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 element of(-1/root 3, 1/root 3), beta(+), beta(-) is an element of R, we construct 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(infinity)(R) is uniquely determined by the boundary conditions <(eta)over tilde>(+/-infinity) = eta+/-. This extends a previous result of [BrK92] by removing the assumption that eta+/- should be close to zero. The existence of the limiting profile <(eta)over tilde> is obtained as an application of the 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.