Dynamical Behavior and Chaotic Nature of M-Fractional Paraxial Wave Equation With Three Analytical Methods

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 Expa-function, improved Kudryashov, and unified solver techniques. These methods yield diverse soliton structures, such as the Expa-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.