Nonlinear modes in finite-dimensional PT-symmetric systems

TL;DR

This paper investigates how nonlinear properties and stability of PT-symmetric waveguide arrays can significantly change under transformations that preserve their linear spectra, highlighting that linear spectral equivalence does not ensure similar nonlinear behavior.

Contribution

It demonstrates that PT-symmetric systems with identical linear spectra can have different nonlinear modes and stability properties, challenging assumptions about their equivalence.

Findings

Linear spectrum equivalence does not guarantee similar nonlinear modes.

Stable nonlinear modes can exist even with non-purely real linear spectra.

Graph representation helps visualize and design PT-symmetric networks.

Abstract

By rearrangements of waveguide arrays with gain and losses one can simulate transformations among parity-time (PT-) symmetric systems not affecting their pure real linear spectra. Subject to such transformations, however, the nonlinear properties of the systems undergo significant changes. On an example of an array of four waveguides described by the discrete nonlinear Schr\"odinger equation with dissipation and gain, we show that the equivalence of the underlying linear spectra implies similarity of neither structure nor stability of the nonlinear modes in the arrays. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of a newly obtained system. Neither the stability is directly related to the PT symmetry: stable nonlinear modes exist even when the spectrum of the linear array is not purely real. We use graph representation of PT-symmetric networks allowing for simple illustration of linearly equivalent networks and indicating on their possible experimental design.