Solitary waves of a PT-symmetric Nonlinear Dirac equation

TL;DR

This paper investigates solitary wave solutions in a PT-symmetric nonlinear Dirac equation, analyzing their stability, bifurcation behavior at the PT-phase transition, and dynamic evolution under parameter quenches.

Contribution

It provides the first detailed analysis of nonlinear solitary waves in a PT-symmetric Dirac model, including stability, bifurcation, and dynamics, with analytical insights into the PT-phase transition.

Findings

Stable solitary waves persist up to the PT-transition threshold.

Wave solutions degenerate into linear waves at the PT threshold.

Oscillatory and exponential growth dynamics depend on parameter quenches.

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 PT-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 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 bi-frequency 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.