Averaging principle for stochastic complex Ginzburg-Landau equations

TL;DR

This paper develops and proves various averaging principles for stochastic complex Ginzburg-Landau equations, demonstrating convergence of solutions, existence of recurrent solutions near averaged solutions, and the attractor's behavior as the oscillation scale diminishes.

Contribution

It introduces three types of averaging principles for stochastic complex Ginzburg-Landau equations, including solution convergence, recurrent solutions, and attractor behavior, advancing understanding of their long-term dynamics.

Findings

Solution converges to averaged solution as ps00 goes to zero.

Existence of a unique recurrent solution near the averaged stationary solution.

Attractor of the original system tends to that of the averaged system in probability measure.

Abstract

Averaging principle is an effective method for investigating dynamical systems with highly oscillating components. In this paper, we study three types of averaging principle for stochastic complex Ginzburg-Landau equations. Firstly, we prove that the solution of the original equation converges to that of the averaged equation on finite intervals as the time scale $\varepsilon$ goes to zero when the initial data are the same. Secondly, we show that there exists a unique recurrent solution (in particular, periodic, almost periodic, almost automorphic, etc.) to the original equation in a neighborhood of the stationary solution of the averaged equation when the time scale is small. Finally, we establish the global averaging principle in weak sense, i.e. we show that the attractor of original system tends to that of the averaged equation in probability measure space as $\varepsilon$ goes to zero.