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

Béthuel, Fabrice; Gravejat, Philippe; Saut, Jean-Claude

Béthuel, Fabrice; Gravejat, Philippe; Saut, Jean-Claude (2008), Existence and properties of travelling waves for the Gross-Pitaevskii equation, dans A. Farina and J.-C. Saut, Stationary and time dependent Gross-Pitaevskii equations, American Mathematical Society : Providence, R.I.

Type

Chapitre d'ouvrage

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http://hal.archives-ouvertes.fr/hal-00363329/en/

Date

2008

Titre de l'ouvrage

Stationary and time dependent Gross-Pitaevskii equations

Auteurs de l’ouvrage

A. Farina and J.-C. Saut

Éditeur

American Mathematical Society

Titre de la collection

Contemporary Mathematics

Ville d’édition

Providence, R.I.

Isbn

978-0-8218-4357-4

Nombre de pages

55-104

Métadonnées

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Auteur(s)

Béthuel, Fabrice
Gravejat, Philippe
Saut, Jean-Claude

Résumé (EN)

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