Solitary Waves of a $\mathcal {P}$ $\mathcal {T}$-Symmetric Nonlinear Dirac Equation

In this study we consider we consider a prototypical example of a
$\mathcal {P}$
$\mathcal {T}$
-symmetric Dirac model. We discuss the underlying linear limit of the model and identify the threshold of the
$\mathcal {P}$
$\mathcal {T}$
-phase transition in an analytical form. We then focus on the examination of the nonlinear model. We consider the continuation in the
$\mathcal {P}$
$\mathcal {T}$
-symmetric model of the solutions of the corresponding Hamiltonian model and find that the solutions can be continued robustly as stable ones all the way up to the
$\mathcal {P}$
$\mathcal {T}$
-transition threshold. In the latter, they degenerate into linear waves. We also examine the dynamics of the model. Given the stability of the waveforms in the
$\mathcal {P}$
$\mathcal {T}$
-exact phase, we consider them as initial conditions for parameters outside of that phase. We find that both oscillatory dynamics and exponential growth may arise, depending on the size of the corresponding “quench”. The former can be characterized by an interesting form of bifrequency solutions that have been predicted on the basis of the SU
$(1,1)$
symmetry. Finally, we explore some special, analytically tractable, but not
$\mathcal {P}$
$\mathcal {T}$
-symmetric solutions in the massless limit of the model.