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DOI: 10.5445/IR/1000085447
Veröffentlicht am 22.08.2018

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
Jahr 2018
Sprache Englisch
Identifikator ISSN: 2365-662X
URN: urn:nbn:de:swb:90-854479
KITopen ID: 1000085447
Verlag KIT, Karlsruhe
Umfang 44 S.
Serie CRC 1173 ; 2018/16
Schlagworte modulation equation, validity, wave trains, long wave approximation, Eckhaus boundary
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