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

## TL;DR

This paper develops a stable PT-symmetric extension of spinor models in (1+1) dimensions, demonstrating integrability, conservation laws, and stable soliton solutions despite symmetry-breaking perturbations.

## Contribution

It introduces a novel PT-symmetric extension of spinor models that avoids instability and shows that these models possess exact, stable soliton solutions with varying conservation properties.

## Key findings

- The PT-symmetric Thirring model is integrable with infinitely many conserved quantities.
- The PT-symmetric Gross-Neveu model conserves energy and momentum but not charge.
- All three models exhibit stable soliton solutions, with stability near the existence boundary.

## Abstract

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.