Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations

  • Maria J. Esteban
  • Vladimir Georgiev
  • Eric Séré


The Maxwell-Dirac system describes the interaction of an electron with its own electromagnetic field. We prove the existence of soliton-like solutions of Maxwell-Dirac in (3+1)-Minkowski space-time. The solutions obtained are regular, stationary in time, and localized in space. They are found by a variational method, as critical points of an energy functional. This functional is strongly indefinite and presents a lack of compactness. We also find soliton-like solutions for the Klein-Gordon-Dirac system, arising in the Yukawa model.

Mathematics subject classification

49S05 81V10 35Q60 35Q51 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Bachelot: Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon. Ann. Inst. H. Poincaré48 (1988) 387–422Google Scholar
  2. 2.
    M. Balabane, T. Cazenave, L. Vazquez: Existence of standing waves for Dirac fields with singular nonlinearities. Comm. Math. Phys.133 (1990) 53–74Google Scholar
  3. 3.
    M. Balabane, T. Cazenave, A. Douady, F. Merle: Existence of excited states for a nonlinear Dirac field. Comm. Math. Phys.119 (1988) 153–176Google Scholar
  4. 4.
    M. Beals, M. Bezard: Solutions faibles sous des conditions d'énergie pour des équations de champ. PreprintGoogle Scholar
  5. 5.
    V. Benci, A. Capozzi, D. Fortunato: Periodic solutions of Hamiltonian systems with superquadratic potential. Ann. Mat. Pura App. (IV), Vol. CXLIII (1986) 1–46Google Scholar
  6. 6.
    V. Benci, P.H. Rabinowitz: Critical point theorems for indefinite functional. Invent. Math.52 (1979) 336–352Google Scholar
  7. 7.
    J.D. Bjorken-S.D. Drell: Relativistic quantum fields. New York, McGraw-Hill, 1965Google Scholar
  8. 8.
    T. Cazenave: On the existence of stationary states for classical nonlinear Dirac fields. In Hyperbolic systems and Mathematical Physics. Textos e Notas 4, CMAF, Lisbonne (1989)Google Scholar
  9. 9.
    T. Cazenave, L. Vazquez: Existence of localized solutions for a classical nonlinear Dirac field. Comm. Math. Phys.105 (1986) 35–47Google Scholar
  10. 10.
    J. Chadam: Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac system in one space dimension. J. Funct. Anal.13 (1973) 173–184Google Scholar
  11. 11.
    J. Chadam, R. Glassey: On the Maxwell-Dirac equations with zero magnetic field and their solutions in two space dimension. J. Math. Anal. Apcl.53 (1976) 495–507Google Scholar
  12. 12.
    Y. Choquet-Bruhat: Solutions globales des équations de Maxwell-Dirac-Klein-Gordon (masses nulles). C.R. Acad. Sci. Paris, Série I292 (1981) 153–158Google Scholar
  13. 13.
    M.J. Esteban, E. Séré: Existence de solutions stationnaires pour l'équation de Dirac non-linéaire et le système de Dirac-Poisson. C.R. Acad. Sci., Série I,319 (1994) 1213–1218Google Scholar
  14. 14.
    M.J. Esteban, E. Séré: Stationary states of the nonlinear Dirac equation: a variational approach. Com. Math. Phys.171 (1995) 323–350Google Scholar
  15. 15.
    M. Flato, J. Simon, E. Taflin: On the global solutions of the Maxwell-Dirac equations. Comm. Math. Physics113 (1987) 21–49Google Scholar
  16. 16.
    A. Garrett Lisi: A solitary wave solution of the Maxwell-Dirac equations. (Preprint)Google Scholar
  17. 17.
    V. Georgiev: Small amplitude solutions of the Maxwell-Dirac equations. Indiana Univ. Math. J.40(3) (1991) 845–883Google Scholar
  18. 18.
    W.T. Grandy, Jr.: Relativistic Quantum Mechanics of Leptons and Fields. Kluwer Acad. Publisher, Fund. Theories of Physics, Vol.41 Google Scholar
  19. 19.
    L. Gross: The Cauchy problem for the coupled Maxwell and Dirac equations. Comm. Pure Appl. Math.19 (1966) 1–5Google Scholar
  20. 20.
    H. Hofer, Wysocki: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. Ann.288 (1990) 483–503Google Scholar
  21. 21.
    L. Hormander: Remarks on the Klein-Gordon equation. Journées E.D.P., Saint Jean des Monts 1 (1987) 1–9Google Scholar
  22. 22.
    S. Klainerman: Uniform decay estimates and the Lorentz invariance of the classical wave equations. Comm. Pure Appl. Math.38 (1985) 301–332Google Scholar
  23. 23.
    S. Klainerman: Global existence of small amplitude solutions to the nonlinear Klein-Gordon equations in four space dimensions. Comm. Pure Appl. Math.38 (1985) 631–641Google Scholar
  24. 24.
    S. Klainerman: The null condition and global existence to nonlinear wave equation Lect. in Appl. Math.23 (1986) 293–326Google Scholar
  25. 25.
    S. Klainerman: Remarks on the global Sobolev inequalities in the Minkowski space ℝn+1. Comm. Pure Appl. Math.40 (1986) 111–117Google Scholar
  26. 26.
    P.-L. Lions: The concentration-compactness method in the Calculus of Variations. The locally compact case. Part. I: Anal, non-linéaire, Ann. IHP1 (1984) 109–145.Google Scholar
  27. 26a.
    P.-L. Lions: The concentration-compactness method in the Calculus of Variations. The locally compact case. Part. II: Anal, non-linéaire, Ann. IHP1 (1984) 223–283Google Scholar
  28. 27.
    F. Merle: Existence of stationary states for nonlinear Dirac equations. J. Differ. Eq.74(1) (1988) 50–68Google Scholar
  29. 28.
    A.F. Rañada: Classical nonlinear Dirac field models of extended particles. In Quantum theory, groups, fields and particles (editor A.O. Barut). Reidel, Amsterdam, 1982Google Scholar
  30. 29.
    E. Séré: Homoclinic orbits on compact hypersurfaces in ℝ2N, of restricted contact type. Com. Math. Phys. (to appear)Google Scholar
  31. 30.
    M. Soler. Phys. Rev. Dl (1970) 2766–2769Google Scholar
  32. 31.
    K. Tanaka: Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonics. J. Differ. Eq.94, 315–339 (1991)Google Scholar
  33. 32.
    M. Wakano: Intensely localized solutions of the classical Dirac-Maxwell field equations. Progr. Theor. Phys.35(6) (1966) 1117–1141Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Maria J. Esteban
    • 1
  • Vladimir Georgiev
    • 2
  • Eric Séré
    • 3
  1. 1.CEREMADE, URA CNRS 749, Université Paris DauphineParis Cedex 16France
  2. 2.Institute of MathematicsBulgarian Academy of SciencesSofiaBulgaria
  3. 3.Courant Institute of Mathematical SciencesNew YorkUSA

Personalised recommendations