Analyzing the soliton waves’ behavior in the (3+1)-dimensional time-fractional Jimbo–Miwa equation

In this study, we investigate an analytical technique for solving the newly developed time-fractional Formula: see text-dimensional Jimbo–Miwa Formula: see text equation in the framework of the conformable fractional derivative. The exact solutions are constructed using the Generalized-Kudryashov-Auxiliary-Jacobian method (GKAJM), which is flexible, efficient, and does not require additional mathematical manipulations. To the best of our knowledge, this method has not previously been employed to obtain exact solutions of the time-fractional Formula: see text-dimensional Formula: see text equation. The proposed approach yields a wide class of exact solutions expressed in logarithmic, exponential, trigonometric, and hyperbolic forms. By choosing appropriate parameter values, the obtained solutions exhibit significantly different dynamical behaviors, including kink-type solitary wave solutions, bell-shaped solitary wave solutions, bright soliton solutions, dark solitary wave solutions, kink-type singular waves, a train of singular solitons, static blow-up-type wave solutions, and rogue-wave-like localized structure solutions. Furthermore, the effectiveness of the method demonstrates its applicability to other nonlinear evolution equations and contributes to a deeper understanding of wave propagation phenomena in water wave dynamics. The results comprehensively describe the perturbed wave dynamics associated with the time-fractional Formula: see text-dimensional Formula: see text equation.