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 periodic stationary states with different wave-numbers
η
±
. These solutions are stable with respect to small perturbations, and approach as
t→+∞
a universal diffusive profile depending only on the values of
η
±
. This extends a previous result of Bricmont and Kupiainen by removing the assumption that
η
±
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.