Nonlinear Fourier Analysis And Geometric Dispersive Equations.

非线性傅里叶分析和几何色散方程。

• 批准号: 0503542
• 负责人: Andrea Nahmod
• 金额: --
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503542&HistoricalAwards=false
• 关键词: Nonlinear; Fourier; Analysis; Geometric; Dispersive

This award will suuport Nahmod's research on geometric partial differential equations and also on the analysis of multilinear pseudodifferential operators. Of special interest are the short and long time behavior of nonlinear waves arising in geometry, ferro-magnetism and gauge field theories; and the development of the time frequency analysis techniques so successfully used to study multi-linear singular operators in one dimension to both develop the $x$-dependent and non-tensorial higher dimensional situations. These two areas come together by way of wave-packet decompositions and frequency interactions estimates needed in the study of nonlinear partial differential equations. The geometric Hamiltonian PDEs to be investigated include the Schroedinger map equation and the Ishimori system - both of which may be characterized variationally and model specific wave-like phenomena. Nahmod will attempt to prove existence results for the Cauchy problem for these systems and also to study stability and blow-up questions of special solutions exploiting the geometric features of the systems. The second topic is to study multi-linear pseudo-differential operators. Their treatment departs from the classical multi-linear theory for in the present situation, the symbols' behavior may be governed by a variety that's allowed to change at each spatial point.The dynamics of many real world physical systems can be described by geometric evolution equations, in particular geometric Hamiltonian partial differential equations. The dynamics of the magnetization field in a ferromagnetic material for example, is described by the Landau-Lifshitz equation. The theory of Schroedinger maps is to model the long wave length limit of an isotropic Heisenberg ferromagnetic lattice. PDEs are the mathematical models of the laws governing much of the phenomena in the physical world. Wave equations model the propagation of different kind of waves - such as light waves- in homogeneous media. Nonlinear models of conservative type arise in quantum mechanics while other variants appear for example, in the study of vibrating systems and semiconductors. The nonlinear Schroedinger equations are fundamental physical equations for they govern the motion of quantum particles, such as electrons. All of these equations have applications to diverse physical problems, e.g. the dynamics of nonlinear waves through optical fibers in which the index of refraction is sensitive to the wave amplitude, and waves at the free surface of an ideal fluid or plasma. Nonlinear Fourier analysis and in particular adapted phase analysis - "wave-packet or time-frequency analysis"- consists in decomposing complex structures into basic building blocks - which are localized and easier to understand - via modulated waveforms. And then piecing them back together in a straightforward manner. It works very similarly to a musical score. These modulated waveforms possess capture amplitude (loudness), scale (duration), frequency (pitch) and position (instant it is played). The objects could be speech, radar signals, as well as oscillatory expressions arising in optics, wave propagation and other phenomena of nonlocal nature.

该奖项将Suuport Nahmod在几何部分微分方程方面以及对多线性伪差操作员的分析方面的研究。特别感兴趣的是在几何形状，铁磁和轨迹场理论中产生的非线性波的短期和长时间行为。时间频率分析技术的开发非常成功地用于研究一个维度的多线性奇异运算符，以开发$ x $依赖性和非敏感性的较高维度。这两个领域通过波包分解和频率相互作用估计在非线性偏微分方程中所需的频率相互作用估计在一起。要研究的几何汉密尔顿PDE包括Schroedinger地图方程和等距系统 - 两者都可以被描述为变异和模型特异性波样现象。 Nahmod将尝试证明这些系统的库奇问题的存在结果，并研究利用系统几何特征的特殊解决方案的稳定性和爆炸问题。第二个主题是研究多线性伪差异操作员。他们的治疗脱离了经典的多线性理论，在当前情况下，符号的行为可以受到一个允许在每个空间上发生变化的多样性的约束。许多现实世界物理系统的动力学可以通过几何进化方程来描述，尤其是几何汉密尔顿部分区分方程。例如，铁磁物质中的磁化场的动力学由Landau-Lifshitz方程描述。 Schroedinger图理论是为了建模各向同性海森堡铁磁晶格的长波长度极限。 PDE是管理物理世界中许多现象的法律的数学模型。波方程模拟了均匀介质中不同类型波的传播（例如光波）。保守类型的非线性模型在量子力学中出现，而其他变体则例如在研究振动系统和半导体的研究中。非线性Schroedinger方程是基本物理方程式，因为它们控制了量子颗粒的运动，例如电子。 所有这些方程都在各种身体问题上应用，例如非线性波通过光纤的动力学，其中折射率对波幅度敏感，在理想的流体或等离子体的自由表面上波动。非线性傅立叶分析，特别是改编的相分析 - “波包或时频分析” - 包括将复杂的结构分解为基本的构建块 - 通过调制波形将复杂的结构分解为基本的构建块，并且易于理解。然后以简单的方式将它们拼凑在一起。它的工作原理与乐谱非常相似。这些调制的波形具有捕获幅度（响度），比例（持续时间），频率（音高）和位置（即时播放）。这些物体可能是语音，雷达信号以及光学，波传播和其他非本地性质现象中产生的振荡表达。