Bifurcation, Multistability, and Soliton Dynamics in the Stochastic Potential Korteweg-de Vries Equation

Abstract

This paper focuses on the dynamical behavior and soliton solutions of the stochastic potential Korteweg-de Vries equation, a crucial model for nonlinear optical solitons, photons, electric circuits, and multicomponent plasmas. Initially, the Lie symmetries of the given equation are determined and employed to convert the model into an ordinary differential equation. Following this, a detailed examination of the equation’s dynamic behavior is conducted from various perspectives. To analyze chaotic behavior, we employ a range of advanced techniques, including time series analysis, bifurcation diagrams, phase portraits, and Lyapunov exponents. Additionally, we derive the solitary wave structures of the system using the new extended direct algebraic method. Through this approach, we identify periodic wave solutions expressed through rational, hyperbolic, and trigonometric functions. Specific parameter values lead to a variety of soliton solutions, including bright soliton, Dark soliton, semi-dark solitons, anti-kink solitons, and kink solitons. Lastly, the model’s multistability is examined under various initial conditions. Understanding the dynamic properties of systems is crucial for predicting outcomes and advancing technological innovations.