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This study investigates the Nonlinear Schrödinger Equation (NLSE) through a combination of analytical and numerical approaches to understand soliton dynamics in nonlinear media. The analytical methods involve the use of the Ansatz technique to derive exact solutions of the NLSE, emphasizing the role of nonlinearity in modulating wave propagation. These solutions illustrate the interplay between dispersive and nonlinear effects essential for soliton formation and stability. Numerical simulations are conducted using the Split Step Fourier method, which effectively handles the temporal evolution of solitons under various initial conditions. The numerical results highlight the stability and robustness of solitons, confirming the analytical predictions. Additionally, the error analysis between analytical and numerical solutions underscores the importance of accurate initial conditions, revealing consistent error patterns attributable to initial guess discrepancies. The comprehensive analysis provided by both methods demonstrates the NLSE's capability to model intricate wave phenomena, making it applicable to diverse physical systems, including optical fibers and Bose-Einstein condensates. This combined approach offers valuable insights into the fundamental characteristics of solitons and their stability criteria, enhancing our understanding of nonlinear wave dynamics.
Muhammad Ananda Riskillah (Tue,) studied this question.