Lie Symmetry Analysis and Exact Solutions of an Extended Estevez–Mansfield–Clarkson Equation

Abstract In this paper, we conduct a comprehensive Lie symmetry investigation of the extended Estevez-Mansfield-Clarkson equation, which generalizes a wide class of nonlinear wave equations. High-order nonlinear partial differential equations (PDEs) remain notoriously difficult to analyze and solve, as their multiple higher-order derivatives and nonlinear terms often lead to non-uniqueness, finite-time blow-up, or chaotic dynamics, while traditional analytical and numerical techniques frequently encounter stability and convergence issues. To overcome these difficulties, we determine the complete Lie symmetry algebra of the equation and construct an optimal system of one-dimensional subalgebras. By means of similarity reductions, the extended equation is systematically transformed into ordinary differential equations (ODEs), enabling the extraction of a variety of exact solutions. Moreover, the associated conservation laws are established, thereby providing new perspectives on the intrinsic properties and physical relevance of the model. Beyond the specific case studied here, our findings underscore the future importance of Lie group methods as powerful tools, since the Lie symmetry method has been employed to identify group invariances and to obtain reductions for certain nonlinear evolution equations, for tackling the increasingly complex high-order nonlinear PDEs that arise in applied mathematics, fluid dynamics, nonlinear optics, and other branches of science and engineering.