This paper presents an analytical investigation of the recently formulated space-time fractional \((2+1)\) -dimensional Calogero–Degasperis \((\textrm{CD})\) equation, utilizing the modified Riemann–Liouville fractional derivative also known as Jumarie’s fractional derivative. This equation plays a pivotal role in modeling wave propagation in nonlinear and dissipative media, particularly within coastal and beachfront ocean engineering, where it is employed to describe the dynamics of shallow-water waves. We apply the generalized Kudryashov–Auxiliary–Jacobian method (GKAJM), a robust and adaptable analytical technique, to derive exact solutions without the need for complex computations or restrictive assumptions. To the best of our knowledge, this is the first time GKAJM has been employed for solving this particular fractional equation. The solutions obtained include a wide range of function types such as logarithmic, exponential, trigonometric, and hyperbolic forms, which effectively describe diverse phenomena. Several of the solutions that have been found are entirely new and might be important for researchers to recognize. Moreover, we graphically illustrate the obtained results through 3D, contour, and 2D plots to visualize various soliton structures, including V-shaped solitons, bright solitons, parabolic solitons, and shock soliton solutions. These graphical interpretations provide deeper insight into the soliton dynamics governed by the fractional \(\textrm{CD}\) equation. Our findings not only contribute novel exact solutions but also demonstrate that the proposed method can be extended to other complex nonlinear fractional evolution equations arising in wave dynamics and mathematical physics. The aforementioned soliton behavior of the space-time fractional \((2+1)\) -dimensional \(\textrm{CD}\) equation is described by these fallouts and depictions.
