UBC Math. Phys. Seminar: Jason Bramburger (Cancelled)

In this talk we will discuss how bistability in a spatially extended system can lead to fascinating localized steady-state solutions. We will primarily focus on the Swift-Hohenberg equation, which is a is known to support a variety of spatially localized steady-states. In one spatial dimensional the Swift-Hohenberg equation exhibits spatially localized steady-state solutions which give way to a bifurcation structure known as snaking. That is, these solutions bounce between two different values of the bifurcation parameter while ascending in norm. The mechanism that drives snaking in one spatial dimension is now well-understood, but recent numerical investigations indicate that upon moving to two spatial dimensions, radially-symmetric and hexagonal spatially-localized solutions take on a significantly different snaking structure. This talk details my recent work on explaining the bifurcation structure of localized solutions in higher space dimensions as well as discussing a number of open problems related to the formation of localized structures in bistable systems.