Nonlinearity, Dispersion, and Dissipation in Water Wave Dynamics: The BL Equation Unraveled

This study presents two novel analytical methodologies for the derivation of solitary wave solutions within the framework of the Benney-Luke (BL) model. The BL equation, characterized as a nonlinear evolution equation, plays a pivotal role in the description of long-wavelength, weakly nonlinear, and dispersive waves in shallow water. It holds particular importance in comprehending the behaviors of water waves in shallow channels and across the surfaces of shallow water bodies. By encompassing nonlinearity, dispersion, and dissipation, this equation encapsulates fundamental factors that significantly impact water wave dynamics, providing invaluable insights into attributes such as wave amplitude, frequency, and propagation speed. The soliton wave solutions obtained are rigorously validated using a contemporary numerical scheme, demonstrating a remarkable congruence with the analytical results. Further analysis of these solutions is carried out through comprehensive graphical representations, effectively highlighting the distinctive and innovative characteristics inherent in the BL model. This investigation makes a substantial contribution to the understanding of the intricate dynamics within the BL equation, thereby expanding the repertoire of available analytical and numerical techniques for the study of similar nonlinear evolution equations.