Analytical studies of standing waves in three NLS models

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Issue Date
2018-08-31Author
Feng, Wen
Publisher
University of Kansas
Format
90 pages
Type
Dissertation
Degree Level
Ph.D.
Discipline
Mathematics
Rights
Copyright held by the author.
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Show full item recordAbstract
In this work, we present analytical studies of standing waves in three NLS models. We first consider the spectral stability of ground states of fourth order semi-linear Schrödinger and Klein-Gordon equations and semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. We use Hamiltonian index counting theory, together with the information from a variational construction to develop sharp conditions for spectral stability for these waves. The second case is about the existence and the stability of the vortices for the NLS in higher dimensions. We extend the existence and stability results of Mizumachi from two-space dimensions to $n$ space dimensions. Finally, the third equation we consider is a nonlocal NLS which comes from modeling nonlinear waves in Parity-time symmetric systems. Here again, we investigate the spectral stability of standing waves of its $\mathcal{PT}$ symmetric solutions.
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