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

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 Alexeeva-Barashenkov-Saxena 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 (CCs) 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 CCs are in agreement with simulations of the nonlinear Dirac equation. The main effect of the imaginary part of the potential 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 & ouml;dinger equation.