# Solitary-like Wave Dynamics in Thin Liquid Films over a Vibrated Inclined Plane
## 4 Theory
The Floquet exponents are related to the monodromy matrix \(\mathbf{M}\) via \(\mathrm{Re}(\lambda)=\frac{1}{2}\ln(|\Lambda|)\), where \(\Lambda\) is the eigenvalue of \(\mathbf{M}\). The \(2\times 2\) complex-valued monodromy matrix \(\mathbf{M}\) is given by the fundamental solution matrix that is obtained by writing Equation (4) as a system of two first-order equations and integrating it over one period \(T\) with the unit \(2\times 2\) matrix as the initial condition. For the inclination angle \(\theta=3^{o}\), we choose the thickness of the water film \(h_{0}=0.6\,\mathrm{mm}\), which is slightly above the critical value for the gravitational instability of \(h_{0}=0.48\,\mathrm{mm}\). The corresponding Reynolds number in the absence of vibration is \(\mathcal{R}e=q_{0}/\
u=\frac{g\sin(\theta)h_{0}^{3}}{3\
u^{2}}\approx 37\). The definition of the Froude number \(\mathcal{F}r\) depends on the source and according to [23] is given by \(\mathcal{F}r=\mathcal{R}e^{2}\
u^{2}/\left(g\sin(\theta)h^{3}\right)\approx 12\) in the absence of vibration. When the vibration is switched on, the characteristic flow speed changes so that the Reynolds and Froude numbers must be redefined. Thus, according to [44], \(\mathcal{F}r=2\pi fh/\sqrt{g\sin(\theta)h}\approx 10\), when vibrated at \(f=48\,\mathrm{Hz}\). The marginal stability curves that correspond to \(\mathrm{Re}(\lambda)=0\) are shown in Figure 6 for four different vibration frequencies \(f=18,20,25,48\,\mathrm{Hz}\). The critical wavevector of the gravitational instability Equation (3) is marked by \(k_{c}\) in Figure 6d and it remains unaffected by the vibration. The shaded regions in Figure 6d indicate the unstable areas. The Faraday instability sets in at the vibration amplitude \(a_{c}\) that corresponds to the tip of the lowest Faraday tongue. The value of \(a_{c}\), as extracted from Figure 6, slightly increases with \(f\), namely: \(a_{c}=0.33\) for \(f=18\,\mathrm{Hz}\), \(a_{c}=0.35\) for \(f=20\,\mathrm{Hz}\), \(a_{c}=0.38\) for \(f=25\,\mathrm{Hz}\) and \(a_{c}=0.48\) for \(f=48\,\mathrm{Hz}\). This observation confirms the earlier statement that, for the range of frequencies between \(30\,\mathrm{Hz}\) and \(50\,\mathrm{Hz}\), the surface waves are much more sensitive to the changes of the vibration amplitude \(a\) than to the changes of the vibration frequency \(f\)Indeed, comparing Figure 6c,d we see only a marginal difference in the critical amplitude \(a_{c}\) when the frequency is doubled. On the other hand, increasing the value of \(a\) from \(a=0.5\) to \(a=1\) will significantly broaden the band of unstable wavevectors of the Faraday instability, thus significantly changing the dynamics of the surface waves. To better understand the temporal signature of the surface waves in response to vibration, we compute the imaginary part of the Floquet exponent \(\mathrm{I}m(\lambda)=\frac{\omega}{2\pi}(\arg(\Lambda))+\omega n\), where, as before, \(\Lambda\) is the eigenvalue of the monodromy matrix and \(n\) is an arbitrary integer. Any neutrally stable wave, i.e., \(\mathrm{Re}(\lambda)=0\), can be represented in the form \(\hbar(x,t)=e^{\mathrm{i}kx}H(t)e^{\mathrm{i}t}=H(t)e^{\mathrm{i}kx+i\mathrm{I }m(\lambda)t}\), where \(H(t)\) is a bounded \(2\pi/\omega\)-periodic function. Therefore, the temporal spectrum of such a neutrally stable wave contains delta peaks located at \(\frac{\omega}{2\pi}(\arg(\Lambda))+\omega n\). The temporal spectrum of a growing wave with \(\mathrm{Re}(\lambda)>0\) contains the same delta peaks that will appear slightly smeared. At this stage, it is important to emphasise that the temporal response of the surface waves that develop on the surface of a liquid layer on a vibrated incline is not necessarily harmonic (frequencies \(\omega n\)) or subharmonic (frequencies \(\omega/2+\omega n\)). This feature is in stark contrast to the standard Faraday instability in horizontal liquid layers, when the neutrally stable waves are always harmonic or subharmonic standing waves [51].