Bi-$\cal{PT}$ symmetry in nonlinearly damped dynamical systems and tailoring $\cal{PT}$ regions with position dependent loss-gain profiles

TL;DR

This paper explores how position-dependent damping influences symmetry breaking in nonlinear $ ext{PT}$-symmetric systems, revealing novel bi-$ ext{PT}$-symmetric systems and demonstrating how tailored damping profiles can control $ ext{PT}$ regions.

Contribution

It introduces the concept of bi-$ ext{PT}$ symmetry in nonlinear systems and shows how position-dependent damping can be used to tailor $ ext{PT}$-symmetric regions and phase transitions.

Findings

Existence of bi-$ ext{PT}$-symmetric systems with twofold symmetry.

Symmetry breaking can occur in pairs or sequentially.

Proper damping design allows tailoring of $ ext{PT}$-symmetric regions.

Abstract

We investigate the remarkable role of position dependent damping in determining the parametric regions of symmetry breaking in nonlinear $\cal{PT}$-symmetric systems. We illustrate the nature of $\cal{PT}$-symmetry preservation and breaking with reference to a remarkable integrable scalar nonlinear system. In the two dimensional cases of such position dependent damped systems, we unveil the existence of a class of novel bi-$\cal{PT}$-symmetric systems which have two fold $\cal{PT}$ symmetries. We analyze the dynamics of these systems and show how symmetry breaking occurs, that is whether the symmetry breaking of the two $\cal{PT}$ symmetries occurs in pair or occurs one by one. The addition of linear damping in these nonlinearly damped systems induces competition between the two types of damping. This competition results in a $\cal{PT}$ phase transition in which the $\cal{PT}$ symmetry is broken for lower loss/gain strength and is restored by increasing the loss/gain strength. We also show that by properly designing the form of the position dependent damping, we can tailor the $\cal{PT}$-symmetric regions of the system.