Stability and Spatial Dynamics

一个系统的特定状态，如沙漠植被的条纹图案或流体流动中的漩涡，是否能被观察到，取决于它对扰动的鲁棒性。如果系统中的一个小的扰动或波动会使它离开那个状态，那么这个状态不太可能持续很长时间，因此不太可能被观察到。这种状态是不稳定的。相反，稳定状态是一种即使存在微小扰动也能持续存在的状态，从而使其在物理上可以观察到。数学模型经常被用来帮助预测现实世界系统的演变。本项目的重点是发展分析这类模型解的稳定性的数学方法。检测数学模型中的给定状态是否稳定是预测其可观察性的关键步骤，从而预测现实世界的行为。在本项目中，使用拓扑、几何和分析工具分析偏微分方程描述的模型的稳定性，重点关注具有多个空间维度的系统。由于许多现有的数学工具仅在一个空间维度中有效，而许多物理系统（如上面提到的那些系统）具有两个或更多空间维度，因此这一重点特别重要。研究生参与该项目的研究。当使用偏微分方程来模拟物理系统时，理解解的长期行为是很重要的。其中一个关键方面是确定模型的某些解决方案或连贯结构，并确定它们是否稳定。这意味着随着时间的增加，它们的小扰动仍然很小，从而使它们在物理上可以观察到。在一个空间维度中，空间动力学允许许多与非线性波有关的问题，特别是它们的稳定性，通过将空间变量视为类时间演化变量，将其置于动力系统框架中。在更高的维度中，空间动力学的概念没有很好的定义，因为一般来说，没有明显的空间变量可以看作是类时变量。最近，利用缩域族和将线性化算子的莫尔斯指数与与缩域族相连的马斯洛夫指数联系起来，对多维稳定性问题进行了重新刻划。这表明马斯洛夫指数可以成为确定多维非线性波谱稳定性的有力工具，并且可以通过索引域的一维变量将空间动力学推广到多维设置。这个项目的重点是开发这两种可能性。研究生参与该项目的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Whether or not a specific state of a system, such as a striped pattern in desert vegetation or a vortex in a fluid flow, is observed depends on how robust it is to perturbations. If a small perturbation, or fluctuation, in the system would drive it away from that state, then the state is unlikely to persist over long times and hence unlikely to be observed. Such a state is unstable. Conversely, a stable state is one that would persist even in the presence of small perturbations, thus rendering it physically observable. Mathematical models are often used to help predict the evolution of real-world systems. This project is focused on the development of mathematical methods for analyzing the stability of solutions of such models. Detecting whether or not a given state in a mathematical model is stable is a key step in predicting its observability, and hence in predicting real-world behavior. In this project, stability in models described by partial differential equations is analyzed using tools from topology, geometry, and analysis, with a focus on systems having more than one spatial dimension. Because many of the existing mathematical tools are valid in only one spatial dimension, while many physical systems, such as those mentioned above, have two or more spatial dimensions, this focus is of particular importance. Graduate students participate in the research of the project.Understanding the long-time behavior of solutions is important when using partial differential equations to model physical systems. A key aspect of this is identifying certain solutions or coherent structures of the model and determining if they are stable. This means that small perturbations of them remain small as time increases, thus rendering them physically observable. In one space dimension, spatial dynamics has allowed many problems related to nonlinear waves, and in particular their stability, to be cast in a dynamical systems framework by viewing the spatial variable as a time-like evolution variable. In higher dimensions, the notion of spatial dynamics is not well-defined, because in general there is no distinguished spatial variable to view as the time-like variable. Recently, multi-dimensional stability problems have been recast using a family of shrinking domains and by relating the Morse index of the linearized operator to a Maslov index connected with this family. This indicates both that the Maslov index could be a powerful tool for determining the spectral stability of multi-dimensional nonlinear waves and that a generalization of spatial dynamics to the multi-dimensional setting could be developed through the one-dimensional variable that indexes the domain. This project is focused on developing both of these possibilities. Graduate students participate in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.