Advisor(s)
Atanas Stefanov
Committee Member(s)
Shangbing Ai
Simon Bortz
Tian Qing
Zoran Grujic
School
College of Arts and Sciences
Document Type
Dissertation
Department (new version)
Applied Mathematics
Date of Award
9-11-2025
Abstract
In this dissertation we prove the existence and stability of solitary waves for the power degenerate non linear Schr ̈odinger equation (2.1.1) and a fourth order wave equation (3.1.1). Construction of the waves are variational in nature. For the NLS we consider a semilinear Schr ̈odinger equation, driven by the power degenerate second order differential operator ∇ · (|x|2a∇), a ∈ (0, 1). We construct the solitary waves, in the sharp range of parameters, as minimizers of the Caffarelli- Kohn-Nirenberg’s inequality. Depending on the parameter a and the nonlinearity, we establish a number of properties, such as positivity, smoothness (away from the origin) and almost exponential decay. Then, and as a consequence of our variational constrcution, we completely characterize the spectral stability of the said solitons. For the wave equation, we concern ourselves with focusing polynomial power non linearity and construct traveling wave solutions for the equation. These equation fall into a class commonly known as beam equation. Additionally, we establish a criteria to ascertain the spectral stability of the waves. We also prove properties such as smoothness and exponential decay for the traveling waves.
ProQuest ID
Recommended Citation
Iyer, Vishnu, "Solitary Waves For Power Degenerate Non Linear Schro ̈Dinger Equation And Fourth Order Wave Equation." (2025). All ETDs from UAB. 7398.
https://digitalcommons.library.uab.edu/etd-collection/7398
