Nucleation of localised pattern in continuous media

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.