Nonlocal interface equations in crystal growth

Amplitude equations are usually partial differential equations which means that the time evolution of the system is local: for sufficiently short times, the dynamics at any given point is determined by the configuration in its neighbourhood only. In particular, this is true for long-wave equations, where the length scales of the system become small in comparison with the wavelength of the pattern. However, there are exceptions to this rule. It turns out that in systems with long-range interactions (e.g. elastic forces, hydrodynamic interactions) the nonlocal nature of the interaction survives the long-wave limit. More generally, nonlocal interface equations arise if nonanalytic terms are present in the dispersion relation of linear stability analysis.