Normal forms for integrable PDEs and billiards

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本课题研究了可积非线性偏微分方程正规形式的构造及其在解的适定性和摄动理论中的各种应用。它还探讨了这些思想在一类可积台球系统的动力学和谱几何中的潜在应用。具体地说，提出研究聚焦和散焦非线性薛定谔方程和聚焦修正Korteweg-de Vries方程。主要研究者的目的是在平方可积的1周期函数的相空间和一定的分布空间中构造这些方程的各种正规形式。对相应频率图的分析将最终导致新的适定性结果，并在微扰理论中得到各种应用。通过引入拟模和Radon变换推广的新技术，提出建立一类具有边界的黎曼流形（称为“Liouville台球桌”）的谱刚性。相应的离散台球系统的动力学性质将在证明所声称的谱刚度方面起关键作用。该项目的范围在于发展新的方法，从哈密顿系统理论到分析、几何和可积非线性偏微分方程理论的基本问题。用这些方法得到的许多结果是用求解此类方程的传统和更一般的方法所不能得到的。除了应用于适定性和摄动理论之外，正规形式通过所谓的Birkhoff坐标提供了对解的非常详细的描述。所提出的活动将有助于更好地理解出现在各种物理系统（如水波、等离子体物理、固态物理、非线性光学和流体力学）中的一类非线性演化方程的动力学和解的稳定性。该活动的最新应用包括统计力学（白噪声的保存）和天体物理学的新结果，即研究在存在宇宙常数的情况下在黑洞背景上移动的光线的问题。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project is concerned with a variety of problems related to the construction of normal forms and their various applications to well-posedness of solutions and perturbation theory for classes of integrable nonlinear partial differential equations. It also explores potential applications of these ideas to the dynamics and the spectral geometry of a class of integrable billiard systems. More specifically, it is proposed to study the focusing and defocusing nonlinear Schrodinger equation and the focusing modified Korteweg-de Vries equation. The principal investigator aims to construct various normal forms for these equations in the phase space of square-integrable, 1-periodic functions and in certain spaces of distributions. The analysis of the corresponding frequency maps would eventually lead to new well-posedness results and to various applications to perturbation theory. The principal investigator proposes to establish spectral rigidity of a class of Riemannian manifolds with boundary (known as "Liouville billiard tables") by introducing new techniques involving quasimodes and a generalization of the Radon transform. The dynamical properties of the corresponding discrete billiard system will play a crucial role in the proof of the claimed spectral rigidity. The scope of the project lies in the development of new methods coming from the theory of Hamiltonian systems to fundamental problems in analysis, geometry, and the theory of integrable nonlinear partial differential equations. Many of the results obtained by these methods cannot be obtained by the traditional and more general methods for solving such equations. Beyond applications to well-posedness and perturbation theory, normal forms provide a very detailed description of the solutions via so-called Birkhoff coordinates. The proposed activity would lead to a better understanding of the dynamics and the stability properties of the solutions of a class of nonlinear evolution equations that arise in various physical systems such as water waves, plasma physics, solid-state physics, nonlinear optics, and fluid mechanics. Recent applications of the activity include new results in statistical mechanics (preservation of white noise) and applications to astrophysics, namely, the problem of studying light rays moving on black hole backgrounds in the presence of a cosmological constant.