Soliton structures of fractional coupled Drinfel’d–Sokolov–Wilson equation arising in water wave mechanics

Abstract This article delves into the dynamic constructions of distinctive traveling wave solutions for wave circulation in shallow water mechanics, specifically addressing the time-fractional couple Drinfel’d–Sokolov–Wilson (DSW) equation. Introducing the previously untapped $$exp(-\phi (\xi ))$$ e x p ( - ϕ ( ξ ) ) -expansion method, we successfully generate a diverse set of analytic solutions expressed in hyperbolic, trigonometric, and rational functions, each with permitted parameters. Visualization through three-dimensional (3D) as well two-dimensional (2D) plots, including contour plots, reveals inherent wave phenomena in the DSW equation. These newly obtained wave solutions serve as a catalyst for refining theories in applied science, offering a means to validate mathematical simulations for the proliferation of waves in shallow water as well as other nonlinear scenarios. Obtained wave solutions demonstrate the bright soliton, periodic, multiple soliton, and kink soliton shape. The simplicity and efficacy of the implemented methods are demonstrated, providing a valuable tool for approximating the considered equation. All figures are devoted to demonstrate the complete wave futures of the attained solutions to the studied equation with the collaboration of specific selections of the chosen parameters. Moreover, it may have summarized that the attained wave solutions and their physical phenomena might be useful to comprehend the various kind of wave propagation in mathematical physics and engineering.