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 Schrödinger equation.

中文翻译：

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具有 PT 对称周期复势的 ABS 旋量模型中的孤子动力学和稳定性

我们研究了引入一个孤子动力学对孤子动力学的影响 PT - (1+1) 维三次狄拉克方程特定族中的对称复势，称为 Alexeeva-Barashenkov-Saxena 模型。利用非线性狄拉克方程承认拉格朗日形式这一事实引入了势。因此，与增益和损耗相关的势的虚部表现为空间周期性阻尼（从正变为负，然后返回），同时作用于两个旋量分量。集体坐标（CCs）理论是通过为移动孤子建立拟像而发展起来的，其中位置、速度、动量、频率和相位都是时间的函数。我们将复势视为扰动，并验证 CC 运动方程的数值解与非线性狄拉克方程的模拟一致。电势虚部的主要作用是引起电荷和能量的振荡（它们对于实电势而言是守恒的），其频率和相位与动量相同。即使电荷和能量振荡非常大，我们也发现了长寿命的孤子。此外，我们将先前在非线性薛定谔方程中成功应用的经验稳定性准则扩展到非线性狄拉克方程。