Solitons, bifurcation, chaos, and stability of the fractional Schrödinger-Hirota equation with beta derivative

The fractional nonlinear Schrödinger-Hirota equation is a dynamic model for describing optical soliton propagation in dispersive fibers, capturing complex higher-order dispersion and fractional scaling effects. In this study, we investigate the underlying dynamics of this framework by incorporating the β-fractional derivative. By employing the extended Riccati scheme, we derive a broad spectrum of novel exact analytical solutions, including kink, dark, periodic, and mixed bright-dark soliton profiles. Beyond exact solutions, this work provides a qualitative assessment of the system’s dynamics using planar dynamical system theory and bifurcation analysis to classify equilibrium states. Furthermore, the introduction of external periodic perturbations reveals a distinct transition from stable oscillatory motion to chaotic behavior. This chaotic transition is precisely quantified through Poincaré sections and the Lyapunov exponent, demonstrating the system’s sensitivity to fractional scaling and perturbation parameters. Finally, linear stability analysis confirms the robustness and physical viability of the derived solitons under disruptive conditions. The results enrich the mathematical understanding of fractional nonlinear dynamics and provide valuable physical insights for optimizing stable, high-capacity optical fiber communication systems and next-generation photonic devices.