Soliton dynamics and stability analysis of the time-fractional Hamiltonian amplitude model: Bifurcation and chaotic behavior scheme

In this study, we investigate the soliton dynamics and stability properties of the time-fractional Hamiltonian amplitude (FHA) equation using the improved F-expansion method. The FHA equation, a fractional extension of the nonlinear Schrödinger equation, governs a wide range of nonlinear physical phenomena, including plasma physics, fluid dynamics, and optical communications. We exploit the beta fractional derivative approach to explore soliton solutions, chaotic behavior, bifurcations, and sensitivity analysis of the model parameters. The attained results reveal a variety of soliton structures, such as quasiperiodic, anti-peakon, and multi-periodic solitons, which are graphically represented to highlight their physical significance. Stability analysis using the linear stability method confirms the robustness of these solutions under certain perturbations. Moreover, bifurcation analysis via phase plane diagrams exposes key insights into the qualitative changes in the dynamical system, including the presence of quasiperiodic and chaotic behavior under external perturbations. These findings contribute to a deeper understanding of complex nonlinear systems and have potential applications in signal processing, optical fiber communications, and materials science.