Step Bunching as Formation of Soliton-Like Pulses in Benney Equation

We study the morphological evolution of a vicinal face that consists of repulsive straight steps moving in a surface diffusion field. Numerical integration of the equations of motion for steps shows that, above the critical undersaturation (or supersaturation, depending on the asymmetry in the step kinetics), unstable density fluctuation grows into periodic step bunches. We construct a continuum model by deriving a non-linear evolution equation for step density from the equations of step motion. Near above the critical point the step density obeys the Benney equation with a large dispersion term, numerical solutions of which reproduce basic features found in the original discrete step model.