Classification of semidiscrete hyperbolic type equations. The case of third order symmetries

TL;DR

This paper classifies semidiscrete hyperbolic equations of the form involving third order symmetries, identifying specific equations that admit such symmetries in both discrete and continuous variables.

Contribution

It provides a systematic classification of semidiscrete hyperbolic equations with third order symmetries, expanding understanding of their symmetry properties.

Findings

List of equations with third order symmetries identified

Conditions for existence of higher symmetries established

Classification framework for semidiscrete hyperbolic equations developed

Abstract

In this paper, a classification of semidiscrete equations of hyperbolic type is carried out. We study the class of equations of the form $$\frac{du_{n+1}}{dx}=f\left(\frac{du_{n}}{dx},u_{n+1},u_{n}\right),$$ here is the unknown function $u_n (x)$ depends on one discrete $n$ and one continuous $x$ variables. The classification is based on the requirement for the existence of higher symmetries in the discrete and continuous directions. The case is considered when the symmetries are of order 3 in both directions. As a result, a list of equations with the required conditions is obtained.