Diffusive Mixing of Stable States in the Ginzburg–Landau Equation

Abstract:

The Ginzburg–Landau equation on the real line has spatially periodic steady states of the form , with and . For , , we construct solutions which converge for all t>0 to the limiting pattern as . These solutions are stable with respect to sufficiently small perturbations, and behave asymptotically in time like , where is uniquely determined by the boundary conditions . This extends a previous result of [BrK92] by removing the assumption that should be close to zero. The existence of the limiting 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.