Some stability results for the complex Ginzburg-Landau equation

TL;DR

This paper investigates the stability and decay of solutions to the complex Ginzburg-Landau equation using dynamical systems methods, establishing conditions for bound-states and their asymptotic stability on the real line.

Contribution

It provides new stability results, decay estimates, and existence proofs for bound-states of the complex Ginzburg-Landau equation under specific parameter conditions.

Findings

Strong solutions exhibit asymptotic decay under certain parameters.

Existence of bound-states is proven for specific domain and parameter conditions.

Bound-states are asymptotically stable when the domain is  and  is sufficiently large.

Abstract

Using some classical methods of dynamical systems, stability results and asymptotic decay of strong solutions for the complex Ginzburg-Landau equation (CGL), $$ \partial_t u = (a + i\alpha) \Delta u - (b + i \beta) |u|^\sigma u + k u, \,\, t > 0,\,\, x\in \Omega, $$ with $a>0, \alpha, b, \beta, k \in \mathbb{R}$, are obtained. Moreover, we show the existence of bound-states under certain conditions on the parameters and on the domain. We conclude with the proof of asymptotic stability of these bound-states when $\Omega=\mathbb{R}$ and $-k$ large enough.