Date of Award

Spring 1-1-2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Congming Li

Second Advisor

Harvey Segur

Third Advisor

James Meiss

Fourth Advisor

Xiaochuan Cai

Fifth Advisor

Alexander Gorokhovsky

Abstract

We mainly study an important and interesting class of nonlinear PDE systems, the Hardy- Littlewood-Sobolev (HLS) type system. In addition, we qualitatively study 3-wave resonance interaction (3WRI). HLS system plays crucial roles in geometric analysis, dynamics analysis of vacuum states, study of nonlinear Schrödinger equations, and many other research areas. 3WRI emerges from nonlinear optics, plasma physics, water wave etc.

Our goal is to develop some new idea and method to qualitatively analyze those systems. This involves many types of problems, e.g. existence and non-existence, asymptotic behavior near singularity or at infinity, stability etc.

Our study shows that nonlinear systems have brought many new challenges to us, where methods and tools in the past may be limited to some special cases or even not applicable. By developing new ideas we can provide insight into these problems and solve some of the challenges.

Included in

Mathematics Commons

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