Waves in liquid films on vibrating substrates

This paper is concerned with liquid films on horizontally vibrating substrates. Using an equation derived by Shklyaev [Phys. Rev. E 79, 051603 (2009)], we show that all periodic and solitary-wave solutions of this equation are unstable regardless of their parameters. Some of the solitary waves, however, are metastable-i.e., still unstable, but with extremely small growth rates-and, thus, can persist without breaking up for a very long time. The crests of these metastable waves are flat and wide, and they all have more or less the same amplitude (determined by the problem's global parameters). The metastable solitary waves play an important role in the evolution of films for which the state of uniform thickness is unstable. Those were simulated numerically, with two basic scenarios observed depending on the parameter A=3 ⁽ ω^/2ν ⁾ 1^/2 U02 /g, where ν is the kinematic viscosity, g is the acceleration due to gravity, and ω and U0 are the frequency and amplitude (maximum velocity) of the substrate's vibration. (i) If A≲25, a small number of metastable solitary waves with flat/wide crests emerge from the evolution and exist without coalescing (or even moving) for an extremely long time. (ii) If A≲25, the solution of the initial-value problem breaks up into a set of noninteracting pulses separated by regions where the film's thickness rapidly tends to zero.