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

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.