Mathematical Sciences: Aspects of Integrability

Fokas PI正在研究可积方程研究中出现的数学方面，以及它们对由此类方程建模的相关物理问题的影响。重点将放在以下两个问题上：(a)关于衰减的初始数据和边界数据的半无限寿命的初始值-边值问题的解；(b)研究了一些已知可积方程的新的可积推广及其新的数学解，其中包括峰子和紧子。近年来，某些非线性方程的研究取得了重要进展。特别地，我们发现大量具有重要物理意义的方程支持相干结构。这种相干结构包括所谓的孤子、子、峰子和紧子。这些研究大多集中在初值问题上。然而，许多物理情况，包括电离层和光开关中出现的问题，都是由初始边值问题来模拟的。对这些问题提出的研究将具有重要的物理意义，例如理解光束与分离具有不同介电介质的两种介质的界面的相互作用。

DMS-9500311 Fokas The PI is investigating mathematical aspects arising in the study of integrable equations, as well as their implications for the associated physical problems modeled by such equations. Emphasis will be placed on the following two problems: (a) The solution of initial- boundary value problems on the half-infinite life for decaying initial and boundary data; (b) Investigation of certain new integrable generalizations of the well known integrable equations and their novel mathematical solutions, which include peakons and compactons. In recent years important advances in the study of certain nonlinear equations have occurred. In particular it has been found that large classes of physically important equations support coherent structures. Such coherent structures include the so called solitons, dromions, peakons, and compactons. Most of these studies have concentrated on initial value problems. However, many physical situations, including problems arising in ionosphere and in optical switches, are modeled by initial-boundary value problems. The proposed investigation of these problems will have important physical implications such as the understanding of the interaction of a light beam with an interface separating two media with different dielectrics.