Modulation Equations Near the Eckhaus Boundary: The KdV Equation

Abstract

We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg-Landau equation ∂ T Ψ = (1 + iα)∂ 2 X Ψ + Ψ − (1 + iβ)Ψ|Ψ| 2 near the Eckhaus boundary, that is, when the wave train is near the threshold of its first instability. Depending on the parameters α, β a number of modulation equations can be derived, such as the KdV equation, the Cahn-Hilliard equation, and a family of Ginzburg-Landau based amplitude equations. Here we establish error estimates showing that the KdV approximation makes correct predictions in a certain parameter regime. Our proof is based on energy estimates and exploits the conservation law structure of the critical mode. In order to improve linear damping we work in spaces of analytic functions.

The KdV approximation for a system with unstable resonances

“…In this section, we collect a number of estimates, which we used in previous sections. Together with their proofs, they can be found as Lemma A.4, Corollary A.5, Corollary A.6, Lemma A.9, and Corollary A.10 in Haas et al We start with some estimates for the nonlinear terms.…”