David M. Winterbottom, University of Nottingham

<H3>Pattern formation in a model of a vibrated granular layer</H3>

A phenomenological model for pattern formation in a vertically vibrated 
granular layer is re-examined. The model, proposed by Tsimring and Aranson 
[Phys. Rev. Lett. \textbf{79} 213], comprises two coupled partial 
differential equations: one describes the evolution of the short-scale 
pattern, while the other enforces conservation of granular material. In a 
layer of moderate horizontal extent, the model predicts that a variety of 
exotic regular patterns may be stable, according to the system parameters. 
We perform a new weakly nonlinear analysis of the model, taking into 
account the influence of conservation of mass which previous investigations 
neglected.  In a sufficiently wide layer, a stability analysis of regular 
one-dimensional roll and two-dimensional square patterns demonstrates that 
each may suffer a modulational instability, which tends to localize the 
pattern. In the regions where squares are unstable, numerical simulations 
reveal worm- or chain-like patterns.