×

Existence of hylomorphic solitary waves in Klein-Gordon and in Klein-Gordon-Maxwell equations. (English) Zbl 1194.35343

The authors study solitary waves whose existence is related to the ratio energy/charge, and which are usually called hylomorphic. The class includes in particular the \(Q\)-balls which are spherically symmetric solutions of the nonlinear Klein-Gordon equation, as well as solitary waves and vortices appearing, by the same mechanism, in the nonlinear Schrödinger equation and the gauge theories. An abstract theorem is proved which allows to derive existence of hylomorphic solitary waves and vortices both for the nonlinear Klein-Gordon and for the nonlinear Klein-Gordon-Maxwell equations.

MSC:

35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
35C08 Soliton solutions
81V10 Electromagnetic interaction; quantum electrodynamics
81Q37 Quantum dots, waveguides, ratchets, etc.
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Abrikosov A. A., On the magnetic properties of superconductors of the second group, Sov. Phys. JETP 5 (1957), 1174-1182.
[2] Ambrosetti A. - Rabinowitz P., Dual variational methods in the critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. · Zbl 0273.49063
[3] Badiale M. - Benci V. - Rolando S., Solitary waves: physical aspects and mathe- matical results, Rend. Sem. Math. Univ. Pol. Torino 62 (2004), 107-154. · Zbl 1120.37045
[4] Badiale M. - Benci V. - Rolando S., A nonlinear elliptic equation with singular potential and applications to nonlinear field equations, J. Eur. Math. Soc. 9 (2007), 355-381. · Zbl 1149.35033
[5] Badiale M. - Benci V. - Rolando S., Three dimensional vortices in the nonlinear wave equation, BUMI, to appear. · Zbl 1178.35263
[6] Bellazzini J. - Benci V. - Bonanno C. - Micheletti A. M., Solitons for the Non- linear Klein-Gordon-Equation, preprint. · Zbl 1200.35248
[7] Bellazzini J. - Benci V. - Bonanno C. - Sinibaldi E., Hylomorphic solitons, pre- print, arXiv:0810.5079. · Zbl 1194.35096
[8] Bellazzini J. - Bonanno C., Nonlinear Schro\"dinger equations with strongly singular potentials, preprint. · Zbl 1197.35263
[9] Bellazzini J. - Bonanno C. - Siciliano G., Magnetostatic vortices in two dimen- sional Abelian gauge theory, Mediterranean J. Math., to appear. · Zbl 1181.35227
[10] Benci V. - Fortunato D., Solitary waves of the nonlinear Klein-Gordon field equation coupled with the Maxwell equations, Rev. Math. Phys. 14 (2002), 409-420. · Zbl 1037.35075
[11] Benci V. - Fortunato D., Solitary waves in Abelian Gauge Theories, Adv. Nonlinear Stud. 3 (2008), 327-352. · Zbl 1157.58005
[12] Benci V. - Fortunato D., Solitary waves in the Nolinear Wave equation and in Gauge Theories, Journal of fixed point theory and Applications, 1, n. 1 (2007) pp. 61-86. · Zbl 1122.35121
[13] Benci V. - Fortunato D., Existence of 3D-Vortices in Abelian Gauge Theories, Med- iterranean Journal of Mathematics 3 (2006), 409-418. · Zbl 1167.35351
[14] Benci V. - Fortunato D., Three dimensional vortices in Abelian Gauge Theories, Nonlinear Analysis 70 (2009), 4402-4421. · Zbl 1173.81013
[15] Benci V. - Fortunato D., Hylomorphic Vortices in Abelian Gauge Theories, pre- print.
[16] Benci V. - Visciglia N., Solitary waves with non vanishing angular momentum, Adv. Nonlinear Stud. 3 (2003), 151-160. · Zbl 1030.35051
[17] Berestycki H. - Lions P. L., Nonlinear scalar field equations, I-Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313-345. · Zbl 0533.35029
[18] Cassani D., Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell’s equations, Nonlinear Anal. 58 (2004), 733-747. · Zbl 1057.35041
[19] Coleman S. - Glaser V. - Martin A., Action minima among Solutions to a class of Euclidean Scalar Field Equation, Comm. Math. Phys, 58, (1978), 211-221.
[20] Coleman S., Q-Balls, Nucl. Phys. B262 (1985) 263-283; erratum: B269 (1986) 744-745.
[21] Crasovan L. C. - Malomed B. A. - Mihalache D., Spinning solitons in cubic- quintic nonlinear media, Pramana 57 (2001), 1041-1059.
[22] D’Aprile T. - Mugnai D., Solitary waves for nonlinear Klein-Gordon-Maxwell and Schro\"dinger-Maxwell equations, Proc. of Royal Soc. of Edinburgh, section A Mathe- matics, 134 (2004), 893-906. 279 · Zbl 1064.35182
[23] D’Aprile T. - Mugnai D., Non-existence results for the coupled Klein-Gordon- Maxwell equations, Advanced Nonlinear studies, 4 (2004), 307-322. · Zbl 1142.35406
[24] Felsager B., Geometry, particles and fields, Odense University press 1981. · Zbl 0489.58001
[25] Gelfand I. M. - Fomin S. V., Calculus of Variations, Prentice-Hall, Englewood Cli\?s, N.J. 1963. · Zbl 0127.05402
[26] Kim C. - Kim S. - Kim Y., Global nontopological vortices, Phys. Review D, 47, (1985), 5434, 5443.
[27] Kusenko A. - Shaposhnikov M., Supersymmetric Q-balls as dark matter, Phys. Lett. B 418 (1998), 46-54.
[28] Nielsen H. - Olesen P., Vortex-line models for dual strings, Nucl. Phys. B 61, (1973), 45-61.
[29] Rajaraman R., Solitons and instantons, North-Holland, Amsterdam 1989.
[30] Rosen G., Particlelike solutions to nonlinear complex scalar field theories with positive- definite energy densities, J. Math. Phys. 9 (1968), 996-998.
[31] Rubakov V., Classical theory of Gauge fields, Princeton University press, Princeton 2002. · Zbl 1036.81002
[32] Ruf B., A sharp Trudinger-Moser type inequality for unbounded domains in R2, J. Functional Analysis 219 (2005), 340-367. · Zbl 1119.46033
[33] Shatah J., Stable Standing waves of Nonlinear Klein-Gordon Equations, Comm. Math. Phys., 91, (1983), 313-327. · Zbl 0539.35067
[34] Strauss W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149-162. · Zbl 0356.35028
[35] Volkov M. S., Existence of spinning solitons in field theory, eprint arXiv:hep-th/ 0401030 (2004).
[36] Volkov M. S. - Woḧnert E., Spinning Q-balls, Phys. Rev. D 66 (2002) 085003.
[37] Struwe M., Variational Methods, Applications to nonlinear partial di\?erential equa- tions and Hamiltonian systems, Springer, New York, Berlin, 1996.
[38] Vilenkin A. - Shellard E. P. S., Cosmic strings and other topological defects, Cam- bridge monographs on mathematical physics, 1994. · Zbl 0978.83052
[39] Yang Y., Solitons in Field Theory and Nonlinear Analysis, Springer, New York, Berlin, 2000.
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.