Stability, Instability and Geometry in Applied Spectral Problems.

在工程和科学中研究的许多模型都承认某些类型的特殊解。这些特殊解，包括行波和其他相干结构，是重要的现象。它们通常与现实世界中观察到的行为相对应，因此理解它们很重要。一个重要的问题是鲁棒性：如果一个人在这些特殊解的一个附近取初始条件，他会想要预测这些解是否会有相似的行为，还是会有不同的行为。这种鲁棒性的概念，在数学中被称为稳定性，决定了哪些解决方案在实践中可能被观察到。由于真实的系统是有噪声的，并且具有不确定性，人们只可能观察到稳定的特殊解。该研究项目旨在理解和量化一些数学模型解决方案的稳定性，这些数学模型控制着各种现象，包括大规模电网络的行为和通道中的波传播。结果将包括证明证明这些解是稳定的定理，或者，如果它们是不稳定的，通过描述不稳定流形，从而给出附近解偏离问题解的某种方式。该项目侧重于在数学和科学的不同领域出现的一些模型，包括非线性薛定谔方程，各种浅水模型，用于振荡器同步的Kuramoto模型，以及考虑神经可塑性和Hebbian相互作用等现象的Kuramoto模型的各种推广。在找到不动点、行波和周期解等相干结构后，研究了相干结构的动力学线性化谱。我们感兴趣的是建立这些解的渐近稳定性或轨道稳定性，或者，在它们不稳定的情况下，计算不稳定流形的维数。我们利用几何和拓扑参数来计算不稳定流形的维数，从而解决了稳定性问题。

Many models studied in engineering and science admit some type of special solutions. These special solutions, including traveling waves and other coherent structures, are important phenomenon. They often correspond to behaviors observed in the real world, and as such it is important to understand them. One important question is that of robustness: if one takes initial conditions near one of these special solutions, one would like to predict if the solutions would have a similar behavior or do something different. This idea of robustness, known in mathematics as stability, governs which solutions are likely to be observed in practice. Since real systems are noisy and have uncertainty, one is only likely to observe special solutions that are stable. This research project is aimed at understanding and quantifying the stability of solutions to a number of mathematical models governing diverse phenomena, including the behavior of a large-scale electrical network and wave propagation in a channel. Results will include proving theorems that either show that these solutions are stable or, if they are unstable, by characterizing the unstable manifold, thereby giving some sense of the manner in which nearby solutions diverge from the solution in question. This project focuses on a number of models that arise in different areas of mathematics and the sciences, including the nonlinear Schrodinger equation, various shallow water models, the Kuramoto model for synchronization of oscillators, and various generalizations of the Kuramoto model that take into account phenomena such as neural plasticity and Hebbian interactions. After finding coherent structures such as fixed points, traveling waves, and periodic solutions, we study the spectrum of the linearization of the dynamics about the coherent structure. We are interested in establishing asymptotic or orbital stability of these solutions or, in the case where they are not stable, in counting the dimension of the unstable manifold. We approach the question on stability using geometric and topological arguments to count the dimension of the unstable manifold.