### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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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