An unusual series of autonomous discrete integrable equations on the square lattice

TL;DR

This paper introduces a novel infinite series of autonomous discrete integrable equations on the square lattice, featuring hierarchies of symmetries and conservation laws, with unique structures and connections to relativistic Toda equations.

Contribution

It presents a new class of discrete equations with hierarchical symmetries and conservation laws, including a novel construction method using master symmetries and a unique incorporation of master symmetry time.

Findings

Hierarchies of symmetries and conservation laws are established for the equations.

The case N=2 reveals a connection to relativistic Toda type equations.

A new construction scheme for symmetries and conservation laws is proposed.

Abstract

We present an infinite series of autonomous discrete equations on the square lattice possessing hierarchies of autonomous generalized symmetries and conservation laws in both directions. Their orders in both directions are equal to $\kappa N$, where $\kappa$ is an arbitrary natural number and $N$ is equation number in the series. Such a structure of hierarchies is new for discrete equations in the case $N>2$. Symmetries and conservation laws are constructed by means of the master symmetries. Those master symmetries are found in a direct way together with generalized symmetries. Such construction scheme seems to be new in the case of conservation laws. One more new point is that, in one of directions, we introduce the master symmetry time into coefficients of discrete equations. In most interesting case $N=2$ we show that a second order generalized symmetry is closely related to a relativistic Toda type integrable equation. As far as we know, this property is very rare in the case of autonomous discrete equations.