Doctoral Dissertations

Abstract

"The cubic nonlinear Schrödinger equation (NLS) is a model of interest in the study of physical problems including nonlinear optics and Bose-Einstein condensates. Of particular interest is the study of cubic NLS with inhomogeneities such as localizations of the nonlinearity or terms introducing potential barriers. We first address some preliminaries and techniques useful in the study of the cubic NLS and its variations. We then consider the cubic NLS with a localized nonlinearity in dimensions d ≥ 2. We show that solutions with data given by small-amplitude wave packets accrue a nonlinear phase that determines the X-ray transform of the nonlinear coefficient. Next, we also consider the dynamics of a boosted soliton evolving under the cubic NLS with an external potential in dimension d = 1. In particular, we study the dynamics of the resulting soliton-potential interactions in the high-velocity regime, and we show that for sufficiently large velocities the soliton is effectively transmitted through the potential" -- Abstract, p. iii

Advisor(s)

Murphy, Jason

Committee Member(s)

Grow, David E.
Le, Vy Khoi
Stutts, Daniel S.
Zhang, Yanzhi

Department(s)

Mathematics and Statistics

Degree Name

Ph. D. in Mathematics

Publisher

Missouri University of Science and Technology

Publication Date

Summer 2024

Pagination

iv, 90 pages

Note about bibliography

Includes_bibliographical_references_(pages 86-89)

Rights

©2024 Christopher Hogan , All Rights Reserved

Document Type

Dissertation - Open Access

File Type

text

Language

English

Thesis Number

T 12386

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Included in

Mathematics Commons

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