Normal forms for integrable PDEs and billiards

该奖项由2009年美国复苏和再投资法案(公法111-5)资助。本项目涉及与正规型的构造有关的各种问题，以及它们在可积非线性偏微分方程解的适定性和扰动理论中的各种应用。它还探索了这些想法在一类可积台球系统的动力学和谱几何中的潜在应用。更具体地说，研究了聚焦和散焦的非线性薛定谔方程和聚焦修正的Korteweg-de Vries方程。主要的研究目的是在平方可积的1周期函数的相空间和某些分布空间中构造这些方程的各种规范形。对相应频率映射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。主要研究人员提出通过引入涉及准模的新技巧和Radon变换的推广来建立一类带边界的黎曼流形(称为“Liouville台球表”)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性方面起到关键作用。该项目的范围在于从哈密顿系统理论到分析、几何和可积非线性偏微分方程组理论中的基本问题的新方法的发展。用这些方法得到的许多结果都不能用传统的、更一般的方法来求解这类方程。除了适定性和微扰理论的应用之外，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。提出的活动将使我们更好地理解一类非线性发展方程的解的动力学和稳定性，这些方程出现在不同的物理系统中，如水波、等离子体物理、固体物理、非线性光学和流体力学。该活动最近的应用包括统计力学(保留白噪声)的新结果和在天体物理学中的应用，即研究在存在宇宙常数的情况下在黑洞背景上运动的光线的问题。

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.