This research work provides a comprehensive investigation of the M‐fractional paraxial wave equation (M‐fPWE) in describing complex optical phenomena in telecommunication systems and nonlinear media, focusing on the dynamical analysis of optical soliton solutions, the impact of M‐fractional parameters, stability, multistability, and the chaotic nature of the proposed model. To examine optical soliton solutions for the time M‐fPWE model, we employ three advanced analytical methods, such as the Exp a ‐function, improved Kudryashov, and unified solver techniques. These methods yield diverse soliton structures, such as the Exp a ‐function technique, which produces kinky periodic waves, kink, and anti‐kink waves, and double periodic waves; the improved Kudryashov method reveals solitary periodic waves, kink, and periodic waves, various periodic breather waves, and interactions such as kink‐periodic lump and anti‐kink–periodic lump waves; the unified solver technique uncovers double periodic waves, periodic breather waves, and kink‐bell shape interactions. Moreover, by employing the Galilean transformation, we formulated the dynamical system of the equation, facilitating a comprehensive chaotic analysis, 2D and 3D phase portraits, Poincaré plots, and multistability analysis that uncovered essential data transmission systems. Finally, we compare our results and outcomes with a published work. The obtained results are significant in understanding key physical phenomena in optical fiber communication.
Roshid et al. (Thu,) studied this question.