This current study presents a precise analytical examination of the generalized third-order nonlinear Schrödinger equation through the application of the new auxiliary equation method. The approach provides several classes of exact solutions, such as V-shaped, dark soliton, periodic, kink, and anti-kink soliton solutions, which prove its effectiveness in solving higher-order nonlinear wave equations. The derived solutions are well depicted through 2D, contour, and 3D plots to show their spatial and temporal evolution features. A complete dynamical system analysis is carried out by Galilean transformation, showing the system behavior through accurate phase portraits and bifurcation diagrams. The analysis offers valuable information on stability of the solutions and transition processes amongst solution types. The system sensitivity analysis to parameters provides significant stability conditions for the solutions obtained. All the outcomes are derived by strict analytical means, and graphical plots are used to support the mathematical analysis. The study makes a contribution to the theoretical basis of nonlinear wave propagation and offers a sound framework for similar nonlinear evolution equations.
Fatima et al. (Tue,) studied this question.