ABS Model: Nonlinear Dirac Equations

• The ABS model is a class of nonlinear Dirac equations in 1+1 dimensions with tunable vector–vector and scalar–scalar self-interactions, enabling analytic solitary wave solutions.
• It generalizes earlier models such as the Gross–Neveu and massive Thirring models by using parameters κ and p to control nonlinearity and soliton profile transitions.
• The model’s exact solutions, stability criteria via the VK method, and nonrelativistic reductions provide a robust framework for analytical and numerical investigations.

The Alexeeva-Barashenkov-Saxena (ABS) model is a class of nonlinear Dirac equations in $1+1$ dimensions characterized by a tunable combination of vector–vector (V–V) and scalar–scalar (S–S) nonlinear self-interactions. It provides a unified analytic framework for constructing and analyzing solitary wave (soliton) solutions with arbitrary nonlinearity and coupling weights, encompassing as special cases the earlier Gross–Neveu and generalized massive Thirring models. The model is parameterized by a nonlinearity index $\kappa>0$ and a weight parameter $p>1$, enabling exhaustive tuning of nonlinear effects. The ABS class possesses exact solitary-wave solutions, a rich variety of stability transitions, and well-defined reduction procedures to effective non-relativistic models.

1. Formal Definition and Parameter Space

The ABS model describes the evolution of a Dirac spinor $\psi(x,t)$ in $1+1$ dimensions, defined by the total Lagrangian

$$\mathcal{L} = \bar\psi \left(i\gamma^\mu \partial_\mu - m\right)\psi + L_I.$$

The interaction term combines S–S and V–V couplings: $$L_I = \frac{g^2}{\kappa+1}\,(\bar\psi\psi)^{\kappa+1} -\frac{g^2}{p(\kappa+1)}\bigl[\bar\psi\gamma^\mu\psi\,\bar\psi\gamma_\mu\psi\bigr]^{(\kappa+1)/2},$$ where $\kappa>0$ controls the nonlinearity and $p>1$ sets the relative V–V/S–S weight ($p=2, \kappa=1$ gives the original ABS model). The Dirac matrices are $\gamma^0 = \sigma_3,\; \gamma^1 = \sigma_1$.

This structure allows for systematic interpolation between distinct nonlinear Dirac models, supporting analytic investigation of families of solitary wave solutions, parameterized by $(\kappa,p)$ (Khare et al., 17 Apr 2025).

2. Solitary Wave Solutions

Stationary rest-frame solitary waves are sought in the form

$$\Psi(x,t) = \Phi(x) e^{-i\omega t}, \quad \Phi(x) = \begin{pmatrix}u(x)\ i\,v(x)\end{pmatrix} = R(x)\begin{pmatrix}\cos\theta(x)\\sin\theta(x)\end{pmatrix}.$$

The system reduces to the first-order equations

$$\frac{d\theta}{dx} = \kappa\left[m\cos(2\theta)-\omega\right],$$

$$H\equiv(\kappa+1)[\omega-m\cos(2\theta)] + g^2 R^{2\kappa}\cos^{\kappa+1}(2\theta) - \frac{g^2}{p} R^{2\kappa} = 0.$$

The phase profile integrates in closed form: $$\theta(x) = \arctan[\alpha \tanh(\kappa\beta x)], \quad \alpha = \sqrt{\frac{m-\omega}{m+\omega}}, \;\beta = \sqrt{m^2-\omega^2}.$$ Solving $H=0$ yields $R(x)$, and one finds that bound-state solutions exist in the parameter range

$\frac{1}{p^{1/(\kappa+1)}} < \frac{\omega}{m} < 1.$

These explicit, analytic soliton solutions generalize the classical solitary waves of the nonlinear Dirac equation, supporting detailed analytical and numerical studies (Khare et al., 17 Apr 2025).

3. Physical Quantities and Scaling Relations

The ABS family features conserved quantities: $$Q = \int_{-\infty}^{\infty} R^2(x) \, dx, \quad E = \kappa H_2 - (\kappa - 1)\omega Q,$$ with

$$H_2 = m\int dx\, R^2\cos(2\theta) = \frac{2m}{\kappa\beta}\left[\frac{p(\kappa+1)(m-\omega)}{g^2}\right]^{1/\kappa} J(\omega,\kappa,p).$$

Both $Q$ and $H_2$ scale as $g^{-2/\kappa}$, and consequentially

$\frac{E}{Q}$

is completely independent of the coupling constant $g$. This property is rare among nonlinear Dirac models and is central to the model’s analytic tractability and application to scaling analyses.

