Existence of hylomorphic solitary waves in Klein–Gordon and in Klein–Gordon–Maxwell equations

Abstract

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. A solitary wave which has a non-vanishing angular momentum is called vortex. We know (at least) three mechanisms which might produce solitary waves and vortices: 1) Complete integrability, (e.g. Kortewg-de Vries equation) 2) Topological constraints, (e.g. Sine-Gordon equation); 3) Ratio energy/charge: (e.g. the nonlinear Klein-Gordon equation). The third type of solitary waves or solitons will be called hylomorphic. This class includes the Q-balls which are spherically symmetric solutions of the nonlinear Klein-Gordon equation (NKG) as well as solitary waves and vortices which occur, by the same mechanism, in the nonlinear Schroedinger equation and in gauge theories. This paper is devoted to an abstract theorem which allows to prove the existence of hylomorphic solitary waves, solitons and vortices in the (NKG) and in the nonlinear Klein-Gordon-Maxwell equations (NKGM)