Coupling Coherent Structures and Linear Radiation

孤立波是一种局域非线性波，它以恒定速度传播而不改变形式。这种孤立波已经在理论和实验上证明存在于光介质中，例如光纤。由于它们的局域性，它们在数字光学系统中是一种天然的比特，并且确实在第一个长距离光学通信系统中被使用。由于光学系统固有的速度和容量优势，目前光学系统（光子学）中许多研究的目标是用全光等效器件取代电子信号处理器件。在这些全光器件中，来自不同信号流的孤立波可以相互作用，导致信号切换和其他基本信号处理功能。用于光纤的玻璃的一个缺点是它具有非常低的非线性响应，因此非线性效应只在长距离上才明显。在罗马的Assanto教授的实验小组进行的一系列开创性实验中，已经表明，由于液晶的高度非线性响应，液晶中的孤波，即所谓的向列子，在短长度尺度上显示出非线性特征。液晶是一种可能的介质，在这种介质中，孤波可以在短长度尺度上相互作用，从而产生更小的光学器件。许多具有孤波解的方程在逆散射方面具有精确解，因此解的所有特征原则上都是已知的。然而，对于描述光信号传播的非线性偏微分方程，特别是在一个以上的空间维度中，这是液晶中孤波的情况，这是不正确的。因此，需要一种新的理论来分析孤波在高维空间中的传播。这一理论的发展及其结果与数值解和实验结果的比较构成了当前研究计划的基础。孤立波在演化过程中释放色散辐射，其演化是由演化中的孤立波与其散射辐射的相互作用驱动和决定的。所提出的研究的主要重点将是开发微扰技术来确定这种相互作用，以便可以准确地预测液晶中的向列演变。控制方程的并行数值解将有助于这些技术的发展。然后将分析和数值工作的结果与Assanto教授的实验结果进行比较，以确定其效用并提出扩展建议。特别地，本实验工作是针对具有非局部材料响应的液晶。由于向列线的横向长度尺度（沿传播方向的横向）远小于液晶变化的横向长度尺度，因此这种非局域响应在分析上存在很大的困难。初步的数值结果表明，向列解是非唯一的，因此对于给定的参数值，可能存在多个孤立波解。这将进一步

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