Analytical Investigation of Nonlinear Wave Dynamics, Stability, and Energy Characteristics in Shallow Water Wave Propagation

In this study, a nonlinear evolution equation is used to model the dynamics of continuous wave motion and soliton bursts, mainly within the fields of fluid mechanics and wave dynamics. Dynamic analytical approaches, such as the enhanced modified extended -expansion method (EMETEM), the $(m+1/G')$-expansion method, and the new Kudryashov method, are used to obtain new soliton solutions, such as dark, bright, singular, periodic, and lump soliton solutions for the equation. An analysis of stability was conducted on several derived wave solutions, and the stability characteristics of these solutions were explored using the Hamiltonian criterion. The energy balance method (EBM) is employed to obtain an approximate periodic solution by using the Hamiltonian invariant to maintain the constancy of kinetic and potential energy. Ultimately, to explore the physical characteristics of several of the derived wave solutions, they have been represented through three-dimensional (3D), contour, and two-dimensional (2D) graphics at particular parameter values.