Dominik Zimmermann

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

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 Bénard–Marangoni problem and short-wave instabilities in the flow down an inclined plane.

Nonlinear Stability at the Eckhaus Boundary

The real Ginzburg-Landau equation possesses a family of spatially periodic equilibria. If the wave number of an equilibrium is strictly below the so called Eckhaus boundary the equilibrium is known to be spectrally and diffusively stable, i.e., stable w.r.t. small spatially localized perturbations. If the wave number is above the Eckhaus boundary the equilibrium is unstable. Exactly at the boundary spectral stability holds. The purpose of the present paper is to establish the diffusive stability of these equilibria. The limit profile is determined by a nonlinear equation since a nonlinear term turns out to be marginal w.r.t. the linearized dynamics.

The Turing Instability in Case of an Additional Conservation Law—Dynamics Near the Eckhaus Boundary and Open Questions

We are interested in spatially extended systems with a diffusively stable background state which becomes unstable via a Turing instability. The Marangoni convection problem is an example for such a system. We discuss the dynamics of such systems close to the instability with the help of effective amplitude equations. We discuss the global existence of solutions, the diffusive stability of the bifurcating Turing rolls, their behavior at the Eckhaus boundary, and a spatially inhomogeneous inhibition of the Turing bifurcation through the diffusive mode. Aside from the presentation of rigorous results we pose a number of open questions.