The Ginzburg-Landau manifold is an attractor

Summary

The Ginzburg-Landau modulation equation arises in many domains of science as a (formal) approximate equation describing the evolution of patterns through instabilities and bifurcations. Recently, for a large class of evolution PDE's in one space variable, the validity of the approximation has rigorously been established, in the following sense: Consider initial conditions of which the Fourier-transforms are scaled according to the so-calledclustered mode-distribution. Then the corresponding solutions of the “full” problem and the G-L equation remain close to each other on compact intervals of the intrinsic Ginzburg-Landau time-variable. In this paper the following complementary result is established. Consider small, but arbitrary initial conditions. The Fourier-transforms of the solutions of the “full” problem settle to clustered mode-distribution on time-scales which are rapid as compared to the time-scale of evolution of the Ginzburg-Landau equation.