## Simulations of a dispersive-dissipative evolution equation

The movies are computer animations of simulations of a dispersive-dissipative evolution equation derived in the context of

liquid film flow down an inclined plane. The dimensionless evolution equation for the film thickness deviation (from its mean), \(\eta\),

is (in a suitable reference frame moving in the downstream direction) $${\eta_t + \eta\eta_z + {\nabla^2}\eta_z - \kappa\eta_{yy} + \varepsilon (\eta_{zz} + {\nabla^4}\eta) = 0},$$ where \({\nabla^2} = {\partial^2}/{\partial y^2} + {\partial^2}/{\partial z^2}\). Here \( y \) and \( z \) are, respectively, the spanwise and streamwise coordinates, \( t \) is time, and the

subscripts indicate the partial derivatives.

We used periodic boundary conditions on the extended spatial domains \(0 \leq y \leq 2{\pi}p \) (with \( p \gg 1 \)) and \(0 \leq z \leq 2{\pi}q \) (with \( q \gg 1)\).

The movies (*.mpg) are in the MPEG format. They can be viewed on any platform if you have the MPEG viewer for that platform.

For \(\kappa = 0\), the evolution equation reduces to a 2D generalization of the KdV equation for \(\varepsilon = 0\) and a 2D generalization of the

Kuramoto-Sivashinsky equation for \(\varepsilon \to \infty\). The movies have been grouped into three categories:

\(\bullet\) Large dispersivity \(\lambda = 1/\varepsilon \ (\varepsilon \ll 1)\).

\(\bullet\) Small dispersivity \((\varepsilon \gg 1)\).

\(\bullet\) Intermediate dispersivity.

The thickness of the film is indicated by the color: blue end of spectrum for larger thickness and the red end for smaller thickness.