Keywords
Nonlinear Schrödinger Equation, Integrable Systems, Inverse Scattering Transform, Solitons, Differential Equations, Random Nonlinear Waves
Abstract
The Nonlinear Schrödinger (NLS) Equation, iψt + 1/2 ψxx ± |ψ|2ψ = 0, is a nonlinear partial differential equation which is used to model several physical phenomena including nonlinear effects inside optical fibers and the formation of rogue waves in shallow water. It is particu- larly difficult to study solutions to this equation due to the nonlinearity, and the nonlinearity leads to incredibly interesting solutions not found in linear PDEs such as solitons. In this thesis, we highlight two methods of obtaining solutions to the (NLS) equation: the Inverse Scattering Transform and the Dressing Method. Furthermore, we study a particular class of solutions called solitons and multi-solitons. Solitons, also called solitary travelling waves, are localized traveling waves which arise as solutions to several nonlinear dispersive partial differential equations. We use the Dressing Method to numerically compute 100-soliton solutions to the NLS equation with 500 digits of precision. We then analyze the statistical properties of these multi-soliton solutions and compare them to some established results computed using other methods.
Thesis Completion Year
2025
Thesis Completion Semester
Spring
Thesis Chair
Jenkins, Robert
College
College of Sciences
Department
Mathematics
Thesis Discipline
Mathematics
Language
English
Access Status
Open Access
Length of Campus Access
None
Campus Location
Orlando (Main) Campus
STARS Citation
Casas-Rocha, Ivan, "The Inverse Scattering Transform for the Nonlinear Schrödinger Equation" (2025). Honors Undergraduate Theses. 258.
https://stars.library.ucf.edu/hut2024/258
Included in
Analysis Commons, Non-linear Dynamics Commons, Ordinary Differential Equations and Applied Dynamics Commons, Partial Differential Equations Commons
Rights Statement
