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

Fabrice Bethuel (1) , Philippe Gravejat (2, 3) , Jean-Claude Saut (4) , Didier Smets (1, 3)

1 LJLL - Laboratoire Jacques-Louis Lions
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 DMA - Département de Mathématiques et Applications - ENS Paris
4 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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.

Mots clés: Gross-Pitaevskii equation; Korteweg-de Vries equation; long-wave limit

Domaines: Equations aux dérivées partielles [math.AP]

Fabrice Bethuel, Philippe Gravejat, Jean-Claude Saut, Didier Smets. On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation II. Communications in Partial Differential Equations, 2010, 35 (1), pp.113-164. ⟨10.1080/03605300903222542⟩. ⟨hal-00371344v2⟩