Advanced methods for studying soliton solutions of cubic Schrödinger model

In numerous branches of physics, the nonlinear cubic Schrödinger equation serves as a fundamental model with profound applications, particularly in optics, plasma physics, fluid dynamics, and quantum mechanics. In this study, four analytical techniques have been employed to explore and derive soliton solutions of the complex Schrödinger equation. The obtained solutions, expressed via hyperbolic, trigonometric, exponential, and rational functions, exhibit rich wave structures and dynamics. To elucidate the physical significance and practical applicability of the proposed model, several graphical simulations have been performed by assigning appropriate values to the key parameters, thereby illustrating the influence of these parameters on wave propagation characteristics. The proposed methods efficiently generate diverse traveling solutions in physics and applied sciences.