Date of Award

5-2010

Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

First Advisor

Floyd Williams

Second Advisor

Panayotis Kevrekidis

Third Advisor

Robert Kusner

Subject Categories

Mathematics | Statistics and Probability

Abstract

We show that Einstein’s gravitational field equations for the Friedmann- Robertson-Lemaître-Walker (FRLW) and for two conformal versions of the Bianchi I and Bianchi V perfect fluid scalar field cosmological models, can be equivalently reformulated in terms of a single equation of either generalized Ermakov-Milne- Pinney (EMP) or (non)linear Schrödinger (NLS) type. This work generalizes or presents an alternative to similar reformulations published by the authors who inspired this thesis: R. Hawkins, J. Lidsey, T. Christodoulakis, T. Grammenos, C. Helias, P. Kevrekidis, G. Papadopoulos and F.Williams. In particular we cast much of these authors’ works into a single framework via straightforward derivations of the EMP and NLS equations from a simple linear combination of the relevant Einstein equations. By rewriting the resulting expression in terms of the volume expansion factor and performing a change of variables, we obtain an uncoupled EMP or NLS equation that is independent of the imposition of additional conservation equations. Since the correspondences shown here present an alternative route for obtaining exact solutions to Einstein’s equations, we reconstruct many known exact solutions via their EMP or NLS counterparts and show by numerical analysis the stability properties of many solutions.

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