Hyperbolic equations with fifth-order symmetries

TL;DR

The paper addresses the classification of hyperbolic PDEs of the form $u_{xy}=F(u_x,u_y,u)$ that admit fifth-order symmetries. It develops a rigorous compatibility analysis between the equation and a fifth-order symmetry, deriving a determining equation and reducing the problem via a lemma that leads to a restricted form $F=F_1(u_y,u) f_1(u_1)+C_2 f_2(u_1)$, with a Wronskian argument narrowing the possibilities. The main result identifies four independent canonical hyperbolic equations (from an initial six up to symmetry) that possess fifth-order symmetries, connecting to known integrable models through transformations and extending prior classifications of lower-order symmetries.

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.