Existence and stability of singular patterns In a Ginzburg-Landau equation coupled with a mean field

Abstract

We study singular patterns in a particular system of parabolic partial differential equations which consist of a Ginzburg-Landau equation and a mean field equation. We prove existence of the three simplest concentrated periodic stationary patterns (single spikes, double spikes, double transition layers) by composing them of more elementary patterns and solving the corresponding consistency conditions. In the case of spike patterns we prove stability for sufficiently large spatial periods by first showing that the eigenvalues do not tend to zero as the period goes to infinity and then passing in the limit to a nonlocal eigenvalue problem which can be studied explicitly. For the two other patterns we show instability by using the variational characterization of eigenvalues.