Bifurcation Analysis and Wave Solutions of the Resonant Nonlinear Schrödinger Equation with Cubic-Quintic Nonlinearity, Dispersion, and Self-Steepening Effects

This paper investigates the resonant nonlinear Schrödinger equation with cubic–quintic nonlinearity, dispersion, and self-steepening effects, which models the propagation of optical pulses in nonlinear optical fibers. The main objective of this study is to construct exact analytical solutions and to analyze the dynamical behavior of the associated system, while demonstrating the effectiveness of the modified extended mapping method for higher-dimensional nonlinear evolution equations. Using this method, several classes of exact solutions are derived, including Weierstrass elliptic periodic function solutions, Jacobi elliptic function solutions, singular periodic solutions, as well as bright, kink, and singular soliton solutions. The influence of physical parameters on the existence and structure of these solutions is examined. In addition, a detailed bifurcation analysis is performed to investigate the stability and qualitative behavior of the equilibrium points, supported by phase portraits corresponding to different parameter regimes. The results confirm that the proposed approach is efficient and robust in capturing complex nonlinear wave structures arising from the resonant nonlinear Schrödinger equation, thereby enriching the spectrum of solitary wave solutions and providing new theoretical insights into nonlinear optical frameworks, plasma wave processes, and higher-dimensional nonlinear wave phenomena.