Solitons, bifurcation analysis and stability analysis for the Drinfeld–Sokolov–Wilson system in dispersive fluids
Abstract A well-known integrable nonlinear model that arises in dispersive fluid dynamics, the Drinfeld–Sokolov–Wilson (DSW) system frequently serves to explain the evolution of dispersive tidal waves. In order to develop new classes of exact solitary wave solutions that are essential to understanding the model’s underlying dispersion and diffusion mechanisms, we examine the dynamical properties of the DSW system in this work. The governing nonlinear system is examined through effective analytical methods that provides mathematical derivation for a wide range of accurate solutions. Periodic wave structures, bright and dark solitons, and singular soliton solutions achieved under suitable parametric constraints are among them. Furthermore, the system is reduced to an analogous planar dynamical system enabling a bifurcation analysis, and phase-plane portraits are used to analyze the qualitative behavior of the solutions. In order to gain a deeper understanding of the DSW system’s nonlinear dynamics and wave propagation characteristics, the stability properties of the generated solitary waves are also examined. Three-dimensional, two-dimensional, and contour plots are used to graphically represent the calculated solutions in order to improve physical interpretation. Mathematica and Maple are used to do all symbolic calculations and visualizations, verifying the precision, effectiveness, and resilience of the analytical techniques used. The findings show that the suggested techniques are quite successful and can be expanded to study increasingly complex nonlinear models that arise in fluid mechanics, engineering, and mathematical physics.