  We study the stability and dynamics of traveling-front solutions of a
modified Kuramoto--Sivashinsky equation arising in the modeling of nanoscale
ripple patterns that form when a nominally flat solid surface is bombarded with
a broad ion beam at an oblique angle of incidence. Structurally, the linearized
operators associated with these fronts have unstable essential
spectrum---corresponding to instability of the spatially asymptotic
states---and stable point spectrum---corresponding to stability of the
transition profile of the front. We show that these waves are linearly
orbitally asymptotically stable in appropriate exponentially weighted spaces.
While the technical device of exponential weights allows us to accommodate the
unstable essential spectrum of individual waves in our linear analysis, it does
not shed light on the long-time pattern formation that is observed
experimentally and in numerical simulations. To begin to address this issue, we
consider a periodic array of unstable front and back solutions. While not an
exact solution of the governing equation, this periodic pattern mimics
experimentally observed phenomena. Our numerical experiments suggest that the
convecting instabilities associated with each individual wave are damped as
they pass through transition layers and that this stabilization mechanism
underlies the pattern formation seen in experiments.

Analysis of PDEs;Pattern Formation and Solitons