# Modulation equations near the Eckhaus boundary - The KdV equation

Haas, Tobias; de Rijk, Björn; Schneider, Guido

##### Abstract:
We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg-Landau equation
$\partial_T\Psi=(1+i\alpha)\partial^2_X\Psi+\Psi-(1+i\beta)\Psi|\Psi|^2$
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.

 Zugehörige Institution(en) am KIT Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht/Preprint Publikationsjahr 2018 Sprache Englisch Identifikator ISSN: 2365-662X urn:nbn:de:swb:90-854479 KITopen-ID: 1000085447 Verlag KIT, Karlsruhe Umfang 44 S. Serie CRC 1173 ; 2018/16 Schlagwörter modulation equation, validity, wave trains, long wave approximation, Eckhaus boundary
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