On the dynamics of traveling fronts arising in nanoscale pattern formation

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. Furthermore, we illustrate that a periodic array of unstable front and back solutions forms a pattern that appears to be spectrally and nonlinearly stable in unweighted spaces.