4. Profile Topology: Hump Transitions

The charge-density profile $\rho(x) = R^2(x)$ exhibits either a single-hump (maximum at $x=0$) or double-hump (symmetrical maxima off-center) structure, determined by a transition point depending on $(\kappa,p)$ and frequency: $$\frac{p\kappa+1}{p(\kappa+1)} \lessgtr \frac{\omega}{m}.$$ For

$$\frac{1}{p^{1/(\kappa+1)}} < \frac{\omega}{m} \le \frac{p\kappa+1}{p(\kappa+1)},$$

the soliton is double-humped, whereas at higher frequencies it is single-humped. This analytically tractable transition captures how the ABS class interpolates between qualitatively distinct solitary wave topologies (Khare et al., 17 Apr 2025).

5. Stability Analysis and Spectral Criteria

Stability of solitary waves is analyzed via the Vakhitov-Kolokolov (VK) criterion,

$\frac{dQ}{d\omega} < 0.$

Numerical evaluations reveal that for $\kappa\le 2$, all solitary waves in the existence band are spectrally stable. For $\kappa > 2$, stability prevails only in a subinterval $\omega_{\min}<\omega<\omega_c(\kappa,p)$; for $\omega>\omega_c$, instability sets in. An additional indicator is that for $\kappa>2$ the curve $E/Q$ develops a maximum exceeding $m$, signaling loss of stability for higher frequencies (Khare et al., 17 Apr 2025).

In external potentials, full PDE simulations demonstrate that most previously observed “numerical instabilities” were artifacts of poor time-stepping or boundary reflections. With rigorous integrators and absorbing boundaries, physically stable solitary waves are reliably obtained for all spectrally stable parameter regimes. Only for $\omega<\omega_c\approx 0.729$ does a true instability emerge, corresponding to exponential growth of perturbations (Mellado-Alcedo et al., 20 Dec 2025).

6. Collective Coordinate Theory and External Potentials

The ABS model admits a collective coordinate reduction in the presence of external potentials $V(x)$, leading to effective dynamical equations for the soliton’s position $q(t)$ and momentum $p(t)$. For a weak, slowly varying $V(x)$, the center-of-mass evolution is governed by

$$\frac{d}{dt}\left[M_0\gamma \dot{q}\right] = \frac{d}{dt}\left[\frac{\partial U}{\partial \dot{q}}\right] - \frac{\partial U}{\partial q},$$

where $M_0$ is the soliton rest mass, $\gamma$ is the Lorentz factor, $U(q,\dot{q})$ is the effective potential, and $Q$ is the charge. These equations reproduce accelerated or oscillatory soliton motion in linear ($V(x) = -V_1 x$) and harmonic ($V(x)=\frac{1}{2}V_2 x^2$) traps with high quantitative agreement to direct simulation. The analytical predictions remain accurate throughout the stable frequency regime (Mellado-Alcedo et al., 20 Dec 2025).

7. Nonrelativistic Reductions and Further Generalizations

In the nonrelativistic limit $\omega \to m$, the ABS model reduces to a modified nonlinear Schrödinger equation (NLSE) of the form

$$\left[-\partial_x^2 + m^2 - \omega^2 - g^2(m+\omega)|u|^{2\kappa}\right] u = 0,$$

with solution $u(x) = A \,\text{sech}^{1/\kappa}(\kappa \beta x)$, $A^{2\kappa} = \frac{(\kappa+1)(m-\omega)}{g^2}$. The full two-parameter generalization produces at the next order an effective NLSE Hamiltonian with both quartic and higher-order nonlinear terms, the structure of which depends on the $(\kappa,p)$ parameters. Stability conditions for the nonrelativistic solitary waves remain equivalent to the relativistic case: Derrick’s theorem and VK criteria both yield stability for $\kappa<2$ (Khare et al., 17 Apr 2025).

The ABS model thus serves as an archetype for analytically tractable generalized nonlinear Dirac equations with rich solitary wave phenomenology and stability properties, providing a rigorous baseline for studies of nonlinear Dirac models under perturbations, external fields, and parameter continuations.