In this article, the (\(1+1\))-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.

在本文中，研究了 ( \(1+1\) ) 维 Pochhammer–Chree (PC) 模型，该模型说明了弹性杆纵波传播的非线性方案，并使用李无穷小生成器确定其解析解。使用 PC 模型的对称性获得的解使人们能够研究弹性杆内纵向变形波的传播。无穷小生成元是通过实施李对称技术和几何方法获得的。在几何方法中，采用Estabrook 和扩展Harrison 微分形式来建立无穷小生成元。使用奥尔弗的传统方法，建立了一个最优子代数系统，因为无穷小生成元有无限种可能的线性组合。此外，通过形式拉格朗日的使用证明了上述模型满足准自共轭准则。此外，与 PC 模型对称性相关的守恒定律是使用准自共轭条件和 Ibragimov 的“新守恒定理”导出的。最后，用相应的密度图和等高线图绘制了所获得解的三维表面。新开发的结果表明了对称技术获得该非线性控制模型不变解的有效性、可靠性和有效性。对称分析的新颖之处在于微分方程保持不变的变换组可以用来简化给定的模型。