Integrability and Symmetries of Difference Equations: the Adler-Bobenko-Suris Case

TL;DR

This paper studies the integrability and symmetry properties of Adler-Bobenko-Suris classified difference equations, revealing their multidimensional consistency, auto-Bäcklund transformations, Lax pairs, and infinite symmetry hierarchies.

Contribution

It provides a comprehensive symmetry analysis and constructs infinite hierarchies of generalized symmetries for these classified difference equations.

Findings

All equations exhibit multidimensional consistency.

Auto-Bäcklund transformations and Lax pairs are constructed.

Infinite hierarchies of generalized symmetries are explicitly derived.

Abstract

We consider the partial difference equations of the Adler-Bobenko-Suris classification, which are characterized as multidimensionally consistent. The latter property leads naturally to the construction of auto-B{\"a}cklund transformations and Lax pairs for all the equations in this class. Their symmetry analysis is presented and infinite hierarchies of generalized symmetries are explicitly constructed.