Averaging Principle for Multiscale Stochastic Fractional Schrödinger–Korteweg-de Vries System

Abstract

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.