Prmblems In Linear & Nonlinear Wave Motion

特别是在强激光和流体动力学建模中出现的非线性双曲型偏微分方程的全局正则性和短波渐近展开是本提案的中心主题。这两种方法的一个目标是更好地理解在强超短激光脉冲中观察到的聚焦成长寿命细丝。关键的目标是推导证明解的正则性或分解的估计，以及研究非线性几何光学在焦点附近的展开。目的是研究描述强波传播和相互作用的基本方程的解。由于这些方程不能精确求解，因此提出了尖锐的近似方法和严格的定性方法。两个特别有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。目标是发展与射线光学经典思想相关的方法。这些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.