Finite-dimensional approximations of random attractor for stochastic discrete complex Ginzburg-Landau equations

TL;DR

This work analyzes a stochastic discrete complex Ginzburg–Landau lattice discretized by an implicit Euler scheme. It proves well-posedness and discretization accuracy for the deterministic case, constructs a numerical attractor with proven upper semicontinuity to the global attractor as the time step vanishes, and develops finite-dimensional and truncated approximations that converge to their infinite-dimensional counterparts. Extending to the stochastic setting, it builds a random attractor via an Ornstein–Uhlenbeck transform, proves its convergence to the deterministic attractor as noise vanishes, and establishes similar truncation convergence and semi-continuity properties. Overall, the paper provides a rigorous framework for approximating global, numerical, and random attractors in stochastic complex Ginzburg–Landau dynamics, with explicit bounds and convergence results in terms of the time step, dimension, and noise level.

Abstract

In this paper, we apply an implicit Euler scheme to discretize the complex Ginzburg-Landau equation and prove the existence of a numerical attractor for the discrete Ginzburg-Landau system. We establish the upper semicontinuity of the numerical attractor with respect to the global attractor as the time step tends to zero. Furthermore, we provide finite-dimensional approximations for three types of attractors (global, numerical, and random), and demonstrate the existence of truncated attractors along with their convergence as the dimension of the state space tends to infinity. Finally, we prove the existence of a random attractor and establish the upper semi-continuity both of the global random attractor and the truncated random attractor.

Theorems & Definitions (41)

• Lemma 2.1
• proof : Proof
• Theorem 2.1
• proof : Proof
• Lemma 2.2
• proof : Proof
• Lemma 2.3
• proof : Proof
• Theorem 2.2
• Theorem 3.1