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、磁、光以及信息通过波运动传播。虽然大约三百年前人们就了解了波传播的基本方面，但技术和科学需要方法来分析与波相关的越来越复杂的现象。例如，互联网的信息以光的形式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，当今互联网通信速度的显着提高是由于玻璃纤维电缆设计的彻底而彻底的变化。当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，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。与该项目主题密切相关的高等数学在此过程中发挥了决定性作用。研究偏微分方程的数学家认识到对学生进行科学培训以满足工业和政府的高要求的重要性。该项目旨在了解各种波动型非线性偏微分方程组解的长期动力学。这通常意味着双曲方程，但也可以指薛定谔方程。虽然在散焦情况（波始终存在并散射到真空状态）方面已经取得了很大进展，但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以及小的数据分散。主要目标是确定全局解是否分散为平稳解（也称为孤子）。如果排除方程的基本不变性（例如由膨胀和平移对称性给出的不变性），则后者似乎是可能的。径向设置下的克莱因-戈登方程正是这种情况。