Peculiar symmetry structure of some known discrete nonautonomous equations

TL;DR

This paper investigates the symmetry structures of certain nonautonomous discrete equations, revealing that their generalized symmetries exist only under periodic coefficient conditions and can form hierarchies with arbitrarily high order.

Contribution

It demonstrates the conditions under which nonautonomous discrete equations possess generalized symmetries, extending understanding of their integrability and symmetry hierarchies.

Findings

Symmetries exist if and only if coefficients are periodic.

The order of symmetries can be arbitrarily high depending on the period.

Hierarchies of symmetries and conservation laws can be constructed for large periods.

Abstract

We study the generalized symmetry structure of three known discrete nonautonomous equations. One of them is the semidiscrete dressing chain of Shabat. Two others are completely discrete equations defined on the square lattice. The first one is a discrete analogue of the dressing chain introduced by Levi and Yamilov. The second one is a nonautonomous generalization of the potential discrete KdV equation or, in other words, the H1 equation of the well-known Adler-Bobenko-Suris list. We demonstrate that these equations have generalized symmetries in both directions if and only if their coefficients, depending on the discrete variables, are periodic. The order of the simplest generalized symmetry in at least one direction depends on the period and may be arbitrarily high. We substantiate this picture by some theorems in the case of small periods. In case of an arbitrarily large period, we show that it is possible to construct two hierarchies of generalized symmetries and conservation laws. The same picture should take place in case of any nonautonomous equation of the Adler-Bobenko-Suris list.