On solitary waves running down an inclined plane

Abstract

We study the existence and the role of solitary waves in the instability of a fluid layer flowing down an inclined plane. The approach presented is fully nonlinear. Solitary waves steady in a moving frame are described by homoclinic trajectories of an associated ordinary differential equation. They are searched numerically for a given value of viscosity and surface tension. Several kinds of solitary waves can exist, characterized by their number n of humps. We investigate the stability of these waves by integrating the initial-value problem directly. Solitary waves with more than 1 hump did not appear in the simulation, and moreover a catastrophic behaviour took place for too large a Reynolds number (R [gsim ] R*1) or too large an amplitude, suggesting a finite-time singularity. The long-term evolution is shown to be a very slow relaxation to a steady state in a moving frame. The relation to the experimental observation of localized wavetrains is also discussed.