β-fractional dynamics of the time-fractional higher-order nonlinear Schrödinger equation: Soliton propagation, bifurcation, and chaos

In this article, we investigate the optical pulse propagation dynamics in nonlinear fiber systems modulated by the time-fractional higher-order nonlinear Schrödinger (HNLS) equation involving the β-fractional derivative. The β-derivative provides a more accurate framework for modeling complex optical phenomena than the classical operator. Using the extended Riccati equation approach, we derive different analytical soliton solutions such as bright, dark, mixed, kink-type, and periodic waveforms in trigonometric, rational, and hyperbolic forms. The fractional order σ succeeds as a pivotal control parameter that modulates dispersion, localization, and soliton sharpness. Graphical analyses, through two- and three-dimensional plots, demonstrate that increasing the fractional parameter σ enhances pulse localization and reduces dispersive broadening, indicating stronger optical confinement. We analyze bifurcation and chaos using planar dynamical theory. The results reveal rich phase structures such as stable centers, saddle points, and periodic oscillations, which show that wave dynamics are sensitive to fractional parameters. This investigation improves the physical understanding of β-fractional soliton dynamics. It provides deeper insight into ultrafast pulse evolution, wave propagation control, and potential applications in optical communication and photonic signal processing.