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Small amplitude solitary waves in the Dirac-Maxwell system

  • * Corresponding author: Andrew Comech

    * Corresponding author: Andrew Comech 
The first author was supported by the Russian Foundation for Sciences (project 14-50-00150).
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  • We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell system, proving the existence of solutions in which the Dirac wave function is of the form $φ(x,ω)e^{-iω t}$, with $ω∈(-m,ω_ *)$ for some $ω_ *>-m$. The solutions satisfy $φ(\,·\,,ω)∈ H^ 1(\mathbb{R}^3,\mathbb{C}^4)$, and are small amplitude in the sense that ${\left\| {φ(\,·\,,ω)} \right\|}^2_{L^ 2} = O(\sqrt{m+ω})$ and ${\left\| {φ(\,·\,,ω)} \right\|}_{L^∞} = O(m+ω)$. The method of proof is an implicit function theorem argument based on the identification of the nonrelativistic limit as the ground state of the Choquard equation. This identification is in some ways unexpected on account of the repulsive nature of the electrostatic interaction between electrons, and arises as a manifestation of certain peculiarities (Klein paradox) which result from attempts to interpret the Dirac equation as a single particle quantum mechanical wave equation.

    Mathematics Subject Classification: Primary: 35C08, 35Q41, 37K40, 81Q05; Secondary: 37N20.


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