Global dynamics for nonlinear dispersive equations

这个项目将研究色散哈密顿偏微分方程解的长期行为，例如半线性波，Klein-Gordon方程和非线性薛定谔方程。这些方程可以是散焦的，也可以是聚焦的，这可以区分非线性是吸引的还是排斥的。在后一种情况下，人们通常会遇到不同的机制，这取决于非线性的威力，这允许丰富的动力学，从长期存在和弥散，到有限时间的爆裂。最近，首席研究员与日本京都大学的中西健二合作，给出了一大类这类聚焦色散波动方程在接近基态能量的能量下所有可能的动力学特征。这种分类是通过结合动力系统方法(双曲动力学、不变流形)和偏微分方程论(如集中紧致性和Kenig-Merle理论)来实现的。几个重要的公开问题仍然存在，其中之一是获得能量临界非线性波动方程的这种类型的分类。考虑到Duyck Aerts-Kenig-Merle关于聚焦方程的相关但互补的研究，这一点特别相关。一个长期的目标是建立孤子分辨猜想。这个猜想可以看作是线性薛定谔演化著名的渐近完备性性质的非线性类比。非线性波动方程在科学中起着核心作用。麦克斯韦的电动力学方程就是这种类型的，它们可以说是现代科学中最有影响力的偏微分方程--无线电波的存在，麦克斯韦在19世纪70年代仅基于这些方程就预测了光和X射线等一般电磁辐射，并在那个世纪晚些时候得到了实验的证实。不用说，从我们的日常生活中移除无线电传输、X射线、激光、微波和许多其他电磁辐射领域是不可想象的。此外，麦克斯韦波动方程的理论影响远远超出了19世纪物理学家和数学家的想象。事实上，它们使光速的恒定变得最自然，麦克斯韦系统的对称性直接导致了爱因斯坦的狭义相对论。将后者与万有引力统一起来，就产生了广义相对论。此外，量子理论提供了更多波动方程的例子，在许多情况下是非线性的。例如，对于通过精心设计的玻璃纤维电缆传输世界互联网流量来说，名为色散管理孤子的特殊解决方案，以及求解非线性光学中产生的某类非线性薛定谔方程的特殊解决方案，在今天是不可或缺的。这些由不同材料交替延伸组成的特殊电缆的引入，大大减少了传输错误和成本，并允许传输的数据量大幅增加。本项目致力于进一步发展和研究在许多物理和工程领域中出现的那种类型的非线性波动方程。一次又一次，数学家们通过纯粹的研究奠定了基础，如果没有这些研究，深刻影响我们日常生活的工程应用是不可能实现的。

This project will investigate the long-term behavior of solutions to dispersive Hamiltonian partial differential equations, such as the semilinear wave, Klein-Gordon, and nonlinear Schroedinger equations. These equations can be either defocusing or focusing, which distinguishes whether the nonlinearity is attractive or repulsive. In the latter case, one typically encounters various regimes depending on the power of the nonlinearity, which allows for rich dynamics ranging from long-term existence and dispersion, to finite-time blowup. Recently, in joint work with Kenji Nakanishi from Kyoto University, Japan, the principal investigator has given a complete characterization of all possible dynamics at energies close to the ground state energy for a large class of these focusing dispersive wave equations. This classification is achieved by a combination of dynamical systems methods (hyperbolic dynamics, invariant manifolds), with partial differential equations arguments such as concentration compactness and the Kenig-Merle theory. Several important open problems remain, among which is to obtain this type of classification for the energy critical nonlinear wave equation. This is particularly relevant in view of the related but complementary research by Duyckaerts-Kenig-Merle on focusing equations. A long-term goal is to establish the soliton-resolution conjecture. This conjecture can be viewed as the nonlinear analogue of the celebrated asymptotic completeness property of the linear Schroedinger evolution. Nonlinear wave equations play a central role in science. Maxwell's equations of electrodynamics are of this type, and they are arguably the most influential partial differential equations of modern science -- the existence of radio waves, and general electromagnetic radiation such as light and X-rays was predicted in the 1870s by Maxwell based on these equations alone and confirmed by experiment later that century. Needless to say, it is unthinkable to remove radio transmission, X-rays, lasers, microwaves, and many other electromagnetic radiation fields from our daily lives. In addition, Maxwell's wave equations have had theoretical impact far beyond anything of which nineteenth-century physicists and mathematicians could have conceived. Indeed, they make the constancy of the speed of light most natural, and the symmetries of Maxwell's system lead directly to Einstein's theory of special relativity. Unifying the latter with gravity then led to the general theory of relativity. In addition, quantum theory has provided many more examples of wave equations, in many cases nonlinear ones. For example, special solutions that go by the name of "dispersion managed solitons," and that solve a certain class of nonlinear Schroedinger equations arising in nonlinear optics, are indispensable today for the transmission of the world's internet traffic through carefully designed glass fiber cables. The introduction of these special cables, which consist of alternating stretches of different materials, drastically reduced transmission errors and cost, and allowed for a huge increase in the data volume being transmitted. This project focuses on the further development and study of nonlinear wave equations of the type that arise in many areas of physics and engineering. Time and time again, mathematicians have laid the foundations through pure research without which the engineering applications that profoundly affect our daily lives could not have been accomplished.