This work concerns the averaging principle for multiscale stochastic fractional Schrodinger–Korteweg-de Vries system. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, the multiscale system can be reduced to a single stochastic fractional Korteweg-de Vries equation (averaged equation) with a modified coefficient, the slow component of multiscale system towards to the solution of the averaged equation in moment. In order to prove the averaging principle, there are two key points: uniform bounds for the solution and Holder continuity of time variable for the slow component. For this, we need a crucial property—smoothing effect of the fractional Korteweg-de Vries semigroup, but the traditional method can’t help us obtain this property. Such analysis does not follow in a straightforward manner from results already available in the mathematical literature. On the contrary, it requires the introduction of some new ideas and techniques. We try to overcome this difficulty by the duality method, the interpolation arguments, the energy estimate method and refined inequality technique.

中文翻译：

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多尺度随机分数 Schrödinger-Korteweg-de Vries 系统的平均原理

这项工作涉及多尺度随机分数 Schrodinger-Korteweg-de Vries 系统的平均原理。随机平均原理是研究不同时间尺度随机动力系统定性分析的有力工具。更准确地说，在合适的条件下，多尺度系统可以简化为具有修正系数的单个随机分数阶 Korteweg-de Vries 方程（平均方程），多尺度系统的慢分量在时刻求平均方程的解. 为了证明平均原理，有两个关键点：解的统一边界和慢分量时间变量的Holder连续性。为此，我们需要一个关键性质——分数阶 Korteweg-de Vries 半群的平滑效应，但是传统的方法不能帮助我们获得这个属性。这种分析并没有直接从数学文献中已有的结果中得出。相反，它需要引入一些新的思想和技术。我们试图通过对偶方法、插值参数、能量估计方法和改进的不等式技术来克服这个困难。