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This article is focused on two related topics within the study of partial differential equations (PDEs) that illustrate a beautiful connection between dynamics, topology, and analysis: stability and spatial dynamics. The first is a property of solutions that describes the extent to which they can be expected to persist, and hence be observed, over long time scales. The second is a perspective that has been used to study various properties, such as stability, of nonlinear waves and coherent structures, the term often used to describe the solutions of interest in the class of PDEs that will be considered here.

Spectral Stability and Spatial Dynamics in Partial Differential Equations