Periodic Solutions of the complex Ginzburg-Landau Equation in bounded domains

TL;DR

This paper investigates the existence of time-periodic solutions to the complex Ginzburg-Landau equation within bounded domains, employing non-monotone perturbation theory to extend understanding of its dynamic behaviors.

Contribution

It introduces a novel approach to analyze periodic solutions of the complex Ginzburg-Landau equation using non-monotone perturbation theory for parabolic equations.

Findings

Established existence of time-periodic solutions in bounded domains.

Extended perturbation theory to handle non-monotone aspects of the equation.

Provided a framework for future analysis of similar nonlinear PDEs.

Abstract

In this paper, we are concerned with complex Ginzburg-Landau (CGL) equations. There are several results on the global existence and smoothing effects of solutions to the initial boundary value problem for (CGL) in bounded or unbounded domains. In this paper, we study the time periodic problem for (CGL) in bounded domains. The main strategy in this paper is to regard (CGL) as a parabolic equation with monotone and non-monotone perturbations and to apply non-monotone perturbation theory of parabolic equations developed by Otani (1984).