Optimal System, Group Invariant Solutions and Conservation Laws of the Non-linear Elastic Wave Equation and Damped Elastic Wave Equation

M. Usman\(^{a,*}\), A. Razzaq\(^a\), Ali Raza\(^b\), F.D. Zaman\(^a\)

\(^a\)Abdus Salam School of Mathematical Sciences, GC University, Lahore Pakistan.

\(^b\)Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore, Pakistan.

Correspondence should be addressed to: musman.awan112@gmail.com, ayesha.razzaq903@gmail.com, dr.aliraza@lahoreschool.edu.pk, f.zaman@sms.edu.pk

Copyright © 2023 M. Usman, A. Razzaq, Ali Raza, F.D. Zaman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: Received: 27 July 2023; Accepted: 29 July 2023; Published Online: 11 August 2023.

Abstract

Non-linear wave equations are created by the elastic wave propagation through inelastic material. We obtain the Lie point symmetries for the non-linear elastic wave equation and the optimal system of the symmetry algebra using Lie symmetry approach. Numerous solutions that are group invariant are obtained under the optimal system of subalgebras of Lie algebra. Additionally, the variational symmetries are obtained via Noether approach and the corresponsing conservation laws are presented. The non-linear elastic wave equation with a damping term is also studied. The local conservation laws using the direct approach are also discussed in this study

Keywords:

Optimal system, Group invariant solutions, Conservation laws, Variational Symmetries, Non-linear elastic waves