LOCALIZED AND CELLULAR-PATTERNS IN A VIBRATED GRANULAR LAYER

We propose a phenomenological model for pattern formation in a vertica lly 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 exp erimental observations [Umbanhowar, Melo, and Swinney, Nature (London) 382, 793 (1996)]. The model is an amplitude equation for the parametr ical instability coupled to the mass conservation law. The structure a nd dynamics of the solutions resemble closely the properties of locali zed and cellular patterns observed in the experiments.