Nonlinear stationary states in PT-symmetric lattices

TL;DR

This paper investigates the linear and nonlinear stationary states of PT-symmetric discrete nonlinear Schrödinger lattices, analyzing eigenvalues, stability, and localized solitons through analytical and numerical methods.

Contribution

It introduces a systematic analysis of nonlinear stationary states in PT-symmetric dNLS systems, including stability intervals and existence of discrete solitons.

Findings

Finite PT-dNLS chains have real eigenvalues within certain parameters.

Stability intervals are wider when a PT-dNLS chain is embedded in an infinite lattice.

Localized stationary states (discrete solitons) exist in the anti-continuum limit.

Abstract

In the present work we examine both the linear and nonlinear properties of two related PT-symmetric systems of the discrete nonlinear Schrodinger (dNLS) type. First, we examine the parameter range for which the finite PT-dNLS chains have real eigenvalues and PT-symmetric linear eigenstates. We develop a systematic way of analyzing the nonlinear stationary states with the implicit function theorem at an analogue of the anti-continuum limit for the dNLS equation. Secondly, we consider the case when a finite PT-dNLS chain is embedded as a defect in the infinite dNLS lattice. We show that the stability intervals of the infinite PT-dNLS lattice are wider than in the case of a finite PT-dNLS chain. We also prove existence of localized stationary states (discrete solitons) in the analogue of the anti-continuum limit for the dNLS equation. Numerical computations illustrate the existence of nonlinear stationary states, as well as the stability and saddle-center bifurcations of discrete solitons.