Analysis of stochastic solitary wave solutions, bifurcation behavior, and modulation instability in the stochastic breaking soliton equation

This paper investigates stochastic solitary wave solutions to the stochastic breaking soliton equation and analyze the bifurcation behavior and modulation instability of the solutions to random perturbations. Even though some research has been done on deterministic soliton models, there has not been adequate research on the stochasticity of the stochastic breaking soliton equation. Despite these studies, the dynamics and stability of its solitary wave solutions are still not fully studied, leaving an important research gap. The modified extended direct algebraic method is used to obtain the exact stochastic solitary waves solutions of the model under consideration to fill this research gap. The novelty of this paper is that it is the first one where the modified extended direct algebraic approach has been employed to the stochastic breaking soliton equation to obtain a series of new analytical stochastic solutions. Moreover, bifurcation analysis and modulation instability are investigated to understand the impact of system parameters and multiplicative noise on the qualitative behavior of the solutions. It is found that the stochastic perturbations have an important influence on the wave dynamics and can cause the changes between the stable and unstable regimes. The main contributions the work has made are the construction of new exact stochastic solutions, study of bifurcation behavior and understanding of modulation instability under noise. The results highlight the effectiveness of the proposed approach and offer deeper insight into stochastic nonlinear wave systems.