Nonlocal Kardar-Parisi-Zhang equation to model interface growth

The dynamics of the growth of interfaces in the presence of noise and when the normal velocity is constant, in the weakly nonlinear limit, are described by the Kardar-Parisi-Zhang (KPZ) equation. In many applications, however, the growth is controlled by nonlocal transport, which is not contained in the original KPZ equation. For these problems we are proposing an extension of the KPZ model, where the nonlocal contribution is expressed through a Hilbert transform and can act to either stabilize or destabilize the interface. The model is illustrated with a specific example from reactive infiltration. The properties of the solution of the resulting equation are studied in one spatial dimension in the linear and the nonlinear limits, for both stable and unstable growth. We find that the early-time behavior has a power-law scaling similar to that of the KPZ equation. However, in the case of stable growth, the scaling of the saturation width is logarithmic, which differs from the power law in the KPZ equation. This dependence reflects the stabilizing effect of nonlocal transport. In the unstable case, we obtain results similar to those of Olami et al. [Phys. Rev. E 55, 2649 (1997)].