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

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Date
2010
Link to item file
http://hal.archives-ouvertes.fr/hal-00371344/en/
Dewey
Analyse
Sujet
Gross-Pitaevskii equation; Korteweg-de Vries equation; long-wave limit
Journal issue
Communications in Partial Differential Equations
Volume
35
Number
1
Publication date
01-2010
Article pages
113-164
Publisher
Taylor & Francis
DOI
http://dx.doi.org/10.1080/03605300903222542
URI
https://basepub.dauphine.fr/handle/123456789/3453
Collections
  • CEREMADE : Publications
Metadata
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Author
Béthuel, Fabrice
Gravejat, Philippe
Saut, Jean-Claude
Smets, Didier
Type
Article accepté pour publication ou publié
Abstract (EN)
In this paper, we proceed along our analysis of the Korteweg-de Vries approximation of the Gross-Pitaevskii equation initiated in a previous paper. At the long-wave limit, we establish that solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation split into two waves with opposite constant speeds $\pm \sqrt{2}$, each of which are solutions to a Korteweg-de Vries equation. We also compute an estimate of the error term which is somewhat optimal as long as travelling waves are considered. At the cost of higher regularity of the initial data, this improves our previous estimate.

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