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

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

Smets, Didier; Saut, Jean-Claude; Gravejat, Philippe; Béthuel, Fabrice (2009), On the Korteweg-de Vries long-wave approximation of the Gross-Pitaevskii equation I, International Mathematics Research Notices, 2009, 14, p. 2700-2748. http://dx.doi.org/10.1093/imrn/rnp031

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00334268/en/

Date

2009

Journal name

International Mathematics Research Notices

Volume

2009

Number

Publisher

Oxford University Press

Pages

2700-2748

Abstract (EN)

The fact that the Korteweg-de-Vries equation offers a good approximation of long-wave solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation was derived several years ago in the physical literature. In this paper, we provide a rigorous proof of this fact, and compute a precise estimate for the error term. Our proof relies on the integrability of both the equations. In particular, we give a relation between the invariants of the two equations, which, we hope, is of independent interest.

Subjects / Keywords

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