Partial differential equations, Hamiltonian dynamical systems, and their applications

该研究方案的总体目标是分析一类偏微分方程组(PDE)的行为，并解决其解的奇异性，这些偏微分方程组控制着涉及波传播的各种物理现象。大多数描述基本物理过程的基本方程都是微分方程式；演化方程形式的守恒定律描述了一个物理系统的未来状态，该物理系统的初始状态是在可能状态的相空间中规定的。当一个过程涉及波传播时，这些方程通常是偏微分方程组，那些具有物理或几何意义的方程通常可以表示为哈密顿动力系统，其相空间自然是无限维态空间。例如，非线性薛定谔方程，多种形式的非线性波动方程，无粘流体动力学的欧拉方程，海浪问题，以及在流体动力学和等离子体中出现的无数的非线性色散方程，如Korteweg DeVries方程和Boussinesq系统。

The overall goals of this research proposal are to analyse the behavior and to address the singularities of solutions of a class of partial differential equations (PDEs) that govern a variety of physical phenomena involving wave propagation. Most of the principal equations which describe fundamental physical processes are differential equations; conservation laws in the form of evolution equations which describe the future state of a physical systems whose initial state is prescribed among a phase space of possible states. When a process involves wave propagation, these equations are generally PDEs, and those of physical or of geometric significance can often be formulated as Hamiltonian dynamical systems, whose phase space is naturally an infinite dimensional space of states. Examples include the nonlinear Schroedinger equation, many versions of nonlinear wave equations, the Euler equations for inviscid fluid dynamics, the problem of ocean waves, and the myriad of nonlinear dispersive equations such as the Korteweg deVries equation and the Boussinesq system that arise in fluid dynamics and plasmas.