Localized states in a model of pattern formation in a vertically vibrated layer

We consider a novel asymptotic limit of model equations proposed to describe the formation of localized states in a vertically vibrated layer of granular material or viscoelastic fluid. In physical terms, the asymptotic limit is motivated by experimental observations that localized states ("oscillons") arise when regions of weak excitation are nevertheless able to expel material rapidly enough to reach a balance with diffusion. Mathematically, the limit enables a novel weakly nonlinear analysis to be performed which allows the local depth of the granular layer to vary by O(1) amounts even when the pattern amplitude is small. The weakly nonlinear analysis and numerical computations provide a robust possible explanation of past experimental results.