Dynamics of the stochastic–fractional drinfel′d–sokolov–wilson equation: Effects of fractional order and noise intensity

In this work, we address the stochastic–fractional Drinfel’d–Sokolov–Wilson (SFDSW) system, a generalized nonlinear evolution model that incorporates fractional-order dynamics and stochastic perturbations to provide a more realistic framework for modeling complex physical processes. For the purpose of obtaining analytical solutions, a generalized Riccati equation mapping method is elaborated and applied to the fractional and stochastic structure of the stochastic–fractional Drinfel′d–Sokolov–Wilson model. Numerical solutions are created by using the differential transform method that allows for reliable approximation of the system. A comparative analysis is conducted with the purpose of examining the impact of fractional derivatives on the behavior of waves, stability, and dissipation, along with sensitivity to different noise intensities. It is determined that analytical mappings and numerical solutions exhibit strong agreement, confirming the efficiency of the approach. Moreover, this work reveals how the fractional order and stochastic forcing jointly shape the qualitative structure of solutions, providing further insight into the dynamics of nonlinear systems of the stochastic–fractional Drinfel′d–Sokolov–Wilson type.