Solitary Waves and Electromagnetic Fields

Fortunato, Donato

Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 767-789

Résumé

Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet; by soliton, we mean a solitary wave which exhibits some form of stability. In this respect solitary waves and solitons have a particle-like behavior and they occur in many questions of mathematical physics, such as superconductivity, phase transition, classical and quantum field theory, non linear optics, (see e.g. [37], [50], [56]). We are not interested in the study of a particular model. Here we shall be concerned with the existence of solitary waves for a class of variational field equations which exhibit suitable symmetry properties, namely equations which are invariant for the Poincarè group and the gauge group. In particular we shall describe two results obtained jointly with V. Benci in [17], [18]. These results state the existence of three dimensional vortices for Abelian gauge theories describing the interaction of electrically charged solitary waves with the electromagnetic field.