Normal forms for integrable PDEs and billiards

可积偏微分方程和台球的范式

• 批准号: 0901443
• 负责人: Peter Topalov
• 金额: --
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901443&HistoricalAwards=false
• 关键词: Normal; forms; integrable; PDEs; billiards

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.

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。该项目关注与规范形式的构建及其在可积非线性偏微分方程类解的适定性和扰动理论中的各种应用相关的各种问题。它还探讨了潜在的应用这些想法的动力学和谱几何的一类可积台球系统。更具体地说，提出了研究聚焦和散焦的非线性薛定谔方程和聚焦修正的Korteweg-de弗里斯方程。主要研究者的目标是在平方可积的相空间，1-周期函数和某些分布空间中为这些方程构建各种规范形式。相应的频率图的分析最终将导致新的适定性结果和各种应用微扰理论。主要研究者提出建立光谱刚性一类黎曼流形的边界（称为“刘维台球桌”）通过引入新的技术，涉及准模和推广的Radon变换。相应的离散台球系统的动力学性质将发挥至关重要的作用，在证明声称的光谱刚性。该项目的范围在于新方法的发展，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中的基本问题。用这些方法得到的许多结果是用传统的和更一般的求解这类方程的方法不能得到的。 除了适定性和微扰理论的应用外，规范形通过所谓的伯克霍夫坐标提供了解的非常详细的描述。拟议的活动将导致更好地了解在各种物理系统，如水波，等离子体物理，固态物理，非线性光学和流体力学中出现的一类非线性演化方程的解的动力学和稳定性。这项活动最近的应用包括统计力学方面的新成果（保留白色噪音）和天体物理学方面的应用，即研究在存在宇宙学常数的情况下在黑洞背景上移动的光线的问题。