Invariant analysis and equivalence transformations for the non-linear wave equation in elasticity

The phenomenon of elastic wave propagation within an inelastic material results in nonlinear wave equations. Our study specifically examines a unidirectional nonlinear elastic wave, incorporating considerations of a sixth-order Murnaghan potential. The problem of elasticity was analyzed using Lie symmetry and equivalence transformations. An equivalence group was determined for the studied wave equations. We constructed an optimal system consisting of non-similar subalgebras of Lie algebra and utilized it to perform symmetry reductions. Exploring the nonlinear elastic wave problem with a damping term involves utilizing a multiplier technique to address the conservation laws associated with these equations.