A high-symmetry crystal surface may undergo a kinetic instability during the growth, such that its late stage evolution resembles a phase separation process. This parallel is rigorous in one dimension, if the conserved surface current is derivable from a free energy. We study the problem in the presence of a physically relevant term breaking the up-down symmetry of the surface and that cannot be derived from a free energy. Following the treatment introduced by Kawasaki and Ohta [Physica A 116, 573 (1982)] for the symmetric case, we are able to translate the problem of the surface evolution into a problem of nonlinear dynamics of kinks (domain walls). Because of the break of symmetry, two different classes $(A$ and $B)$ of kinks appear and their analytical form is derived. The effect of the adding term is to shrink a kink $A$ and to widen the neighboring kink $B$ in such a way that the product of their widths keeps constant. Concerning the dynamics, this implies that kinks $A$ move much faster than kinks $B$. Since the kink profiles approach exponentially the asymptotical values, the time dependence of the average distance $L\left(t\right)\mathrm{}$ between kinks does not change: $L\left(t\right)\mathrm{}\sim \ln t$ in the absence of noise, and $L\left(t\right)\mathrm{}\sim t^{1/3}$ in the presence of (shot) noise. However, the crossover time between the first and the second regime may increase even of some orders of magnitude. Finally, our results show that kinks $A$ may be so narrow that their width is comparable to the lattice constant: in this case, they indeed represent a discontinuity of the surface slope, that is, an angular point, and a different approach to coarsening should be used.

• Received 16 December 1997

DOI:https://doi.org/10.1103/PhysRevE.58.281