Georgia Institute of Technology College of Sciences

In this talk, I will discuss localized stationary 1D and 2D structures such as hexagon patches, localized radial target patterns, and localized 1D rolls in the Swift-Hohenberg equation and other models. Some of these solutions exhibit snaking: in parameter space, the localized states lie on a vertical sine-shaped bifurcation curve so that the width of the underlying periodic pattern, such as hexagons or rolls, increases as we move up along the bifurcation curve. In particular, snaking implies the coexistence of infinitely many different localized structures. I will give an overview of recent analytical and numerical work in which localized structures and their snaking or non-snaking behavior is investigated.