Hyperbolic equations with fifth-order symmetries

TL;DR

This paper classifies a specific class of hyperbolic equations based on the existence of fifth-order symmetries, resulting in a list of four special equations with particular symmetry properties.

Contribution

It provides a new classification of hyperbolic equations with fifth-order symmetries, identifying four equations that satisfy these symmetry conditions.

Findings

Identified four hyperbolic equations with fifth-order symmetries.

Established criteria for the existence of higher-order symmetries in hyperbolic equations.

Contributed to the mathematical understanding of symmetry properties in differential equations.

Abstract

This paper examines the classification of hyperbolic equations. We study a class of equations of the form $$\frac{\partial^2 u}{\partial x\partial y}=F\left(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},u\right),$$ where $u(x,y)$ is the unknown function and $x,y$ are independent variables. The classification is based on the requirement for the existence of higher fifth-order symmetries. As a result, a list of four equations with the required conditions was obtained.