Optimal system of Lie subalgebras and exact solutions of a nonlinear elastic rod equation

The objective of this article is to find new traveling wave solutions of a celebrated nonlinear elastic rod (NER) equation with lateral inertia, describing the propagation of longitudinal waves through a nonlinear elastic rod by applying both the Lie point symmetry analysis and the first integral method. In order to determine the similarity reductions and invariant solutions, the Lie point symmetry analysis is applied to the NER equation and to this purpose, the invariants of the Lie algebra as well as a one-dimensional optimal system of subalgebras are constructed. Based on the members of the optimal system, similarity reductions and corresponding invariant solutions of the NER equation are derived. In addition, kink-shaped, anti-kink-shaped solitons, and a parabolic structure solution are obtained by solving the similarity reductions. Furthermore, we have found two new traveling wave solutions of the NER equation by applying the well-known first integral method. Using numerical results, the parametric dependence of the obtained solutions is also presented.