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Studies on Lattice Systems Motivated by PT-Symmetry and Granular Crystals

Abstract
This dissertation aims to study some nonlinear lattice dynamical systems arising in various areas, especially in nonlinear optics and in granular crystals. At first, we study the 2-dimensional PT-symmetric square lattices (of the discrete non-linear Schr¨odinger (dNLS) type) and identify the existence, stability and dynamical evolu- tion of stationary states, including discrete solitons and vortex configurations. To enable the analytical study, we consider the so-called anti-continuum (AC) limit of lattices with uncoupled sites and apply the Lyapunov–Schmidt reduction. Numerical experiments will also be provided accordingly. Secondly, we investigate the nonlinear waves in the granular chains of elastically inter- acting (through the so-called Hertzian contacts) beads. Besides the well-understood standard one-component granular chain, the traveling waves and dynamics of its variants such as heterogeneous granular chains and locally resonant granular crystals (otherwise known as mass-in-mass (MiM) or mass-with-mass (MwM) systems) are also studied. One of our goals is to systematically understand the propagation of traveling waves with an apparently non-decaying oscillating tail in MiM/MwM systems and the antiresonance mechanisms that lead to them, where Fourier transform and Fourier series are utilized to obtain integral reformulations of the problem. Additionally, we study finite mixed granular chains with strong precompression and classify different families of systems based on their well-studied linear limit. Motivated by the study of isospectral spring-mass systems (i.e., spring-mass systems that bear the same eigenfrequencies), we present strategies for building granular chains that are isospectral in the linear limit and their nonlinear dynamics will also be considered.
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