Solitary waves in focussing and defocussing nonlinear, nonlocal optical media

Nonlinear, nonlocal optical media has emerged as an ideal setting for experimentally observing and studying spatial optical solitary waves which otherwise cannot occur in Kerr media. Of particular interest is the eventual application to all-optical circuits. However, there is considerable work left to do on the theoretical end before this is a possibility. In this thesis we consider three problems. The first is how to solve the governing equations for optical beam propagation in the particular medium of the nematic liquid crystal (NLC), which is used as a prototypical example, exactly and approximately. In this respect we provide the first known, explicit solutions to the model as well as a comprehensive assessment on how to use variational, or modulation theory, in this context. This leads to the discovery of a novel form of bistability in the system, which shows there are two stable solitary wave solutions for a fixed power or L2 norm. We then consider how to approximate solutions for optical solitary waves propagating in a more general class of nonlocal nonlinear media using asymptotic methods. This is a long open problem and is resolved in the form of a simple to implement method with excellent accuracy and general applicability to previously intractable models. We conclude with the discovery and characterization of an instability mechanism in a coupled, defocussing nonlinear Schrodinger system. We show there is no stable, coupled, localized solution. This result is compared with the more well-studied bright solitary wave system and physical and mathematical explanations are offered