Travelling waves for the Gross-Pitaevskii equation II

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

1 LJLL - Laboratoire Jacques-Louis Lions
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results, where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy.

Mots clés: Travelling waves; Gross-Pitaevksii equation; Travelling waves.

Domaines: Equations aux dérivées partielles [math.AP]

Fabrice Bethuel, Philippe Gravejat, Jean-Claude Saut. Travelling waves for the Gross-Pitaevskii equation II. 2007. ⟨hal-00187672v1⟩