Spinning Q-balls for the Klein-Gordon-Maxwell equations. (English) Zbl 1194.78009

The paper deals with the nonlinear Klein-Gordon-Maxwell equations \(\frac{\partial^2\psi}{\partial t^2} - \nabla\psi +W'(\psi)=0\), where the nonlinear term \(W\) is assumed to be positive and \(W(0)=0\). The vortices in the nonlinear Klein-Gordon equation are often considered in the Physics literature with the name of spinning \(Q\)-balls. The main result of the paper is the proof of existence of spinning \(Q\)-balls for the nonlinear Klein-Gordon-Maxwell equations, that is, three-dimensional vortex-solutions which are finite energy, stationary solutions such that the matter field has a nontrivial angular momentum and the magnetic field looks like the field created by a finite solenoid.


78A25 Electromagnetic theory (general)
47J30 Variational methods involving nonlinear operators
35J50 Variational methods for elliptic systems
81V10 Electromagnetic interaction; quantum electrodynamics
Full Text: DOI


[1] Abrikosov A.A.: On the magnetic properties of superconductors of the second group. Sov. Phys. JETP 5, 1174–1182 (1957)
[2] Anagnostopoulos, K.N., Axenides, M., Floratos, E.G., Tetradis, N.: Large gauged Q-Balls. Phys. Rev. D64 (2001)
[3] Badiale, M., Benci, V., Rolando, S.: Three dimensional vortices in the nonlinear wave equation. Boll. Unione Mat. Ital., Ser. IX, in press · Zbl 1178.35263
[4] Bellazzini, J., Benci, V., Bonanno, C., Sinibaldi, E.: Hylomorphic solitons in the nonlinear Klein-Gordon equation. http://arxiv.org/abs/0810.5079v1[math.Ap] , 2008 · Zbl 1307.35169
[5] Bellazzini, J., Bonanno, C.: Nonlinear Schrödinger equations with strongly singular potentials. http://arxiv.org/abs/0903.3301v1[math-ph] , 2009 · Zbl 1197.35263
[6] Benci V.: Hylomorphic solitons. Milan J. Math. 77, 271–332 (2009) · Zbl 1205.35040
[7] Benci V., Fortunato D.: Solitary waves of the nonlinear Klein-Gordon field equation coupled with the Maxwell equations. Rev. Math. Phys. 14, 409–420 (2002) · Zbl 1037.35075
[8] Benci V., Fortunato D.: Solitary waves in Abelian gauge theories. Adv. Nonlinear Stud. 3, 327–352 (2008) · Zbl 1157.58005
[9] Benci V., Fortunato D.: Solitary waves in the nolinear wave equation and in Gauge theories. J. Fixed Point Th and Appl. 1(1), 61–86 (2007) · Zbl 1122.35121
[10] Benci V., Fortunato D.: Existence of 3D-vortices in abelian Gauge theories. Med. J. Math. 3, 409–418 (2006) · Zbl 1167.35351
[11] Benci V., Fortunato D.: Three dimensional vortices in abelian Gauge theories. Nonlinear Analysis 70, 4402–4421 (2009) · Zbl 1173.81013
[12] Benci V., Fortunato D.: Existence of hylomorphic solitary waves in Klein-Gordon and in Klein-Gordon-Maxwell equations. Rend. Accad. Naz. Lincei, Mat. Appl. 20, 243–279 (2009) · Zbl 1194.35343
[13] Benci V., Visciglia N.: Solitary waves with non vanishing angular momentum. Adv. Nonlinear Stud. 3, 151–160 (2003) · Zbl 1030.35051
[14] Berestycki H., Lions P.L.: Nonlinear scalar field equations, I - Existence of a ground state. Arch. Rat. Mech. Anal. 82, 313–345 (1983) · Zbl 0533.35029
[15] Cassani D.: Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell’s equations. Nonlinear Anal. 58, 733–747 (2004) · Zbl 1057.35041
[16] Campanelli L., Ruggieri M.: Spinning supersymmetric Q balls. Phys. Rev. D 80, 036006 (2009)
[17] Coleman S., Glaser V., Martin A.: Action minima among solutions to a class of Euclidean Scalar field equation. Commun. Math. Phys. 58, 211–221 (1978)
[18] Coleman, S.: Q-Balls. Nucl. Phys. B262, 263–283 (1985); erratum: B269, 744–745 (1986)
[19] D’Aprile T., Mugnai D.: Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger -Maxwell equations. Proc. of Royal Soc. of Edinburgh, Sect. A Math. 134, 893–906 (2004) · Zbl 1064.35182
[20] D’Aprile T., Mugnai D.: Non-existence results for the coupled Klein-Gordon- Maxwell equations. Adv. Nonlinear Stud. 4, 307–322 (2004) · Zbl 1142.35406
[21] D’Avenia P., Pisani L.: Nonlinear Klein-Gordon equations coupled with Born-Infeld equations. Electronics J. Diff. Eqs. 26, 1–13 (2002) · Zbl 0993.35083
[22] Enqvist K., McDonald J.: Q-Balls and Baryogenesis in the MSSM. Phys. Lett. B 425, 309–321 (1998)
[23] Esteban M., Lions P.L.: A compactness lemma. Nonlinear Anal. 7, 381–385 (1983) · Zbl 0512.46035
[24] Felsager, B.: Geometry, Particles and Fields. Odense: Odense University Press, 1981 · Zbl 0489.58001
[25] Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Englewood Cliffs, NJ: Prentice-Hall, 1963 · Zbl 0127.05402
[26] Kim C., Kim S., Kim Y.: Global nontopological vortices. Phys. Rev. D 47, 5434–5443 (1985)
[27] Lee K., Stein-Schabes J.A., Watkins R., Widrow L.M.: Gauged Q balls. Phys. Rev. D 39, 1665–1673 (1989)
[28] Kusenko A., Shaposhnikov M.: Supersymmetric Q-balls as dark matter. Phys. Lett. B 418, 46–54 (1998)
[29] Landau, L., Lifchitz, E.: Théorie du Champ. Moscow: Editions Mir, 1966
[30] Nielsen H., Olesen P.: Vortex-line models for dual strings. Nucl. Phys. B 61, 45–61 (1973)
[31] Rajaraman, R.: Solitons and Instantons. Amsterdam: North-Holland, 1989 · Zbl 0493.35074
[32] Rosen G.: Particlelike solutions to nonlinear complex scalar field theories with positive-definite energy densities. J. Math. Phys. 9, 996–998 (1968)
[33] Rubakov, V.: Classical Theory of Gauge Fields. Princeton, NJ: Princeton University Press, 2002 · Zbl 1036.81002
[34] Strauss W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977) · Zbl 0356.35028
[35] Struwe, M.: Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. NewYork-Berlin: Springer, 1996 · Zbl 0864.49001
[36] Vilenkin, A., Shellard, E.P.S.: Cosmic Strings and other Topological Defects. Cambridge: Cambrige University Press, 1994 · Zbl 0978.83052
[37] Volkov, M.S., Wöhnert, E.: Spinning Q-balls. Phys. Rev. D 66, 085003 (2002)
[38] Yang, Y.: Solitons in Field Theory and Nonlinear Analysis. NewYork-Berlin: Springer, 2000
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