Perturbative Symmetry Approach for Differential-Difference Equations

TL;DR

This paper introduces a new symbolic method to determine integrability of differential-difference equations, enabling classification and discovery of new integrable equations of arbitrary order.

Contribution

It develops a symbolic representation and a quasi-local extension of the difference ring to formulate necessary integrability conditions and classify integrable equations.

Findings

Classified 17 integrable equations of order (-3,3)

Produced infinite families of higher order integrable equations

Identified some new integrable equations

Abstract

We propose a new method for solution of the integrability problem for evolutionary differential-difference equations of arbitrary order. It enables us to produce necessary integrability conditions, to determine whether a given equation is integrable or not, and to advance in classification of integrable equations. In this paper we define and develop symbolic representation for the difference polynomial ring, difference operators and formal series. In order to formulate necessary integrability conditions, we introduce a novel quasi-local extension of the difference ring. It enables us to progress in classification of integrable differential-difference evolutionary equations of arbitrary order. In particular, we solve the problem of classification of integrable equations of order $(-3,3)$ for the important subclass of quasi-linear equations and produce a list of 17 equations satisfying the necessary integrability conditions. For every equation from the list we present an infinite family of integrable higher order relatives. Some of the equations obtained are new.