Spinor solitons and their $\mathcal{PT}$-symmetric offspring

摘要: Although the spinor field in (1+1) dimensions has right structure to model a dispersive bimodal system with gain and loss, plain addition of one component loss other results an unstable dispersion relation. In this paper, we advocate different recipe for $\mathcal{PT}$-symmetric extension models --- that does not produce instability linear Dirac equation. Having exemplified physical origins $\mathcal P$- T$-breaking terms, consider extensions three U(1)-invariant cubic nonlinearity. Of these, \PT-symmetric Thirring is shown be completely integrable possess infinitely many conserved quantities. The Gross-Neveu equation conserves energy momentum but conserve charge. third introduced purpose comparison previous two; its no conservation laws at all. Despite dramatic difference integrability properties, all are have exact soliton solutions. Similar solitons extended equations, new found stable except narrow band frequencies adjacent existence boundary. persistence under perturbations as well prevalence stability highlight remarkable sturdiness dimensions.