Anisotropic Kuramoto-Sivashinsky Equation for Surface Growth and Erosion

We study the anisotropic two-dimensional Kuramoto-Sivashinsky equation ${{∂}}ₜh=[{-}{{∂}}ₓ²{-}{α}{{∂}}y²{-}{({{∂}}ₓ²+{{∂}}y²)}²]h+1/2[{({{∂}}ₓh)}²+{β}{({{∂}}yh)}²]$, with real parameters ${α}$, ${β}$, 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 ${β}>min(0, {α})$. 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.