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.