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

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.

Automated summary

In this article, the integrability of differential-difference lattices of hyperbolic type is investigated with a focus on constructing generalized symmetries. A method for solving functional equations using characteristic Lie-Rinehart algebras of semi-discrete models is proposed, leading to a classification method for integrable semi-discrete lattices. An interesting result includes a new example of an integrable equation, the semi-discrete analogue of the Tzizeica equation.