Stability in Discrete and Continuous Dynamical Systems

PI将运用现代分析技术研究离散Schr\ odinger方程和Klein-Gordon方程环境下呼吸稳定性理论中一些突出的开放性问题，以及色散偏微分方程的特解。该项目集中于稳定性的基本问题，以及具有少数不稳定方向的解的稳定流形的存在性和精确描述。在过去20年左右的时间里，稳定方案受到了大量关注，而接近不稳定方案的行为却没有得到很好的研究。一个原因是，不稳定的结构在必须使用的数学技术方面呈现出更具挑战性的环境。在存在边缘稳定谱、基本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所在。该项目将重点研究非线性色散方程，该方程在数学上模拟重要的过程，例如光在光学介质中的传播。其中一些模型是在量子力学系统和非线性光学的研究中出现的，而另一些则是在流体动力学中找到根源的。这些问题在稳定理论中也可以表现为控制问题。粗略地说，如果一个人开始接近一个不稳定的构型，他如何在这个过程中只做很小的调整，以保持接近初始构型？更好的数学描述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，将极大地提高我们对基础物理的理解，并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.