Long-Term Dynamics of Nonlinear Evolution Partial Differential Equations

电、磁、光以及信息都是通过波动传播的。虽然大约三百年前人们就了解了波传播的基本方面，但技术和科学需要分析越来越复杂的波现象的方法。例如，互联网的信息以光的形式沿着玻璃光纤电缆传递，也通过太空中的卫星通过电磁波传递。手机技术基本上也是如此。例如，与20世纪90年代中期相比，今天互联网通信速度的急剧提高是由于玻璃光纤电缆设计的彻底改变。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应发挥作用。这种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。与本课题密切相关的高等数学在这个过程中起到了决定性的作用。研究偏微分方程的数学家认识到培养学生在科学方面的重要性，以满足工业和政府的高要求。本项目旨在了解各种波型非线性偏微分方程系统解的长期动力学。这通常意味着双曲方程，但它也可以指薛定谔方程。在散焦的情况下，波一直存在并散射到真空状态，虽然已经取得了很大进展，但对聚焦方程的研究却少得多。这种类型的方程可以表现出有限时间爆炸和小数据散射。主要目标是确定全局解是否散射到一个固定解，也称为孤子。如果排除方程的基本不变性，如由膨胀和平移对称给出的不变性，后者似乎是可能的。这正是克莱因-戈登方程在径向环境下的情况。

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.