Qualitative Dynamics, Analytical Solutions, Chaos, and Soliton Structures of the β-Fractional Kuralay Equation

In this study, we investigate the closed-form soliton solutions and qualitative analysis of the β space-time fractional Kuralay-II equation within the mathematical framework. The model is formulated using the recently developed beta fractional derivative, which provides greater flexibility for modeling complex systems. The exact traveling wave and soliton solutions are constructed using the (ψ′/ψ, 1 / ψ)-expansion method. The solutions include rational, hyperbolic, trigonometric, and mixed soliton structures, like periodic, bell-shaped, anti-bellshaped, kink, V-shaped, etc., for the suitable values of the arbitrary parameters. We have studied the effects of key physical and fractional parameters on wave propagation using graphical illustrations through two-dimensional profiles to display the impact of the parameters on solitons. The associated reduced dynamical system is also studied using phase portraits, energy analysis, and bifurcation theory. This study reveals the supercritical and subcritical pitchfork bifurcations. Also, numerical simulations and time-series analysis demonstrate that chaos appears under external periodic excitation. The results focus on the decisive role of the fractional-order parameters in controlling stability transitions, nonlinear wave patterns, and chaos generation. This study presents novel insights into fractional integrable systems and enriches the theoretical understanding of nonlinear wave dynamics relevant to optical fibers, ferromagnetic spin chains, and related intricate physical systems.