Skip to main content
SpringerLink
Log in

Travelling Waves for the Gross-Pitaevskii Equation II

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results (see [3,4,8]) where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Almeida L.: Topological sectors for Ginzburg-Landau energy. Rev. Mat. Iber. 15(3), 487–546 (1999)

    MATH  MathSciNet  Google Scholar 

  2. Berloff, N.: Quantum vortices, travelling coherent structures and superfluid turbulence. Preprint

  3. Béthuel F., Orlandi G., Smets D.: Vortex rings for the Gross-Pitaevskii equation. J. Eur. Math. Soc. 6(1), 17–94 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Béthuel F., Saut J.-C.: Travelling waves for the Gross-Pitaevskii equation I. Ann. Inst. Henri Poincaré, Physique Théorique. 70(2), 147–238 (1999)

    MATH  Google Scholar 

  5. Béthuel, F., Saut, J.-C.: Vortices and sound waves for the Gross-Pitaevskii equation. In: Nonlinear PDE’s in Condensed Matter and Reactive Flows, Volume 569 of NATO Science Series C. Mathematical and Physical Sciences, Dordrecht, Kluwer Academic Publishers, 2002, pp. 339–354

  6. Bona J.L., Li Y.A.: Analyticity of solitary-wave solutions of model equations for long waves. SIAM J. Math. Anal. 27(3), 725–737 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Bona J.L., Li Y.A.: Decay and analyticity of solitary waves. J. Math. Pures Appl. 76(5), 377–430 (1997)

    MATH  MathSciNet  Google Scholar 

  8. Chiron D.: Travelling waves for the Gross-Pitaevskii equation in dimension larger than two. Nonlinear Anal. 58(1-2), 175–204 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. de Bouard, A., Saut, J.-C.: Remarks on the stability of generalized KP solitary waves. In: Mathematical problems in the theory of water waves (Luminy, 1995), Volume 200 of Contemp. Math., Providence, RI: Amer. Math. Soc., 1996, pp. 75–84

  10. de Bouard A., Saut J.-C.: Solitary waves of generalized Kadomtsev-Petviashvili equations. Ann. Inst. Henri Poincaré, Analyse Non Linéaire. 14(2), 211–236 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. de Bouard A., Saut J.-C.: Symmetries and decay of the generalized Kadomtsev-Petviashvili solitary waves. SIAM J. Math. Anal. 28(5), 1064–1085 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ekeland I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  13. Farina A.: From Ginzburg-Landau to Gross-Pitaevskii. Monatsh. Math. 139, 265–269 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gallo, C.: The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity. Preprint

  15. Gérard P.: The Cauchy problem for the Gross-Pitaevskii equation. Ann. Inst. Henri Poincaré, Analyse Non Linéaire. 23(5), 765–779 (2006)

    Article  MATH  ADS  Google Scholar 

  16. Gérard, P.: The Gross-Pitaevskii equation in the energy space. Preprint

  17. Goubet O.: Two remarks on solutions of Gross-Pitaevskii equations on Zhidkov spaces. Monatsh. Math. 151(1), 39–44 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gravejat P.: Limit at infinity for travelling waves in the Gross-Pitaevskii equation. C. R. Math. Acad. Sci. Paris. 336(2), 147–152 (2003)

    MATH  MathSciNet  Google Scholar 

  19. Gravejat P.: A non-existence result for supersonic travelling waves in the Gross-Pitaevskii equation. Commun. Math. Phys. 243(1), 93–103 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Gravejat P.: Decay for travelling waves in the Gross-Pitaevskii equation. Ann. Inst. Henri Poincaré, Analyse Non Linéaire. 21(5), 591–637 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gravejat P.: Limit at infinity and nonexistence results for sonic travelling waves in the Gross-Pitaevskii equation. Differ. Int. Eqs. 17(11–12), 1213–1232 (2004)

    MATH  MathSciNet  Google Scholar 

  22. Gravejat P.: Asymptotics for the travelling waves in the Gross-Pitaevskii equation. Asymptot. Anal. 45(3–4), 227–299 (2005)

    MATH  MathSciNet  Google Scholar 

  23. Gravejat P.: First order asymptotics for the travelling waves in the Gross-Pitaevskii equation. Adv. Differ. Eqs. 11(3), 259–280 (2006)

