A discrete Darboux-Lax scheme for integrable difference equations

TL;DR

This paper introduces a discrete Darboux-Lax scheme to derive transformations and solutions for integrable difference equations, including systems lacking 3D consistency, exemplified by the Adler-Yamilov system related to the NLS equation.

Contribution

It presents a novel discrete Darboux-Lax scheme applicable to non-3D consistent quad-graph equations, enabling the construction of auto-Bäcklund transformations and soliton solutions.

Findings

Constructed auto-Bäcklund transformation for Adler-Yamilov system

Developed superposition principle for the scheme

Derived one- and two-soliton solutions

Abstract

We propose a discrete Darboux-Lax scheme for deriving auto-B\"acklund transformations and constructing solutions to quad-graph equations that do not necessarily possess the 3D consistency property. As an illustrative example we use the Adler-Yamilov type system which is related to the nonlinear Schr\"odinger (NLS) equation [19]. In particular, we construct an auto-B\"acklund transformation for this discrete system, its superposition principle, and we employ them in the construction of the one- and two-soliton solutions of the Adler-Yamilov system.