Snakes and ghosts in a parity-time-symmetric chain of dimers

TL;DR

This paper investigates parity-time symmetric chains of dimers modeled by coupled nonlinear Schrödinger equations, revealing snaking bifurcation diagrams, symmetry-breaking phenomena, and the existence of ghost states through analytical and numerical methods.

Contribution

It introduces the analysis of snaking bifurcations and ghost states in a PT-symmetric dimer chain, extending understanding of localized solutions and symmetry-breaking in such systems.

Findings

Bifurcation diagrams exhibit snaking behavior for localized solutions.

Critical gain/loss value marks PT-symmetry breaking and merging of bifurcations.

Ghost states continue to exhibit snaking bifurcations beyond symmetry breaking.

Abstract

We consider linearly coupled discrete nonlinear Schr\"odinger equations with gain and loss terms and with a cubic-quintic nonlinearity. The system models a parity-time ($\cal{PT}$)-symmetric coupler composed by a chain of dimers. Particularly we study site-centered and bond-centered spatially-localized solutions and present that each solution has a symmetric and antisymmetric configuration between the arms. When a parameter is varied, the resulting bifurcation diagrams for the existence of standing localized solutions have a snaking behaviour. The critical gain/loss coefficient above which the $\cal{PT}-$symmetry is broken corresponds to the condition when bifurcation diagrams of symmetric and antisymmetric states merge. Past the symmetry breaking, the system no longer has time-independent states. Nevertheless, equilibrium solutions can be analytically continued by defining a dual equation that leads to so-called ghost states associated with growth or decay, that are also identified and examined here. We show that ghost localized states also exhibit snaking bifurcation diagrams. We analyse the width of the snaking region and provide asymptotic approximations in the limit of strong and weak coupling where good agreement is obtained.