  One of the open problems of the formation and evolution of the nanostructures
on solid surfaces that are driven by particle irradiation is whether all the
nano-patterns can be accounted for by the continuum theory, since the
Cuerno-Barabasi continuum theory explanations have thus far focused mainly on
ripple patterns and rough self-affine scaling. In this article, we extend
continuum theoretical calculations based on the nonlinear Cuerno-Barabasi
theory to regions yet unexplored in the continuum theory literature but
recently shown, in Monte Carlo simulations, to be devoid of ripple patterns. We
obtained results of weighted ion penetrations and for different impingement
angles. Our results show that the balance of sputtering coefficients required
for ripple patterns was never attained in this region, which confirmed that
ripple patterns are indeed absent and that the continuum theory does not fail
as previous studies indicated. In particular our results show a remarkable
agreement between the continuum and the discrete (solid-on-solid) models; by
calculations based on the continuum theory cascade parameters alone, we
obtained the same topographic phase diagram predicted in the recent Monte Carlo
simulation studies.
