Soliton solutions, bifurcations, and sensitivity analysis to the higher-order nonlinear fractional Schrödinger equation in optical fibers

In this article, we present new exact soliton solutions for the space-time fractional higher-order nonlinear Schrödinger equation, which describes the propagation of ultra-short pulses in nonlinear optical fibers. We apply a traveling wave transformation with the Beta derivative to convert the nonlinear fractional differential equation into a standard nonlinear differential equation. To find the exact analytical solutions, we implement the extended Riccati equation method, which led to a variety of soliton and soliton-like solutions, including trigonometric, hyperbolic, and rational functions. The graphical representations of these solutions show different physical forms, such as kink, periodic, bright bell, and dark bell structures. Additionally, we used planar dynamical systems theory to investigate the bifurcation phenomena in the derived system. A detailed sensitivity analysis was also performed on the dynamical system utilizing the Runge-Kutta method. These exact solitons play a vital role in understanding wave propagation and are essential for validating both numerical simulations and experimental findings in areas such as quantum mechanics, nonlinear optics and engineering.