Regular patterns on a periodic rectangle and their stability

Optimizing a pairwise potential given by Laplacian's Green's function is a classical problem that occurs in numerous applications, including spike patterns in reaction-diffusion systems, brownian motion in the presence of small traps, and oxygen transport flow. Here, we present recent results for a periodic rectangle. We classify regular lattice patterns and study their stability up to 200 particles. For example, there are exactly 27 distinct regular patterns with 30 spikes, of which only two are stable on a square, including a presumed global minimizer (there are also local minimizers that are non-regular). Stability of a lattice pattern on a long rectangle is analysed in detail, including instability thresholds leading to a low-mode deformation of a double-stripe pattern.

Prof. Theodore Kolokolnikov obtained his PhD from UBC in 2004 under supervision of Michael Ward. His research interests include pattern formation in PDEs, dynamical systems, mathematical modelling, stochastic processes, multi-particle systems and graph theory. He has been at Dalhousie University since 2006. He is the recipient of CAIMS/PIMS Earlier Career Award in Applied Mathematics prize in 2012. He was awarded NSERC Accelearator award in 2014, and Killam Professorship in 2018.