# Solitary-like Wave Dynamics in Thin Liquid Films over a Vibrated Inclined Plane
## 4 Theory
To capture rolling waves, we use a one-dimensional version of the Shkadov model, which is formulated as a set of two coupled nonlinear equations for \(h(x,t)\) and the local flux across the layer \(q(x,t)=\int_{0}^{h(x,t)}u(x,z,t)\,dz\), where \(u(x,z,t)\) is the longitudinal flow velocity and \(z\)-axis is perpendicular to the incline  
\[\rho\bigg{[}\partial_{t}q+\frac{6}{5}\partial_{x}\bigg{(}\frac{ q^{2}}{h}\bigg{)}\bigg{]} = -\frac{3\mu q}{h^{2}}+\sigma h\partial_{x}^{3}h-\rho g(t)\cos( \theta)h\partial_{x}h+\rho g(t)\sin(\theta)h,\] \[\partial_{t}h+\partial_{x}q = 0, \tag{1}\]  
where \(\mu\) is the dynamic viscosity, \(\sigma\) is the liquid-air surface tension, and the time-dependent gravity acceleration due to vibration is \(g(t)=g(1+a\cos(\omega t))\). The inclination of the plate leads to a re-distribution of the vertical vibration into a longitudinal \(g(t)\sin(\theta)\) and an orthogonal \(g(t)\cos(\theta)\) components, respectively. The base flow corresponds to a time-periodic spatially homogeneous flux \(q_{0}(t)\) and a flat film surface \(h_{0}=const\). From Equation (1) we obtain the following expression by setting \(\partial_{x}q_{0}(t)=\partial_{x}h_{0}=0\):  
\[q_{0}(t)=\frac{g\sin(\theta)h_{0}^{3}}{3\
u}\Bigg{[}1+\frac{a\cos(\omega t)}{(2 h_{0}^{2}/3l_{\rm ac}^{2})^{2}+1}+\frac{2h_{0}^{2}}{3l_{\rm ac}^{2}}\frac{a\sin( \omega t)}{(2h_{0}^{2}/3l_{\rm ac}^{2})^{2}+1}\Bigg{]}, \tag{2}\]  
where \(l_{\rm ac}=\sqrt{2\
u/\omega}\) represents the length of the acoustic boundary layer. In the absence of vibration, i.e., when \(a=0\), the base flow is the time-independent Nusselt flow, where the linear stability is well-known in the case of a falling film, i.e., at \(\theta=90^{o}\)[22; 23]. For an arbitrary inclination angle \(\theta\), the instability sets in when \(\mathcal{R}e>\cot(\theta)\), where \(\mathcal{R}e=q_{0}/\
u=\frac{g\sin(\theta)h_{0}^{3}}{3\
u^{2}}\) is the Reynolds number. To put this condition into perspective, for a water film on a \(\theta=3^{o}\) incline, the flow is unstable when \(h_{0}>0.48\,\mathrm{mm}\). The corresponding instability is called gravitational instability and it leads to the onset of long surface waves propagating downstream. The wavelength of unstable waves is longer than \(\lambda_{c}=2\pi/k_{c}\), where \(k_{c}\) is the critical wavevector of the gravitational instability  
\[k_{c}=\Bigg{(}\frac{\rho g\sin(\theta)}{\sigma}\Bigg{(}\frac{g\sin(\theta)h_{0 }^{3}}{3\
u^{2}}-\cot(\theta)\Bigg{)}\Bigg{)}^{1/2}. \tag{3}\]  
Neutrally stable waves with the wavelength \(\lambda_{c}=2\pi/k_{c}\) propagate downstream with a speed \(c\), which is twice as large as the surface speed in the Nusselt flow, i.e., \(c=g\sin(\theta)h_{0}^{2}/\
u\). When the vibration is switched on, the Faraday instability mode develops and it competes with the gravitational instability mode. To study the stability of the base flow Equation (2) we use the standard plane-wave ansatz \(q(x,t)=q_{0}(t)+\hat{q}(t)e^{ikx}\) and \(h(x,t)=h_{0}+\hat{h}(t)e^{ikx}\), where \(k\) is the wavevector of the small-amplitude perturbation. By differentiating the second equation in Equation (1) with respect to time and the first equation with respect to \(x\), the flux perturbation \(\tilde{q}\) can be eliminated to yield a complex-valued Mathieu-like equation for the film thickness perturbation \(\tilde{h}\)  
\[\partial_{tt}\tilde{h}+A(t)\partial_{t}\tilde{h}+B(t)\tilde{h}=0, \tag{4}\]  
with \(A(t)=\frac{3\
u}{h_{0}^{2}}+\frac{12}{5}ik\frac{\partial_{0}(t)}{h_{0}}\) and \(B(t)=ikg(t)\sin(\theta)+\frac{\rho}{\rho}h_{0}k^{4}+g(t)\cos(\theta)h_{0}k^{2 }+ik\delta\
u\frac{q_{0}(t)}{h_{0}^{2}}-\frac{6}{5}k^{2}\frac{q_{0}(t)^{2}}{h_ {0}^{2}}\). According to the Floquet theory, the solution of Equation (4) is given by \(\tilde{h}(t)=H(t)e^{\lambda t}\), where \(H(t)\) is some bounded periodic function with the period \(T=2\pi/\omega\) and \(\lambda\) is the Floquet exponent. The solution is stable when the real part of the largest Floquet exponent is negative, i.e., \(\mathrm{Re}(\lambda)<0\) and it is unstable otherwise.