Group classification, optimal system and optimal reductions of a class of Klein Gordon equations

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.