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Group classification, optimal system and optimal reductions of a class of Klein Gordon equations. (English) Zbl 1221.35023

Summary: Complete symmetry analysis is presented for non-linear Klein Gordon equations \(u_{tt}=u_{xx}+f(u)\). A group classification is carried out by finding \(f(u)\) that give larger symmetry algebra. One-dimensional optimal system is determined for symmetry algebras obtained through group classification. The subalgebras in one-dimensional optimal system and their conjugacy classes in the corresponding normalizers are employed to obtain, up to conjugacy, all reductions of equation by two-dimensional subalgebras. This is a new idea which improves the computational complexity involved in finding all possible reductions of a PDE of the form \(F(x,t,u,u_{x},u_{t},u_{xx},u_{tt},u_{xt})=0\) to a first order ODE. Some exact solutions are also found.

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
35L71 Second-order semilinear hyperbolic equations
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