Towards the classification of integrable differential-difference equations in 2 + 1 dimensions

TL;DR

This paper develops a classification framework for integrable differential-difference equations in 2+1 dimensions using hydrodynamic reductions, identifying key scalar integrable equations like the intermediate long wave and Toda types.

Contribution

It introduces a novel application of hydrodynamic reductions to classify integrable equations with discrete variables in higher dimensions.

Findings

Classified several scalar integrable equations in 2+1 dimensions.

Extended hydrodynamic reduction methods to dispersive equations.

Identified key equations such as intermediate long wave and Toda types.

Abstract

We address the problem of classification of integrable differential-difference equations in 2+1 dimensions with one/two discrete variables. Our approach is based on the method of hydrodynamic reductions and its generalisation to dispersive equations. We obtain a number of classification results of scalar integrable equations including that of the intermediate long wave and Toda type.