric reduction, and introduces a tau/sigma formulation for simplification and connection to classical systems.

Findings

Construction of the discrete non-commutative Darboux system with sources

Presentation of a symmetric reduction of the system

Introduction of tau/sigma form for simplified equations

Abstract

The discrete non-commutative Darboux system of equations with self-consistent sources is constructed, utilizing both the vectorial fundamental (binary Darboux) transformation and the method of additional independent variables. Then the symmetric reduction of discrete Darboux equations with sources is presented. In order to provide a simpler version of the resulting equations we introduce the $\tau/\sigma$ form of the (commutative) discrete Darboux system. Our equations give, in continuous limit, the version with self-consistent sources of the classical symmetric Darboux system.