Long-Term Dynamics of Nonlinear Evolution Partial Differential Equations

电、磁、光，因此信息都是通过波动传播的。虽然人们在大约三百年前就了解了波传播的基本方面，但技术和科学需要方法来分析越来越复杂的与波有关的现象。例如，互联网的信息以光的形式通过玻璃纤维电缆传输，也通过空间中的卫星通过电磁波传输。手机技术基本上也是以同样的方式运作。例如，与20世纪90年代中期相比，今天互联网通信速度的大幅提高是由于玻璃纤维电缆设计的彻底和彻底的改变。目前的设计不是在数百或数千英里内使用相同的材料，而是在不同的材料之间交替使用，从而允许微妙的非线性效果发挥作用。这一革命性的设计是工程师、应用数学家和材料科学家相互作用的结果。与本项目主题密切相关的高等数学在这一过程中发挥了决定性作用。从事偏微分方程式研究的数学家认识到，为了满足工业和政府的高要求，培养学生在科学方面的重要性。这个项目的目的是了解各种波型非线性偏微分方程组的解的长期动力学。这通常指的是双曲方程，但也可以指薛定谔方程。虽然在散焦情况下已经取得了很大的进展，在这种情况下，波一直存在，并且散射到真空态，但对聚焦方程的研究要少得多。这种类型的方程可以表现出有限时间的爆破和小的数据散射。主要目标是确定整体解是否分散到一个静止的解，也称为孤子。如果排除方程的基本不变性，例如由伸缩和平移对称给出的基本不变性，则后一种情况似乎是可能的。径向背景下的Klein-Gordon方程就是这种情况。

Electricity, magnetism, light, and therefore information propagates by means of wave motion. While basic aspects of wave propagation were understood about three hundred years ago, technology and science demand methods to analyze more and more sophisticated phenomena relating to waves. For example, information for the internet is passed along glass fiber cables in the form of light as well as via satellites in space through electromagnetic waves. Cell phone technology operates essentially the same way. The dramatic increase in speed in internet communication today as compared to the mid 1990s, for example, is due to a complete and radical change in the design of glass fiber cables. Instead of using the same material for hundreds or thousands of miles, the current design alternates between different materials thus allowing for subtle nonlinear effects to come into play. This revolutionary design is the result of interactions between engineers, applied mathematicians, and material scientists. Advanced mathematics very closely related to the subject matter of this project played a decisive role in the process. Mathematicians working in partial differential equations are cognizant of the importance of training students in the sciences in order to meet the high demands of industry and government. This project aims at understanding the long-term dynamics of solutions to various systems of nonlinear partial differential equation of wave type. This typically means hyperbolic equations, but it can also refer to the Schroedinger equation. While much progress has been made on the defocusing case, where waves exist for all times and scatter to the vacuum state, focusing equations are much less studied. This type of equation can exhibit finite time blowup as well as small data scattering. The main goal is to determine whether or not global solutions scatter to a stationary solution also known as a soliton. The latter seems likely if basic invariances of the equations, such those given by dilation and translation symmetries, are excluded. This is precisely the case for Klein-Gordon equations in the radial setting.