Lie Symmetry Reduction, Series Solutions, Solitary Wave Structures and Dynamical Analysis of a Nonlinear Water Wave Model

In this work, we study an extended (3+1)-dimensional nonlinear shallow water wave equation arising in fluid dynamics and systematically derive a variety of solitary wave solutions. A rich class of exact solutions is achieved using similarity variable reduction via Lie symmetry analysis. The optimal system of one-dimensional subalgebras from one class is constructed. This is followed by the Kumar-Malik method that includes bright, dark, periodic and singular solutions. Graphical simulations in 2-dimensional, 3-dimensional surface and contour plots show the dynamical features and structural diversity of the solutions. Also, series solutions of a reduced partial differential equation and the other ordinary differential equation are obtained to have more analytical insight into the behavior of the proposed model. Furthermore, the stability properties and qualitative transitions of the corresponding reduced nonlinear dynamical system are studied through bifurcation analysis. Chaotic behavior is studied as well to show complex and sensitive dependence on the system parameters. The results significantly enlarge the solution space of the proposed model and shows its effectiveness in capturing complex multidimensional wave phenomena and nonlinear dynamical behaviours in fluid systems.