Soliton dynamics and stability in the ABS spinor model with a PT-symmetric periodic potential

TL;DR

This work analyzes soliton dynamics in the ABS nonlinear Dirac model under a PT-symmetric complex periodic potential. By leveraging a Lagrangian formulation and a five-parameter collective-coordinate ansatz, the authors derive variational equations that govern soliton position, rapidity, momentum, frequency, and phase, and they obtain expressions for energy and momentum in terms of these coordinates. They show that the imaginary part of the potential mainly induces oscillations in charge and energy, which are in phase with the momentum, and they validate the variational predictions against full spinor simulations, including regimes with multiple oscillation frequencies when the real and imaginary potentials have different periods. Importantly, they extend an empirical stability criterion from the nonlinear Schrödinger setting to the nonlinear Dirac equation, confirming its usefulness for predicting instabilities in this PT-symmetric Dirac system. The results demonstrate robust, long-lived solitons across a range of parameters and highlight rich dynamical behavior arising from the interaction of PT-symmetric perturbations with Dirac solitons, with potential implications for controlled soliton dynamics in relativistic-like wave equations.

Abstract

We investigate the effects on solitons dynamics of introducing a PT-symmetric complex potential in a specific family of the cubic Dirac equation in (1+1)-dimensions, called the ABS model. The potential is introduced taking advantage of the fact that the nonlinear Dirac equation admits a Lagrangian formalism. As a consequence, the imaginary part of the potential, associated with gains and losses, behaves as a spatially periodic damping (changing from positive to negative, and back) that acts at the same time on the two spinor components. A collective coordinates theory is developed by making an ansatz for a moving soliton where the position, rapidity, momentum, frequency, and phase are all functions of time. We consider the complex potential as a perturbation and verify that numerical solutions of the equation of motions for the collective coordinates are in agreement with simulations of the nonlinear Dirac equation. The main effect of the imaginary part of the potencial is to induce oscillations in the charge and energy (they are conserved for real potentials) with the same frequency and phase as the momentum. We find long-lived solitons even with very large charge and energy oscillations. Additionally, we extend to the nonlinear Dirac equation an empirical stability criterion, previously employed successfully in the nonlinear Schrödinger equation.

Figures (8)

• Figure 1: Critical velocity, $v_c$, versus amplitude, $W_0$, of the imaginary part of the potential, computed from the collective coordinate equations. For initial velocities above (below) $v_c$, the soliton motion is unidirectional (oscillatory). In the blue shaded region, the stability curve $\tilde{P}(\dot{q})$ has at least one branch with negative slope and thus instability is expected. The dashed line between A and B is a guide for the eyes. Parameters: $V_0=0.001$, $k=l=\pi/32$. Initial conditions: $\omega(0)=0.9$, $q(0)=0$, and $\phi(0)=0$.
• Figure 2: Soliton oscillatory motion in the complex potential (\ref{['potV']})-(\ref{['potW']}) for $V_0=W_0=0.001$, and $k=l=\pi/32$. Evolutions of $q(t)$, $P(t)$, $Q(t)$, and $E(t)$ are shown. Solid lines and red points correspond to simulations and collective coordinate results, respectively. Left-hand panels: dashed lines represent approximate analytical solutions at zero-order for $q(t)$ and $P(t)$, given by Eqs. (\ref{['eq:pos']}) and (\ref{['eq:Psuper0']}), respectively. Right-hand panels: dashed lines show approximate analytical solutions for the charge and the energy up to the first-order correction. Initial conditions: $\omega(0)=0.9$, $q(0)=0$, $\phi(0)=0$, and $\dot{q}(0)=0.01$.
• Figure 3: Unbounded motion of the soliton in the complex potential (\ref{['potV']})-(\ref{['potW']}) for $V_0=W_0=0.001$, and $k=l=\pi/32$. Evolutions of $q(t)$, $P(t)$, $Q(t)$, and $E(t)$ are shown. Solid line and red points correspond to simulations and collective coordinate results, respectively. Left-hand panels: dashed lines represent approximate analytical solutions at zero-order for $q(t)$ and $P(t)$, given by Eqs. (\ref{['eq:pos']}) and (\ref{['eq:Psuper0']}), respectively. Right-hand panels: dashed lines show approximate analytical solutions for the charge and the energy up to the first-order correction. Initial conditions: $\omega(0)=0.9$, $q(0)=0$, $\phi(0)=0$, and $\dot{q}(0)=0.07$.
• Figure 4: Soliton dynamics for a lower frequency $\omega(0)=0.74$. The rest of parameters are the same as in Fig. \ref{['fig3']}.
• Figure 5: Soliton dynamics for $V_0=0.001$, $W_0=0.002$, and $k=l=\pi/32$ (point C in Fig. \ref{['fig1']}). In the top and middle panels, simulation results for the soliton position $q(t)$, momentum $P(t)$, charge $Q(t)$, and energy $E(t)$ (solid lines) are compared with CC results (red points). The dashed vertical line at $t_b=770$ corresponds to point $b$ in the bottom right-hand panel at which the curve $\tilde{P}(\dot{q})$ changes its curvature. The bottom left-hand panel presents the evolution in time of the charge density. Initial conditions: $q(0)=0$, $\omega(0)=0.9$, $\phi(0)=0$, and $\dot{q}(0)=0.05 \gtrapprox v_c$.