Nonlinear Fourier Analysis And Geometric Dispersive Equations.

该奖项将支持Nahmod对几何偏微分方程的研究以及对多线性伪微分算子的分析。特别感兴趣的是几何、铁磁和规范场论中产生的非线性波的短时间和长时间行为；而时频分析技术的发展使其成功地应用于一维多线性奇异算子的研究，从而发展出与x无关和非张量的高维情况。这两个领域通过波包分解和非线性偏微分方程研究中需要的频率相互作用估计的方式结合在一起。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-两者都可以变分表征并模拟特定的波状现象。Nahmod将尝试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稳定性和特解的爆破问题。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，符号的行为可能由允许在每个空间点上改变的多样性所控制。许多现实世界物理系统的动力学可以用几何演化方程，特别是几何哈密顿偏微分方程来描述。例如，铁磁材料中磁化场的动力学可以用朗道-利夫希茨方程来描述。薛定谔映射理论是为了模拟各向同性海森堡铁磁晶格的长波长度极限。偏微分方程是控制物理世界中许多现象的定律的数学模型。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。在量子力学中出现了保守型的非线性模型，而在振动系统和半导体的研究中也出现了其他的变体。非线性薛定谔方程是基本的物理方程，因为它们控制着量子粒子（如电子）的运动。所有这些方程都适用于各种物理问题，例如，通过光纤的非线性波的动力学，其中折射率对波幅很敏感，以及理想流体或等离子体自由表面的波。非线性傅立叶分析，特别是适应相位分析——“波包或时频分析”——通过调制波形将复杂的结构分解成基本的构建块——这些构建块是局部化的，更容易理解。然后用一种简单的方式把它们拼接起来。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（它被播放的瞬间）。对象可以是语音、雷达信号，也可以是光学、波传播等非局域现象中产生的振荡表达式。

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.