## Introduction to movies of small-dispersivity regimes

**1. Initial conditions in modeling physical experiments**

This group of simulations were motivated by the experiments of Gollub and coworkers [See Liu et al., Phys. Fluids, 7, pp. 55-67 (1995)] on film flows down slightly inclined planes, and were done with corresponding values of parameters. We see, for corresponding initial conditions, the same patterns and transitions between them as in the physical experiments. The initial conditions are of the form $$\eta(y,z;t=0) = A_f \cos(n_f \tilde{z}) + A_h[ \cos(2n_f \tilde{z} - \phi_1) + \cos(3n_f \tilde{z} - \phi_2) ]$$ $$+ A_h[ \cos(n_f \tilde{z} - \phi_3) + \cos(2n_f \tilde{z} - \phi_4) + \cos(3n_f \tilde{z} - \phi_5) ] \cos \tilde{y}$$ $$+ A_{sub}[ \cos(0.5n_f \tilde{z} - \phi_6) + \cos(0.5n_f \tilde{z} - \phi_7) \cos \tilde{y} ] $$ $$ + small \ amplitude \ white \ noise,$$ where \(\tilde{y} = y/p\) and \(\tilde{z} = z/q\). They model the nearly monochromatic forcing in the physical experiments.

**2. Checkerboard patterns:**

The first two movies, numbers 10 and 11, show the period-doubling transition to checkerboard patterns. (Such transitions were observed in the physical experiments for sufficiently high frequencies of forcing.) The parameters \(\kappa\) and \(\varepsilon\) are 0.56 and 1/0.057 respectively.

For movie 10, we used \(p=8, \ q=16, \ A_f=1.2, \ A_h=0.0, \ A_{sub}=0.05 \), and \(\ n_f=15\); no white noise was added (a "simplified" initial condition).

For movie 11, we used \(p=8, \ q=23, \ A_f=1.2, \ A_h=0.05, \ A_{sub}=0.05 \), and \(\ n_f=22\); white noise was added (a more realistic initial condition).

**3. Synchronous instability:**

At smaller forcing frequencies, no period doubling appears. The movie 12, with the "frequency" \(n_f=9\) (other parameters are: \(\kappa = 0.18, \ \varepsilon = 1/0.05, \ p=8, \ q=16, \ A_f = 1.2, \ A_h = 0.05, \) and \(\ A_{sub} = 0.0\)) shows the synchronous instability, similar to those observed in physical experiments at intermediate frequencies of forcing. In this transition, the wave phase develops dependence on the spanwise coordinate.

**4. Solitary Waves**

At small forcing frequencies, solitary waves appeared in physical experiments. Correspondingly, movie 13 (\(n_f=2,\); other parameters are: \(\kappa = 0.56, \ \varepsilon = 1/0.057, \ p=8, \ q=16, \ A_f = 1.2, \ A_h = 0.0, \) and \(\ A_{sub} = 0.0\)) shows the transition to solitary waves.