Understanding fully localised 2D patterns with dihedral symmetry #

Dan J Hill, Jason J Bramburger, David J B Lloyd

14:30 Monday in 2Q49.

Part of the Dynamics of reaction-transport systems session.

Abstract #

The emergence of small-amplitude localised patterns is well understood in one spatial dimension, thanks in part to the theory of spatial dynamics developed by Kirchgässner in the 1980s. A key concept in this approach is treating an unbounded spatial direction as a time-like' variable and using tools from dynamical systems to prove the existence of localised (in time’) solutions. However, a key question emerges when considering higher spatial dimensions; how can one use such an approach to study patterns that are localised in multiple directions? Such patterns are often found in experiments and yet there is very little analytic theory regarding them, beyond a handful of examples.

In this talk, we will focus on 2D patterns that are localised in the radial direction and use radial spatial dynamics, introduced by Scheel in 2003, to study approximations for fully localised patterns with dihedral symmetries. We present various new classes of solutions that emerge in this framework, along with several new questions/phenomena for future study.