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In this study we consider we consider a prototypical example of a <inline-formula><tex-math notation="LaTeX">$\mathcal {P}$</tex-math> </inline-formula><inline-formula><tex-math notation="LaTeX">$\mathcal {T}$</tex-math></inline-formula>-symmetric Dirac model. We discuss the underlying linear limit of the model and identify the threshold of the <inline-formula><tex-math notation="LaTeX">$\mathcal {P}$ </tex-math></inline-formula><inline-formula><tex-math notation="LaTeX">$\mathcal {T}$</tex-math></inline-formula>-phase transition in an analytical form. We then focus on the examination of the nonlinear model. We consider the continuation in the <inline-formula><tex-math notation="LaTeX">$\mathcal {P}$</tex-math></inline-formula><inline-formula><tex-math notation="LaTeX">$\mathcal {T}$</tex-math> </inline-formula>-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 <inline-formula><tex-math notation="LaTeX">$\mathcal {P}$</tex-math> </inline-formula><inline-formula><tex-math notation="LaTeX">$\mathcal {T}$</tex-math></inline-formula>-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 <inline-formula><tex-math notation="LaTeX">$\mathcal {P}$</tex-math></inline-formula><inline-formula><tex-math notation="LaTeX"> $\mathcal {T}$</tex-math></inline-formula>-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 &#x201C;quench&#x201D;. The former can be characterized by an interesting form of bifrequency solutions that have been predicted on the basis of the SU<inline-formula><tex-math notation="LaTeX">$(1,1)$</tex-math></inline-formula> symmetry. Finally, we explore some special, analytically tractable, but not <inline-formula><tex-math notation="LaTeX">$\mathcal {P}$</tex-math> </inline-formula><inline-formula><tex-math notation="LaTeX">$\mathcal {T}$</tex-math></inline-formula>-symmetric solutions in the massless limit of the model.

ieee journal of selected topics in quantum electronics

Solitary Waves of a <inline-formula><tex-math notation="LaTeX">$\mathcal {P}$</tex-math></inline-formula><inline-formula> <tex-math notation="LaTeX">$\mathcal {T}$</tex-math></inline-formula>-Symmetric Nonlinear Dirac Equation

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