<p>The Drinfeld–Sokolov–Wilson (DSW) equation arises in physics as a fundamental model describing the interaction of solitonic waves in nonlinear dispersive systems, such as fluids, plasmas, and optical fibres. Since it is non-linear and has intricate wave dynamics, rigorous mathematical analysis is required to understand its underlying mechanisms and predict how it will behave. In this study, we extend the classical DSW system to its fractional-order form, leveraging the concept of fractional calculus to incorporate memory effects and nonlocal interactions. This introduces a richer dynamical structure and enables modelling real-world phenomena with greater flexibility. A numerical simulation is performed to investigate the influence of the fractional parameter on the evolution of the system, which highlights its role in understanding how dispersion and nonlinearity interact using a semi-analytical technique. Several key features are discussed, including wave propagation, energy dissipation, long-term memory effects, and modulation instability. The analysis of modulation instability provides insight into the conditions under which small perturbations in the wave profile grow, leading to the formation of complex wave structures and potentially turbulent behaviors. This is crucial for predicting the stability of solitonic waves in various physical contexts. To demonstrate the efficacy and accuracy of the obtained solution using the semi-analytical technique, we conducted a comparison with other numerical and analytical solutions available in the literature.</p>