Coupling Coherent Structures and Linear Radiation

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A solitary wave is a localised nonlinear wave that propagates at constant velocity without change of form. Such solitary waves have been shown both theoretically and experimentally to exist in optical media, such as optical fibres. Due to their localised nature, they are a natural bit in digital optical systems and are indeed being used as such in the first long distance optical telecommunication systems. The current goal of much research in the science of optical systems, photonics, is to replace electronic signal processing devices with all-optical equivalents, due to the inherent speed and capacity advantages of optical systems. In these all-optical devices, solitary waves from different signal streams can interact, leading to signal switching and other basic signal processing functions. One draw-back of the glass used in optical fibres is that it has a very low nonlinear response and so nonlinear effects are only apparent over long distances. In a groundbreaking series of experiments by the experimental group of Professor Assanto in Rome, it has been shown that due to the high nonlinear response of liquid crystals, solitary waves in liquid crystals, so-called nematicons, display nonlinear features over short length scales. Liquid crystals then are a possible medium in which solitary waves can interact over short length scales, leading to smaller optical devices.Many equations having solitary wave solutions possess exact solutions in terms of inverse scattering, and so all features of the solutions are in principle known. However this is not true of the nonlinear partial differential equations describing optical signal propagation, particularly in more than one space dimension, which is the case for solitary waves in liquid crystals. Hence there is a need for a new theory to analyse solitary waves propagation in higher space dimensions. The development of this theory and the comparison of its results with numerical solutions and experimental results forms the basis of the current research proposal. As solitary waves evolve, they shed dispersive radiation and their evolution is driven and determined by the interaction between the evolving solitary wave and its shed radiation. The main emphasis of the proposed research will be to develop perturbation techniques to determine this interaction so that nematicon evolution in liquid crystals can be accurately predicted. The development of these techniques will be helped by the concurrent numerical solution of the governing equations. The results of the analytical and numerical work will then be compared with experimental results of Professor Assanto to determine their utility and to suggest extensions. In particular, this experimental work is for liquid crystals which have a non-local material response. This non-local response presents significant analytical difficulties as the transverse length scale (transverse to the direction of propagation) of the nematicon is much smaller than the transverse length scale of the variation of the liquid crystal. Preliminary numerical results suggest that the nematicon solution is non-unique, so that multiple solitary wave solutions for given parameter values are possible. This will be further