Anisotropic Kuramoto-Sivashinsky Equation for Surface Growth and Erosion

Abstract

We study the anisotropic two-dimensional Kuramoto-Sivashinsky equation ${\ensuremath{\partial}}_{t}h=[\ensuremath{-}{\ensuremath{\partial}}_{x}^{2}\ensuremath{-}\ensuremath{\alpha}{\ensuremath{\partial}}_{y}^{2}\ensuremath{-}{({\ensuremath{\partial}}_{x}^{2}+{\ensuremath{\partial}}_{y}^{2})}^{2}]h+\frac{1}{2}[{({\ensuremath{\partial}}_{x}h)}^{2}+\ensuremath{\beta}{({\ensuremath{\partial}}_{y}h)}^{2}]$, with real parameters $\ensuremath{\alpha}$, $\ensuremath{\beta}$, which arises, e.g., in sputter erosion and epitaxial growth on vicinal surfaces. The nonlinearities stabilize the linear instability, leading to a state of bounded spatiotemporal chaos, only if $\ensuremath{\beta}>min(0, \ensuremath{\alpha})$. Otherwise the equation exhibits two symmetry-related families of exponentially growing solutions for which the nonlinearities cancel. The competition between the two families gives rise to a coarsening pattern of rippled domains.