Institute for Mathematics and Its Applications
Mary Silber, Northwestern University
``Superlattice patterns'' are characterized as having spatial structure on two disparate length scales; they are spatially-periodic on the large scale, and have beautiful intricate structure on the smaller scale. Such patterns were recently observed in experiments on parametrically-excited surface waves by Kudrolli, Pier and Gollub. I will review results of equivariant bifurcation theory that show that patterns similar to those seen in the experiments arise in a generic symmetry-breaking bifurcation of a spatially-uniform state. These bifurcation results will then be applied to a general two-component reaction-diffusion system in the vicinity of a Turing instability. By considering a particular degenerate bifurcation problem we are able to show that the transitions observed in the hydrodynamic problem cannot be reproduced in the chemical reaction-diffusion system.