Dynamics of Bifurcation, Chaos, Sensitivity and Diverse Soliton Solution to the Drinfeld-Sokolov-Wilson Equations Arise in Mathematical Physics

In this study, we present an amalgamation of solitary wave solutions for the fractional coupled Drinfeld-Sokolov-Wilson (FCDSW) equations, a versatile mathematical model with significant applications in fluid dynamics, plasma physics, and nonlinear dynamics. By leveraging the two recently developed computational approaches, namely the modified Sardar sub-equation (MSSE) method and the improved \(\mathbb {F}\)- expansion method, we manifested the novel soliton solutions in the form of dark, dark-bright, bright-dark, singular, periodic, exponential, and rational forms. Furthermore, we also extracted W-shape, V-shape, mixed trigonometric, and hyperbolic soliton wave solutions, which are not reported in previously. Also, the modulation instability (MI) of the proposed model is also examined. To further understand the dynamics of the FCDSW equations, we conducted a detailed bifurcation analysis to investigate the bifurcation events exhibited by these equations. A sensitivity analysis was also performed to assess the model’s robustness against variations in initial conditions and parameters, offering valuable insights into the system’s susceptibility to perturbations. We proved the effectiveness of our suggested techniques in the analysis of the FCDSW equations using analytical tools and numerical simulations. Our results provide new insights into the behavior and solutions of the FCDSW equations and enhance the mathematical tools for investigating nonlinear partial differential equations (NLPDEs). These findings have potential applications in fields like fluid flow modeling, wave propagation in plasmas, and the study of nonlinear optical phenomena in physics and applied mathematics.