WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity

Thomas Alazard; Rémi Carles

Annales de l'I.H.P. Analyse non linéaire (2009)

  • Volume: 26, Issue: 3, page 959-977
  • ISSN: 0294-1449

How to cite


Alazard, Thomas, and Carles, Rémi. "WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity." Annales de l'I.H.P. Analyse non linéaire 26.3 (2009): 959-977. <http://eudml.org/doc/78876>.

author = {Alazard, Thomas, Carles, Rémi},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {semi-classical analysis; Gross-Pitaevskii equation; Zhidkov spaces; cubic-quintic nonlinearity},
language = {eng},
number = {3},
pages = {959-977},
publisher = {Elsevier},
title = {WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity},
url = {http://eudml.org/doc/78876},
volume = {26},
year = {2009},

AU - Alazard, Thomas
AU - Carles, Rémi
TI - WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2009
PB - Elsevier
VL - 26
IS - 3
SP - 959
EP - 977
LA - eng
KW - semi-classical analysis; Gross-Pitaevskii equation; Zhidkov spaces; cubic-quintic nonlinearity
UR - http://eudml.org/doc/78876
ER -


  1. [1] Abdullaev F.Kh., Gammal A., Tomio L., Frederico T., Stability of trapped Bose–Einstein condensates, Phys. Rev. A63 (4) (2001) 043604. 
  2. [2] Alazard T., Carles R., Semi-classical limit of Schrödinger–Poisson equations in space dimension n 3 , J. Differential Equations233 (1) (2007) 241-275. Zbl1107.35018MR2290279
  3. [3] T. Alazard, R. Carles, Super-critical geometric optics for nonlinear Schrödinger equations, Preprint, arXiv:0704.2488. Zbl1179.35302
  4. [4] R. Anton, Cubic nonlinear Schrödinger equation on three dimensional balls with radial data, Comm. Partial Differential Equations, in press. Zbl1157.35101MR2475322
  5. [5] Anton R., Global existence for defocusing cubic NLS and Gross–Pitaevskii equations in exterior domains, J. Math. Pures Appl.89 (4) (2008) 335-354. Zbl1148.35081MR2401142
  6. [6] Bethuel F., Saut J.-C., Travelling waves for the Gross-Pitaevskii equation. I, Ann. Inst. H. Poincaré Phys. Théor.70 (2) (1999) 147-238. Zbl0933.35177MR1669387
  7. [7] Brenier Y., Convergence of the Vlasov–Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations25 (3–4) (2000) 737-754. Zbl0970.35110MR1748352
  8. [8] Carles R., Geometric optics and instability for semi-classical Schrödinger equations, Arch. Ration. Mech. Anal.183 (3) (2007) 525-553. Zbl1134.35098MR2278414
  9. [9] Carles R., WKB analysis for nonlinear Schrödinger equations with potential, Commun. Math. Phys.269 (1) (2007) 195-221. Zbl1123.35062MR2274468
  10. [10] Colin T., Soyeur A., Some singular limits for evolutionary Ginzburg–Landau equations, Asymptotic Anal.13 (4) (1996) 361-372. Zbl0885.35127MR1425274
  11. [11] Dalfovo F., Giorgini S., Pitaevskii L.P., Stringari S., Theory of Bose–Einstein condensation in trapped gases, Rev. Mod. Phys.71 (3) (1999) 463-512. 
  12. [12] Gallo C., Schrödinger group on Zhidkov spaces, Adv. Differential Equations9 (5–6) (2004) 509-538. Zbl1103.35093MR2099970
  13. [13] C. Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, Preprint. Zbl1156.35086MR2424376
  14. [14] Gammal A., Frederico T., Tomio L., Chomaz Ph., Atomic Bose–Einstein condensation with three-body interactions and collective excitations, J. Phys. B33 (2000) 4053-4067. 
  15. [15] Gasser I., Lin C.-K., Markowich P.A., A review of dispersive limits of (non)linear Schrödinger-type equations, Taiwanese J. Math.4 (4) (2000) 501-529. Zbl0972.35112MR1799752
  16. [16] P. Gérard, Remarques sur l'analyse semi-classique de l'équation de Schrödinger non linéaire, in: Séminaire sur les Équations aux Dérivées Partielles, 1992–1993, École polytech., Palaiseau, 1993, Exp. No. XIII. Zbl0874.35111
  17. [17] Gérard P., The Cauchy problem for the Gross–Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire23 (5) (2006) 765-779. Zbl1122.35133
  18. [18] Grenier E., Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc.126 (2) (1998) 523-530. Zbl0910.35115
  19. [19] Josserand C., Pomeau Y., Nonlinear aspects of the theory of Bose–Einstein condensates, Nonlinearity14 (5) (2001) R25-R62. Zbl1037.82031
  20. [20] Lannes D., Sharp estimates for pseudo-differential operators with symbols of limited smoothness and commutators, J. Funct. Anal.232 (2) (2006) 495-539. Zbl1099.35191
  21. [21] Lin F., Zhang P., Semiclassical limit of the Gross–Pitaevskii equation in an exterior domain, Arch. Rational Mech. Anal.179 (1) (2006) 79-107. Zbl1079.76016MR2208290
  22. [22] Majda A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. Zbl0537.76001MR748308
  23. [23] Métivier G., Remarks on the well-posedness of the nonlinear Cauchy problem, in: Geometric Analysis of PDE and Several Complex Variables, Contemp. Math., vol. 368, Amer. Math. Soc., Providence, RI, 2005, pp. 337-356. Zbl1071.35074MR2127041
  24. [24] Michinel H., Campo-Táboas J., García-Fernández R., Salgueiro J.R., Quiroga-Teixeiro M.L., Liquid light condensates, Phys. Rev. E65 (2002) 066604. 
  25. [25] Pitaevskii L., Stringari S., Bose–Einstein Condensation, International Series of Monographs on Physics, vol. 116, The Clarendon Press, Oxford University Press, Oxford, 2003. Zbl1110.82002MR2012737
  26. [26] Sideris T., Formation of singularities in three-dimensional compressible fluids, Commun. Math. Phys.101 (1985) 475-485. Zbl0606.76088MR815196
  27. [27] Taylor M., Partial Differential Equations. III, Nonlinear Equations, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Zbl0869.35004MR1477408
  28. [28] Thomann L., Instabilities for supercritical Schrödinger equations in analytic manifolds, J. Differential Equations245 (1) (2008) 249-280. Zbl1157.35107MR2422717
  29. [29] P.E. Zhidkov, The Cauchy problem for a nonlinear Schrödinger equation, JINR Commun., P5-87-373, Dubna, 1987 (in Russian). 
  30. [30] Zhidkov P.E., Korteweg–de Vries and Nonlinear Schrödinger Equations: Qualitative Theory, Lecture Notes in Mathematics, vol. 1756, Springer-Verlag, Berlin, 2001. Zbl0987.35001MR1831831

Citations in EuDML Documents

  1. Fabrice Béthuel, Raphaël Danchin, Philippe Gravejat, Jean-Claude Saut, Didier Smets, Les équations d’Euler, des ondes et de Korteweg-de Vries comme limites asymptotiques de l’équation de Gross-Pitaevskii
  2. Rémi Carles, Bijan Mohammadi, Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime
  3. Rémi Carles, Bijan Mohammadi, Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime

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