Fully-localised planar patterns with dihedral symmetry, including cellular hexagons and squares, have been found experimentally and numerically in various continuum models; for example, in nonlinear optics, semi-arid vegetation, and on the surface of a ferrofluid (a magnetic fluid). However, there is currently no mathematical theory for the emergence of these types of patterns. In this talk, I will present recent progress regarding the existence of localised dihedral patterns (not necessarily hexagon or square) emerging from a Hamiltonian--Hopf bifurcation for a general class of two-component reaction-diffusion systems.

The planar problem is approximated through a Galerkin scheme, where a finite-mode Fourier decomposition in polar coordinates yields a large, but finite, system of coupled radial differential equations. We then apply techniques from radial spatial dynamics to prove the existence of a zoo of localised dihedral patterns in the finite-mode reduction, subject to solving an (N+1)-dimensional algebraic matching condition. We conclude by studying this matching condition for various finite-mode reductions, and present a computer-assisted proof for the existence of localised patches with 6m-fold symmetry for arbitrarily large Fourier decompositions.

This work is in collaboration with Jason Bramburger (Concordia University) and David Lloyd (University of Surrey).