Infinitely many symmetries and conservation laws for quad-graph equations via the Gardner method

TL;DR

This paper demonstrates how the Gardner method, leveraging multidimensional consistency, can generate infinitely many conservation laws and symmetries for ABS equations and beyond, including asymmetric cases.

Contribution

It shows that Bäcklund transformations and initial conservation laws derive from multidimensional consistency and extends the Gardner method to generate symmetries for discrete integrable equations.

Findings

All ABS equations admit infinitely many conservation laws.

The Gardner method can generate symmetries for discrete KdV and other ABS equations.

An extension of the Gardner method to asymmetric equations is developed.

Abstract

The application of the Gardner method for generation of conservation laws to all the ABS equations is considered. It is shown that all the necessary information for the application of the Gardner method, namely B\"acklund transformations and initial conservation laws, follow from the multidimensional consistency of ABS equations. We also apply the Gardner method to an asymmetric equation which is not included in the ABS classification. An analog of the Gardner method for generation of symmetries is developed and applied to discrete KdV. It can also be applied to all the other ABS equations.