  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.
