Stability band structure for periodic states in periodic potentials

TL;DR

This paper investigates the spectral stability of periodic solutions in nonlinear Schrödinger equations with non-Hermitian potentials, revealing stability conditions within specific Floquet-Bloch bands through analytical and numerical methods.

Contribution

It introduces a method to construct complex potentials supporting desired periodic solutions and analyzes their stability bands using the plane-wave-expansion method.

Findings

Periodic solutions can be stable in certain Floquet-Bloch bands.

Stability bands are identified through spectral analysis.

Numerical simulations confirm stability predictions.

Abstract

A class of periodic solutions of the nonlinear Schrodinger equation with non- Hermitian potentials are considered. The system may be implemented in planar nonlinear optical waveguides carrying an appropriate distribution of local gain and loss, in a combination with a photonic-crystal structure. The complex potential is built as a solution of the inverse problem, which predicts the potential supporting required periodic solutions. The main subject of the analysis is the spectral structure of the linear (in)stability for the stationary spatially periodic states in the periodic potentials. The stability and instability bands are calculated by means of the plane-wave-expansion method, and verified in direct simulations of the perturbed evolution. The results show that the periodic solutions may be stable against perturbations in specific Floquet-Bloch bands, even if they are unstable against small random perturbations.