Comptes Rendus
Partial Differential Equations
Non-existence for travelling waves with small energy for the Gross–Pitaevskii equation in dimension N3
Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 375-380.

We prove that the Ginzburg–Landau energy of non-constant travelling waves of the Gross–Pitaevskii equation has a lower positive bound, depending only on the dimension, in any dimension larger or equal to three. In particular, we conclude that there are no non-constant travelling waves with small energy.

On démontre que l'énergie de Ginzburg–Landau des ondes progressives non constantes de l'équation de Gross–Pitaevskii est bornée inférieurement par une constante positive qui ne dépend que de la dimension, pour toute dimension supérieure ou égale à trois. En particulier, on en déduit qu'il n'existe pas d'onde progressive non constante d'énergie petite.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.02.006

André de Laire 1

1 UPMC Université Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
@article{CRMATH_2009__347_7-8_375_0,
     author = {Andr\'e de Laire},
     title = {Non-existence for travelling waves with small energy for the {Gross{\textendash}Pitaevskii} equation in dimension $ N\ensuremath{\geqslant}3$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {375--380},
     publisher = {Elsevier},
     volume = {347},
     number = {7-8},
     year = {2009},
     doi = {10.1016/j.crma.2009.02.006},
     language = {en},
}
TY  - JOUR
AU  - André de Laire
TI  - Non-existence for travelling waves with small energy for the Gross–Pitaevskii equation in dimension $ N⩾3$
JO  - Comptes Rendus. Mathématique
PY  - 2009
SP  - 375
EP  - 380
VL  - 347
IS  - 7-8
PB  - Elsevier
DO  - 10.1016/j.crma.2009.02.006
LA  - en
ID  - CRMATH_2009__347_7-8_375_0
ER  - 
%0 Journal Article
%A André de Laire
%T Non-existence for travelling waves with small energy for the Gross–Pitaevskii equation in dimension $ N⩾3$
%J Comptes Rendus. Mathématique
%D 2009
%P 375-380
%V 347
%N 7-8
%I Elsevier
%R 10.1016/j.crma.2009.02.006
%G en
%F CRMATH_2009__347_7-8_375_0
André de Laire. Non-existence for travelling waves with small energy for the Gross–Pitaevskii equation in dimension $ N⩾3$. Comptes Rendus. Mathématique, Volume 347 (2009) no. 7-8, pp. 375-380. doi : 10.1016/j.crma.2009.02.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.02.006/

[1] F. Béthuel; P. Gravejat; J.-C. Saut Travelling waves for the Gross–Pitaevskii equation II, Comm. Math. Phys., Volume 285 (2009) no. 2, pp. 567-651

[2] A. Farina From Ginzburg–Landau to Gross–Pitaevskii, Monatsh. Math., Volume 139 (2003) no. 4, pp. 265-269

[3] P. Gravejat A non-existence result for supersonic travelling waves in the Gross–Pitaevskii equation, Comm. Math. Phys., Volume 243 (2003) no. 1, pp. 93-103

[4] E. Gross Hydrodynamics of a superfluid condensate, J. Math. Phys., Volume 4 (1963) no. 2, pp. 195-207

[5] C.A. Jones; P.H. Roberts Motions in a Bose condensate IV. Axisymmetric solitary waves, J. Phys. A: Math. Gen., Volume 15 (1982) no. 8, pp. 2599-2619

[6] C.A. Jones; S.J. Putterman; P.H. Roberts Motions in a Bose condensate V. Stability of solitary wave solutions of non-linear Schrödinger equations in two and three dimensions, J. Phys. A: Math. Gen., Volume 19 (1986) no. 15, pp. 2991-3011

[7] P.I. Lizorkin (Lp,Lq)-multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR, Volume 152 (1963), pp. 808-811

[8] L.P. Pitaevskii Vortex lines in an imperfect Bose gas, Sov. Phys. JETP, Volume 13 (1961) no. 2, pp. 451-454

[9] E. Tarquini A lower bound on the energy of travelling waves of fixed speed for the Gross–Pitaevskii equation, Monatsh. Math., Volume 151 (2007) no. 4, pp. 333-339

Cited by Sources:

Comments - Policy