Classification of evolutionary equations on the lattice. I. The general theory

TL;DR

This paper extends the symmetry approach to classify integrable differential-difference equations on the lattice, allowing for a broader set of functions and thus a more comprehensive understanding of such equations.

Contribution

It introduces a modified symmetry approach that considers an infinite set of functions, broadening the classification of integrable lattice equations beyond previous methods.

Findings

The approach encompasses well-known equations like Volterra and Toda.

It allows for classification with a more general set of functions.

Provides a framework for identifying new integrable lattice equations.

Abstract

A modification of the symmetry approach for the classification of integrable differential-difference equations of the form $$ u_{n,t} = f_n(u_{n-1}, u_n, u_{n+1}), $$ where $n$ is a discrete integer variable, is presented (the well-known Volterra and Toda equations can be written in this form). If before, in the framework of the symmetry approach, only equations similar to $$ u_{n,t} = f(u_{n-1}, u_n, u_{n+1}), $$ i.e. defined by a function $f$, were considered, now we have an infinite set $f_n$ of a priori quite different functions.