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

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Abstract

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.

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References

  1. A. Bachelot: Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon. Ann. Inst. H. Poincaré48 (1988) 387–422

    Google Scholar 

  2. M. Balabane, T. Cazenave, L. Vazquez: Existence of standing waves for Dirac fields with singular nonlinearities. Comm. Math. Phys.133 (1990) 53–74

    Google Scholar 

  3. M. Balabane, T. Cazenave, A. Douady, F. Merle: Existence of excited states for a nonlinear Dirac field. Comm. Math. Phys.119 (1988) 153–176

    Google Scholar 

  4. M. Beals, M. Bezard: Solutions faibles sous des conditions d'énergie pour des équations de champ. Preprint

  5. V. Benci, A. Capozzi, D. Fortunato: Periodic solutions of Hamiltonian systems with superquadratic potential. Ann. Mat. Pura App. (IV), Vol. CXLIII (1986) 1–46

    Google Scholar 

  6. V. Benci, P.H. Rabinowitz: Critical point theorems for indefinite functional. Invent. Math.52 (1979) 336–352

    Google Scholar 

  7. J.D. Bjorken-S.D. Drell: Relativistic quantum fields. New York, McGraw-Hill, 1965

    Google Scholar 

  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. T. Cazenave, L. Vazquez: Existence of localized solutions for a classical nonlinear Dirac field. Comm. Math. Phys.105 (1986) 35–47

    Google Scholar 

  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) 173–184

    Google Scholar 

  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) 495–507

    Google Scholar 

  12. Y. Choquet-Bruhat: Solutions globales des équations de Maxwell-Dirac-Klein-Gordon (masses nulles). C.R. Acad. Sci. Paris, Série I292 (1981) 153–158

    Google Scholar 

  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) 1213–1218

    Google Scholar 

  14. M.J. Esteban, E. Séré: Stationary states of the nonlinear Dirac equation: a variational approach. Com. Math. Phys.171 (1995) 323–350

    Google Scholar 

  15. M. Flato, J. Simon, E. Taflin: On the global solutions of the Maxwell-Dirac equations. Comm. Math. Physics113 (1987) 21–49

    Google Scholar 

  16. A. Garrett Lisi: A solitary wave solution of the Maxwell-Dirac equations. (Preprint)

  17. V. Georgiev: Small amplitude solutions of the Maxwell-Dirac equations. Indiana Univ. Math. J.40(3) (1991) 845–883

    Google Scholar 

  18. W.T. Grandy, Jr.: Relativistic Quantum Mechanics of Leptons and Fields. Kluwer Acad. Publisher, Fund. Theories of Physics, Vol.41

  19. L. Gross: The Cauchy problem for the coupled Maxwell and Dirac equations. Comm. Pure Appl. Math.19 (1966) 1–5

    Google Scholar 

  20. H. Hofer, Wysocki: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Math. Ann.288 (1990) 483–503

    Google Scholar 

  21. L. Hormander: Remarks on the Klein-Gordon equation. Journées E.D.P., Saint Jean des Monts 1 (1987) 1–9

    Google Scholar 

  22. S. Klainerman: Uniform decay estimates and the Lorentz invariance of the classical wave equations. Comm. Pure Appl. Math.38 (1985) 301–332

    Google Scholar 

  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) 631–641

    Google Scholar 

  24. S. Klainerman: The null condition and global existence to nonlinear wave equation Lect. in Appl. Math.23 (1986) 293–326

    Google Scholar 

  25. S. Klainerman: Remarks on the global Sobolev inequalities in the Minkowski space ℝn+1. Comm. Pure Appl. Math.40 (1986) 111–117

    Google Scholar 

  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. P.-L. Lions: The concentration-compactness method in the Calculus of Variations. The locally compact case. Part. II: Anal, non-linéaire, Ann. IHP1 (1984) 223–283

    Google Scholar 

  28. F. Merle: Existence of stationary states for nonlinear Dirac equations. J. Differ. Eq.74(1) (1988) 50–68

    Google Scholar 

  29. A.F. Rañada: Classical nonlinear Dirac field models of extended particles. In Quantum theory, groups, fields and particles (editor A.O. Barut). Reidel, Amsterdam, 1982

    Google Scholar 

  30. E. Séré: Homoclinic orbits on compact hypersurfaces in ℝ2N, of restricted contact type. Com. Math. Phys. (to appear)

  31. M. Soler. Phys. Rev. Dl (1970) 2766–2769

    Google Scholar 

  32. K. Tanaka: Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonics. J. Differ. Eq.94, 315–339 (1991)

    Google Scholar 

  33. M. Wakano: Intensely localized solutions of the classical Dirac-Maxwell field equations. Progr. Theor. Phys.35(6) (1966) 1117–1141

    Google Scholar 

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Supported by Contract MM-31 with Bulgarian Ministry of Culture, Science and Education and Alexander Von Humboldt Foundation.

Partially supported by NSF grant DMS-9114456.

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Esteban, M.J., Georgiev, V. & Séré, E. Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations. Calc. Var 4, 265–281 (1996). https://doi.org/10.1007/BF01254347

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