In this paper, we propose a novel integrable system named the stochastic mKdV equation, along with its corresponding Lax pair. We aim to extend the methodology of deterministic integrable systems to construct and solve stochastic integrable systems. The Darboux transformation effectively obtains analytic solutions for the integrable stochastic mKdV equation. Using the Darboux transformation, soliton solutions incorporating stochastic terms are obtained as Wronskian determinants. Furthermore, we conduct an in-depth analysis of the dynamics exhibited by the stochastic one-soliton and the two-soliton solutions.

中文翻译：

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高斯白噪声和维纳过程下的mKdV方程：达布变换和随机孤子解

在本文中，我们提出了一种新颖的可积系统，称为随机 mKdV 方程及其相应的 Lax 对。我们的目标是将确定性可积系统的方法扩展到构造和解决随机可积系统。达布变换有效地获得了可积随机 mKdV 方程的解析解。使用达布变换，获得包含随机项的孤子解作为朗斯基行列式。此外，我们对随机单孤子和双孤子解所表现出的动力学进行了深入分析。