Analytic exact solutions to the nonlinear Dirac equation

TL;DR

This paper provides exact analytical solutions to the nonlinear Dirac equation, highlighting singularities specific to different nonlinearities and their relation to the Compton length.

Contribution

It introduces explicit exact solutions for the nonlinear Dirac equation with different nonlinearities, revealing the nature and size of singularities.

Findings

Solutions exhibit ring singularity for Nambu--Jona-Lasinio nonlinearity.

Solutions show shell singularity for the Soler nonlinearity.

Singular regions are of the order of the Compton length.

Abstract

We present analytic exact solutions to the nonlinear Dirac equation: they display a ring singularity for the Nambu--Jona-Lasinio nonlinearity and a shell singularity for the Soler nonlinearity. For both cases the size of the singular region is of the order of the Compton length.