Steady states of the conserved Kuramoto-Sivashinsky equation

Details Steady states of the conserved Kuramoto-Sivashinsky equation Steady states of the conserved Kuramoto-Sivashinsky equation

2006-09-21

arxiv.org/abs/cond-mat/0609545v1

Physics

Condensed Matter

Statistical Mechanics

11 pages

Scientific paper

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 =0: a(lambda(t))=lambda^2(t)/48.

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