Mathematical Sciences: Generation and Propagation of Dispersive Waves in Integrable Systems:

本课题将继续研究近可积系统，重点研究四个主要问题：小色散KdV方程的解、Toda激波问题、非线性薛定谔方程的小色散时间极限、逆散射理论与近可积演化方程的关系。偏微分方程的可积系统通常有其性质可以精确建立的解。这和通常的非线性偏微分方程很不一样。这里重要的是，当系统受到扰动时，这些属性通常保持不变。因此，人们可以对其他棘手的问题有很多了解。这些问题出现在非线性光学中（例如，在使用光纤的长距离通信网络的设计中），在非线性声学介质中的波的运动中，以及在薄机翼周围的流动中。

This project continues studies in near-integrable systems, concentrating on four main problems: solution of the small-dispersion KdV equation, the Toda shock problem, the small-dispersion long-time limit of the nonlinear Schrodinger equation, and relations of inverse scattering theory to nearly integrable evolution equations. Integrable systems of partial differential equations often have solutions whose properties can be exactly established. This is quite unlike the usual case for nonlinear partial differential equations. What is important here is that often the properties remain when the system is perturbed. Thus one can tell a great deal about otherwise intractable problems. Such problems arise in nonlinear optics (for example, in the design of long-distance communication networks using fiber optics), in the motion of waves in nonlinear acoustic media, and in flows around thin wings.