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 Kuramoto-Sivashinsky (CKS) equation: \partial_t u = -\partial_{xx}(u+u_{xx}+u_x^2), 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 (lambda-A), with d(lambda)/dA>0. For large wavelength, lambda=4\sqrt{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 \partial_t <u(x,t)>=0: a(lambda(t))=lambda^2(t)/48.