Symmetry analysis and closed-form invariant solutions of the nonlinear wave equations in elasticity using optimal system of Lie subalgebra

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.