On the integrability of PT-symmetric dimers

TL;DR

This paper explores the integrability of PT-symmetric dimers modeled by coupled nonlinear Schrödinger equations, deriving phase portraits and analyzing solution behaviors including unbounded solutions and phase-plane topology changes.

Contribution

It extends the understanding of PT-symmetric dimers by constructing phase portraits and analyzing solution behaviors using an integrability-based approach.

Findings

Derived a pendulum equation explaining unbounded solutions

Analyzed phase-plane topological changes

Identified critical parameters for solution behavior

Abstract

The coupled discrete linear and Kerr nonlinear Schrodinger equations with gain and loss describing transport on dimers with parity-time PT symmetric potentials are considered. The model is relevant among others to experiments in optical couplers and proposals on Bose-Einstein condensates in PT symmetric double-well potentials. It is known that the models are integrable. Here, the integrability is exploited further to construct the phase-portraits of the system. A pendulum equation with a linear potential and a constant force for the phase-difference between the fields is obtained, which explains the presence of unbounded solutions above a critical threshold parameter. The behaviour of all solutions of the system, including changes in the topological structure of the phase-plane, is then discussed.