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Modulation equations near the Eckhaus boundary - The KdV equation

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


We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg-Landau equation
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.

Volltext §
DOI: 10.5445/IR/1000085447
Veröffentlicht am 22.08.2018
Cover der Publikation
Zugehörige Institution(en) am KIT Sonderforschungsbereich 1173 (SFB 1173)
Publikationstyp Forschungsbericht/Preprint
Publikationsjahr 2018
Sprache Englisch
Identifikator ISSN: 2365-662X
KITopen-ID: 1000085447
Verlag Karlsruher Institut für Technologie (KIT)
Umfang 44 S.
Serie CRC 1173 ; 2018/16
Schlagwörter modulation equation, validity, wave trains, long wave approximation, Eckhaus boundary
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