Math. Z. 232, 73–102 (1999) c  Springer-Verlag 1999 Solitons and the electromagnetic field ⋆ V. Benci 1 , D. Fortunato 2 , A. Masiello 2 , L. Pisani 2 1 Dip. di Matematica Applicata “U. Dini”, Universit` a degli Studi di Pisa, Via Bonanno 25/b, I-56126 Pisa, Italy 2 Dip. Interuniversitario di Matematica, Universit` a e Politecnico di Bari, Via Orabona 4, I-70125 Bari, Italia Received November 11, 1997; in final form July 3, 1998 1. Introduction In a recent paper [4], it has been introduced a Lorentz invariant equation in three space dimensions, having soliton like solutions. We recall that, roughly speaking, a soliton is a solution whose energy travels as a localized packet and which preserves this form of localization under small perturbations (see [6], [15], [13], [10]). The equation introduced in [4] is the Euler Lagrange equation of an action functional S 1 (ψ)=  t 1 t 0  R 3 L 1 dxdt where L 1 = L 1 (ψ, ∇ψ,ψ t ) is a suitable Lagrangian density (see Sub- sect. 1.1) for the 4-dimensional vector field ψ =(ψ 1 ,ψ 2 ,ψ 3 ,ψ 4 ) defined in the space-time R 4 . Here ∇ψ (resp. ψ t ) denotes the derivatives of ψ with respect to the space variable x ∈ R 3 (resp. the time variable t). One of the main features of these soliton solutions is that they behave as relativistic particles. In fact, by using the No¨ ether theorem, we can introduce the energy E(ψ) and the mass m(ψ) and it can be proved (see [4]) that the celebrated Einstein relation E(ψ)= m(ψ) c 2 ⋆ Sponsored by M.U.R.S.T. (40% and 60% funds); the authors from Bari were sponsored also by E.E.C., Program Human Capital Mobility (Contract ERCBCHRXCT 940494).