Nucleation of localised pattern in continuous media
- Date: 09/14/2006
David Lloyd (University of Surrey, UK)
University of British Columbia
The formation of patterns from quiescence under the continuous
variation of a parameter has long been of interest across the physical
and life sciences since the pioneering work of Alan Turing. We describe
how spatially localised patches of pattern arise spontaneously in
experiments in a wide variety of nonlinear media including liquid
crystals, autocatalytic chemical reactions, gas discharge systems,
optical crystals and in solidification. Perhaps the most intriguing
examples of such patterns are small circularly symmetric spatially
localised subharmonic excitations (dubbed emph{oscillons}) that occur
in vertically vibrated granular materials, viscous fluids and plasmas.
Oscillons tend to form tightly packed strongly interacting clusters
which coexist with an undeformed background and cannot be captured by
theories of weakly interacting localised atoms. Here we present a
predictive theory for the nucleation and pattern selection of
emph{multi-dimensional} localised structures in quite general
continuous media, via the interplay between linear instability and
nonlinear bistability. We show how specific kinds of localised patterns
(spots, targets, hexagonal arrays etc.) are selected and emerge
subcriticality depending on the amount of bistability between the
background and finite-amplitude cellular patterns. These parameter
regions of localised pattern overlap as the amount of hysteresis in the
system is increased, explaining experimental results showing
competition between different localised patterns. Furthermore, using a
Maxwell point argument that goes well beyond one-dimensional theory, we
reveal a complex {em snaking} transition diagram that provides the
mechanism by which larger localised clusters form and self completion
occurs.
