Solitary Waves in the Dirac–Klein–Gordon System

• Solitary waves in the DKG system are localized, time-harmonic solutions coupling a Dirac spinor and a scalar field through Yukawa interaction, exhibiting particle-like behavior.
• They are constructed using rigorous variational techniques, ODE reductions, and shooting methods, ensuring precise error control and adherence to conservation laws.
• The inherent SU(1,1) Bogoliubov symmetry underpins a bi-frequency structure and spectral stability, reinforcing the model's relevance in relativistic quantum field theory.

Solitary waves in the Dirac–Klein–Gordon (DKG) system are localized, time-harmonic solutions of a coupled nonlinear system comprising a Dirac spinor field and a real scalar (Klein–Gordon) field. These objects generalize solitary-wave phenomena from the nonlinear Schrödinger and Klein–Gordon frameworks to the relativistic, spinor-valued context with Yukawa-type coupling. The DKG system serves as a canonical relativistic model for describing particle-like entities in quantum field theory and mathematical physics, linking variational structure, spectral analysis, and relativistic invariance.

1. Mathematical Formulation of Dirac–Klein–Gordon Solitary Waves

The DKG system in $n$ spatial dimensions couples a spinor $\psi(t,x)\in\mathbb{C}^N$ and a scalar field $\phi(t,x)\in\mathbb{R}$ via the field equations

$$i\gamma^\mu\partial_\mu\psi - m\psi - g\phi\psi = 0,\qquad \Box\phi + M^2\phi - g\,\bar\psi\psi = 0,$$

where $\gamma^\mu$ are Dirac matrices ($\mu = 0,\ldots,n$), $\bar\psi = \psi^\dagger\gamma^0$, $m$ and $M$ are the Dirac and scalar masses, $g$ is the Yukawa coupling, and $\Box = \partial_t^2 - \Delta_x$ denotes the d'Alembertian.

Solitary waves are sought as time-harmonic, spatially localized solutions: $$\psi(t,x) = e^{-i\omega t}u(x),\qquad \phi(t,x) = \Phi(x),\quad |u(x)|, |\Phi(x)| \to 0\ \text{as}\ |x|\to\infty,$$ with $\omega\in(-m,m)$ the internal frequency. This ansatz reduces the system to a coupled stationary elliptic system for $u$ and $\Phi$: $$\left(-i\alpha\cdot\nabla + \beta m - \omega\right)u = g\,\Phi\,u,\qquad (-\Delta + M^2)\Phi = g\,\bar u u,$$ where the standard representation of $\gamma$-matrices is assumed ($\gamma^0=\beta$, $\gamma^j=\beta\alpha^j$).

A significant generalization involves constructing bi-frequency solitary waves of the form

$$\psi(t,x) = a\,\phi_\xi(x)\,e^{-i\omega t} + b\,\chi_\eta(x)\,e^{+i\omega t},\quad |a|^2 - |b|^2 = 1,$$

where $\phi_\xi$ and $\chi_\eta$ are spinor profiles related by charge-conjugation, and the scalar profile $\Phi$ (identical for both modes) solves the corresponding stationary equation (Boussaid et al., 2017).

2. Existence and Construction Methods

Rigorous proofs of existence for solitary waves in DKG leverage variational techniques and ODE theory:

• In the radial, spin-symmetric case, a reduction via spherical harmonics and a first Wakano Ansatz yields an ODE system for real-valued functions $v(r)$, $u(r)$, along with $\Phi(r)$. Existence of solutions follows by critical-point theory (concentration-compactness, mountain-pass) under Soler-type assumptions on possible additional nonlinearity, and uniqueness within the radial ansatz class is established by ODE shooting arguments (Dudnikova, 2014).
• For the bi-frequency family, the construction invokes charge-conjugation symmetry and an underlying Bogoliubov $\mathrm{SU}(1,1)$ action, producing a $2$-parameter family of exact solutions sharing a single scalar profile (Boussaid et al., 2017).

A precise summary of the procedure for constructing solitary waves is as follows (Comech et al., 31 Dec 2025):

• Fix $\omega\in(0,m)$; solve the nonlinear Dirac equation (ignoring the scalar field) to provide initial spinor profiles.
• For the prescribed spinor profile and a guessed $g$, solve the scalar Poisson (or Yukawa) equation to determine the scalar profile.
• Adjust $g$ iteratively, reconstructing the spinor and scalar fields until normalization and coupling are simultaneously met.
• For $M=0$ (massless scalar), a shooting method in a single parameter suffices, exploiting the invariance under constant shifts in the scalar potential.

Numerical studies confirm convergence of these schemes, with virial identities serving as high-precision error diagnostics; in practice, errors in the virial are below $10^{-4}\%$ for iterative procedures and $10^{-12}$ for shooting (Comech et al., 31 Dec 2025).

