Existence and properties of travelling waves for the Gross-Pitaevskii equation

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

This paper presents recent results concerning the existence and qualitative properties of travelling wave solutions to the Gross-Pitaevskii equation posed on the whole space R^N. Unlike the defocusing nonlinear Schrödinger equations with null condition at infinity, the presence of non-zero conditions at infinity yields a rather rich and delicate dynamics. We focus on the case N = 2 and N = 3, and also briefly review some classical results on the one-dimensional case. The works we survey provide rigorous justifications to the impressive series of results which Jones, Putterman and Roberts established by formal and numerical arguments.

Mots clés: Gross-Pitaevskii equation; travelling wave; Kadomtsev-Petviashvili equation

Domaines: Equations aux dérivées partielles [math.AP]

Fabrice Bethuel, Philippe Gravejat, Jean-Claude Saut. Existence and properties of travelling waves for the Gross-Pitaevskii equation. Alberto Farina and Jean-Claude Saut. Stationary and time dependent Gross-Pitaevskii equations, 473, American Mathematical Society, pp.55-104, 2008, Contemporary Mathematics, 978-0-8218-4357-4. ⟨10.1090/conm/473⟩. ⟨hal-00363329⟩