Randomness in Waves and Fluids

该项目解决了非线性波传播一般领域的一些问题。考虑的大多数问题是试图评估传播介质中的随机非均匀性对非线性结构（如孤子）的影响。在线性介质的情况下，众所周知，随机不均匀性的影响是抑制波的传播，这种效应被称为安德森局域化。该提议将研究非均匀性的影响与非线性的影响之间的竞争，非均匀性往往会破坏局部脉冲，如孤子，而非线性的作用是使脉冲保持在一起。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”，其中一种效果或另一种效果占主导地位。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，用于解决无源标量的输运问题，如染料或示踪剂，通过流体流动。在这个项目中解决的问题，最好地说明了一个重要的应用，激光在光纤中传播。光纤中的强光有一个令人惊讶的特性，它可以与自身相互作用，并且可以聚焦自己。这种自聚焦的过程导致了被称为孤子的亮点的形成，这些亮点沿着光纤传播而不改变形状。这些孤子非常有趣，因为它们有可能用作光通信系统的基础。本项目研究这些结构如何受到纤维性能变化的影响。最根本的问题是：当光纤的特性发生变化时，这些孤子是否会持续存在，还是会被破坏？

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?