Randomness in Waves and Fluids

点击翻译按钮获取中文摘要

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?