# Solitary-like Wave Dynamics in Thin Liquid Films over a Vibrated Inclined Plane
## 4 Theory
In some special cases, however, such as that discussed in [45] for transversally vibrated falling liquid films, the magnitude of \(\frac{\omega}{2\pi}(\arg(\Lambda))\) may be close to zero or \(\omega/2\), leading to an almost harmonic or subharmonic response. For the fluid parameters used in the present study, the frequency of the Faraday mode is significantly shifted from \(\omega\) or \(\omega/2\), as shown in Figure 4b. Next, we simulate the experimental conditions at which the results shown in Figure 4b were obtained to gain a better understanding of how the vibration changes the dynamics of the waves in the early stages of evolution. Thus, we numerically integrate Equation (1) over the time interval of \(3\,\mathrm{s}\) in the system of length of \(60\,\mathrm{cm}\) with periodic boundaries. The vibration amplitude is \(a=0.8\), and the other parameters are the same as in Figure 6d. Figure 6: Marginal stability curves of the base flow Equation (2) for a \(0.6\,\mathrm{mm}\) water film on a \(\theta=3^{\circ}\) incline vibrated at (**a**) \(f=18\,\mathrm{Hz}\), (**b**) \(f=20\,\mathrm{Hz}\), (**c**) \(f=25\,\mathrm{Hz}\) and (**d**) \(f=48\,\mathrm{Hz}\). The shaded regions in panel (**d**) highlight the unstable areas, \(k_{c}\) indicates the critical wavevector of the gravitational long-wave instability Equation (3), and \(a_{c}\) marks the critical vibration amplitude when the Faraday instability sets in. As the initial conditions, we use zero flux and random initial perturbation of the flat film surface with the amplitude of \(10^{-3}\,\)mm. The dispersion map is obtained by taking the two-dimensional Fourier transformation of the solution \(h(x,t)\). The contour lines of the dispersion map that correspond to the level of 3% of its maximum are shown by the thick lines in Figure 7. The thin solid lines in Figure 7 correspond to the dispersion curves \(\text{Im}(\lambda)(k)\), computed from Equation (4) for \(a=0.8\) and \(f=48\,\)Hz. It can be seen that the results of the direct simulation of the full system Equation (1) are in perfect agreement with the dispersion curves of the small-amplitude surface waves. The dispersion map in Figure 7 is dominated by the delta peaks located at \(f=\pm 14\) and \(f=\pm 34\,\)Hz, thus confirming that the primary response of the liquid film to a harmonic vibration is neither harmonic, nor subharmonic. Qualitatively, the shift of the response frequency away from the standard for the Faraday instability subharmonic mode can be explained as follows. In a horizontal layer vibrated at frequency \(\omega\), the Faraday instability sets in the form of a standing wave oscillating at the subharmonic frequency \(\omega/2\). Any standing wave can be represented as a superposition of two plane waves travelling at the phase speed of \(c=\omega/(2k)\) in the opposite directions, i.e., \(h(x,t)=Ae^{i\omega t/2+ikx}+Ae^{i\omega t/2-ikx}+cc\). When the layer is slightly inclined with the positive direction pointing downstream, it would be reasonable to assume that the plane wave propagating downstream will increase its phase speed by some amount \(\delta c\), but the wave propagating upstream will decrease its phase speed by the same amount \(\delta c\). Assuming that the wavevector remains unaffected by a small inclination angle, the resulting solution is represented by \(h(x,t)=Ae^{i(\omega/2-k\delta c)t/2+ikx}+Ae^{i(\omega/2+k\delta c)t/2-ikx}+cc\). Therefore, the temporal spectrum of \(h(x,t)\) will contain delta peaks located at \(\pm(\omega/2+k\delta c)\) and \((\pm(\omega/2-k\delta c))\), in agreement with Figure 4.0  
Alongside the delta peaks, the dispersion map in Figure 7 also contains a band of linearly unstable plane waves with the wavevectors \(k<k_{c}\). These long waves are amplified as the result of the gravitational instability mode. It can be seen that the gravitational band falls perfectly on the central dispersion curve that passes through the origin.