Three-dimensional vortices in Abelian gauge theories. (English) Zbl 1173.81013

The theory of Abelian gauge theories in \(\mathbb{R}^4\) equipped with the Minkowski metric leads to a system of KGM (Klein-Gordon-Maxwell) equations, which provide models for the interaction between the electromagnetic field and matter. The authors show that a suitable choice of the term \(W\) guarantees the existence of finite energy vortices in tree space dimensions. Here, a vortex is a finite energy solution in which the magnetic field looks like the field created by a solenoid.


81T13 Yang-Mills and other gauge theories in quantum field theory
51B20 Minkowski geometries in nonlinear incidence geometry
47J30 Variational methods involving nonlinear operators
35J50 Variational methods for elliptic systems
81V10 Electromagnetic interaction; quantum electrodynamics
35Q60 PDEs in connection with optics and electromagnetic theory
Full Text: DOI arXiv


[1] Abrikosov, A.A., On the magnetic properties of superconductors of the second group, Sov. phys. JETP, 5, 1174-1182, (1957)
[2] Ambrosetti, A; Rabinowitz, P., Dual variational methods in the critical point theory and applications, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[3] 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
[4] V. Benci, D. Fortunato, Hylomorphic vortices in Abelian gauge theories, preprint, 2008 · Zbl 1157.58005
[5] Benci, V.; Fortunato, D., Solitary waves in abelian gauge theories, Adv. nonlinear stud., 3, 327-352, (2008) · Zbl 1157.58005
[6] Benci, V.; Fortunato, D., Solitary waves in the nonlinear wave equation and in gauge theories, J. fixed point theory appl., 1, 61-86, (2007) · Zbl 1122.35121
[7] 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
[8] D’Aprile, T.; Mugnai, D., Solitary waves for nonlinear klein – gordon – maxwell and schrödinger – maxwell equations, Proc. roy. soc. Edinburgh, sec. A math., 134, 893-906, (2004) · Zbl 1064.35182
[9] 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
[10] D’Avenia, P.; Pisani, L., Nonlinear klein – gordon equations coupled with born – infeld equations, Electron. J. differential equations, 26, 1-13, (2002) · Zbl 0993.35083
[11] Esteban, M.; Lions, P.L., A compactness lemma, Nonlinear anal., 7, 381-385, (1983) · Zbl 0512.46035
[12] Felsager, B., Geometry, particles and fields, (1981), Odense University Press · Zbl 0489.58001
[13] Nielsen, H.; Olesen, P., Vortex-line models for dual strings, Nuclear phys. B, 61, 45-61, (1973)
[14] Rajaraman, R., Solitons and instantons, (1989), North-Holland Amsterdam · Zbl 0493.35074
[15] Rubakov, V., Classical theory of gauge fields, (2002), Princeton University Press Princeton · Zbl 1036.81002
[16] Struwe, M., Variational methods, applications to nonlinear partial differential equations and Hamiltonian systems, (1996), Springer New York, Berlin · Zbl 0864.49001
[17] Yang, Y., Solitons in field theory and nonlinear analysis, (2000), Springer New York, Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.