Justification of the Ginzburg–Landau Approximation in Case of Marginally Stable Long Waves

Abstract

The Ginzburg–Landau equation can be derived via multiple-scaling analysis as a universal amplitude equation for the description of bifurcating solutions in spatially extended pattern-forming systems close to the first instability. Here we are interested in approximation results showing that there are solutions of the pattern-forming system which behave as predicted by the Ginzburg–Landau equation. In the classical case the proof of the approximation result is based on the fact that the quadratic interaction of the critical modes, i.e., of the modes with positive or zero growth rates, gives only non-critical modes, i.e., modes which are damped with some exponential rate. It is the purpose of this paper to develop a method to handle a situation when this condition is violated by an additional curve of stable eigenvalues which possesses a vanishing real part at the Fourier wave number k=0 for all values of the bifurcation parameter. The investigations are motivated by the Benard–Marangoni problem and short-wave instabilities in the flow down an inclined plane.