I could find 1D and 2D form of Kuramoto-Sivashinski equation. See e.g. 2D equation.
$ u_{t} + \nabla^{2}u + \nabla^{4}u + \frac{1}{2}|\nabla u|^2 = 0 $
But I could not find explicit components expression of this equation. I am quite sure that biharmonic operator should be in 2D wikipedia definition
$ \Delta^{2}= \nabla^{4} = \partial_{xxxx} + 2\partial_{xxyy} + \partial_{yyyy} $.
But I am not sure how $ |\nabla u|^{2} $ term looks like. If we denote x and y components of "velocity" $ u(x,y,t) $ as $ w $ and $ v $ I guess system of partial equations should be like
$ w_{t} = -ww_{x}-vw_{y} - w_{xx} - w_{yy} - w_{xxxx} - 2w_{xxyy}-w_{yyyy}$ $ v_{t} = -wv_{x}-vv_{y} - v_{xx} - v_{yy} - v_{xxxx} - 2v_{xxyy}-v_{yyyy}$
1D Kuramoto equation should be then (I am sure about that) $ w_{t} = -ww_{x} - w_{xx} - w_{xxxx}$. Can somebody show, if component expansion is correct?