PT-symmetric sine-Gordon breathers

TL;DR

This paper investigates the existence and stability of sine-Gordon breathers in a $ ext{PT}$-symmetric medium, revealing they only persist at the gain-loss interface and are destabilized by Hopf bifurcations.

Contribution

It demonstrates that sine-Gordon breathers can only survive at the gain-loss interface in a $ ext{PT}$-symmetric setting, providing new insights into their stability and dynamics.

Findings

Breathers only persist at the gain-loss interface.

Breathers decay or generate kink-antikink pairs outside the interface.

Breathers are destabilized via Hopf bifurcation.

Abstract

In this work, we explore a prototypical example of a genuine continuum breather (i.e., not a standing wave) and the conditions under which it can persist in a $\mathcal{P T}$-symmetric medium. As our model of interest, we will explore the sine-Gordon equation in the presence of a $\mathcal{P T}$- symmetric perturbation. Our main finding is that the breather of the sine-Gordon model will only persist at the interface between gain and loss that $\mathcal{P T}$-symmetry imposes but will not be preserved if centered at the lossy or at the gain side. The latter dynamics is found to be interesting in its own right giving rise to kink-antikink pairs on the gain side and complete decay of the breather on the lossy side. Lastly, the stability of the breathers centered at the interface is studied. As may be anticipated on the basis of their "delicate" existence properties such breathers are found to be destabilized through a Hopf bifurcation in the corresponding Floquet analysis.