Abstract

The evolution of surface waves in deep water is governed by the nonlinear Schrodinger (NLS) equation. Spatially periodic breathers (SPBs) and rational solutions of the NLS equation are used as typical models for rogue waves since they exhibit many features of rogue waves. A major component of the dissertation is the stability of solutions of the NLS equation. We address the stability of the rational solutions of the NLS equation used to model rogue waves using squared eigenfunctions of the associated Lax Pair. This allows us to contrast to the existing results for SPBs. The stability of the constant amplitude solution of the higher order NLS (HONLS) equation with additional novel perturbations, relevant to our subsequent study on downshifting, is considered next. In addition to the higher order perturbations, we include linear effects and nonlinear damping of the mean flow to the HONLS equation. In addition to stability, we discuss rogue waves and downshifting. Permanent downshifting occurs when energy if permanently transferred from the initially dominant mode to lower modes and is observed in physical experiments and field studies of deep water waves. Although these experimental observations are well documented, neither NLS nor HONLS equations describe this behavior. Nonlinear damping of the mean flow, included in our studies, is shown to model permanent downshifting. We examine the interaction of rogue waves and downshifting in a sea state with both nonlinear and linear effects. We show that there are no rogue waves after permanent downshifting. Analytical and numerical analysis are provided to support the findings.

Notes

If this is your thesis or dissertation, and want to learn how to access it or for more information about readership statistics, contact us at STARS@ucf.edu

Graduation Date

2016

Semester

Summer

Advisor

Schober, Constance

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematics

Format

application/pdf

Identifier

CFE0006402

URL

http://purl.fcla.edu/fcla/etd/CFE0006402

Language

English

Release Date

August 2019

Length of Campus-only Access

3 years

Access Status

Doctoral Dissertation (Open Access)

Download

Included in

Mathematics Commons

Share

COinS