Donato Fortunato

Solitons and the electromagnetic field

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

The General Principles

This chapter concerns the very general principles which are at the base of the existence of hylomorphic solitons such as the Variational Principle, the Invariance Principle, Noether’s theorem, the Hamilton-Jacobi theory. A recent historical and epistemological analysis of these principles can be found in [44].

The Nonlinear Klein-Gordon-Maxwell Equations

In this chapter, the Maxwell equations are deduced from the general principles of Chap. 1. If these equations are coupled with the nonlinear Klein-Gordon equation, we get the simplest gauge theory with “matter”. After having analyzed the general features of these equations, we apply the abstract theory of Chap. 2 and we prove the existence of hylomorphic solitons.

Solitary Waves and Solitons: Abstract Theory

In this chapter we construct a functional abstract framework which allows to define solitary waves, solitons and hylomorphic solitons (Sects. 2.1.1 and 2.1.3). Then, we will give some abstract existence theorems (Sect. 2.2). These theorems are based on two general minimization principles related to the concentration compactness techniques (see Sects. 2.2.3 and 2.2.4). These results are able to cover all the situations considered in the rest of this book and in most of the present literature on this subject. In the last two Sects. 2.3.1 and 2.3.2, we will discuss the meaning, the structure and possible interpretations of hylomorphic solitons.

The Nonlinear Schrödinger-Maxwell Equations

In the preceding chapter we have seen that the use of the covariant derivative provides a very elegant way to combine the action related to the nonlinear Klein- Gordon equation with the action related to the Maxwell equations. It is possible to use this procedure to couple Schrodinger and Maxwell equations. This situation describes the interaction between a charged “matter field” with the electromagnetic field when the relativistic effects are negligible.

A “Birkhoff-Lewis” Type Result for Nonautonomous Hamiltonian Systems

This paper contains results concerning the existence of long periodic solutions of nonautonomous Hamiltonian systems near the origin.