Nonlinear Fourier Analysis and Partial Differential Equations

非线性傅里叶分析和偏微分方程

• 批准号: 0803160
• 负责人: Andrea Nahmod
• 金额: $ 15万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0803160&HistoricalAwards=false
• 关键词: Nonlinear; Fourier; Analysis; Partial; Differential

This project will involve research in partial differential equations, geometry, and nonlinear Fourier analysis. Its intent is twofold. On the one hand, it is concerned with the behavior of nonlinear waves and solutions to nonlinear dispersive equations arising in physics, nonlinear optics, and ferromagnetism. On the other, it is focused on wave-packet analysis techniques and the study of multilinear singular operators, in both the non-translation-invariant and nontensorial settings. These are two areas that are intimately related to one another by way of decompositions, frequency interaction analysis, and nonlinear estimates. The first part of the project concentrates on the study of certain nonlinear partial differential equations and systems, including the spin-model known as the hyperbolic Ishimori system, which plays a central role in the theory of ferromagnetism. This system arises naturally from the Landau-Lifshitz equation governing both the static and the dynamic properties of magnetization when coupling to a mean field is taken into account. The global-in-time behavior of solutions with special symmetries and initially carrying small energy will be studied. In particular, one would like to know whether such systems are close to equilibrium as time evolves. The principal investigator will also study soliton solutions of the associated hyperbolic cubic nonlinear Schroedinger equation. Of special interest here is the existence of symbions, which are solutions of symbiotic form to dark and bright solitons. A longer term goal is to understand the blow-up dynamics associated with large energy data. In a slightly different direction, the principal investigator plans to obtain sharpened local well-posedness and almost sure global existence results (i.e., for "generic data") for certain periodic nonlinear equations for which there remains a gap between the local-in-time results and those that could be globally achieved for all solutions. The approach is to construct and to exploit the invariance of the associated Gibbs measure that, just like typical conserved quantities, controls the growth in time of the solutions through its support. The second major component of the project is part of a comprehensive program to develop wave packet analysis and time frequency techniques to study multilinear pseudo-differential operators. Their treatment departs from the classical multilinear theory because the behavior of the associated symbols may be governed by a variety that is allowed to change at each spatial point or curvature assumptions are not necessarily imposed in certain directions.Wave phenomena in physics such as light, sound, and gravity, are mathematically modeled using partial differential equations. Nonlinear wave models arise in quantum mechanics and ferromagnetism, as well as in the study of vibrating systems, semiconductors, and optical fibers. Nonlinear Schroedinger equations are fundamental physical equations, for they govern the motion of quantum particles, such as electrons. Some of the topics that the project will explore are of basic interest in connection to both the theory of vortex filaments in three-dimensional fluids and aerodynamics -- a vortex filament can be visualized as a thin tube in which the flow has vorticity -- and to current work in nonlinear fiber optics that is of fundamental importance in today's telecommunication systems. The hyperbolic nonlinear Schroedinger equation has recently received increased attention by physicists and applied mathematicians studying the evolution of optical pulses in normally dispersive nonlinear array structures. Nonlinear Fourier analysis in general (and adapted wave-packet analysis in particular) consists in decomposing complex structures via modulated waveforms into basic building blocks that are localized and thus relatively easy to understand. These blocks can then be put back together in a straightforward manner. The modulated waveforms capture amplitude, scale, frequency, and position, just like a musical score. The objects to which the technique applies include speech, radar signals, oscillatory expressions arising in optics, wave propagation, and other phenomena of a nonlocal nature. This analysis is thus well adapted to study the nonlinear effects that allow waves to interact and produce new modified propagation patterns.

本计画将涉及偏微分方程式、几何学及非线性傅立叶分析之研究。其意图是双重的。一方面，它关注非线性波的行为以及物理学、非线性光学和铁磁性中产生的非线性色散方程的解。另一方面，它是集中在波包分析技术和多线性奇异算子的研究，在非平移不变和非张量设置。这两个领域通过分解、频率交互作用分析和非线性估计彼此密切相关。该项目的第一部分集中于研究某些非线性偏微分方程和系统，包括被称为双曲Ishimori系统的自旋模型，该模型在铁磁性理论中起着核心作用。这个系统自然产生的Landau-Lifshitz方程的静态和动态特性的磁化耦合时，考虑到。我们将研究具有特殊对称性和初始携带小能量的解的整体时间行为。特别是，人们想知道随着时间的推移，这些系统是否接近平衡。主要研究者也将研究相关的双曲立方非线性薛定谔方程的孤子解。这里特别感兴趣的是共生体的存在，这是共生形式的暗孤子和亮孤子的解决方案。更长期的目标是了解与大量能量数据相关的爆破动力学。在一个稍微不同的方向上，主要研究者计划获得尖锐的局部适定性和几乎肯定的全局存在性结果（即，对于“通用数据”），对于某些周期性非线性方程，在局部时间结果和对于所有解可以全局实现的结果之间仍然存在差距。 该方法是构造和利用相关吉布斯测度的不变性，就像典型的守恒量一样，通过其支持控制解决方案的时间增长。该项目的第二个主要组成部分是一个综合计划的一部分，开发波包分析和时间频率技术，研究多线性伪微分算子。它们的处理方法与经典的多重线性理论不同，因为相关符号的行为可能受各种各样的影响，这些变化在每个空间点都可以改变，或者曲率假设不一定强加在某些方向上。物理学中的波动现象，如光、声和重力，都是用偏微分方程来数学建模的。非线性波模型出现在量子力学和铁磁性，以及振动系统，半导体和光纤的研究中。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。该项目将探讨的一些主题与三维流体和空气动力学中的涡丝理论（涡丝可以被视为流动具有涡度的细管）以及当前在非线性光纤中的工作有关，这在当今的电信系统中具有根本的重要性。双曲型非线性薛定谔方程近年来越来越受到物理学家和应用数学家的关注，他们研究光脉冲在正常色散非线性阵列结构中的演化。非线性傅立叶分析一般（特别是自适应波包分析）在于通过调制波形将复杂结构分解为局部化的基本构建块，因此相对容易理解。然后，这些块可以以简单的方式重新组合在一起。调制波形捕获幅度、比例、频率和位置，就像乐谱一样。该技术适用的对象包括语音、雷达信号、光学中产生的振荡表达式、波传播和其他非局部性质的现象。 因此，这种分析非常适合研究非线性效应，使波相互作用，并产生新的修改后的传播模式。