Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations
type: Item
instance of: scholarly article
title: Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations
zbMATH Open document ID: 0869.35105
DOI: 10.1007/BF01254347
author: Maria J. Esteban
author: Vladimir Georgiev
author: Éric Séré
published in: Calculus of Variations and Partial Differential Equations
publication date: 1997-08-25T00:00:00Z
review text: Interaction of an electron with its own electromagnetic field is described by the Maxwell-Dirac (M-D) equations. They are written as: \[ (i\gamma^\mu\partial_\mu- i\gamma^\mu A_\mu)\psi- m\psi= 0,\quad \partial_\mu A^\mu=0,\quad 4\pi\partial_\mu\partial^\mu A^\nu= J^\nu\quad\text{in }\mathbb{R}\times\mathbb{R}^3, \] where \(\gamma^\mu=\left[\begin{matrix} 0 &\sigma^\mu\\ -\sigma^\mu & 0\end{matrix}\right]\) for \(\mu=1,2,3\), and \(\gamma^0=\left[\begin{matrix} I & 0\\ 0 &-I\end{matrix}\right]\). Here \(\sigma^\mu\) are the Pauli matrices, \(I\) is the identity matrix. An extensive literature exists concerning special cases of the M-D equations. For example, Georgiev proved existence of global solutions for a certain class of initial values. Using approximations to solutions of M-D equations Wakano proved in 1996 the existence of stationary, i.e. soliton-like standing waves. The present authors consider exact equations (not approximations) and apply a variational principle of Esteban and Séré to the energy of the M-D system. The solution is the critical point of the energy functional. A similar argument works also for the Klein-Gordon-Dirac equations, again showing existence of soliton-like solutions. These results are important, since as is well-known soliton waves have many properties of both waves and particles.
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 35Q75
Mathematics Subject Classification ID: 83C15
Mathematics Subject Classification ID: 49S05
zbMATH DE Number: 881149
zbMATH Keywords: Maxwell-Dirac equations
zbMATH Keywords: Klein-Gordon-Dirac equations
zbMATH Keywords: soliton-like solutions
MaRDI profile type: Publication
cites work: Q3825451
cites work: Existence of standing waves for Dirac fields with singular nonlinearities
cites work: Existence of excited states for a nonlinear Dirac field
cites work: Q5286629
cites work: Periodic solutions of Hamiltonian systems with superquadratic potential
cites work: Critical point theorems for indefinite functionals
cites work: Q5576969
cites work: Existence of localized solutions for a classical nonlinear Dirac field
cites work: Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension
cites work: On the Maxwell-Dirac equations with zero magnetic field and their solution in two space dimensions
cites work: Q3963445
cites work: Q4318680
cites work: Stationary states of the nonlinear Dirac equation: A variational approach
cites work: On global solutions of the Maxwell-Dirac equations
cites work: Q3992010
cites work: The cauchy problem for the coupled maxwell and dirac equations
cites work: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems
cites work: Uniform decay estimates and the lorentz invariance of the classical wave equation
cites work: Global existence of small amplitude solutions to nonlinear klein-gordon equations in four space-time dimensions
cites work: Remarks on the global sobolev inequalities in the minkowski space Rn+1
cites work: Existence of stationary states for nonlinear Dirac equations
cites work: Homoclinic orbits in a first order superquadratic Hamiltonian system: Convergence of subharmonic orbits
full work available at URL: https://doi.org/10.1007/bf01254347
OpenAlex ID: W2027267123
description: scientific article; zbMATH DE number 881149