Exact soliton solutions and soliton diffusion of two kinds of stochastic KdV equations with variable coefficients

Abstract

In this paper, we consider the exact solutions and soliton diffusion phenomenon of two kinds of stochastic KdV equations with variable coefficients. Firstly, according to symmetric reduction, stochastic KdV equations with variable coefficients are transformed into a coupled system of a deterministic KdV-type equation with variable coefficients and a solvable stochastic ordinary differential equation. Then, by the generalized wave transformation and the Clarkson-Kruskal direct method, we obtain the exact solutions of the deterministic KdV-type equation with variable coefficients. By coupling with the exact solutions of the stochastic ordinary differential equation, the exact solutions of stochastic KdV equations with variable coefficients are obtained. Compared to Wick-type stochastic KdV equations, our research work does not require additional inverse transformations, but solves stochastic partial differential equations more concisely, systematically, and directly. Secondly, two examples are given to verify the correctness of the theoretical analysis, and the soliton diffusion phenomenon of the system is discussed. Finally, by the Zabusky-Kruskal finite difference scheme, numerical simulations are provided to demonstrate the effectiveness of the analytic methods. The results indicate that the soliton diffusion phenomenon is subject to noise influence. In particular, the wave speed accelerates the soliton diffusion over time in the multiplicative noise background, and the wave speed slows down the soliton diffusion over time in the additive noise background.