Step motion, patterns, and kinetic instabilities on crystal surfaces

We study a mesoscopic kinetic model for step flow on crystal surfaces in the presence of impurities. The evolution of the system of steps in the small line tension limit leads to the formation of complex highly connected step patterns. We identify repeating features in the patterns and calculate them analytically by analyzing a simplified mean-field-like model. For the same model we also calculate analytically the coarsening with time of the typical length scale associated with the patterns. All the analytical results are in excellent quantitative agreement with numerical simulations.