Communications in Mathematical Physics

, Volume 297, Issue 3, pp 733–758

Bubbling Solutions for Relativistic Abelian Chern-Simons Model on a Torus

Article

Abstract

We prove the existence of bubbling solutions for the following Chern-Simons-Higgs equation:
$$ \Delta u +\frac{1}{\varepsilon^2} e^u(1-e^u) =4\pi \sum_{j=1}^N \delta_{p_j},\quad {\rm in} \, \Omega, $$
where Ω is a torus. We show that if N > 4 and p1pj, j = 2, . . . , N, then for small ε > 0, the above problem has a solution uε, and as ε → 0, uε blows up at the vertex point p1, and satisfies
$$ \frac{1}{\varepsilon^2} e^u(1-e^u)\to 4\pi N \delta_{p_1}. $$
This is the first result for the existence of a solution which blows up at a vertex point.

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References

  1. 1.
    Caffarelli L.A., Yang Y.S.: Vortex condensation in the Chern-Simons Higgs model: an existence theorem. Commun. Math. Phys. 168(2), 321–336 (1995)MATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Chae D., Imanuvilov O.Y.: Non-topological multivortex solutions to the self-dual Maxwell-Chern-Simons-Higgs systems. J. Funct. Anal. 196(1), 87–118 (2002)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Choe K.: Asymptotic behavior of condensate solutions in the Chern-Simons-Higgs theory. J. Math. Phy. 48, 103501 (2007)CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    Choe K., Kim N.: Blow-up solutions of the self-dual Chern-Simons-Higgs vertex equation. Ann. I.H. Poincaré Anal., Non Linéaire 25, 313–338 (2008)MATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Chan H., Fu C.-C., Lin C.-S.: Non-topological multi-vortex solutions to the self-dual Chern-Simons-Higgs equation. Commun. Math. Phys. 231(2), 189–221 (2002)MATHCrossRefMathSciNetADSGoogle Scholar
  6. 6.
    Chern, J.-L., Chen, Z.-Y., Lin, C.-S.: Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles. PreprintGoogle Scholar
  7. 7.
    Dunne, G. V.: Aspects of Chern-Simons theory. In: Aspects topologiques de la physique en basse dimension/Topological aspects of low dimensional systems (Les Houches, 1998), Les Ulis: EDP Sci., 1999, pp. 177–263Google Scholar
  8. 8.
    Dziarmaga J.: Low energy dynamics of [U(1)]N Chern-Simons solitons and two dimensional nonlinear equations. Phys. Rev. D 49, 5469–5479 (1994)CrossRefADSGoogle Scholar
  9. 9.
    Hong J., Kim Y., Pac P.: Multivortex solutions of the abelian Chern-Simons-Higgs theory. Phys. Rev. Lett. 64, 2230–2233 (1990)MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Jackiw R., Weinberg E.: Self-dual Chern-Simons vortex. Phy. Rev. Lett. 64, 2234–2237 (1990)MATHCrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Jaffe A., Taubes, C.: Vortices and Monopoles: Progr. Phys. Vol. 2, Boston, MA: Birkhäuser Boston, 1990Google Scholar
  12. 12.
    Kim C., Lee C., Ko P., Lee B.-H: Schrödinger fields on the plane with [U(1)]N Chern-Simons interactions and generalized self-dual solitons. Phys. Rev. D (3) 48, 1821–1840 (1993)CrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Lin C.-S., Ponce A.C., Yang Y.: A system of elliptic equations arising in Chern-Simons field theory. J. Funct. Anal. 247(2), 289–350 (2007)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lin C.-S., Prajapat J.V.: Vortex condensates for relativistic Abelian Chern-Simons model with two Higgs scalar fields and two gauge fields on a torus. Commun. Math. Phys. 288, 311–347 (2009)MATHCrossRefMathSciNetADSGoogle Scholar
  15. 15.
    Lin, C.-S., Wang, C.-L.: Elliptic functions, Green functions and the mean field equation on tori. Ann. of Math., to appear, available at http://pjm.math.berkeley.edu/editorial/uploads/annals/accepted/080814-Wang-VL.pdf
  16. 16.
    Tarantello G.: Multiple condensate solutions for the Chern-Simons-Higgs theory. J. Math. Phys. 37(8), 3769–3796 (1996)MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Tarantello G.: Self-dual gauge field vortices: an analytical approach. Springer, Berlin-Heidelberg-NewYork (2007)Google Scholar
  18. 18.
    Nolasco M., Tarantello G.: On a sharp Sobolev-type inequality on two dimensional compact manifolds. Arch. Rat. Mech. Anal. 154, 161–195 (1998)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Nolasco M., Tarantello G.: Double vortex condensates in the Chern-Simons-Higgs theory. Calc. Var. and PDE 9, 31–94 (1999)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Nolasco M., Tarantello G.: Vortex condensates for the SU(3) Chern-Simons theory. Commun. Math. Phys. 213(3), 599–639 (2000)MATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Yang, Y.: Solitons in field theory and nonlinear analysis. Springer Monographs in Mathematics. New York: Springer-Verlag, 2001Google Scholar

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Taida Institute of Mathematical SciencesNational Taiwan UniversityTaipeiTaiwan
  2. 2.Department of MathematicsThe University of New EnglandArmidaleAustralia

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