On the validity of the degenerate Ginzburg-Landau equation

Abstract

The Ginzburg{Landau equation which describes nonlinear modulation of the amplitude of the basic pattern does not give a good approximation when the Landau constant (which describes the in uence of the nonlinearity) is small. In this paper a derivation of the so{called degenerate (or generalized) Ginzburg{Landau (dGL) equation is given. It turns out that one can understand the dGL{equation as an example of a normal form of a co{dimension two bifurcation for parabolic PDEs. The main body of the paper is devoted to the proof of the validity of the dGL as an equation whose solution approximate the solution of the original problem.