Three-dimensional solitary waves in the presence of additional surface effects

Abstract

Bifurcations from the quiescent state of three dimensional water wave solutions of a sixth order model equation are analysed. The equation in question is a generalization of the Kadomtsev-Petviashvili equation, and is obtained due to the presence of certain surface effects. These effects are caused either by a surface tension with Bond number close to 1 3 , or by an elastic ice-sheet floating on the water surface. The equation describing travelling waves is reduced to a system of ordinary differential equations on a center manifold. Solutions having the form of a solitary wave with damped oscillations, propagating in a channel, are obtained. In the direction transverse to the propagation they satisfy boundary conditions which are either periodic or of Dirichlet type. In the periodic case we find both asymmetric and symmetric waves. In particular, some of these solutions fill a gap in the speeds of the travelling waves where no two-dimensional solitary waves exist. We show that the critical spectra of the linear operators of the model equation and of the full water wave problem are identical.