One-mode solitary solutions for capillary-gravity waves on deep water

An approximate one-mode theory of capillary-gravity waves of solitary type on the surface of infinitely deep water is considered. In this theory, the stream function is described by the mth-order elementary solutions psi similar to cos(mpsi - psi(0))/R-m to Laplace's equation in the polar coordinates R and psi. Analytical solutions for symmetric (psi(0) = 0, pi) and asymmetric (psi(0) = pi/2) solitary waves (solitons) are obtained for in = 2, 1, and 1/2. The solitons of limiting form are shown to be unstable in all the cases under consideration, and their profiles have either one or two angular points, where the surface forms a 90degrees corner.