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.

该项目将涉及部分微分方程，几何和非线性傅立叶分析的研究。它的意图是双重的。一方面，它与物理，非线性光学和非线性分散方程的非线性波和解决方案的行为有关。另一方面，它专注于波包分析技术和多线性奇异算子的研究，在非翻译不变和非偏见的环境中。这是通过分解，频率相互作用分析和非线性估计的两个区域相互关联的领域。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的研究，包括称为双曲线等肌系统的自旋模型，该系统在铁磁性理论中起着核心作用。该系统自然源于landau-lifshitz方程，当耦合到平均场时，磁化的静态和动态特性既有静态和动态属性。将研究具有特殊对称性和最初携带少量能量的解决方案的全球时间行为。特别是，人们想知道随着时间的发展，这种系统是否接近平衡。主要研究者还将研究相关双曲线非线性schroedinger方程的孤子溶液。这里特别感兴趣的是Symbions的存在，它们是对黑暗和明亮的孤子的共生形式的解决方案。一个长期的目标是了解与大能量数据相关的爆破动力学。在略有不同的方向上，主要研究者计划获得一些较尖锐的本地供应良好，并且几乎确定全球存在的结果（即，对于“通用数据”），对于某些周期性的非线性方程式，该方程与局部及时结果之间存在差异与所有解决方案可以在全球范围内保持差距。 该方法是构建和利用相关Gibb的不变性测量，就像典型的保守量一样，通过其支持来控制解决方案时间的增长。该项目的第二个主要组成部分是开发波数据包分析和时间频率技术的综合计划的一部分，以研究多线性伪差异操作员。他们的治疗与经典的多线性理论背道而驰，因为相关符号的行为可能受到允许在每个空间点或曲率假设上更改的多样性的控制，不一定会施加在某些方向上。在某些方向上，光，声音和引力等物理学现象是使用部分差分方程进行了数学模型的。非线性波模型出现在量子力学和铁磁学以及振动系统，半导体和光纤的研究中。非线性Schroedinger方程是基本的物理方程，因为它们控制了量子颗粒的运动，例如电子。该项目将探讨的一些主题是与三维流体和空气动力学的涡流理论相关的基本兴趣 - 可以将涡流丝视为流量具有涡度的细管 - 以及当前非线性光纤在当今电信系统中具有基本重要性的作品。双曲线非线性Schroedinger方程最近受到物理学家和应用数学家的关注，研究了光脉冲在正常的分散非线性阵列结构中的演变。通常，非线性傅立叶分析（尤其是改编的波包分析）包括通过调制波形分解复杂的结构，为基本的构建块，这些构件局部化，因此相对易于理解。然后可以简单地将这些块放回原处。调制波形捕获幅度，比例，频率和位置，就像音乐得分一样。该技术应用的对象包括语音，雷达信号，光学中产生的振荡表达，波浪传播以及非本地性质的其他现象。 因此，该分析非常适合研究允许波浪相互作用并产生新的修饰传播模式的非线性效应。