Existence and Stability of Radially Symmetric Solutions to the Swift--Hohenberg Equation

Applied and Computational Mathematics Seminar
Monday, March 2, 2015 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 005
Professor Scott McCalla – Montana State University
Martin Short
The existence, stability, and bifurcation structure of localized radially symmetric solutions to the Swift--Hohenberg equation is explored both numerically through continuation and analytically through the use of geometric blow-up techniques. The bifurcation structure for these solutions is elucidated by formally treating the dimension as a continuous parameter in the equations. This reveals a family of solutions with an anomalous amplitude scaling that is far larger than expected from a formal scaling in the far field. One key advantage of the geometric blow-up techniques is that a priori knowledge of this scaling is unnecessary as it naturally emerges from the construction. The stability of these patterned states will also be discussed.