Linear and Nonlinear PT-symmetric Oligomers: A Dynamical Systems Analysis

TL;DR

This paper analyzes the existence, stability, and bifurcations of solutions in PT-symmetric oligomers with linear and nonlinear gain/loss, revealing new asymmetric solutions and complex dynamics in dimers and trimers.

Contribution

It introduces the analysis of nonlinear gain/loss effects in PT-symmetric oligomers, uncovering asymmetric solutions and bifurcations absent in linear models.

Findings

Existence of solutions beyond the linear PT-symmetry critical point

Presence of asymmetric solutions with complex eigenvalues

Nonlinear gain/loss enables bifurcations between symmetric and asymmetric states

Abstract

In the present work we focus on the cases of two-site (dimer) and three-site (trimer) configurations, i.e. oligomers, respecting the parity-time (PT) symmetry, i.e., with a spatially odd gain-loss profile. We examine different types of solutions of such configurations with linear and nonlinear gain/loss profiles. Solutions beyond the linear PT-symmetry critical point as well as solutions with asymmetric linearization eigenvalues are found in both the nonlinear dimer and trimer. The latter feature is absent in linear PT-symmetric trimers, while both of them are absent in linear PT symmetric dimers. Furthermore, nonlinear gain/loss terms enable the existence of both symmetric and asymmetric solution profiles (and of bifurcations between them), while only symmetric solutions are present in the linear PT-symmetric dimers and trimers. The linear stability analysis around the obtained solutions is discussed and their dynamical evolution is explored by means of direct numerical simulations. Finally, a brief discussion is also given of recent progress in the context of PT-symmetric quadrimers.