Abstract
Nonlinear wave equations are generated by the elastic wave propagation through inelastic material. We studied such a unidirectional nonlinear elastic wave by considering potential having geometrical nonlinearity. The nonlinear problem in elasticity is studied by symmetry method. Optimal system of non-similar one dimensional subalgebras of Lie algebra is constructed. Symmetry reductions are performed using these subalgebras and corresponding group invariant solutions are found. The problem of nonlinear elastic wave with a damping term is also studied. Multiplier approach is used for finding conservation laws of these equations.