In this article, the (1+1\documentclass[12pt]{minimal}
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\begin{document}$$1+1$$\end{document})-dimensional Pochhammer–Chree (PC) model, which illustrates a nonlinear scheme of longitudinal wave propagation of elastic bars, is investigated to determine its analytical solutions using Lie infinitesimal generators. The solutions obtained using the symmetries of the PC model allow one to investigate the propagation of longitudinal deformation waves within an elastic rod. The infinitesimal generators are obtained by implementing the Lie symmetry technique and the geometric method. In the geometric method, the Estabrook and extended Harrison differential forms are used for establishing the infinitesimal generators. Using Olver’s conventional method, a system of optimal subalgebras has been built because there are an infinite number of possible linear combinations of infinitesimal generators. Furthermore, it is demonstrated by the use of formal Lagrangian that the aforementioned model satisfies the quasi-self-adjointness criteria. Additionally, the conservation laws associated with the symmetries of the PC model are derived using the quasi-self-adjointness condition and Ibragimov’s ‘new conservation theorem’. Finally, the three-dimensional surfaces of the acquired solutions have been plotted with the corresponding density and contour plots. The newly developed results show the effectiveness, dependability and validity of the symmetry techniques for obtaining invariant solutions to this nonlinear governing model. The novelty of symmetry analysis is that the group of transformations under which the differential equations remain invariant can be used to simplify the given model.
