Linear and Nonlinear Stability for Infinite-Dimensional Dynamical Systems

无限维动力系统的线性和非线性稳定性

• 批准号: 1211315
• 负责人: Milena Stanislavova
• 金额: $ 21.52万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1211315&HistoricalAwards=false
• 关键词: Linear; Nonlinear; Stability; Infinite; Dimensional

The proposal focuses on the stability of waves for several models both in one and higher spatial dimensions. The common feature of these equations is that they can be viewed as dynamical systems on an infinite-dimensional space. These Hamiltonian PDEs support coherent structures such as solitary waves as well as periodic wave solutions. The PI will address open questions by developing a systematic approach to the spectral stability of waves for second order in time PDEs. An advantage of this method is that it treats both spatially periodic and solitary waves, which provides a link between the well studied results for solitary waves and the mostly open questions on periodic waves. These models are at the frontiers of current research and present lots of challenges at the spectral and linear stability level, but virtually nothing is known for their asymptotic stability. The PI studies will provide for better theoretical understanding of various nonlinear systems such as the beam equation, water waves and the Klein-Gordon model. Any progress on these questions will not only be important mathematically, but will find immediate applications in physical sciences. The PI will train the future researches by working with graduate and undergraduate students on the topics of the proposal. A special freshman seminar on nonlinear waves is being designed by the PI as part of a pilot First Year Seminar program at the University of Kansas. The PI has worked with students from local schools and is the main organizer for several mathematics competitions for school age kids. The students preparing for the competitions have been exposed to applied mathematics problems and have gained a better understanding of the role mathematics plays in modeling real life phenomena.

该提案的重点是在一个和更高的空间维度的几个模型的波的稳定性。这些方程的共同特点是它们可以被看作是无限维空间上的动力系统。 这些哈密顿偏微分方程支持相干结构，如孤立波以及周期波的解决方案。 PI将通过开发一种系统的方法来解决二阶时间偏微分方程波的谱稳定性问题。这种方法的一个优点是，它处理空间周期和孤立波，这提供了孤立波的研究结果和最开放的问题周期波之间的联系。 这些模型处于当前研究的前沿，在谱和线性稳定性水平上提出了许多挑战，但实际上对它们的渐近稳定性一无所知。PI的研究将提供更好的理论理解的各种非线性系统，如梁方程，水波和Klein-Gordon模型。在这些问题上的任何进展不仅在数学上很重要，而且将在物理科学中找到直接的应用。 PI将通过与研究生和本科生就提案的主题进行合作来培训未来的研究。PI正在设计一个关于非线性波的特别大一研讨会，作为堪萨斯大学一年级研讨会试点项目的一部分。PI与当地学校的学生合作，是几个学龄儿童数学竞赛的主要组织者。准备参加竞赛的学生接触了应用数学问题，并对数学在模拟真实的生活现象中所起的作用有了更好的理解。