Date of Award

2-2010

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

First Advisor

Panayotis G. Kevrekidis

Second Advisor

Nathaniel Whitaker

Third Advisor

Hans Johnston

Keywords

DNLS, NLS, Non-square lattices, Nonlinear waves, Photonic lattices, Saturable nonlinearity

Subject Categories

Mathematics | Statistics and Probability

Abstract

The focus of this dissertation is the existence, stability, and resulting dynamical evolution of localized stationary solutions to Nonlinear Schr¨odinger (NLS) equations with periodic confining potentials in 2(+1) dimensions. I will make predictions about these properties based on a discrete lattice model of coupled ordinary differential equations with the appropriate symmetry. The latter has been justified by Wannier function expansions in a so-called tight-binding approximation in the appropriate parametric regime. Numerical results for the full 2(+1)-D continuum model will be qualitatively compared with discrete model predictions as well as with nonlinear optics experiments in optically induced photonic lattices in photorefractive crystals. The predictions are also relevant for BECs (Bose-Einstein Condensates) in optical lattices.