  A thin liquid film falling on a uniformly heated horizontal plate spreads
into fingering ripples that can display a complex dynamics ranging from
continuous waves, nonlinear spatially localized periodic wave patterns (i.e.
rivulet structures) to modulated nonlinear wavetrain structures. Some of these
structures have been observed experimentally, however conditions under which
they form are still not well understood. In this work we examine profiles of
nonlinear wave patterns formed by a thin liquid film falling on a uniformly
heated horizonal plate. In this purpose, the Benney model is considered
assuming a uniform temperature distribution along the film propagation on the
horizontal surface. It is shown that for strong surface tension but relatively
small Biot number, spatially localized periodic-wave structures can be
analytically obtained by solving the governing equation under appropriate
conditions. In the regime of weak nonlinearity, a multiple-scale expansion
combined with the reductive perturbation method leads to a complex
Ginzburg-Landau equation, the solutions of which are modulated periodic pulse
trains which amplitude, width and period are expressed in terms of
characteristic parameters of the model.
