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

TL;DR

This work addresses the existence and stability of solitary waves in the Dirac–Klein–Gordon system with Yukawa coupling in 1D and 3D. It introduces a robust iterative method that starts from nonlinear Dirac solitons, computes the accompanying scalar field, and tunes the coupling to obtain localized DKG solutions, with virial identities serving as rigorous error checks. The authors compute the energy E(ω) and charge Q(ω) across frequencies and masses, and establish scaling and validation frameworks, including a massless scalar field analysis via shooting that corroborates the iterative results. Their findings suggest a broad spectral-stability region for the DKG solitary waves, with the stability boundary shifting with the scalar mass M and approaching the nonrelativistic Choquard limit as M→0, highlighting important implications for nonperturbative field theories and potential dark matter couplings.

Abstract

We use numerics to construct solitary waves in Dirac--Klein--Gordon (in one and three spatial dimensions) and study the dependence of energy and charge on $ω$. For the construction, we use the iterative procedure, starting from solitary waves of nonlinear Dirac equation, computing the corresponding scalar field, and adjusting the coupling constant. We also consider the case of massless scalar field, when the iteration procedure could be compared with the shooting method. We use the virial identities to control the error of simulations. We also discuss possible implications from the obtained results for the spectral stability of solitary waves.

Figures (6)

• Figure 1: (1+1)D. Energy $E$ of solitary waves as a functions of $\omega$ for the Dirac--Klein--Gordon system. Dashed line corresponds to the solitary waves of the cubic nonlinear Dirac equation.
• Figure 2: (1+1)D, $m=M=1$; $\omega=0.5$ (left) and $\omega=0.91$ (right). Solitary waves of cubic NLD (dashed) and iterations of solitary waves of DKG system: first and second (dotted) and twentieth (solid lines) iterations are plotted. The value of the coupling constant $\mathrm{g}$ corresponds to the limiting values from Table \ref{['table-1-1']}.
• Figure 3: 3D. Energy $E$ of solitary waves (left) as a functions of $\omega$ for the Dirac--Klein--Gordon system. Dashed line corresponds to the solitary waves of the cubic nonlinear Dirac equation. Right: magnified region corresponding to $\omega\in(0.98,1)$. (The iterative method for $M>0$; the shooting method for $M=0$.)
• Figure 4: 3D, $m=M=1$; $\omega=0.5$ (left) and $\omega=0.9$ (right). Solitary waves of cubic NLD (dashed) and iterations of solitary waves of DKG system: first and second (dotted) and twentieth iterations are plotted.
• Figure 5: 3D, $m_0=1$, $M=0$; $\omega_0=0.5$ (left) and $\omega_0=0.9$ (right); the iterative method. Solitary waves of cubic NLD (dashed) and iterations of solitary waves of DKG system: first, second, third (dotted), and twentieth iterations are plotted. One can see from these plots that $h_*$ intersects the $r$-axis, hence $\lim_{r\to\infty}h_*(r)<0$. One concludes from \ref{['def-h-ast']} that the corresponding effective values of the mass are given by $\mu=m_0-h_*(+\infty)>m_0$ (cf. \ref{['dkg']}), which are approximately $1.3$ and $1.04$. As a result, these solitary waves correspond to solitary waves of the system \ref{['dkg']} with $m=1$ with frequencies $\omega=\omega_0/\mu$ (cf. \ref{['rg']}) which are approximately $0.5/1.3\approx 0.38$ and $0.9/1.03\approx 0.86$.

Theorems & Definitions (14)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Remark 3.1
• Remark 3.2
• Remark 4.1