# American Institute of Mathematical Sciences

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Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models
July  2018, 17(4): 1349-1370. doi: 10.3934/cpaa.2018066

## Small amplitude solitary waves in the Dirac-Maxwell system

 1 Texas A & M University, College Station, TX 77843, USA 2 IITP, Moscow 127051, Russia 3 St. Petersburg State University, St. Petersburg 199178, Russia 4 University of Cambridge, Cambridge CB3 0WA, UK

* Corresponding author: Andrew Comech

Received  December 2016 Revised  August 2017 Published  April 2018

Fund Project: The first author was supported by the Russian Foundation for Sciences (project 14-50-00150).

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.

Citation: Andrew Comech, David Stuart. Small amplitude solitary waves in the Dirac-Maxwell system. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1349-1370. doi: 10.3934/cpaa.2018066
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