Diffusive mixing of stable states in the Ginzburg-Landau equation

The Ginzburg-Landau equation partial derivative(t)u = partial derivative(x) (2)u + u - \u\(2)u on the real line has spatially periodic steady states of the form U-eta,U-beta(x) = root 1-eta(2) e(i(eta x+beta)), with \eta\ less than or equal to 1 and beta is an element of R For eta(+), eta(-) is an el ement of(-1/root 3, 1/root 3), beta(+), beta(-) is an element of R, we cons truct solutions which converge for all t > 0 to the limiting pattern U-eta/-,U-beta+/- as x --> +/- infinity. These solutions are stable with respect to sufficiently small H-2 perturbations, and behave asymptotically in time like (1 - <(eta)over tilde>(x/root t)(2))(1/2) exp(i root t (N) over tilde (x/root t)), where (N) over tilde' = <(eta)over tilde> is an element of c(i nfinity)(R) is uniquely determined by the boundary conditions <(eta)over ti lde>(+/-infinity) = eta+/-. This extends a previous result of [BrK92] by re moving the assumption that eta+/- should be close to zero. The existence of the limiting profile <(eta)over tilde> is obtained as an application of th e theory of monotone operators, and the longtime behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term.