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

doi:10.1016/S0997-7546(98)80023-X

European Journal of Mechanics - B/Fluids

Volume 17, Issue 5, September–October 1998, Pages 739-768

https://doi.org/10.1016/S0997-7546(98)80023-X Get rights and content

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 $13$ , 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.

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^b: Permanent address: Mathematical Steklov Inst., Gubkina str. 8, 117966 Moscow GSP-1, Russia

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

Abstract

Physica D

Physica D

Physica D

Physica D

J. Diff. Eqns

Adv. Appl. Mech.

J. Appl. Math. Mech.

Radiation and modulational instability described by the fifth-order Korteweg-de Vries equation

Contemp. Math.

Hamiltonian Spatial Structure for Three-Dimensional Water Waves in a Moving Frame of Reference

J. Nonlinear Sci.

Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system

J. Dyn. Stab. Sys.

Model equations for water waves in the presence of surface tension

Eur. J. Mech. B/Fluids

Breaking the Dimension of a Steady Wave: Some Examples