Stability, Instability and Geometry in Applied Spectral Problems.

在工程和科学中研究的许多模型都允许某些类型的特殊解。这些特殊解，包括行波和其他相干结构，都是重要的现象。它们通常对应于在现实世界中观察到的行为，因此理解它们是很重要的。一个重要的问题是稳健性：如果一个人把初始条件放在这些特殊解的附近，人们会想要预测这些解是否会有类似的行为或做一些不同的事情。这种稳健性的概念在数学上被称为稳定性，它决定了在实践中可能会观察到哪些解决方案。由于真实系统是有噪声和不确定性的，人们只可能观察到稳定的特殊解。这一研究项目旨在了解和量化管理各种现象的一些数学模型的解的稳定性，包括大规模电子网络的行为和波在通道中的传播。结果将包括证明定理，这些定理要么表明这些解是稳定的，或者如果它们是不稳定的，则通过刻画不稳定的流形，从而给出附近的解与所讨论的解发散的方式的某种意义。这个项目集中在数学和科学的不同领域中出现的一些模型，包括非线性薛定谔方程，各种浅水模型，振子同步的Kuramoto模型，以及考虑到神经可塑性和Hebbian相互作用等现象的Kuramoto模型的各种推广。在找到了不动点、行波和周期解等相干结构后，我们研究了相干结构动力学线性化的谱。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度。我们用几何和拓扑学的方法来计算不稳定流形的维度来讨论稳定性问题。

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.