Steady states of the conserved Kuramoto-Sivashinsky equation

Recent work on the dynamics of a crystal surface [T. Frisch and A. Verga, Phys. Rev. Lett. 96, 166104 (2006)] has focused the attention on the conserved KuramotoSivashinsky (CKS) equation: ∂tu = −∂xx(u+ uxx + ux), which displays coarsening. For a quantitative and qualitative understanding of the dynamics, the analysis of steady states is particularly relevant. In this paper we provide a detailed study of the stationary solutions and their explicit form is given. Periodic configurations form an increasing branch in the space wavelength–amplitude (λ–A), with dλ/dA > 0. For large wavelength, λ = 4 √ A and the orbits in phase space tend to a separatrix, which is a parabola. Steady states are found up to an additive constant a, which is set by the dynamics through the conservation law ∂t〈u(x, t)〉 = 0: a(λ(t)) = λ2(t)/48.