Communications in Mathematical Physics

, Volume 359, Issue 1, pp 265–295 | Cite as

Scattering for the 3D Gross–Pitaevskii Equation

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Abstract

We study the Cauchy problem for the 3D Gross–Pitaevskii equation. The global well-posedness in the natural energy space was proved by Gérard (Ann. Inst. H. Poincaré Anal. Non Linéaire 23(5):765–779, 2006). In this paper we prove scattering for small data in the same space with some additional angular regularity, and in particular in the radial case we obtain small energy scattering.

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References

  1. 1.
    Béthuel F., Saut J.-C.: Travelling waves for the Gross–Pitaevskii equation I. Ann. Inst. H. Poincaré Phys. Théor. 70(2), 147–238 (1999)MathSciNetMATHGoogle Scholar
  2. 2.
    Chiron D.: Travelling waves for the Gross–Pitaevskii equation in dimension larger than two. Nonlinear Anal. 58(1–2), 175–204 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bethuel F., Gravejat P., Saut J.C.: Travelling waves for the Gross–Pitaevskii equation II. Commun. Math. Phys. 285(2), 567–651 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Fetter A.L., Svidzinsky A.A.: Vortices in a trapped dilute Bose–Einstein condensate. J. Phys. Condens. Matter 13, R135–R194 (2001)ADSCrossRefGoogle Scholar
  5. 5.
    Gérard P.: The Cauchy problem for the Gross–Pitaevskii equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(5), 765–779 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Guo Z.: Sharp spherically averaged Strichartz estimates for the Schrödinger equation. Nonlinearity 29, 1668–1686 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Guo Z., Nakanishi K.: Small energy scattering for the Zakharov system with radial symmetry. Int. Math. Res. Not. 9, 2327–2342 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Guo Z., Nakanishi K., Wang S.: Small energy scattering for the Klein–Gordon–Zakharov system with radial symmetry. Math. Res. Let. 21(4), 733–755 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Guo Z., Lee S., Nakanishi K., Wang C.: Generalized Strichartz estimates and scattering for 3D Zakharov system. Commun. Math. Phys. 331(1), 239–259 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Guo Z., Peng L., Wang B.: Decay estimates for a class of wave equations. J. Funct. Anal. 254, 1642–1660 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Gustafson S., Nakanishi K., Tsai T.-P.: Scattering theory for the Gross–Pitaevskii equation. Math. Res. Lett. 13(2), 273–285 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gustafson S., Nakanishi K., Tsai T.-P.: Global dispersive solutions for the Gross–Pitaevskii equation in two and three dimensions. Ann. Henri Poincaré 8, 1303–1331 (2007)ADSMathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gustafson S., Nakanishi K., Tsai T.-P.: Scattering theory for the Gross–Pitaevskii equation in three dimensions. Commun. Contem. Math. 4, 657–707 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Stein E., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)MATHGoogle Scholar
  15. 15.
    Stein E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)MATHGoogle Scholar
  16. 16.
    Watson, G.: A treatise on the theory of Bessel functions, Reprint of the second (1944) edition. Cambridge University Press, Cambridge (1995)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityMelbourneAustralia
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  3. 3.Department of Pure and Applied Mathematics, Graduate School of Information Science and TechnologyOsaka UniversitySuita, OsakaJapan

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