Numerical study of solitary waves in Dirac--Klein--Gordon system

TL;DR

This paper numerically constructs solitary waves in the Dirac--Klein--Gordon system across different dimensions, analyzing their energy and charge dependence, and explores stability implications using virial identities.

Contribution

It introduces two numerical procedures for constructing solitary waves in the Dirac--Klein--Gordon system and examines their properties in various scenarios.

Findings

Successful numerical construction of solitary waves in 1D and 3D.

Analysis of energy and charge dependence on frequency.

Application of virial identities to assess simulation accuracy.

Abstract

We use numerics to construct solitary waves $\phi_\omega(x) e^{-\mathrm{i}\omega t}$ in Dirac--Klein--Gordon (in one and three spatial dimensions) and study the dependence of energy and charge of $\omega$. To construct solitary waves, we use two different procedures: the iterative method and the nested shooting method. We also consider the case of massless scalar field where we show that the standard shooting method becomes available. We use the virial identities to control the error of simulations. We discuss possible implications for the stability of solitary waves.