Nonlinear Wave Equations and Hamiltonian Systems

9706273 克雷格 本文研究的是一类偏微分方程 这些方程因其波动方程的性质而统一起来 而且它们都可以写成 Hamilton系统此类的重要示例包括 线性和非线性薛定谔方程，非线性波 Klein-Gordon方程，Korteweg deVries方程， 流体动力学中控制自由表面问题的欧拉方程。 调查员的第一个项目是考虑以下问题： 的初值问题的存在性和正则性 方程，它将继续研究卡姆托里和其他 无限维的主要特征的构造 这些方程所处的相空间。第二个项目 是色散线性方程的分析，主要是 薛定谔方程，解决细节的地方和 基本解的微局部正则性及其 的双特征的全局行为的关系， 主要符号第三个研究项目包括一个特殊的 着重于自由表面问题的数学分析， 流体动力学 在这个项目中研究的非线性微分方程发生在数学物理和工程在一个显着的各种设置，从宇宙学的海洋流体动力学的非线性光学。这一领域的工作涉及到一些 这些方程解的基本特征， 这是对我们物理世界知识的贡献， 它能以非平凡的方式 在先进的技术。该项目还特别追求 纯粹的数学目标是发展一个美丽的类比， 上世纪经典力学理论与 现代偏微分方程理论

9706273 Craig The proposed research is on a class of partial differential equations which are unified by their character as wave equations and by the fact that they can each be written in the form of a Hamiltonian system. Important examples of this class are the linear and nonlinear Schroedinger equations, the nonlinear wave and Klein-Gordon equations, the Korteweg deVries equation, and the Euler equations governing free surface problems in fluid dynamics. The first project of the investigator is to consider issues of existence and regularity of the initial value problem for these equations, and it will continue a study of KAM tori and other constructions of the principal features of the infinite dimensional phase space in which these equations are posed. The second project is an analysis of dispersive linear equations, principally the Schroedinger equation, addressing details of the local and microlocal regularity of the fundamental solution and its relationship to the global behavior of bicharacteristics of the principal symbol. The third research project includes a special focus on the mathematical analysis of free surface problems of fluid dynamics. The nonlinear differential equations studied in this project occur in mathematical physics and engineering in a remarkable variety of settings, from cosmology to fluid dynamics of the ocean to nonlinear optics. Work in this field addresses some of the basic characteristics of solutions of these equations, which is a contribution to our knowledge of the physical world, and which can contribute in nontrivial ways to developments in advanced technology. This project also specifically pursues the purely mathematical goal of developing the beautiful analogy between the theory of classical mechanics of last century and the modern theory of partial differential equations.