Long-Term Dynamics of Nonlinear Evolution Partial Differential Equations

电、磁、光以及信息都是通过波动来传播的。虽然大约三百年前人们就已经了解了波浪传播的基本方面，但技术和科学需要分析与波浪有关的越来越复杂的现象的方法。例如，互联网的信息以光的形式沿着沿着玻璃纤维电缆传递，以及通过电磁波通过太空中的卫星传递。 手机技术的运作方式基本相同。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增加是由于玻璃纤维电缆设计的彻底和根本性变化。而不是使用相同的材料为数百或数千英里，目前的设计交替不同的材料，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之间相互作用的结果。高等数学与这一项目的主题密切相关，在这一过程中起了决定性的作用。从事偏微分方程研究的数学家认识到培养学生的科学能力以满足工业和政府的高要求的重要性。本计画旨在了解各种波动型非线性偏微分方程系统解的长期动态。这通常意味着双曲方程，但它也可以指薛定谔方程。虽然在散焦情况下已经取得了很大进展，其中波一直存在并散射到真空状态，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爆破以及小数据散射。主要目标是确定全局解是否分散到一个固定的解，也称为孤立子。后者似乎很可能，如果基本的不变性的方程，如由膨胀和平移对称，被排除在外。这正是径向设置的克莱因-戈登方程的情况。

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.