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P.G. Kevrekidis, D.E. Pelinovsky, and D.Y.Tyugin

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Nonlinear stationary states in PT-symmetric lattices

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SIAM Journal of Applied Dynamical Systems 12, 1210-1236 (2013)

**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.

**Keywords:**

PT-symmetry, discrete Schrodinger equation, existence and stability of stationary states