A Kuramoto–Sivashinsky equation in two space
dimensions arising in thin film flow is considered on
doubly periodic domains. In the absence of dispersive
effects, this anisotropic equation admits chaotic
solutions for sufficiently large length scales with
fully two-dimensional profiles; the one-dimensional
dynamics observed for thin domains are structurally
unstable as the transverse length increases. We
find that, independent of the domain size, the
characteristic length scale of the profiles in the
streamwise direction is about 10 space units, with that
in the transverse direction being approximately three
times larger. Numerical computations in the chaotic
regime provide an estimate for the radius of the
absorbing ball in
L
2
in terms of the length scales, from
which we conclude that the system possesses a finite
energy density. We show the property of equipartition
of energy among the low Fourier modes, and report
the disappearance of the inertial range when solution
profiles are two-dimensional. Consideration of the
high frequency modes allows us to compute an
estimate for the analytic extensibility of solutions
in
C
2
. We examine the addition of a physically
derived third-order dispersion to the problem; this
has a destabilising effect, in the sense of reducing
analyticity and increasing amplitude of solutions.
However, sufficiently large dispersion may regularise
the spatiotemporal chaos to travelling waves. We
focus on dispersion where chaotic dynamics persist,
and study its effect on the interfacial structures,
absorbing ball, and properties of the power spectrum.