A note on KAM for gravity-capillary water waves

Three-dimensional travelling gravity-capillary water waves

1.1 The hydrodynamic problem The classical water-wave problem concerns the irrotational flow of a perfect fluid of unit density subject to the forces of gravity and surface tension. The fluid motion is described by the Euler equations in a domain bounded below by a rigid horizontal bottom {y = −h} and above by a free surface which is described as a graph {y = η(x, z, t)}, where the function η d...

Three-dimensional Solitary Gravity-capillary Water Waves

The existence of solitary-wave solutions to the three-dimensional water-wave problem with strong surface-tension effects is predicted by the KP-I model equation. The term solitary wave describes any solution which has a pulse-like profile in its direction of propagation, and the KP-I equation admits explicit solutions for three different types of solitary wave. A line solitary wave is spatially...

Model Equations for Gravity-capillary Waves in Deep Water

The Euler equations for water waves in any depth have been shown to have solitary wave solutions when the effect of surface tension is included. This paper proposes three quadratic model equations for these types of waves in infinite depth with a two-dimensional fluid domain. One model is derived directly from the Euler equations. Two further simpler models are proposed, both having the full gr...

A plethora of generalised solitary gravity-capillary water waves

Quasi-periodic standing wave solutions of gravity-capillary water waves

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension.