Consider the generalized Kuramoto–Sivashinsky (gKS) equation. It is a model prototype for a wide variety of physical systems, from flame-front propagation, and more general front propagation in reaction– diffusion systems, to interface motion of viscous film flows. Our aim is to develop a systematic and rigorous low-dimensional representation of the gKS equation. For this purpose, we approximat...

A continuum model, based on the damped Kuramoto-Sivashinsky equation, is shown to reproduce the morphology evolution during ion sputtering quite successfully. In a very narrow range of the damping parameter a , the alignment of the structures into hexagonal domains is obtained under normal incidence of ions with striking resemblance to the experimentally observed dot patterns. The origin of thi...

A new theorem is applied to the Kuramoto-Sivashinsky equation with L-periodic boundary conditions, proving the existence of an asymptotically complete inertial manifold attracting all initial data. Combining this result with a new estimate of the size of the globally absorbing set yields an improved estimate of the dimension, N ∼ L.