Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (leaves 72-80). Also available in print.

Thesis (Ph. D.)--University of Washington, 1990. / Vita. Includes bibliographical references (leaves [217]-225).

The derivation of nonlinear dispersive PDE, such as the nonlinear Schrödinger (NLS) or nonlinear Hartree equations, from many body quantum dynamics is a central topic in mathematical physics, which has been approached by many authors in a variety of ways. In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDE. In this thesis we present two types of results related to obtaining NLS via the GP hierarchy. In the first part of the thesis, we derive a NLS with a linear combination of power type nonlinearities in R[superscript d] for d = 1, 2. In the second part of the thesis, we focus on considering solutions to the cubic GP hierarchy and we prove unconditional uniqueness of low regularity solutions to the cubic GP hierarchy in R[superscript d] with d ≥ 1: the regularity of solution in our result coincides with known regularity of solutions to the cubic NLS for which unconditional uniqueness holds. / text

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (Ph.D.)--Auburn University, 2007. / Abstract. Includes bibliographic references (ℓ. 163-)

Thesis (Ph. D.)--University of Washington, 1998. / Vita. Includes bibliographical references (leaves [114]-119).

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Leonard F. Register. Vita. Includes bibliographical references. Also available from UMI.

Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 152-162). Available also in a digital version from Dissertation Abstracts.

We study Schrödinger operators perturbed by non-compact (spatially extended) defects. We consider two models: a one-dimensional (1D) dimer structure with a global phase shift, and a two-dimensional (2D) honeycomb structure with a line-defect or "edge''. In both the 1D and 2D settings, the non-compact defects are modeled by adiabatic, domain wall modulations of the respective dimer and honeycomb structures. Our main results relate to the rigorous construction of states via bifurcations from continuous spectra. These bifurcations are controlled by asymptotic effective (homogenized) equations that underlie the protected or non-protected character of the states. In 1D, the states we construct are localized solutions. In 2D, they are "edge states'' - time-harmonic solutions which are propagating (plane-wave-like) parallel to a line-defect or "edge'' and are localized transverse to it. The states are described as protected if they persist in the presence of spatially localized (even strong) deformations of the global phase defect (in 1D) or edge (in 2D). The protected states bifurcate from "Dirac points'' (linear/conical spectral band-crossings) in the continuous spectra and are seeded by an effective Dirac equation. The (more conventional) non-protected states bifurcate from spectral band edges are seeded by an effective Schrödinger equation. Our 2D model captures many aspects of the phenomenon of topologically protected edge states observed in honeycomb structures such as graphene and "artificial graphene''. The protected states we construct in our 1D dimer model can be realized as highly robust TM- electromagnetic modes for a class of photonic waveguides with a phase-defect. We present a detailed computational study of an experimentally realizable photonic waveguide array structure.

Thesis (Ph. D.)--University of Washington, 2001. / Includes bibliographical references (leaves 156-168).