Date of Award

Spring 1-1-2011

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Mark J. Ablowitz

Second Advisor

Keith Julien

Third Advisor

Gregory Beylkin

Abstract

This thesis examines the effects of small perturbation to soliton solutions of the nonlinear Schrödinger (NLS) equation on two fronts: the development of a direct perturbation method for dark solitons, and the application of perturbation theory to the study of nonlinear optical systems including the dynamics of ultra-short pulses in mode-locked lasers.

For dark soliton solutions of the NLS equation a direct perturbation method for approximating the influence of perturbations is presented. The problem is broken into an inner region, where core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton which propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the NLS equation are used to determine the properties of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated including linear and nonlinear dissipative perturbations.

In the study of mode-locking lasers the power-energy saturation (PES) equation is a variant of the nonlinear NLS equation, which incorporates gain and filtering saturated with energy, and loss saturated with power (intensity). Solutions of the PES equation are studied using adiabatic perturbation theory. In the anomalous regime individual soliton pulses are found to be well approximated by soliton solutions of the unperturbed NLS equation with the key parameters of the soliton changing slowly as they evolve. Evolution equations are found for the pulses’ amplitude, velocity, position, and phase using integral relations derived from the PES equation. It is shown that the single soliton case exhibits mode-locking behavior for a wide range of parameters. The results from the integral relations are shown to agree with the secularity conditions found in multi-scale perturbation theory.

In the normal regime both bright and dark pulses are found. Here the NLS equation does not have bright soliton solutions, and the mode-locked pulse are wide and strongly chirped. For dark pulses there are two interpretations of the PES equation. The existence and stability of mode-locked dark pulses are studied for both cases.

Soliton strings are found in both the constant dispersion and dispersion-managed systems in the (net) anomalous and normal regimes. Analysis of soliton interactions show that soliton strings can form when pulses are a certain distance apart relative to their width. Anti-symmetric bi-soliton states are also obtained. Initial states mode-lock to these states under evolution.

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