Stability and Spatial Dynamics

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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.