Long-Term Dynamics of Nonlinear Evolution Partial Differential Equations

非线性演化偏微分方程的长期动力学

• 批准号: 1842197
• 负责人: Wilhelm Schlag
• 金额: $ 2.39万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1842197&HistoricalAwards=false
• 关键词: Long; Term; Dynamics; Nonlinear; Evolution

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.

电力，磁性，光线，因此信息通过波运动传播。虽然大约三百年前就了解了波传播的基本方面，但技术和科学需求方法是分析与波浪有关的越来越复杂的现象。例如，互联网的信息以光线以及通过电磁波在太空中的卫星形式传递。 手机技术的运行方式基本相同。例如，与1990年代中期相比，当今互联网通信速度的急剧提高是由于玻璃纤维电缆设计的完整而根本变化。当前的设计没有使用数百或数千英里的相同材料，而是在不同材料之间交替，因此可以发挥微妙的非线性效果。这种革命性的设计是工程师，应用数学家和物质科学家之间相互作用的结果。高级数学与该项目的主题密切相关，在此过程中起了决定性的作用。在部分微分方程中工作的数学家意识到在科学中培训学生的重要性，以满足行业和政府的高度要求。该项目旨在理解波动类型非线性偏微分方程各种系统解决方案的长期动力学。这通常意味着双曲方程，但也可以指施罗丁格方程。尽管在散落的情况下已经取得了很大进展，但在所有时间都存在波浪并散布到真空状态的情况下，焦点方程的研究要少得多。这种类型的方程式可以表现出有限的时间爆炸以及小数据散射。主要目标是确定全球解决方案是否散射到固定溶液中也称为孤子。后者似乎很可能排除了方程式的基本不变，例如扩张和翻译对称性的敌人。在径向环境中，klein-gordon方程正是这种情况。