  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 presence of a
physically relevant term breaking the up-down symmetry of the surface and which
can not be derived from a free energy. Following the treatment introduced by
Kawasaki and Ohta [Physica 116A, 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 neighbouring 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 $L(t)$ between kinks does
not change: $L(t)\sim\ln t$ in absence of noise, and $L(t)\sim t^{1/3}$ in
presence of (shot) noise. However, the cross-over 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.
