Analytical construction of exact traveling wave solutions to the nonlinear water wave equation

Abstract

The Benney–Luke model, which is a well-known nonlinear water wave equation, plays a significant role in describing bidirectional (two-way) propagation of long surface gravity waves in shallow water. In this study, exact traveling-wave solutions of the Benney–Luke model are derived using five distinct analytical techniques, namely the extended simple equation method, the modified extended auxiliary equation mapping method, $(G^{\prime }/G)$-expansion method, $Exp(-\Psi \left(\varphi \right))$-expansion method, and the modified F-expansion method, respectively. By employing these analytical approaches, a wide variety of exact traveling wave solutions is obtained, expressed in terms of hyperbolic, trigonometric, exponential, and rational functions. These solutions reveal diverse wave structures, including solitary waves, periodic waves, and rational wave profiles. To elucidate the physical characteristics and dynamical behavior of the model, selected exact solutions are illustrated through graphical representations by assigning suitable values to the involved parameters. The graphical analysis provides deeper insight into the influence of system parameters on wave propagation and shape. Furthermore, the analytical methods employed in this work are shown to be straightforward, efficient, and highly versatile, making them applicable to a wide range of nonlinear partial differential equations arising in water wave theory and other areas of mathematical physics.