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

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

Béthuel, Fabrice; Gravejat, Philippe; Saut, Jean-Claude; Smets, Didier (2010), On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation II, Communications in Partial Differential Equations, 35, 1, p. 113-164. http://dx.doi.org/10.1080/03605300903222542

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00371344/en/

Date

2010

Journal name

Communications in Partial Differential Equations

Volume

Number

Publisher

Taylor & Francis

Pages

113-164

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.

Subjects / Keywords

Gross-Pitaevskii equation; Korteweg-de Vries equation; long-wave limit