String models of interfacial pattern formation

We introduce a new approach to describe the dynamics of interfacial pattern formation. The central idea is that local properties may dominate the dynamics of the interface in many realistic situations. Here we have chosen to present a model for dendritic solidification where the basic mechanism which allows pattern formation - the Mullins-Sekerka instability - is well understood. In this and many other cases where the dynamics is driven by an additional field such as thermal or chemical diffusion, the coupling of this field to the interface may be phenomenologically described by a boundary layer with its own dynamics. With the inclusion of the non-equilibrium effect of crystalline anisotropy, our preliminary results appear to reproduce the observed dynamics of dendritic growth.