Recursion operators, conservation laws and integrability conditions for difference equations

TL;DR

This paper develops a framework for integrability in difference equations by adapting recursion operators, enabling the generation of symmetries and conservation laws, and applies it to classify integrable equations like Viallet and ABS.

Contribution

It introduces a consistent theory of recursion operators and conservation laws for difference equations, extending integrability concepts from PDEs to discrete systems.

Findings

Recursion operators for Viallet and ABS equations identified

Infinite symmetries and conservation laws generated

Provides integrability conditions for difference equations

Abstract

In this paper we make an attempt to give a consistent background and definitions suitable for the theory of integrable difference equations. We adapt a concept of recursion operator to difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. Similar to the case of partial differential equations these canonical densities can serve as integrability conditions for difference equations. We have found the recursion operators for the Viallet and all ABS equations.