PT-symmetric branched optical lattices: Spectral properties and stability of solitons

TL;DR

This paper explores the spectral properties and stability of solitons in PT-symmetric branched optical lattices, revealing how boundary conditions influence the real or complex nature of the spectrum and soliton stability.

Contribution

It introduces a vertex boundary condition framework for PT-symmetric lattices, linking spectral properties and soliton stability to these conditions in both linear and nonlinear regimes.

Findings

PT symmetry enforces real spectra when boundary conditions are satisfied.

Violation of boundary conditions leads to complex eigenvalues.

Stable solitons correspond to boundary conditions satisfying PT symmetry.

Abstract

We investigate branched PT-symmetric optical lattices. We consider both the linear and nonlinear Schr\"odinger equations with a PT-symmetric periodic potential on the graph and solve them by imposing weighted vertex boundary conditions. A constraint derived from these vertex conditions determines the exceptional point of the system. In the PT unbroken phase, this constraint enforces PT-symmetric boundary conditions at the vertices, ensuring a purely real spectrum; its violation leads to the emergence of complex eigenvalues in the linear regime. In the nonlinear regime, the same constraint determines the linear stability of solitons: satisfying the constraint yields stable solitons, whereas violating it corresponds to unstable solitons.