Nonlinear evolution of the step meandering instability of a growing crystal surface

The growth of crystal surfaces, under non-equilibrium conditions, involves the displacement of mono-atomic steps by atom diffusion and atom incorporations into steps. The time-evolution of the growing crystal surface is thus governed by a free boundary value problem [known as the Burton--Cabrera--Franck model]. In the presence of an asymmetry of the kinetic coefficients [Erlich--Schwoebel barriers], ruling the rates of incorporation of atoms at each step, it has been shown that a train of straight steps is unstable to two dimensional transverse perturbations. This instability is now known as the Bales-Zangwill instability (meandering instability). We study the non-linear evolution of the step meandering instability that occurs on a crystalline vicinal surface under growth, in the absence of evaporation, in the limit of a weak asymmetry of atom incorporation at the steps. We derive a nonlinear amplitude equation displaying spatiotemporal coarsening. We characterize the self-similar solutions of this equation.