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

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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.

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.