The Klein-Gordon Equation and Differential Substitutions of the Form $v=\varphi(u,u_x,u_y)$

TL;DR

This paper classifies Klein-Gordon equations and their related equations via differential substitutions of a specific form, providing a comprehensive understanding of their interconnections.

Contribution

It offers a complete classification of equations connected by differential substitutions of the form v=φ(u,u_x,u_y), expanding the understanding of Klein-Gordon equations.

Findings

Classification of equations connected by differential substitutions

Explicit forms of substitutions for Klein-Gordon equations

Conditions for the existence of such substitutions

Abstract

We present the complete classification of equations of the form $u_{xy}=f(u,u_x,u_y)$ and the Klein-Gordon equations $v_{xy}=F(v)$ connected with one another by differential substitutions $v=\varphi(u,u_x,u_y)$ such that $\varphi_{u_x}\varphi_{u_y}\neq 0$ over the ring of complex-valued variables.