Anisotropic Kuramoto-Sivashinsky Equation for Surface Growth and Erosion

Abstract

We study the anisotropic two-dimensional Kuramoto-Sivashinsky equation ${\partial }_{t}h=[-{\partial }_x^2-\alpha {\partial }_y^2-{({\partial }_x^2+{\partial }_y^2)}^2]h+\frac{1}{2}[{\left({\partial }_{x}h\right)}^2+\beta {\left({\partial }_{y}h\right)}^2]$, with real parameters $\alpha$, $\beta$, 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 $\beta >min(0, \alpha )$. 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.