Hamiltonian formulation of the Klein–Gordon–Maxwell equations

Abstract

Abstract. -The nonlinear Klein-Gordon-Maxwell equations (NKGM) provide models for the interaction between the electromagnetic field and matter. The relevance of NKGM relies on the fact that they are the ''simplest'' gauge theory which is invariant under the group of Poincaré. These equations present the interesting phenomenon of solitons. In this paper, we show that NKGM present an Hamiltonian structure and hence they can be written as equations of the first order in t. This fact is not trivial since the Lagrangian does not depend on q t j (see section 3.3) and a suitable analysis of its structure is necessary. In the last section, we recall a recent result which states the existence of solitons by using the particular structure of the Hamiltonian.

Hylomorphic solitons and charged Q-balls: Existence and stability

“…Actually, in the literature there are few results relative to this problem (we know only [26], [32], [34]) and we do not know which are the assumptions that W should satisfy. Also we refer to [14] for a discussion and some partial results on this issue.…”

“…The seminorm u ♯ defined in (75) satisfies the property (14), namely {u n is a T −vanishing sequence} ⇒ u n ♯ → 0.…”

A minimization method and applications to the study of solitons

“…Then Λ(u n ) = J δ (u n ) − δΦ(u n ) is bounded below and so (13) is proved. By (12) and (13) we have, for some subsequence, that…”

On the existence of stable charged Q-balls

“…Actually, in the literature there are few results relative to this problem (we know only [15]) and we do not know which are the assumptions that W should satisfy. Also, we refer to [10] for a discussion and some partial result on this issue.…”

“…We get the conclusion if we show that V (u n ) → 0. We have by (10), that d(u n , v n ) → 0 and hence d(g n u n , g n v n ) → 0 and so, since g n v n → w, we have g n u n → w ∈ Γ. Therefore, by the continuity of V and since w ∈ Γ, we have V (g n u n ) → V ( w) = 0 and we can conclude that V (u n ) → 0.…”