Soliton dynamics of the fourth-order nonlinear Boussinesq equation arising in shallow-water via advanced analytical techniques

This paper focuses on the fourth-order nonlinear Boussinesq equation (FONBE), which describes the interaction mechanisms of solutions in shallow-water waves. The FONBE has broad applications across various fields of physics and engineering, including heat transfer, fluid convection, and modeling of underwater volcanic activity, ocean currents, seafloor movements, and other hydrothermal processes. The Kumar-Malik method (KMM) and the modified exponential function method (MEFM) are employed to derive different types of soliton solutions. Analytical forms expressed through Jacobi elliptic, hyperbolic, trigonometric, exponential, and rational functions are obtained. Graphical representations further support the analytical findings, providing a clear understanding of the solution behaviors. These results highlight the effectiveness of the proposed methodology in addressing nonlinear challenges in mathematics and engineering, offering improvements over previous studies. The applied techniques are robust, efficient, and adaptable to a wide class of nonlinear partial differential equations. The novelty of this work lies in the generation of several new soliton solutions through the application of analytical approaches that have not been previously applied to this equation.