Mathematical Sciences: Microlocal Analysis and Nonlinear Waves

本项目涉及使用和发展微局域技术来研究三类非线性波动现象：光滑障碍物的绕射、非光滑障碍物的绕射和从一种介质到另一种介质的传播。在所有这三种情况下，我们都关注承载奇点的射线的行为以及行进波。其目的是产生正则性定理的传播以及演示这些定理的最优性的例子。在后两种情况下，本研究解决了一些关于存在的基本问题，这些问题即使在线性情况下也是开放的。

This project involves the use and development of microlocal techniques to study three categories of nonlinear wave phenomena: diffraction by smooth obstacles, diffraction by nonsmooth obstacles, and transmission from one medium to another. In all three cases we are concerned with the behavior of singularity- bearing rays as well as with progressing waves. The aim is to produce propagation of regularity theorems along with examples demonstrating the optimality of those theorems. In the latter two categories, the research addresses some basic questions concerning existence which remain open even in the linear case.