title: Stability of ring patterns arising from two-dimensional particle interactions.
abstract: Pairwise particle interactions arise in diverse physical systems ranging from insect swarms to self-assembly of nanoparticles. In the presence of long-range attraction and short-range repulsion, such systems can exhibit bound states. We use linear stability analysis of a ring equilibrium to classify the morphology of patterns in two dimensions. Conditions are identified that assure the well-posedness of the ring. In addition, weakly nonlinear theory and numerical simulations demonstrate how a ring can bifurcate to more complex equilibria including triangular shapes, annuli, and spot patterns with N-fold symmetry. Many of these patterns have been observed in nature, although a general theory has been lacking, in particular how small changes to the interaction potential can lead to large changes in the self-organized state.
authors.affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada.
authors.forename: Theodore
authors.initials: T
authors.lastname: Kolokolnikov
authors.forename: Hui
authors.initials: H
authors.lastname: Sun
authors.forename: David
authors.initials: D
authors.lastname: Uminsky
authors.forename: Andrea L
authors.initials: AL
authors.lastname: Bertozzi
mesh_terms: D008962:Models, Theoretical; D017711:Nonlinear Dynamics