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上的多种模型的波的稳定性。这些方程式的共同特征是它们可以在无限维空间上被视为动态系统。 这些Hamiltonian PDE支持相干结构，例如孤立波和周期性波解。 PI将通过开发系统的频谱稳定性来解决二阶PDES的波谱稳定性，以解决开放问题。该方法的一个优点是它同时处理空间周期性和孤立波，这在研究的孤立波和周期性波的大多数开放问题之间提供了联系。 这些模型位于当前研究的前沿，并在光谱和线性稳定性水平上提出了许多挑战，但实际上，几乎没有任何以其渐近稳定性而闻名的。 PI研究将提供对各种非线性系统（例如梁方程，水波和克莱因 - 戈登模型）的更好理论理解。这些问题上的任何进展不仅在数学上都很重要，而且会在物理科学中找到立即应用。 PI将与研究生和本科生有关该提案的主题，培训未来的研究。 PI是一项针对非线性波浪的特别新生研讨会，作为堪萨斯大学试点第一年研讨会计划的一部分。 PI曾与当地学校的学生合作，并且是几个数学比赛的主要组织者。为比赛做准备的学生已经暴露于应用数学问题，并更好地了解了数学在建模现实生活现象中的作用。