  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.
