Towards the theory of integrable hyperbolic equations of third order

TL;DR

This paper explores integrable third-order hyperbolic equations with two variables, identifying specific equations with symmetries related to well-known integrable systems and discussing variable choices.

Contribution

It introduces new integrable third-order hyperbolic equations and analyzes their symmetries, advancing the understanding of their structure and variable selection.

Findings

Identified a third-order hyperbolic equation with Krichever--Novikov symmetry.

Found a hyperbolic equation with modified Landau--Lifshitz symmetry.

Discussed the importance of variable choice in integrable hyperbolic equations.

Abstract

The examples are considered of integrable hyperbolic equations of third order with two independent variables. In particular, an equation is found which admits as evolutionary symmetries the Krichever--Novikov equation and the modified Landau--Lifshitz system. The problem of choice of dynamical variables for the hyperbolic equations is discussed.