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Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

2014

First Advisor

Panayotis Kevrekidis

Second Advisor

Nathaniel Whitaker

Third Advisor

Hans Johnston

Subject Categories

Applied Mathematics | Mathematics

Abstract

In this thesis we summarize the classical cases of one-dimensional oligomers and two dimensional plaquettes, respecting the parity-time (PT ) symmetry. We examine different types of solutions of such configurations with linear and nonlinear gain or loss profiles. For each configuration, we develop a dynamical model and examine its PT symmetry. The corresponding nonlinear modes are analyzed starting from the Hamiltonian limit, with zero value of the gain-loss coefficient, γ. Once the relevant waveforms have been identified (analytically or numerically), their stability as well as those of the ghost states in certain regimes is examined by means of linearization in the vicinity of stationary points. This reveals diverse and, occasionally, fairly complex bifurcations. The evolution of unstable modes is explored by means of direct simulations.

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