A unified framework for deriving and visualizing soliton solutions in the paraxial nonlinear Schrödinger equation

This research investigates the paraxial nonlinear Schrödinger equation commonly used in quantum mechanics, plasma physics, and nonlinear fiber optics. Employing the extended modified auxiliary equation mapping method, we obtained different soliton solutions, which were tested via Hamiltonian method of stability analysis. The dynamic behavior of the solutions was realized by making use of Stream Density graphs, 3D slice contour graphs, Linear graphs, Density linear graphs, and 2D graphs. The results obtained were tabulated systematically to ensure accuracy; therefore, this research would be of practical use in soliton dynamics and nonlinear wave propagation and can be useful in furtherance of mathematics and bio-mathematics as well as industrial research. The given model and soliton solutions can be efficaciously used to model pulse propagation in optical fibers, investigate energy localization in plasmas, and examine wave packet dynamics in quantum systems. Such uses highlight the practical relevance of paraxial nonlinear Schrödinger equation to promote technologies in telecommunications, fusion science, and nanoscale materials science.