Fractional Nonlinear Dynamics and Solitary Waves in the β- Fractional mKdV-gMEW Equation

This paper examines the space-time β-fractional modified Korteweg-de Vries generalized modified equal-width (mKdV-gMEW) equation, a nonlinear dispersive wave model governing ion-acoustic wave propagation in plasma, shallow-water dynamics, and nonlinear optics, extended to the β-fractional framework to encode memory effects and spatial nonlocality inaccessible to classical integer-order formulations. Exact traveling wave solutions comprising bright and dark solitons, kink and anti-kink waves, parabolic solitons, peakon solitons, and rational solutions are derived by simultaneously applying the modified extended tanh expansion method and the improved one-variable expansion method, with the effect of the β-fractional order on soliton amplitude, width, and energy localization examined in detail. A thorough dynamic analysis of the traveling-wave reduction shows periodic, quasi-periodic, and chaotic regimes using phase portraits, Poincaré sections, time series, and the largest Lyapunov exponent. This is backed up by bifurcation analysis, sensitivity analysis, multi-stability exploration, and Lyapunov stability, which shows that the solutions are stable even with small changes. In contrast to previous research limited to single-method integer-order models devoid of a cohesive dynamical framework, this study represents the inaugural dual-method β-fractional analysis of the mKdV-gMEW equation, providing a comprehensive analytical dynamical framework with significant ramifications for plasma physics, shallow-water hydrodynamics, and nonlinear optics.