Generalized symmetries and integrability conditions for hyperbolic type semi-discrete equations

TL;DR

This paper develops a method for constructing generalized symmetries in hyperbolic semi-discrete equations, introduces a classification approach for integrable lattices, and presents a new semi-discrete Tzizeica equation example.

Contribution

It introduces a novel classification method for integrable semi-discrete lattices using characteristic Lie-Rinehart algebras and provides the first example of a semi-discrete Tzizeica equation.

Findings

Effective use of Lie-Rinehart algebras for symmetry construction

Classification scheme for integrable semi-discrete lattices

Discovery of a new semi-discrete Tzizeica equation

Abstract

In the article differential-difference (semi-discrete) lattices of hyperbolic type are investigated from the integrability viewpoint. More precisely we concentrate on a method for constructing generalized symmetries. This kind integrable lattices admit two hierarchies of generalized symmetries corresponding to the discrete and continuous independent variables $n$ and $x$. Symmetries corresponding to the direction of $n$ are constructed in a more or less standard way while when constructing symmetries of the other form we meet a problem of solving a functional equation. We have shown that to handle with this equation one can effectively use the concept of characteristic Lie-Rinehart algebras of semi-discrete models. Based on this observation, we have proposed a classification method for integrable semi-discrete lattices. One of the interesting results of this work is a new example of an integrable equation, which is a semi-discrete analogue of the Tzizeica equation. Such examples were not previously known.