Solitons and the electromagnetic field
This paper deals with a model of solitons, in three space dimensions, recently introduced by Benci, Fortunato and Pisani. These solitons are traveling waves with a topological constraint, which is called charge. Since these soliton-like solutions behave as relativistic particles, we introduce a suitable model of interaction between waves and an electromagnetic field. The Lagrangian density of the system is the sum of three terms: the first is that of the free soliton, the second is the classical Lagrangian density of an electromagnetic field, the third is due to the interaction. In particular the last term, which is gauge invariant, is chosen so that the electric charge coincides with the topological charge; moreover, the current density vanishes whenever the charge carried out by the soliton does not move. We prove the existence of a non-trivial static solution (with magnetic field equal to zero). This solution is characterized as a saddle point of the energy functional, which is strongly unbounded from above, as well as from below. In order to deal with a functional which is bounded from below, a suitable variational principle is used. After this reduction, since the domain is unbounded, we use a concentration-compactness argument, which allows to find non-trivial minima.
scientific article; zbMATH DE number 1351166