Analytical optical soliton solution & stability analysis of the dimensionless time-dependent paraxial equation

This study investigates the analytical solutions of optical solitons governed by the a time-dependent Paraxial Equation incorporating Kerr law nonlinearity. A rigorous analytical method is developed to obtain these solutions, offering insights into the behavior of optical solitons in nonlinear media. Here, we have applied the eMETEM approach for the first time to create robust solutions to the time-dependent Paraxial Equation. Building an efficient plan to solve the governing model has been our main goal. The Kerr law nonlinearity, which describes the intensity-dependent refractive index, profoundly affects the propagation characteristics of optical pulses. Through systematic analysis, key properties such as soliton formation, stability, and evolution are elucidated. The derived analytical solutions provide a valuable framework for understanding the intricate dynamics of optical solitons in nonlinear media, facilitating advancements in various applications including optical communication and signal processing.