Diffusive Mixing of Stable States in the Ginzburg-Landau Equation

Abstract

For the time-dependent Ginzburg-Landau equation on the real line, we construct solutions which converge, as $x \to \pm\infty$, to periodic stationary states with different wave-numbers $η_\pm$. These solutions are stable with respect to small perturbations, and approach as $t \to +\infty$ a universal diffusive profile depending only on the values of $η_\pm$. This extends a previous result of Bricmont and Kupiainen by removing the assumption that $η_\pm$ should be close to zero. The existence of the diffusive profile 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.