Nonlinear Fourier Analysis and Partial Differential Equations

该项目将涉及偏微分方程、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，它关注非线性波的行为和在物理、非线性光学和铁磁性中出现的非线性色散方程的解。另一方面，它的重点是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。这是两个通过分解、频率交互分析和非线性估计密切相关的领域。该项目的第一部分集中于某些非线性偏微分方程和系统的研究，包括自旋模型，即双曲石森系统，它在铁磁性理论中起着核心作用。当考虑到与平均场的耦合时，该系统由控制磁化的静态和动态特性的朗道-利夫希茨方程自然产生。本文将研究具有特殊对称性且初始携带能量小的解的全局时态。特别是，人们想知道这样的系统是否随着时间的推移而接近平衡。首席研究员还将研究相关双曲三次非线性薛定谔方程的孤子解。这里特别有趣的是共生的存在，这是暗孤子和亮孤子的共生形式的解。长期目标是了解与大能量数据相关的爆炸动力学。在一个稍微不同的方向上，主要研究者计划获得某些周期非线性方程的尖锐的局部适定性和几乎肯定的全局存在性结果（即，对于“一般数据”），这些方程的局部及时结果与那些可以全局实现所有解的结果之间仍然存在差距。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，通过它的支持来控制解的时间增长。该项目的第二个主要组成部分是开发波包分析和时频技术以研究多线性伪微分算子的综合计划的一部分。它们的处理方法与经典的多线性理论不同，因为相关符号的行为可能由允许在每个空间点上改变的各种符号所控制，或者曲率假设不一定在某些方向上施加。物理学中的波动现象，如光、声和重力，都是用偏微分方程进行数学建模的。非线性波动模型出现在量子力学和铁磁性中，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。该项目将探索的一些主题与三维流体中的涡丝理论和空气动力学（涡丝可以被看作是一根细管，其中的流动具有涡度）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重要意义。双曲型非线性薛定谔方程近年来受到了物理学家和应用数学家的越来越多的关注，这些物理学家和应用数学家正在研究通常色散非线性阵列结构中光脉冲的演化。一般来说，非线性傅立叶分析（特别是适应波包分析）包括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，这些块可以以一种简单的方式重新组合在一起。调制波形捕获幅度，尺度，频率和位置，就像一个乐谱。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，这种分析很好地适用于研究允许波相互作用并产生新的修正传播模式的非线性效应。

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.