Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations

TL;DR

This paper constructs and classifies autonomous quad-equations that admit known five-point differential-difference equations as symmetries, revealing new sine-Gordon and Liouville type examples and situating results within existing literature.

Contribution

It introduces a classification of autonomous quad-equations with specific symmetry properties, including new examples of integrable equations of sine-Gordon and Liouville types.

Findings

Classification of autonomous quad-equations admitting known symmetries

Identification of new sine-Gordon and Liouville type equations

Analysis within the context of existing integrability literature

Abstract

In this paper we construct the autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous transformations. We discuss our results in the framework of the known literature. There are among them a few new examples of both sine-Gordon and Liouville type equations.