  We investigate nonlinear periodic and solitary two-dimensional rolling waves
in a falling two-layer liquid film in the regime of non-zero Reynolds numbers.
At any flow rate, a falling two-layer liquid film is known to be linearly
unstable with respect to long-wave deformations of the liquid-air surface and
liquid-liquid interface. Two different types of zero-amplitude neutrally stable
waves propagate downstream without growing or shrinking: a zig-zag surface mode
and a thinning varicose interface mode. Using a boundary-layer reduction of the
Navier-Stokes equation, we investigate the onset, possible bifurcations and
interactions of nonlinear periodic travelling waves. Periodic waves are
obtained by continuation as stationary periodic solutions in the co-moving
reference frame starting from small-amplitude neutrally stable waves. We find a
variety of solitary waves that appear when a periodic solution approaches a
homoclinic loop. Wave interactions are studied using direct numerical
simulations of the boundary-layer model. We reveal that in the early stages of
temporal evolution coarsening is dominated by an inelastic collision and
merging of waves that travel at different speeds. Eventually, coarsening
becomes arrested when the waves have reached the largest admissible amplitude.
Bifurcation analysis confirms the existence of an upper bound of possible
solitary wave amplitudes, thus explaining the arrest of coarsening. In the
mixed regime, when both mode types are unstable, the temporal dynamics becomes
highly irregular due to the competition between a faster-travelling zig-zag
mode and a slower-travelling varicose mode. A quintessentially two-layer
dynamical regime is found, which corresponds to a ruptured second layer. In
this r
