SIGMA 8 (2012), 090, 37 pages      arXiv:1111.7255      https://doi.org/10.3842/SIGMA.2012.090
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”

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

Mariya N. Kuznetsova ^a, Aslı Pekcan ^b and Anatoliy V. Zhiber ^c
a) Ufa State Aviation Technical University, 12 K. Marx Str., Ufa, Russia
^b) Department of Mathematics, Istanbul University, Istanbul, Turkey
^c) Ufa Institute of Mathematics, Russian Academy of Science, 112 Chernyshevskii Str., Ufa, Russia

Received April 25, 2012, in final form November 14, 2012; Published online November 26, 2012

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.

Key words: Klein-Gordon equation; differential substitution.

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