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

Abstract

This paper is devoted to the study of solitary waves whose existence is related to the ratio energy/charge. These solitary waves are 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. It is proved an abstract theorem which allows to show the existence of hylomorphic solitary waves and vortices in the (NKG) and in the nonlinear Klein-Gordon-Maxwell equations (NKGM).

Standing-wave solutions to a system of non-linear Klein–Gordon equations with a small energy/charge ratio

“…By 1 (ℝ ; ℝ ) we denote the space of radially symmetric functions. We assume that is continuously di erentiable and The idea underlying the variational method used in [6] that we wish to extend to ≥ 2 is the following: a Palais-Smale sequence ( , ) such that lim < is pre-compact (this is shown in Proposition 2.2). A minimizing sequence as above arises from the quantity…”

“…The paper is focused on the extension of the variational method used in [6] to obtain a solution to (1.1) when = 1. We obtain solutions to (1.1) as minima of the energy functional where > 0 for any 1 ≤ ≤ .…”

“…However, each of the equations in (3.7) can be addressed separately and the existence of a solution follows from the arguments used in the scalar case in[6], or by applying Theorem 1.1 when = 1. to you by | New York University Bobst Library Technical Services Authenticated Download Date | 6/16/15 8:00 AM We de ne the non-linearity * , for every ≥ 1 and ≥ 3; 2 satis es the assumptions (A 0 ) and (A 2 ). It is not hard to check that there are positive constants and such that (A 1 ), (A 3 ) and (A 4 ) hold too.…”

Interaction of Solitons with the Electromagnetic Field in Classical Nonlinear Field Models

“…Substantial work has been done regarding existence of solitons in the NKGM system [43][44][45], in some cases even with nonzero angular momentum [16]. Stability of solitons in the NKGM system is a difficult subject, but some progress has been made [17,21,44], especially in the case of small coupling of the NKG equation to the electromagnetic field.…”

Hamiltonian formulation of the Klein–Gordon–Maxwell equations

“…[2], [3], [4], [7], [9], [10]), the notion of hylomorphic soliton has been introduced and analyzed. The existence and the properties of hylomorphic solitons are guaranteed by the interplay between energy E and another integral of motion which is called hylenic charge and it will be denoted by C. More precisely, a soliton u 0 a X is hylomorphic if…”