Solitary waves in a layer of viscous liquid

Abstract

Solitary waves in a thin layer of viscous liquid which is running down a vertical surface under the action of gravity are investigated. The existence of such waves was demonstrated in the experiments of [1, 2]. The difficulties that must be faced in a theoretical computation were also noted in these studies. Below a solution of the problem of stationary waves is obtained by the method of expansion in the small parameter in two regions with subsequent matching and also by a numerical integration method. It is shown that in each case a solution of solitary wave type exists along with the single-parameter family of periodic solutions (parameter—the wave number α). On decreasing the wave number, the periodic waves go over into a succession of solitary waves.