Nonlinear Stationary States in PT-Symmetric Lattices

Abstract

In the present work we examine both the linear and nonlinear properties of two related parity-time (PT)-symmetric systems of the discrete nonlinear Schrödinger (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. Second, 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 for a finite PT-dNLS defect in the infinite dNLS lattice are wider than in the case of an isolated PT-dNLS chain. We also prove existence of localized stationary states (discrete solitons) in the analogue of the anticontinuum 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.