# Solitary-like Wave Dynamics in Thin Liquid Films over a Vibrated Inclined Plane
## 4 Theory  
The theoretical description of a non-steady flow in the presence of deformable interfaces, such as the flow in a thin liquid layer down a vibrated incline, is a notoriously difficult hydrodynamic problem. The exact analysis is only available in the linear case, i.e., when the deformation amplitude of the liquid-air interface is much smaller than the  
Figure 5: Close-ups of the dispersion maps presented in Figure 4. The linear fits of the dispersion bands and the corresponding wave velocities are shown (**a,b**). average film thickness [42]. In the general case, when a harmonic vibration is applied both in the perpendicular and parallel directions with respect to the inclined plate, the unperturbed base flow is given by a superposition of a steady Nusselt flow and of an additional harmonically oscillating flow parallel to the incline with a flat free surface [42]. The Navier-Stokes equation for an incompressible fluid can be linearised about the base flow and the Floquet theory-based stability analysis determines if the flow is stable or unstable. The linear stability of a flat film flowing down an incline under the combined action of the longitudinal and orthogonal vibration has been investigated in Ref. [43] using a theoretical approach based on the exact linearisation of the Navier-Stokes equation [42]. The full nonlinear problem with a large amplitude deformation of the film surface can only be studied approximately using simplified hydrodynamic models. Here we use the well-known Shkadov model [52; 53], which can be derived from the Navier-Stokes equation by assuming a self-similar parabolic longitudinal velocity profile. The model developed by [PERSON] was used earlier to study nonlinear solitary waves in falling liquid films in the absence of vibration [22; 23] and to investigate the onset of Faraday waves in vertically vibrated isolated liquid drops [57]. Despite a long history of the boundary layer models and other simplified hydrodynamic models in thin liquid films dynamics, the nonlinear behaviour of the surface waves in a liquid film flowing down an incline in the presence of both the longitudinal and orthogonal vibrations remains largely unexplored. Indeed, earlier studies have mainly focused either on the nonlinear Faraday waves in liquid films on a horizontal plate subjected to horizontal and vertical vibrations [58] or on liquid films falling down a vertical plate with the vibration applied orthogonally to the plate surface [44; 45]. In this work, we use the Shkadov model [52; 53] to study linear stability and nonlinear dynamics of the surface waves in liquid layers on a slightly inclined plate mounted on a vertically vibrated table. In this situation, the driving force due to gravity is not as dominating as in falling liquid films and the inclination of the plate leads to a redistribution of the vibration into the longitudinal and orthogonal components. Our theoretical analysis presented below is used to validate the experimental results and to explain the nonlinear interaction between the long-wave gravitational instability mode and the short-wave Faraday mode. Thus, we consider a liquid film with the local film thickness \(h(x,t)\) flowing down an inclined solid plate that makes an angle \(\theta\) with the horizontal, as shown in Figure 1. In our model, the \(x\)-axis is chosen to be parallel to the plate with the positive direction pointing down the incline.