Solitary waves in the nolinear wave equation and in gauge theories

Roughly speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. This paper is an introduction to the study of solitary waves relative to the nonlinear wave equation and to the Abelian gauge theories. Abelian gauge theories consist of a class of field equations obtained by coupling in a suitable way the nonlinear wave equation with the Maxwell equations. They provide a model for the interaction of matter with the electromagnetic field. One of the motivations of this study lies in the fact that the nonlinear wave equation and the Abelian gauge theories are the simplest equations which satisfy the basic principles of modern physics. Dedicated to the memory of Jean Leray