Randomness in Waves and Fluids

这个项目解决了非线性波传播的一般领域中的一些问题。所考虑的大多数问题都是试图评估传播介质中的随机不均匀对诸如孤子之类的非线性结构的影响。在线性介质的情况下，众所周知，随机不均匀的效应是抑制波的传播，这种效应被称为安德森局部化。该提出者将研究非均匀效应和非线性效应之间的竞争，前者倾向于破坏像孤子这样的局域脉冲，后者起到将脉冲保持在一起的作用。在早期的工作中，他发现了有趣而复杂的行为，具有不同的“阶段”，其中一个或另一个是主导的。他还将开始一个项目，涉及一些波传播工具的应用，最重要的是费曼路径积分，用于通过流体传输被动标量的问题，如染料或示踪剂。这个项目中解决的这类问题通过一个重要的应用得到了最好的说明，那就是激光在光纤中的传输。光纤中非常强的光具有某种令人惊讶的特性，它可以与自己相互作用，并且实际上可以聚焦自己。这种自聚焦过程会形成被称为孤子的亮点，这些亮点在不改变形状的情况下沿光纤传播。这些孤子由于有可能被用作光通信系统的基础而引起人们的极大兴趣。本项目研究纤维特性的变化对这些结构的影响。根本的问题是：当光纤的性质允许变化时，这些孤子是持续存在的，还是被摧毁了？

This project addresses a number of problems in the general area of nonlinear wave propagation. Most of the problems considered are attempts to assess the effects of random inhomogeneities in the medium of propagation on nonlinear structures such as solitons. In the case of linear media, it is well known that the effect of random inhomogeneities is to inhibit the propagation of waves, an effect known as Anderson localization. The proposer will study the competition between the effect of the inhomogeneities, which tend to destroy a localized pulse such as a soliton, with the effect of nonlinearity,which acts to hold a pulse together. In earlier work he found interesting and complicated behavior, with different ``phases'' in which one effect or the other is dominant. He will also begin a project involving the application of some of the tools of wave propagtion, most importantly the Feynman path integral, to problems of the transport of a passive scalar, such as a dye or tracer, by a fluid flow. The kind of problems addressed in this project are best illustrated by one important application, that of laser light propagating in an optical fiber. Very intense light in an optical fiber has the somewhat surprising property that it can interact with itself, and can actually focusitself. This process of self-focusing leads to the formation of bright spots, called solitons, which propagate along the fiber without changing shape. These solitons are of great interest due to the possibility of using them as the basis for optical communications systems. This project studies how these kinds of structures are effected by variations in the fiber properties. The fundamental question is this: Do these solitons persist when the properties of the fiber are allowed to vary, or are they destroyed?