Analytical Investigation of Trajectory Evolution and Bifurcation Behavior in the Fractional Perturbed Gerdjikov–Ivanov Equation With External Modulation

In this work, an efficient Bäcklund-enhanced unified analytical framework is developed to study the fractional perturbed Gerdjikov–Ivanov (pGI) equation that describes the propagation of nonlinear pulses in dispersive optical media. A fractional formulation is developed with Jumarie’s modified Riemann–Liouville derivative to include memory-dependent dynamics. The proposed method involves the unified method and the Bäcklund transformation and is systematically used to obtain a wide class of exact analytical solutions. The spatiotemporal dynamics generated are later investigated in detail by spatiotemporal simulations and nonlinear dynamical diagnostics with maximum Lyapunov exponent and phase portraits, as well as energy landscapes. The results show that the fractional-order parameter can be used as an effective control tool for setting the wave propagation properties and the stability range of the system, which can give valuable insight for the design and optimization of future high-capacity photonic crystal fiber communication systems.