Prmblems In Linear & Nonlinear Wave Motion

DMS-9500823 Rauch全球正则性和非线性双曲型偏微分方程的短波渐近展开式，特别是在强激光和流体动力学的模拟中出现的，是该提案的中心主题。这两种方法的一个目标是更好地理解在强而短的激光脉冲中观察到的聚焦成长寿命灯丝的过程。证明解的正则性或崩溃的估计的推导，以及焦点附近的非线性几何光学展开的研究是关键目标。其目的是研究描述强波传播和相互作用的基本方程的解。这些方程不能精确求解，因此发展了精辟的近似方法和严格的定性方法。两个特别令人感兴趣的问题是强激光与物质的相互作用以及复杂的流体流动。一个目标是发展与射线光学的经典思想相关的方法。这种方法已经在发现和分析新现象方面发挥了重要作用。

DMS-9500823 Rauch Global regularity and short wavelength asymptotic expansions of nonlinear hyperbolic partial differential equations arising, in particular, in the modelling of intense lasers and fluid dynamics are the central themes of the proposal. One goal for both methods is to understand better the focussing into long lived filaments that is observed in intense ultra brief laser pulses. The derivations of estimates proving regularity or breakdown of solutions, and, the study of nonlinear geometric optics expansions near focal points are key objectives. The objective is to study the solutions of fundamental equations describing the propagation and interaction of intense waves. These equations cannot be solved exactly so incisive approximate methods together with rigorous qualitative methods are developed. Two problems of particular interest are the interaction of intense lasers with matter, and, complex fluid flows. A goal is to develope methods related to the classical ideas of ray optics. Such methods have already played an important role in the discovery and analysis of new phenomena.