Some stability results for the complex Ginzburg-Landau equation

Abstract

Using some classical methods of dynamical systems, stability results and asymptotic decay of strong solutions for the complex Ginzburg-Landau equation (CGL), $${\partial }_{t}u=(a+i\alpha )\Delta u-(b+i\beta )|u{|}^{\sigma }u+ku,t>0,x\in \Omega ,$$ 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.

16 pages