Attraction properties of the Ginzburg-Landau manifold

Abstract

We consider solutions of weakly unstable PDE on an unbounded spatial domain. It has been shown earlier by the first author that the set of modulated solutions (called "Ginzburg-Landau manifold") is attracting. We seek to understand "how big" is the domain of attraction. Starting with general initial conditions of order " for the Fourier-transformed version of the given PDE we find that on the time-scale T " ; 2 (that is long in the terms of the original "physical" time t, but shorter than the natural time for the Ginzburg-Landau) the corresponding solutions evolve to the scaling of the clustered modes-distribution peaked at the integer multiples of the critical wave number, with the amplitudes sensitively dependent on such that for arbitrary close to zero after the time T " ; 2 solutions get on the Ginzburg-Landau manifold.