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.

该奖项将支持Nahmod在几何偏微分方程方面的研究，以及对多线性伪微分算子的分析。特别感兴趣的是短期和长期的非线性波所产生的几何，铁磁性和规范场理论的行为;和时间频率分析技术的发展，以便成功地用于研究多线性奇异运营商在一维发展的$x$依赖和非张量高维的情况。这两个领域走到一起的方式波包分解和频率相互作用的估计需要在非线性偏微分方程的研究。要研究的几何哈密顿偏微分方程包括薛定谔映射方程和Ishimori系统-这两个系统的特点是变分和模型特定的波动现象。Nahmod将试图证明这些系统的柯西问题的存在性结果，并利用系统的几何特征研究特殊解的稳定性和爆破问题。第二个主题是研究多线性伪微分算子。它们的处理方式与经典的多线性理论不同，因为在目前的情况下，符号的行为可能由在每个空间点上允许变化的变量所支配。许多真实的世界物理系统的动力学可以用几何演化方程，特别是几何Hamilton偏微分方程来描述。例如，在铁磁材料中的磁化场的动力学由朗道-李夫希茨方程描述。薛定谔映射理论是模拟各向同性海森堡铁磁晶格的长波极限。偏微分方程是物理世界中许多现象的数学模型。波动方程模拟了不同种类的波（如光波）在均匀介质中的传播。保守型非线性模型出现在量子力学中，而其他变体出现在例如振动系统和半导体的研究中。非线性薛定谔方程是控制电子等量子粒子运动的基本物理方程。 所有这些方程都可以应用于不同的物理问题，例如通过光纤的非线性波的动力学，其中折射率对波的振幅敏感，以及理想流体或等离子体的自由表面处的波。非线性傅立叶分析，特别是适应相位分析-“波包或时间-频率分析”-包括通过调制波形将复杂结构分解为基本构建块-这些基本构建块是局部的并且更容易理解。然后再以简单的方式把它们拼在一起。它的工作原理与乐谱非常相似。这些调制波形具有捕获幅度（响度）、尺度（持续时间）、频率（音高）和位置（播放的瞬间）。对象可以是语音、雷达信号，以及光学、波传播和其他非局部性质的现象中产生的振荡表达式。