Available at: https://digitalcommons.calpoly.edu/theses/2836
Date of Award
6-2024
Degree Name
MS in Mathematics
Department/Program
Mathematics
College
College of Science and Mathematics
Advisor
Efstathios Charalampidis
Advisor Department
Mathematics
Advisor College
College of Science and Mathematics
Abstract
The study of large nonlinear waves that seem to “appear out of nowhere and disappear without a trace”, known as rogue or freak waves, began largely in response to observations of catastrophic ocean waves. However, the study of rogue waves has since been expanded to a wider collection of physical scenarios, including discrete systems, such as those that appear in optics, as opposed to the continuous system of water waves. Waves in these discrete settings can be modeled as solutions to lattice wave equations.
The nonlinear Schrodinger equation (NLSE) is one of the most ubiquitous continuous wave models for physical systems where rogue waves emerge. This thesis focuses on the two discrete analogs of the NLSE: a non-integrable model called the discrete nonlinear Schrodinger equation (DNLS) and its integrable sibling called the Ablowitz-Ladik (AL) equation. The physical relevance of DNLS model motivates the search for its rogue wave solutions; a search that is impeded by its lack of integrability. However, it is homotopically paired with the integrable AL equation through the Salerno model, providing a potential outlet to find numerically exact solutions. This threefold investigation will look at: (i) finding time-periodic solutions to the DNLS atop a constant non-zero background, (ii) proximity of solutions to the AL and DNLS equations over time, and (iii) time-periodic solutions to the defocusing AL model.