Bounded and unbounded patterns of the Benney equation

Abstract

The boundedness of 2‐D liquid film flows on an inclined plane in the context of the regularized Benney, u_τ+λu²u_x+[(μu⁶−νu³)u_x]_x +σ{u³[u_xx/(1+ε²u_x²)^3/2]_x}_x=0, and the Benney (ε=0) equation are studied. Here u, x, τ are the rescaled film thickness, the longitudinal coordinate, and time, respectively; λ, μ, and ν are non‐negative constants determined at equilibrium; and ε is the parameter related to the film aspect ratio. For a vertical plane (ν=0) a critical curve λ=λ_c(μ) has been found bifurcating from the point (λ,μ)=(0,1) which divides the λ‐μ space into two domains. When λ≳λ_c(μ) the initial data evolves into modulating traveling waves similar to the solutions of the Kuramoto–Sivashinsky equation. However, when λ<λ_c(μ), either an infinite spike forms in the solution in finite time and the original Benney model breaks down or the solution of the regularized Benney equation forms an infinite slope when the wavelike solution attempts to become multivalued. In a tilted plane (ν≳0) the boundedness of the emerging pattern is sensitive to the choice of initial data. It is also found that the Benney equation does not describe wave breaking where the solution develops an infinite slope.