Numerical Investigation of Solitary Wave Behavior of Certain Types of Nonlinear Partial Differential Equations

Abstract

The objective of this work is to obtain accurate numerical results of certain types of partial differential equations with boundary conditions by integrating the Fourier Spectral Method (FSM) for spatial discretization and the Central Finite Difference Method (CFDM) for time integration. The novelty of this study lies in this combination, which enhances accuracy and stability for solving nonlinear dispersive shallow water wave equations while preserving physical invariants such as momentum and energy. Five distinct applications are solved, each representing solitary wave phenomena. The accuracy and efficiency of the used method are measured through evaluating the error norms and conservation properties. To demonstrate the potency and nature of the raised solutions, the 2D and 3D graphical representations and the tables are introduced, showing that the proposed approach achieves high numerical precision and successfully reproduces the physical behavior of solitary waves, preserving their amplitude and energy during propagation.