Localized and cellular patterns in a vibrated granular layer

We propose a phenomenological model for pattern formation in a vertically vibrated layer of granular material. This model exhibits a variety of stable cellular patterns, including standing rolls and squares as well as localized objects (oscillons and worms), similar to recent experimental observations [Umbanhowar, Melo, and Swinney, Nature (London) 382, 793 (1996)]. The model is an amplitude equation for the parametrical instability coupled to the mass conservation law. The structure and dynamics of the solutions resemble closely the properties of localized and cellular patterns observed in the experiments.