Steady states of the conserved Kuramoto-Sivashinsky equation [article]

Paolo Politi, Ruggero Vaia
2006 arXiv   pre-print
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: ∂_t u = -∂_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
more » ... n increasing branch in the space wavelength-amplitude (lambda-A), with d(lambda)/dA>0. For large wavelength, lambda=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 =0: a(lambda(t))=lambda^2(t)/48.
arXiv:cond-mat/0609545v1 fatcat:qav4mtgi4fatzoge47kzfz5f5e