All ETDs from UAB

Advisory Committee Chair

Yulia Karpeshina

Advisory Committee Members

Ryoichi Kawai

Boris Kunin

Hyunkyoung Kwon

Marius Nkashama

Document Type

Dissertation

Date of Award

2016

Degree Name by School

Doctor of Philosophy (PhD) College of Arts and Sciences

Abstract

Gross-Pitaevskii equation (GPE), which was introduced in early 1960s, is widely used in several areas, such as condensed matter physics, and nonlinear optics. The purpose of this thesis is to carry out mathematical studies for GPE. We concentrate on extended solutions of GPE. This thesis mainly contains three parts. The first part is to investigate explicit perturbation formulas for stationary solutions close to a plane wave and their corresponding energies for the nonlinear polyharmonic equation when 2l > n, where n is dimension and l is a power of the Laplacian. The main idea is to consider a non-linear term as an unknown potential W and look for solutions of the linear system by making a sequence Wm and using successive approximation method. The second part is devoted to the perturbation formulas for the nonlinear polyharmonic equation when 4l > n + 1. The case 4l > n + 1 includes the case of the classical Gross-Pitaevskii equation in dimension two. In the case 4l > n + 1, the investigation of the sequence Wm is much more technical. The final part is to show an important technical result (namely estimates for Fourier coefficients of Wm), which plays a crucial role in proving the results for the case 4l > n+1.

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