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Three-dimensional solitary waves in the presence of additional surface effects. (English) Zbl 0933.76012
(Authors’ abstract.) We analyze bifurcations from the quiescent state of three-dimensional water wave solutions of a sixth-order model equation. The equation in question is a generalization of the Kadomtsev-Petviashvili equation, and is obtained by taking into account certain surface effects. These effects are caused either by the 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. We obtain solutions having the form of a solitary wave with damped oscillations, propagating in a channel. 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.

MSC:
76B25 Solitary waves for incompressible inviscid fluids
76B45 Capillarity (surface tension) for incompressible inviscid fluids
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
35Q51 Soliton equations
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