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On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation I

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Date
2009
Link to item file
http://hal.archives-ouvertes.fr/hal-00334268/en/
Dewey
Probabilités et mathématiques appliquées
Sujet
long-wave approximation; Korteweg-de Vries equation; Gross-Pitaevskii equation
Journal issue
International Mathematics Research Notices
Volume
2009
Number
14
Publication date
2009
Article pages
2700-2748
Publisher
Oxford University Press
DOI
http://dx.doi.org/10.1093/imrn/rnp031
URI
https://basepub.dauphine.fr/handle/123456789/3438
Collections
  • CEREMADE : Publications
Metadata
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Author
Smets, Didier
Saut, Jean-Claude
Gravejat, Philippe
Béthuel, Fabrice
Type
Article accepté pour publication ou publié
Abstract (EN)
The fact that the Korteweg-de-Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation was derived several years ago in the physical literature. In this paper, we provide a rigorous proof of this fact, and compute a precise estimate for the error term. Our proof relies on the integrability of both the equations. In particular, we give a relation between the invariants of the two equations, which, we hope, is of independent interest.

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