Annals of PDE

, 4:4 | Cite as

Leapfrogging Vortex Rings for the Three Dimensional Gross-Pitaevskii Equation

  • Robert L. Jerrard
  • Didier SmetsEmail author


Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date. In this paper, we rigorously derive the corresponding leapfrogging motion for the axially symmetric three-dimensional Gross-Pitaevskii equation.


Vortex rings Leapfrogging Gross-Pitaevskii equation Incompressible Euler equation 



The research of RLJ was partially supported by the National Science and Engineering Council of Canada under operating Grant 261955. The research of DS was partially supported by the Agence Nationale de la Recherche through the Project ANR-14-CE25-0009-01. Both authors wish to warmly thank a referee for his very careful reading of the manuscript and his judicious remarks.


  1. 1.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions, U.S. Government Printing Office, Washington D.C. (1964)Google Scholar
  2. 2.
    Bethuel, F., Brezis, H., Hélein, F.: Ginzburg-Landau Vortices. Birkhäuser, Boston (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bethuel, F., Gravejat, P., Smets, D.: Stability in the energy space for chains of solitons of the one-dimensional Gross-Pitaevskii equation. Ann. Inst. Fourier 64, 19–70 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bethuel, F., Orlandi, G., Smets, D.: Vortex rings for the Gross-Pitaevskii equation. J. Eur. Math. Soc. (JEMS) 6, 17–94 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benedetto, D., Caglioti, E., Marchioro, C.: On the motion of a vortex ring with a sharply concentrated vorticity. Math. Methods Appl. Sci. 23, 147–168 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dyson, F.W.: The potential of an anchor ring. Philos. Trans. R. Soc. Lond. A 184, 43–95 (1893)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Helmholtz, H.: Über Integrale der hydrodynamischen Gleichungen welche den Wirbelbewegungen entsprechen. J. Reine Angew. Math. 55, 25–55 (1858)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Helmholtz, H.: (translated by P.G. Tait) On the integrals of the hydrodynamical equations which express vortex-motion. Phil. Mag. 33, 485–512 (1867)Google Scholar
  9. 9.
    Hicks, W.M.: On the mutual threading of vortex rings. Proc. R. Soc. Lond. A 102, 111–131 (1922)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Jackson, J.D.: Classical Electrodynamics. Wiley, New York (1962)zbMATHGoogle Scholar
  11. 11.
    Jerrard, R.L., Smets, D.: Vortex dynamics for the two dimensional non homogeneous Gross-Pitaevskii equation. Annali Scuola Normale Sup. Pisa Cl. Sci. 14, 729–766 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Jerrard, R.L., Soner, H.M.: The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differ. Equ. 14, 151–191 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jerrard, R.L., Spirn, D.: Refined Jacobian estimates for Ginzburg-Landau functionals. Indiana Univ. Math. J. 56, 135–186 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jerrard, R.L., Spirn, D.: Refined Jacobian estimates and Gross-Pitaevsky vortex dynamics. Arch. Ration. Mech. Anal. 190, 425–475 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kurzke, M., Marzuola, J.L., Spirn, D.: Gross-Pitaevskii vortex motion with critically-scaled inhomogeneities. SIAM J. Math. Anal. 49, 471–500 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Love, A.E.H.: On the motion of paired vortices with a common axis. Proc. Lond. Math. Soc. 25, 185–194 (1893)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Marchioro, C., Negrini, P.: On a dynamical system related to fluid mechanics. NoDEA Nonlinear Differ. Equ. Appl. 6, 473–499 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Martel, Y., Merle, F., Tsai, T.-P.: Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gKdV equations. Commun. Math. Phys. 231, 347–373 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Laboratoire Jacques-Louis LionsUniversité Pierre and Marie CurieParis Cedex 05France

Personalised recommendations