Bifurcation, Soliton Solutions, and Planar Dynamical System Analysis of the Generalised Resonant Nonlinear Schrödinger Equation

Abstract

This paper investigates optical soliton solutions and bifurcation in the generalised resonant nonlinear Schr & ouml;dinger equation (NSE). The generalised resonant NSE portrays wave propagation in optical fibers. The modified F-expansion technique (MFT) is utilized to construct solitary, periodic, singular, dark, and kink-type wave profiles. By employing a travelling wave transformation, the generalised resonant NSE is simplified to a singular dynamical system, which is subsequently converted into a regular dynamical system via variable substitution. The results show that the first integrals of both the singular and regular systems are mathematically equivalent, ensuring topological consistency between their trajectories. This equivalence facilitates a comprehensive examination of their phase portraits, highlighting their topological and geometric characteristics. Graphical representations in 2D and 3D of the solutions are provided to demonstrate their features physically, with parameters selected to emphasize the impacts of the generalised resonant NSE. These findings elucidate the kinetics of nonlinear wave propagation in optical systems.