    MATH  MathSciNet  Google Scholar 

  24. Gravejat P.: Asymptotics of the solitary waves for the generalised Kadomtsev-Petviashvili equations. Disc. Cont. Dynam. Syst. 21(3), 835–882 (2008)

    MATH  MathSciNet  Google Scholar 

  25. Gross E.P.: Hydrodynamics of a superfluid condensate. J. Math. Phys. 4(2), 195–207 (1963)

    Article  ADS  Google Scholar 

  26. Gustafson, S., Nakanishi, K., Tsai, T.-P.: Scattering theory for the Gross-Pitaevskii equation in three dimensions. Preprint, available at http://arxiv.org/abs/0803.3208vi[math.AP], 2008

  27. Gustafson S., Nakanishi K., Tsai T.-P.: Scattering for the Gross-Pitaevskii equation. Math. Res. Lett. 13(2), 273–285 (2006)

    MATH  MathSciNet  Google Scholar 

  28. Gustafson S., Nakanishi K., Tsai T.-P.: Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions. Ann. Henri Poincaré. 8(7), 1303–1331 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  29. Iordanskii S.V., Smirnov A.V.: Three-dimensional solitons in He II. JETP Lett. 27(10), 535–538 (1978)

    ADS  Google Scholar 

  30. Jones C.A., Putterman S.J., Roberts P.H.: Motions in a Bose condensate V. Stability of solitary wave solutions of nonlinear Schrödinger equations in two and three dimensions. J. Phys. A, Math. Gen. 19, 2991–3011 (1986)

    Article  ADS  Google Scholar 

  31. Jones C.A., Roberts P.H.: Motions in a Bose condensate IV. Axisymmetric solitary waves. J. Phys. A, Math. Gen. 5, 2599–2619 (1982)

    Article  ADS  Google Scholar 

  32. Kato K., Pipolo P.-N.: Analyticity of solitary wave solutions to generalized Kadomtsev-Petviashvili equations. Proc. Roy. Soc. Edinb. A. 131(2), 391–424 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  33. Lizorkin P.I.: On multipliers of Fourier integrals in the spaces L p,θ . Proc. Steklov Inst. Math. 89, 269–290 (1967)

    MATH  Google Scholar 

  34. Lopes O.: A constrained minimization problem with integrals on the entire space. Bol. Soc. Bras. Mat. 25(1), 77–92 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  35. Maris M.: Analyticity and decay properties of the solitary waves to the Benney-Luke equation. Differ. Int. Eqs. 14(3), 361–384 (2001)

    MATH  MathSciNet  Google Scholar 

  36. Maris M.: On the existence, regularity and decay of solitary waves to a generalized Benjamin-Ono equation. Nonlinear Anal. 51(6), 1073–1085 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  37. Nakanishi, K.: Scattering theory for the Gross-Pitaevskii equation. Preprint

  38. Pitaevskii L.P.: Vortex lines in an imperfect Bose gas. Sov. Phys. JETP. 13(2), 451–454 (1961)

    MathSciNet  Google Scholar 

  39. Stein, E.M.: Harmonic analysis : Real-variable methods, orthogonality, and oscillatory integrals. Volume 43 of Princeton Mathematical Series. Monographs in Harmonic Analysis. Princeton, NJ: Princeton Univ. Press, 1993 (With the assistance of T.S. Murphy)

  40. Tarquini É.: A lower bound on the energy of travelling waves of fixed speed for the Gross-Pitaevskii equation. Monatsh. Math. 151(4), 333–339 (2007)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte Courrier 187, 75252, Paris Cedex 05, France

    Fabrice Béthuel

  2. Centre de Recherche en Mathématiques de la Décision, Université Paris Dauphine, Place du Maréchal De Lattre de Tassigny, 75775, Paris Cedex 16, France

    Philippe Gravejat

  3. Laboratoire de Mathématiques, Université Paris Sud, Bâtiment 425, 91405, Orsay Cedex, France

    Jean-Claude Saut

Authors
  1. Fabrice Béthuel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Philippe Gravejat
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jean-Claude Saut
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Philippe Gravejat.

Additional information

Communicated by P. Constantin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Béthuel, F., Gravejat, P. & Saut, JC. Travelling Waves for the Gross-Pitaevskii Equation II. Commun. Math. Phys. 285, 567–651 (2009). https://doi.org/10.1007/s00220-008-0614-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0614-2

Keywords

Search

Navigation