Th. Gallay (Paris XI) and A. Mielke (Hannover)
Diffusive Mixing of Stable States in the Ginzburg-Landau Equation
(168K, Postscript (gzipped and uuencoded))

ABSTRACT.  The Ginzburg-Landau equation $\partial_t u = \partial_x^2u+u-|u|^2u$ 
on the real line has spatially periodic steady states of the the form 
$U_{\eta,\beta}(x)=(1{-}\eta^2)^{1/2}\,{\mathrm e}^{{\mathrm i}
(\eta x+\beta)}$, with $|\eta| \leq 1$ and $\beta \in {\mathbb R}$. 
For $\eta_+,\eta_-{\in} (-1/\sqrt{3},1/\sqrt{3})$, $\beta_+,\beta_-{\in}
{\mathbb R}$, we construct solutions which converge for all $t>0$ to 
the limiting pattern $U_{\eta_\pm,\beta_\pm}$ as $x\to \pm \infty$. 
These solutions are stable with respect to sufficiently small 
${\mathrm H}^2$ perturbations, and behave asymptotically in time 
like $(1-\widetilde\eta(x/\sqrt t)^2)^{1/2}\,\exp({\mathrm i}\sqrt t
\,\widetilde N(x/ \sqrt t\,))$, where $\widetilde N'=\widetilde\eta 
\in {\mathcal C}^\infty({\mathbb R})$ is uniquely determined by the 
boundary conditions $\widetilde\eta(\pm\infty) = \eta_\pm$. This 
extends a previous result of Bricmont and Kupiainen by removing 
the assumption that $\eta_\pm$ should be close to zero. The existence 
of the limiting profile $\widetilde\eta$ is obtained as an application 
of the theory of monotone operators, and the long-time behavior of 
our solutions is controlled by rewriting the system in scaling variables 
and using energy estimates involving an exponentially growing damping 
term.