Exact Solutions and Qualitative Analysis of the Stochastic Model for Embedded Solitons with \(\chi ^{(2)}\) and \(\chi ^{(3)}\) Nonlinear Susceptibilities

The exact solutions and qualitative analysis of the stochastic governing model for embedded solitons with \(\chi ^{(2)}\) and \(\chi ^{(3)}\) nonlinear susceptibilities are investigated in this study. The model introduces a stochastic term-white noise for the first time, bringing the model closer to reality. The trial equation method is used for mathematical analysis and the complete discriminant system for polynomial method is used for qualitative analysis. Using the bifurcation theory and the complete discriminant system for polynomial method, the existence of the soliton and periodic solutions is confirmed and the exact travelling wave solutions are generated to validate our findings. Furthermore, we explore the various sorts of exact solutions by illustrating the associated phase diagrams and providing two-dimensional diagrams to demonstrate the model’s dynamical behavior. The plethora of exact solutions shows that the effect of white noise exists only in the phase component of the solitons, providing insight into the optical solitons of stochastic nonlinear models.