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Date of Award

9-2012

Document Type

Campus Access

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics and Statistics

First Advisor

Panayotis G. Kevrekidis

Second Advisor

Nathaniel Whitaker

Third Advisor

Qian-Yong Chen

Subject Categories

Mathematics

Abstract

The short-pulse equation (SPE) was first derived for ultra short pulse propagating in nonlinear optics. In this thesis, we consider short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. Assuming general Kramers-Kronig or Sellmeier formulas for the permittivity and permeability, we derive two SPEs for the high- and low-frequency band gap respectively in one space dimension (1D). Then we generalize this model into two dimensional (2D) case, and give two 2D SPEs. For 1D SPE, we discuss the connection with the nonlinear Schrödinger equation (NLS), and the robustness of various solutions emanating from the sine-Gordon equation and their periodic generalizations, then we consider the wellposedness of generalized Ostrovsky equation with small initial data. For the 2D SPEs, we discuss the Hamiltonian structures, robustness of line breather solutions, and evolution of localized 2D initial data.

Then we focus on an experiment on a left-handed nonlinear transmission line, which is a good way to study left-handed materials. We develop a discrete model that emulates the relevant circuit and benchmark its linear properties against the experimental dispersion relation. Using a perturbation method, we derive, from the discrete model, a NLS that predicts accurately the focusing (defocusing) carrier frequency threshold. We use numerical simulations to corroborate our experimental and theoretical findings and monitor the space-time evolution of the discrete solitons.

Finally, we consider a setting of optical models outside of the regime of SPE type equations. We introduce a generalized model which accounts for phenomena of nonlinear diffraction within the one-dimensional NLS equation from the perspective of the self-consistent Lagrangian/Hamiltonian formulation. A detailed analysis of the fundamental solitary waves is reported. The Vakhitov-Kolokolov (VK) criterion is used to precisely predict the stability border for the solitons, along with the largest total power that the waves may possess. Past a critical point, collapse effects are observed, caused by suitable perturbations. Interactions between two identical parallel solitary beams are explored by dint of direct numerical simulations. It is found that in-phase solitons merge into robust or collapsing pulsons, depending on the strength of the nonlinear diffraction.

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