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

Abstract

Abstract-In the present work, we consider a prototypical example of a PT -symmetric Dirac model. We discuss the underlying linear limit of the model and identify the threshold of the PT -phase transition in an analytical form. We then focus on the examination of the nonlinear model. We consider the continuation in the PT -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 PTtransition 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 PT -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 PT -symmetric solutions in the massless limit of the model.

Spinor solitons and their PT-symmetric offspring

“…The model (2.16) has originally appeared in [40] as a PT -symmetric free-fermion quantum theory, outside the gain and loss context. We should also note two recent publications [41] and [42] where a closely related PT -symmetric extension of the Dirac equation was introduced.…”

Dirac equation with self interaction induced by torsion

“…In Minkowski space-time, it contains a familiar nonlinear Dirac equation whose solution has been considered in the 1 + 1 dimensions. Such equation has recently received increasing attention both in physics and mathematics (see, e.g., [7,4] and references therein).…”

Dynamics of Dirac solitons in networks

“…[27]- [49]). In [27,28,31,32,33,34], a detailed study of soliton solutions for different types of nonlinearity, their stability and discussions of conservation laws were presented. These developments have, in turn, had an impact also on the mathematical literature where the stability of solutions for different forms of the nonlinear Dirac equation was explored in [29,30,36].…”