Stability, Instability and Geometry in Applied Spectral Problems.

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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.