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

TL;DR

This paper studies how PT-symmetric complex potentials affect soliton behavior in a nonlinear Dirac equation, revealing oscillatory dynamics and conditions for long-term stability.

Contribution

It introduces a collective coordinates approach to analyze soliton dynamics under PT-symmetric potentials in the ABS model, extending stability criteria to the nonlinear Dirac context.

Findings

Imaginary potential induces charge and energy oscillations.

Long-lived solitons persist despite large oscillations.

Extended stability criterion from NLS to nonlinear Dirac equation.

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 Schr\"odinger equation.