The solutions with recurrence property for stochastic linearly coupled complex cubic-quintic Ginzburg–Landau equations

Abstract

Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. In this paper, we will establish an averaging principle for multiscale stochastic linearly coupled complex cubic-quintic Ginzburg-Landau equations with slow and fast time scales. Under suitable conditions, the existence of an averaging equation eliminating the fast variable for this coupled system is proved, and as a consequence, the system can be reduced to a single stochastic complex cubic-quintic Ginzburg-Landau equation with a modified coefficient.where T > 0, I = (0, 1), Q = I × (0, T ), the stochastic perturbations are of additive type, W 1 and W 2 are mutually independent Wiener processes on a complete stochastic basis (Ω, F, F t , P), which will be specified later, we denote by E the expectation with respect to P. The noise coefficients σ 1 and σ 2 are positive constants and the parameter ε is small and positive, which describes the ratio of time scale between the process A ε and B ε . With this time scale the variable A ε is referred as slow component and B ε as the fast component.Many problems in the natural sciences give rise to singularly perturbed systems of stochastic partial differential equations. In the past four decades, singularly perturbed systems have been the focus of extensive research within the framework of averaging methods. The separation of scales is then taken to advantage to derive a reduced equation, which approximates the slow components. Conditions under which the averaging principle can be applied to this kind of system are well known in the classical literature.Multiscale stochastic partial differential equations(SPDEs) arise as models for various complex systems, such model arises from describing multiscale phenomena in, for example, nonlinear oscillations, material sciences, automatic control, fluids dynamics, chemical kinetics and in other areas leading to mathematical description involving "slow" and "fast" phase variables. The study of the asymptotic behavior of such systems is of great interest. In this respect, the question of how the physical effects at large time scales influence the dynamics of the system is arisen. We focus on this question and show that, under some dissipative conditions on fast variable equation, the complexities effects at large time scales to the asymptotic behavior of the slow component can be omitted or neglected in some sense.The theory of stochastic averaging principle provides an effective approach for the qualitative analysis of stochastic systems with different time-scales and is relatively mature for stochastic dynamical systems. The theory of averaging principle serves as a tool in study of the qualitative behaviors for complex systems with multiscales, it is essential for describing and understanding the asymptotic behavior of dynamical systems with fast and slow variables. Its basic idea is to approximate the original system by a reduced system. The theory of averaging for deterministic dynamica...