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

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 $\mathcal{P}\mathcal{T}$-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.

33 pages, 2 figures, to appear in Annals of Physics

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Cited by in corpus (11)

• Darboux transformation and soliton solutions of the semi-discrete massive Thirring model
• Solitons in $\mathcal{P}\mathcal{T}$-symmetric systems with spin-orbit coupling and critical nonlinearity
• Integrable semi-discretization of the massive Thirring system in laboratory coordinates
• Dirac Solitons and Topological Edge States in the $\beta$-Fermi-Pasta-Ulam-Tsingou dimer lattice
• Study of instability of the Fourier split-step method for the massive Gross--Neveu model
• Integrable coupled massive Thirring model with field values in a Grassmann algebra
• Stability of nonlinear Dirac solitons under the action of external potential
• Soliton dynamics in the ABS nonlinear spinor model with external fields
• Stability of parametrically driven, damped nonlinear Dirac solitons
• Soliton dynamics and stability in the ABS spinor model with a PT-symmetric periodic potential
• Classical Hamiltonian Systems with Balanced loss and gain