Small amplitude solitary waves in the Dirac-Maxwell system

TL;DR

This paper proves the existence of small amplitude solitary wave solutions in the Dirac-Maxwell system, linking their properties to the ground state of the Choquard equation through an implicit function theorem.

Contribution

It establishes the existence of nonlinear bound states in the Dirac-Maxwell system with small amplitude, connecting the solutions to the nonrelativistic limit and the Choquard equation.

Findings

Existence of solitary waves with amplitude proportional to (m - |ω|)

Wave functions are in H^1 with norms scaling as (m - |ω|)

Solutions are connected to the ground state of the Choquard equation

Abstract

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 $\phi(x,\omega)e^{-i\omega t}$, $\omega\in(-m,\omega_*)$, with some $\omega_*>-m$, such that $\phi_\omega\in H^1(\mathbb{R}^3,\mathbb{C}^4)$, $\Vert\phi_\omega\Vert^2_{L^2}=O(m-|\omega|)$, and $\Vert\phi_\omega\Vert_{L^\infty}=O(m-|\omega|)$. The method of proof is an implicit function theorem argument based on an identification of the nonrelativistic limit as the ground state of the Choquard equation.