Classification of five-point differential-difference equations

TL;DR

This paper classifies integrable five-point differential-difference equations using symmetry methods, identifying 17 equations including known and potentially new ones, and explores transformations linking them.

Contribution

It provides a comprehensive classification of a subclass of integrable five-point differential-difference equations, including new equations and their interrelations.

Findings

Identified 17 integrable equations, some potentially new.

Established transformations linking the equations and their symmetries.

Included well-known equations like Itoh-Narita-Bogoyavlensky and Sawada-Kotera.

Abstract

Using the generalized symmetry method, we carry out, up to autonomous point transformations, the classification of integrable equations of a subclass of the autonomous five-point differential-difference equations. This subclass includes such well-known examples as the Itoh-Narita-Bogoyavlensky and the discrete Sawada-Kotera equations. The resulting list contains 17 equations some of which seem to be new. We have found non-point transformations relating most of the resulting equations among themselves and their generalized symmetries.