Bifurcations of a weighted-residual integral boundary-layer model for nonlinear dynamics of falling liquid films

The nonlinear dynamics of thin liquid films falling on a vertical plane is investigated numerically using the first-order time-dependent weighted-residual integral boundary layer (WRIBL) equations derived by Ruyer-Quil and Manneville (2000). We find that sufficiently close to the stability threshold of the system with periodic boundary conditions, the emerging waves are of γ1-type. However, beyond a secondary bifurcation threshold, γ2-type waves emerge and can coexist with γ1 waves. The analysis of the WRIBL equations reveals the existence of both periodic traveling wave (TW) and aperiodic non-stationary wave (NSW) flows. The bifurcation structure of WRIBL equations is found to include three distinct regions: (i) linearly stable, (ii) bounded wave flows, (iii) reverse-flow solutions