Exploring solitary wave solutions of the Drinfeld-Sokolov-Wilson equation

Understanding shallow water wave dynamics is essential for coastal engineering, oceanography, and nonlinear physics. In this study, I explored the nonlinear dynamics of the coupled Drinfeld-Sokolov-Wilson system, a key nonlinear partial differential equation model that plays a crucial role in understanding shallow water wave behavior. To investigate the solitary wave solutions of this model, four analytical tools were employed: the Kumar-Malik method, the generalized Arnous method, the Riccati modified extended simple equation method, and the logarithmic transformation. These techniques yielded a variety of soliton solutions, including dark, bright, mixed, kink, bright-dark, breather, multiwave, and mixed-form solitons. Moreover, the periodic, hyperbolic and the exponential function solutions were recovered. This study not only confirms known solutions but significantly enriches the literature on the Drinfeld-Sokolov-Wilson system through the application of novel analytical methods that yield an exceptionally diverse family of soliton solutions, complemented by a comprehensive graphical investigation that illustrates the profound influence of physical parameters on wave dynamics. This study affirms the practical utility of these modern methods while delivering novel insights that advance the theoretical understanding of higher-dimensional wave dynamics.