Stability in Discrete and Continuous Dynamical Systems

PI将把现代分析技术应用于离散薛定谔方程和Klein-Gordon方程中呼吸子稳定性理论中的一些突出的公开问题，以及色散偏微分方程组的特解。该项目集中于关于稳定性的基本问题以及关于解的稳定流形的存在和精确描述，以及几个不稳定方向。虽然在过去二十年左右的时间里，稳定情景引起了人们的极大关注，但对接近不稳定解的行为的研究却很少。其中一个原因是，就必须使用的数学技术而言，不稳定的结构呈现在一个更具挑战性的环境中。这在存在边缘稳定的光谱、基本光谱的共振边缘和低维情况下尤其如此，所有这些都将是本项目的主要兴趣。该项目将专注于研究非线性色散方程，该方程对重要的数学过程进行建模，例如光在光学介质中的传播。其中一些模型产生于对量子力学系统和非线性光学的研究，而另一些模型则起源于流体动力学。这些问题也可以表现为镇定理论中的控制问题。粗略地说，如果一开始接近不稳定的配置，如何在整个过程中只进行微小的调整，以保持与初始配置的接近？对这些方程的解的行为，特别是它们在时间和空间上的渐近行为的更好的数学描述，将极大地提高我们对基本物理的理解，这将有助于使用它们的技术的发展。

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.