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The exact solutions and qualitative analysis of the stochastic governing model for embedded solitons with χ (2) and χ (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. [ABSTRACT FROM AUTHOR]

In this paper, the perturbed Gerdjikov–Ivanov equation with spatio-temporal dispersion is investigated. The trial equation method, the complex envelope travelling wave transformation, and the complete discriminant system for polynomial method, respectively, are all utilized. As a consequence, twenty-eight exact chirped solutions are obtained, and some representative two-dimensional patterns under concrete parameters are presented. The results acquired are conducive to studying the dynamic behavior of solitons. [ABSTRACT FROM AUTHOR]

This paper secures exact solutions from perturbed complex Ginzburg–Landau equation that is taken into account with Kerr law and cubic–quintic–septic nonlinearity. Two approaches are used, namely the trial equation method and complete discriminant system for polynomial method. The abundant exact solutions obtained can better analyze the complex optical phenomena and further demonstrate their essence.

This paper devotes to investigating the chirped solutions for the nonlinear Schrödinger equation with Kudryashov's law with dual form of generalized non-local nonlinearity in optical fibers. By implementing the trial equation method and the complete discriminant system for polynomial method, we obtain nineteen exact chirped solutions, whose types include elliptic function doubly-periodic solutions, solitary wave solutions and singular solutions. After selecting specific parameters, 2 D and 3 D graphs of several typical solutions are drawn to demonstrate their accurate physical behaviors. These solutions recovered abundant patterns of the chirped wave propagations of the model. [ABSTRACT FROM AUTHOR]

This paper secures exact solutions from perturbed complex Ginzburg–Landau equation that is taken into account with Kerr law and cubic–quintic–septic nonlinearity. Two approaches are used, namely the trial equation method and complete discriminant system for polynomial method. The abundant exact solutions obtained can better analyze the complex optical phenomena and further demonstrate their essence. • Exact solutions from complex Ginzburg–Landau equation are obtained. • The trial equation method and complete discriminant system for polynomial method are used. • The novelty of this work is that it does not set up constraints in advance for the purpose of integrability like other papers. • These results are more abundant than the existed solutions. [ABSTRACT FROM AUTHOR]