旋量孤子及其$\mathcal{PT}$对称子体
Spinor solitons and their $\mathcal{PT}$-symmetric offspring
摘要：
虽然（1+1）维的旋量场具有合适的结构来模拟具有增益和损耗的色散双峰系统，但是简单地将增益添加到场的一个分量并且将损耗添加到另一个分量导致不稳定的色散关系。在本文中，我们主张旋量模型的$\mathcal{PT}$对称扩展的一种不同的方法-不产生线性狄拉克方程不稳定性的方法。在举例说明了$\mathcal P$ -和$\mathcal T$ -破缺项的物理起源之后，我们考虑了三个具有立方非线性的U（1）-不变旋量模型的推广.其中，Thirring模型的\PT对称扩展被证明是完全可积的，并具有无穷多个守恒量. \PT对称的Gross-Neveu方程保持能量和动量，但不保持电荷。第三个模型是为了与前两个模型进行比较而引入的;它的\PT对称扩展完全没有守恒律。尽管这种戏剧性的差异，在可积性和守恒性质，所有三个\PT对称模型被证明有精确的孤子解。类似于扩展的Thirring和Gross-Neveu方程的孤子，新模型的孤子被发现是稳定的-除了一个窄带的频率附近的孤子存在的边界。在$\mathcal P$ -和$\mathcal T$ -破缺扰动下的持续性以及稳定性的普遍存在突出了（1+1）维旋量孤子的显著坚固性。
英文摘要：
Although the spinor field in (1+1) dimensions has the right structure to model a dispersive bimodal system with gain and loss, the plain addition of gain to one component of the field and loss to the other one results in an unstable dispersion relation. In this paper, we advocate a different recipe for the $\mathcal{PT}$-symmetric extension of spinor models --- the recipe that does not produce instability of the linear Dirac equation. Having exemplified the physical origins of the $\mathcal P$- and $\mathcal T$-breaking terms, we consider the extensions of three U(1)-invariant spinor models with cubic nonlinearity. Of these, the \PT-symmetric extension of the Thirring model is shown to be completely integrable and possess infinitely many conserved quantities. The \PT-symmetric Gross-Neveu equation conserves energy and momentum but does not conserve charge. The third model is introduced for the purpose of comparison with the previous two; its \PT-symmetric extension has no conservation laws at all. Despite this dramatic difference in the integrability and conservation properties, all three \PT-symmetric models are shown to have exact soliton solutions. Similar to the solitons of the extended Thirring and Gross-Neveu equations, the solitons of the new model are found to be stable --- except for a narrow band of frequencies adjacent to the soliton existence boundary. The persistence under the $\mathcal P$- and $\mathcal T$-breaking perturbations as well as the prevalence of stability highlight a remarkable sturdiness of spinor solitons in (1+1) dimensions.
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