Magneto-Static Vortices in Two Dimensional Abelian Gauge Theories

  • Jacopo BellazziniEmail author
  • Claudio Bonanno
  • Gaetano Siciliano


We study the existence of vortices of the Klein-Gordon-Maxwell equations in the two dimensional case. In particular we find sufficient conditions for the existence of vortices in the magneto-static case, i.e. when the electric potential \({\phi = 0}\). This result, due to the lack of suitable embedding theorems for the vector potential A is achieved with the help of a penalization method.

Mathematics Subject Classification (2000)

Primary 35J50 Secondary 35Q60 


Klein-Gordon-Maxwell equations two dimensional vortex 


  1. 1.
    Abrikosov A.A.: On the magnetic properties of superconductors of the second group. Soviet Physics. JETP 5, 1174–1182 (1957)Google Scholar
  2. 2.
    Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Azzollini A., Benci V., D’Aprile T., Fortunato D.: Existence of static solutions of the semilinear Maxwell equations. Ric. Mat. 55, 283–297 (2006)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Benci V., d’Avenia P., Fortunato D., Pisani L.: Solitons in several space dimensions: Derrick’s problem and infinitely many solutions. Arch. Ration. Mech. Anal. 154, 297–324 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Benci V., Fortunato D.: Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations. Rev. Math. Phys. 14, 409–420 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Benci V., Fortunato D.: Towards a unified field theory for Classical Electrodynamics. Arch. Rat. Mech. Anal. 173, 379–414 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Benci V., Fortunato D.: Solitary waves in the nolinear wave equation and in gauge theories. J. Fixed Point Theory Appl. 1, 61–86 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    V.Benci, D.Fortunato, Three dimensional vortices in Abelian Gauge Theories. ArXiv:0711.3351v1 [math.AP].Google Scholar
  9. 9.
    D’Aprile T., Mugnai D.: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schroedinger-Maxwell equations. Proc. Roy. Soc. Edinburgh Sect. A 134, 893–906 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Felsager B.: Geometry, Particles and Fields. Odense University Press, Odense (1981)zbMATHGoogle Scholar
  11. 11.
    Fetter A.L., Walecka J.D.: Quantum Theory of Many-Particle Systems. Dover, New York (2003)Google Scholar
  12. 12.
    Nielsen H., Olesen P.: Vortex-line models for dual strings. Nuclear Phys. B 61, 45–61 (1973)CrossRefGoogle Scholar
  13. 13.
    Palais R.S.: The principle of symmetric criticality. Comm. Math. Phys. 79, 19–30 (1979)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Rajaraman R.: Solitons and Instantons. North-Holland, Amsterdam (1989)Google Scholar
  15. 15.
    V. Rubakov, Classical Theory of Gauge Fields. Princeton University Press, 2002.Google Scholar
  16. 16.
    Yang Y.: Solitons in Field Theory and Nonlinear Analysis. Springer, New York, Berlin (2000)Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Jacopo Bellazzini
    • 1
    Email author
  • Claudio Bonanno
    • 1
  • Gaetano Siciliano
    • 2
  1. 1.Dipartimento di Matematica ApplicataUniversità di PisaPisaItaly
  2. 2.Dipartimento di MatematicaUniversità di BariBariItaly

Personalised recommendations