Stability in Discrete and Continuous Dynamical Systems

PI将应用现代分析技术来解决离散Schr\“odinger方程和Klein-Gordon方程背景下呼吸子稳定性理论中的一些悬而未决的问题，以及色散偏微分方程的特解。该项目集中在有关稳定性的基本问题，以及存在性和精确描述的稳定流形的解决方案，与一些不稳定的方向。在过去的二十年里，稳定的情形得到了大量的关注，而接近不稳定解的行为却没有得到很好的研究。一个原因是，不稳定结构本身在一个更具挑战性的环境中，就必须使用的数学技术而言。这是特别真实的存在下，一个边缘稳定的频谱，共振边缘的基本频谱和低维的情况下，所有这些都将是本项目的主要兴趣。该项目将侧重于非线性色散方程的研究，这些方程在数学上模拟了重要的过程，例如光在光学介质中的传播。其中一些模型出现在量子力学系统和非线性光学的研究中，而另一些则在流体动力学中找到了它们的根源。这些问题也可以在稳定化理论中表现为控制问题。粗略地说，如果一个人开始接近一个不稳定的配置，如何只做沿着小的调整，以保持接近初始配置？对这些方程的解的行为进行更好的数学描述，特别是它们在时间和空间中的渐近行为，将大大提高我们对底层物理的理解，并有助于开发使用它们的技术。

The PI will apply the techniques of modern analysis to some outstanding open problems in the stability theory of breathers in the discrete Schr\"odinger equation and Klein-Gordon equation context as well as special solutions for dispersive partial differential equations. The project concentrates on the basic questions concerning stability as well as the existence and the precise description of stable manifolds for solutions, with a few unstable directions. While the stable scenarios have received a great deal of attention in the last twenty years or so, the behavior close to unstable solutions has been less well-studied. One reason is that unstable structures present themselves in a more challenging environment in terms of the mathematical techniques that must be used. This is especially true in the presence of a marginally stable spectrum, resonant edges of the essential spectrum and in the low dimensional cases, all of which will be of primary interest in this project. The project will focus on the study of nonlinear dispersive equations, which model mathematically important processes, such as propagation of light in optical medium. Some of these models arise in the study of quantum mechanical systems and nonlinear optics, while others find their roots in fluid dynamics. These problems can also present themselves as a control problem in stabilization theory. Roughly speaking, if one starts close to an unstable configuration, how does one make only small adjustments along the way, in order to stay close to the initial configuration? Better mathematical description of the behavior of the solutions of these equations, especially their asymptotic behavior in time and space, will greatly improve our understanding of the underlying physics and it will help in the development of technologies that use them.