98-26 Th. Gallay (Paris XI) and A. Mielke (Hannover)

Diffusive Mixing of Stable States in the Ginzburg-Landau Equation (168K, Postscript (gzipped and uuencoded)) Jan 23, 98

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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.

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