Vortex Dynamics for the Nonlinear Maxwell–Klein–Gordon Equation

We derive the vortex dynamics for the nonlinear Maxwell–Klein–Gordon equation with the Ginzburg–Landau type potential. In particular, we consider the case when the external electric fields are of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${O( | \log \epsilon |^{\frac{1}{2}})}$$\end{document}. We study the convergence of the space–time Jacobian \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\partial_t \psi \cdot i \nabla \psi}$$\end{document} as an interaction term between the vortices and electric fields. An explicit form of the limiting vector measure is shown.