Date of Award

Spring 1-1-2013

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Applied Mathematics

First Advisor

Mark Ablowitz

Second Advisor

Christopher Curtis

Third Advisor

Bengt Fornberg

Abstract

In recent years, large amplitude rogue waves have been studied in water and optical fibers. These large waves occur more frequently than suggested by conventional linear models and nonlinear phenomena are considered by many to be responsible for these waves.

The nonlinear Schrödinger equation (NLS) models a slowly modulated, monochromatic, deep water wave train. Moreover, perturbed plane wave solutions of the nonlinear Schrödinger equation experience growth due to modulational instability. Through repeated numerical simulations, wave height statistics are determined for both NLS and an equation which incorporates the full linear water wave dispersion relation. The latter equation prevents unbounded spectral broadening present in 2D NLS. All equations studied lead to non-Gaussian wave statistics that are well-described by Rayleigh distributions, and support rogue waves with amplitudes up to five times the initial amplitude. The differences between the one and two dimensional results are not substantial, with two dimensional equations leading to wave height distributions with smaller variance and higher mean. This suggests that studying, the one dimensional nonlinear Schrödinger equation plus suitable perturbations may be sufficient for a basic understanding of rogue waves, without having to turn to higher dimensional equations.

Finally, NLS-type equations that model pulse propagation in zero dispersion nonlinear fibers are also studied. In addition to modulational instability, it appears that certain parameter regimes are governed by a nonlinear instability. Both processes cause significant growth that can lead to large amplitudes events.

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