There are several methods and techniques available for finding exact solutions to nonlinear differential equations, including the Darboux transformation, Ricatti method, Kudryashov method, Hirota bilinear transformation method, Lie symmetry method, extended tanh function method, G’/G expansion method, G’/G² expansion method, and the sine-Gordon approach, among others. Regarding the Drinfel’d–Sokolov–Wilson (DSW) equation, modified extended direct algebraic methods yielded solutions in the form of bell, anti-bell, periodic, and dark solitary waves in 2017, while series solutions were obtained using the Adomian decomposition method in 2022. In the field of fractional calculus, various types of fractional derivatives have been applied, such as the Beta derivative, Caputo fractional derivative, conformable fractional derivative, Riemann–Liouville derivative, and truncated M-fractional derivative. Many researchers are focused on constructing exact solutions for fractional differential equations. In 2023, singular bright, dark, periodic, bell, and lump-type water wave solutions to the coupled nonlinear fractional Drinfel’d–Sokolov–Wilson (FDSW) model with the Beta derivative were explored using the generalized rational exponential function method. However, the exact form of Weierstrass function-type solutions has not yet been established. Therefore, in this study, we aim to construct new travelling wave solutions for the FDSW model using the complex method and compare our results to those that are already known.