3. Symmetry Structures and SU(1,1) Bogoliubov Invariance

The Bogoliubov $\mathrm{SU}(1,1)$ symmetry is central to the theory of multi-frequency solitary waves in DKG. Under the action of

$$G_{\mathrm{Bogoliubov}} = \left\{a + b\,BK\,\mid\,|a|^2 - |b|^2 = 1\right\} \cong \mathrm{SU}(1,1),$$

where $B = -i\gamma^2$ and $K$ is complex conjugation, the set of solitary wave solutions is mapped onto itself, preserving energy, charge, and DKG dynamics. This symmetry is responsible for the existence of a two-complex-parameter family of bi-frequency solitary waves, with the charge-conjugate structure tightly controlled by the algebraic properties of Dirac matrices.

Linearizing about a one-frequency solitary wave, the $\mathrm{SU}(1,1)$ symmetry enforces the presence of eigenvalues $\lambda = \pm 2i\omega$ in the spectrum, which correspond to tangent directions along the manifold of bi-frequency states.

4. Energy–Momentum Relation and Lorentz Covariance

Solitary waves of the DKG system satisfy the exact relativistic energy–momentum relation of classical point particles. Given a rest energy $E_0$ (computed from the stationary profiles), a Lorentz boost with velocity $v$ yields

$$E_v = \gamma E_0,\quad P_v = \gamma v E_0,\quad \gamma = (1 - |v|^2)^{-1/2},$$

and hence

$E_v^2 - |P_v|^2 = E_0^2 = M^2.$

This demonstrates that the solitary wave manifold is Lorentz-invariant, with rest mass $M = E_0 > 0$, and transforms precisely in accordance with special relativity. All conserved quantities—including charge, energy, and momentum—obey the same transformation laws as those for relativistic particles (Dudnikova, 2014).

5. Numerical Methodologies and Parameter Dependence

Numerical construction of solitary waves in DKG employs iterative procedures anchored in variational and ODE reductions, as well as direct shooting for particular scalar mass regimes. The structure of energy $E(\omega)$ and charge $Q(\omega)$ curves depends crucially on the scalar mass $M$ and the spatial dimension:

• For $n=1$ and $n=3$, $E(\omega)$ displays a U-shaped dependence, with the minimum shifting as $M$ varies.
• In $3$D, as $M \to \infty$, the critical frequency at which instability appears in the NLD (Soler) model is located at $\omega_0 \approx 0.936\,m$; as $M \to 0$, the lower edge of the solitary wave curve approaches $m$.
• In the nonrelativistic limit $\omega \to m$, solitary wave profiles converge to solutions of the Schrödinger–Poisson (Choquard) system, with $v(r) \sim \epsilon^2 Q_0(\epsilon r)$ and $u(r)=O(\epsilon^3)$, $\epsilon = \sqrt{m^2 - \omega^2}$ (Comech et al., 31 Dec 2025).

Numerical accuracy is monitored via virial identities relevant to the dimension and coupling regime. In the massless scalar case, shooting provides extremely accurate characterization of the full one-parameter family.

6. Spectral and Orbital Stability

Spectral stability of DKG solitary waves is governed by the structure of the linearized operator about a stationary solution. Key results are:

• The spectrum contains symmetry-induced eigenvalues (at $\lambda = 0$ due to phase and translation invariance; at $\lambda= \pm 2i\omega$ due to Bogoliubov symmetry).
• Absence of nonimaginary eigenvalues away from these points implies spectral stability.
• For bi-frequency waves, spectral stability reduces strictly to the stability of the underlying one-frequency solitary wave: no new instability arises from bi-frequency perturbations (Boussaid et al., 2017).
• In the limit $M\to 0$, coupling to light or massless scalars increases the domain of spectral stability ($\omega_0 \to m$), as confirmed by numerics and nonrelativistic reductions.
• Under suitable conditions (e.g., subcritical charge), spectral stability implies orbital (nonlinear) stability, with the solitary wave manifold persisting under small perturbations (Comech et al., 31 Dec 2025, Boussaid et al., 2017).

The Vakhitov–Kolokolov condition, interpreted as $Q'(\omega) = 0$, delineates the boundary of the stability region; failure of this condition coincides with real eigenvalue collisions and onset of instability in the spectrum.

7. Physical and Mathematical Significance

Solitary waves in DKG provide models for relativistic, particle-like objects exhibiting both Dirac and scalar interactions and serve as mathematical analogues for extended field-theoretic objects with precise energy-momentum relations. The richness of the symmetry structure, particularly the $\mathrm{SU}(1,1)$ Bogoliubov invariance, enables rigorous analysis of multi-frequency excitations and their stability spectra. Lorentz covariance is not only manifest in the equations but is preserved by the solitary wave solutions and their nonlinear dynamics.

The precise mapping between the profile dynamics, conservation laws, and relativistic invariance positions the DKG solitary waves as foundational objects in the mathematical study of relativistic nonlinear wave equations, with implications for quantum field model-building and the theory of nonlinear bound states.

References:

• “Spectral stability of bi-frequency solitary waves in Soler and Dirac–Klein–Gordon models” (Boussaid et al., 2017)
• “Energy-momentum relation for solitary waves of nonlinear Dirac equations” (Dudnikova, 2014)
• “Numerical study of solitary waves in Dirac–Klein–Gordon system” (Comech et al., 31 Dec 2025)