The Klein-Gordon Equation and Differential Substitutions of the Form v =φ( u , u x , u y )

[摘要] 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.

[关键词] Klein–Gordon equation;dif ferential substitution