Invariant formulation of nonclassical symmetries and explicit solutions of Rosenau-Hyman equation along with bifurcation analysis

This study focuses on the Rosenau–Hyman equation, which is a fundamental model in nonlinear wave dynamics, and investigates it through the lens of nonclassical symmetry analysis. The approach employs symbolic computation to derive determining equations and uncover new invariant formulations, from which several explicit exact solutions are constructed. To further understand the system’s behavior, dynamical tools such as bifurcation analysis, sensitivity tests, Lyapunov exponents, and phase portraits are applied, highlighting the presence of stability transitions, multistability, and chaotic regimes. In addition, travelling wave solutions are obtained using the enhanced modified extended tanh function method (eMETFM), providing complementary wave structures. The findings deepen our understanding of nonlinear dispersive wave propagation and soliton interactions, with particular relevance to shallow water dynamics. More broadly, the developed solutions and their graphical interpretations contribute valuable insights for theoretical studies and applied research in fluid dynamics and wave modeling.