Comptes Rendus
no. 2
Partial Differential Equations
Limit at infinity for travelling waves in the Gross–Pitaevskii equation
[Limite à l'infini des ondes progressives dans l'équation de Gross–Pitaevskii]

Présenté par : Haïm Brèzis

Philippe Gravejat1
1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris-6), BC 187, 4, place Jussieu, 75252 Paris cedex 05, France
Comptes Rendus. Mathématique, Volume 336 (2003) no. 2, pp. 147-152.

Nous étudions la limite à l'infini des ondes progressives d'énergie finie dans l'équation de Gross–Pitaevskii en dimension supérieure ou égale à trois et nous montrons leur convergence uniforme vers une constante de module un.

We study the decay of the travelling waves of finite energy in the Gross–Pitaevskii equation in dimension greater than three and prove their uniform convergence to a constant of modulus one at infinity.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00007-4
Philippe Gravejat 1

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris-6), BC 187, 4, place Jussieu, 75252 Paris cedex 05, France 
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Philippe Gravejat. Limit at infinity for travelling waves in the Gross–Pitaevskii equation. Comptes Rendus. Mathématique, Volume 336 (2003) no. 2, pp. 147-152. doi : 10.1016/S1631-073X(03)00007-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00007-4/

[1] F. Béthuel, G. Orlandi, D. Smets, Vortex rings for the Gross–Pitaevskii equation, Preprint

[2] F. Béthuel; J.C. Saut Travelling waves for the Gross–Pitaevskii equation I, Ann. Inst. H. Poincaré Phys. Théor., Volume 70 (1999) no. 2, pp. 147-238

[3] F. Béthuel, J.C. Saut, Travelling waves for the Gross–Pitaevskii equation II, in preparation

[4] A. Cianchi; L. Pick Sobolev embeddings into BMO, VMO and L∞, Ark. Mat., Volume 36 (1998), pp. 317-340

[5] A. Farina, From Ginzburg–Landau to Gross–Pitaevskii, Preprint

[6] P. Gravejat, Decay for travelling waves in the Gross–Pitaevskii equation, in preparation

[7] C.A. Jones; P.H. Roberts Motions in a Bose condensate IV, Axisymmetric solitary waves, J. Phys. A, Volume 15 (1982), pp. 2599-2619

[8] P.I. Lizorkin Multipliers of Fourier integrals, Proc. Steklov Inst. Math., Volume 89 (1967), pp. 269-290

[9] M. Maris Analyticity and decay properties of the solitary waves to the Benney–Luke equation, Differential Integral Equations, Volume 14 (2001) no. 3, pp. 361-384

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