{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T16:06:31Z","timestamp":1762445191183,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,4,27]],"date-time":"2021-04-27T00:00:00Z","timestamp":1619481600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We present a simple model of interaction of the Maxwell equations with a matter field defined by the Klein\u2013Gordon equation. A simple linear interaction and a nonlinear perurbation produces solutions to the equations containing hylomorphic solitons, namely stable, solitary waves, whose existence is related to the ratio energy\/charge. These solitons, at low energy, behave as poinwise charged particles in an electromagnetic field. The basic points are the following ones: (i) the matter field is described by the Nonlinear Klein\u2013Gordon equation with a suitable nonlinear term; (ii) the interaction is not described by the equivariant derivative, but by a very simple coupling which preseves the invariance under the Poincar\u00e9 group; (iii) the existence of soliton can be proved using the tecniques of nonlinear analysis and, in particular, the Mountain Pass Theorem; (iv) a suitable choice of the parameters produces solitons with a prescribed electric charge and mass\/energy; (v) thanks to the point (ii), the dynamics of these solitons at low energies is the same of classical charged particles.<\/jats:p>","DOI":"10.3390\/sym13050760","type":"journal-article","created":{"date-parts":[[2021,4,27]],"date-time":"2021-04-27T21:18:20Z","timestamp":1619558300000},"page":"760","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Model for the Maxwell Equations Coupled with Matter Based on Solitons"],"prefix":"10.3390","volume":"13","author":[{"given":"Vieri","family":"Benci","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica, Universit\u00e0 degli Studi di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2021,4,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"996","DOI":"10.1063\/1.1664693","article-title":"Particlelike solutions to nonlinear complex scalar field theories with positive-definite energy densities","volume":"9","author":"Rosen","year":"1968","journal-title":"J. Math. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"119","DOI":"10.4208\/jpde.v31.n2.2","article-title":"Klein-Gordon-Maxwell-Proca type systems in the electro-magneto-static case","volume":"31","author":"Hebey","year":"2018","journal-title":"J. Partial Differ. Equ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1515\/ans-2017-6018","article-title":"Klein-Gordon-Maxwell systems with nonconstant coupling coefficient","volume":"18","author":"Lazzo","year":"2018","journal-title":"Adv. Nonlinear Stud."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1007\/s10883-019-09451-8","article-title":"Standing waves for nonautonomous Klein-Gordon-Maxwell systems","volume":"26","author":"Lazzo","year":"2020","journal-title":"J. Dyn. Control Syst."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00526-014-0739-4","article-title":"Spinning Q-balls in Abelian gauge theories with positive potentials: Existence and non existence","volume":"53","author":"Mugnai","year":"2015","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"ref_6","unstructured":"Azzollini, A. (2020). Finite energy standing waves for the Klein-Gordon-Maxwell system: The limit case, preprint. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1515\/anona-2017-0039","article-title":"Boundary layers to a singularly perturbed Klein-Gordon-Maxwell-Proca system on a compact Riemannian manifold with boundary","volume":"8","author":"Clapp","year":"2019","journal-title":"Adv. Nonlinear Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"041503","DOI":"10.1063\/1.4982038","article-title":"Vortex ground states for Klein-Gordon-Maxwell-Proca type systems","volume":"58","author":"Mederski","year":"2017","journal-title":"J. Math. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/s00032-009-0105-8","article-title":"Hylomorphic solitons","volume":"77","author":"Benci","year":"2009","journal-title":"Milan J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Benci, V., and Fortunato, D. (2014). Variational Methods in Nonlinear Field Equations, Springer. Springer Monographs in Mathematics.","DOI":"10.1007\/978-3-319-06914-2"},{"key":"ref_11","unstructured":"Dodd, R.K., Eilebeck, J.C., Gibbon, J.D., and Morris, H.C. (1982). Solitons and Nonlinear Wave Equations, Academic Press."},{"key":"ref_12","unstructured":"Rajaraman, R. (1989). Solitons and Instantons, North-Holland."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Yang, Y. (2000). Solitons in Field Theory and Nonlinear Analysis, Springer.","DOI":"10.1007\/978-1-4757-6548-9"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1007\/BF01626517","article-title":"Existence of solitary waves in higher dimensions","volume":"55","author":"Strauss","year":"1977","journal-title":"Commun. Math. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/BF00250555","article-title":"Nonlinear scalar field equations, I\u2014Existence of a ground state","volume":"82","author":"Berestycki","year":"1983","journal-title":"Arch. Rational Mech. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/0550-3213(85)90286-X","article-title":"Q-Balls","volume":"B262","author":"Coleman","year":"1985","journal-title":"Nucl. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1142\/S0129055X02001168","article-title":"Solitary waves of the nonlinear Klein-Gordon field equation coupled with the Maxwell equations","volume":"14","author":"Benci","year":"2002","journal-title":"Rev. Math. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"093701","DOI":"10.1063\/1.3629848","article-title":"On the existence of stable charged Q-balls","volume":"52","author":"Benci","year":"2011","journal-title":"J. Math. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1016\/0022-1236(73)90051-7","article-title":"Dual variational methods in critical point theory","volume":"14","author":"Ambrosetti","year":"1973","journal-title":"J. Funct. Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"311","DOI":"10.4310\/DPDE.2009.v6.n4.a2","article-title":"Hylomorphic solitons in the nonlinear Klein-Gordon equation","volume":"6","author":"Bellazzini","year":"2009","journal-title":"Dyn. Partial Differ. Equ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1252","DOI":"10.1063\/1.1704233","article-title":"Comments on Nonlinear Wave Equations as Model for Elementary Particles","volume":"5","author":"Derrick","year":"1964","journal-title":"J. Math. Phys."},{"key":"ref_22","first-page":"1408","article-title":"Eigenfunctions of the equation \u0394u+\u03bbf(u)=0","volume":"165","author":"Pohozaev","year":"1965","journal-title":"Soviet Math. Dokl."},{"key":"ref_23","unstructured":"De Broglie, L. (1960). Un Tentative d\u2019Interpr\u00e9tation Causule et non Lin\u00e9aire de la Mecanique Ondulatoire: La Theorie de la Double Soution, Elsevier. English Translation: Non-Linear Wave Mechanics, a Causal Interpretation."},{"key":"ref_24","unstructured":"Benci, V., D\u2019Avenia, P., and Masiello, A. A a model for the electron in classical mechanics, in preparation."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/760\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:53:23Z","timestamp":1760162003000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/5\/760"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,27]]},"references-count":24,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["sym13050760"],"URL":"https:\/\/doi.org\/10.3390\/sym13050760","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,4,27]]}}}