### Three-dimensional water waves

Groves, M *(Loughborough University)*

Wednesday 23 July 2014, 16:15-17:00

Seminar Room 1, Newton Institute

#### Abstract

The existence of solitary-wave solutions to the three-dimensional
water-wave problem with is predicted by the Kadomtsev-Petviashvili (KP)
equation for strong surface tension and Davey-Stewartson (DS) equation
for weak surface tension.The term solitary wave describes any solution which
has a pulse-like profile in its direction of propagation, and these model
equations admit three types of solitary waves. A line solitary wave
is spatially homogeneous in the direction transverse to its direction of
propagation, while a periodically modulated solitary wave is periodic
in the transverse direction. A fully localised solitary wave on the
other hand decays to zero in all spatial directions.
In this talk I outline mathematical results which confirm the existence
of all three types of solitary wave for the full gravity-capillary water-wave
problem in its usual formulation as a free-boundary problem for
the Euler equations. Both strong and weak surface tension are treated.
The line solitary waves are found by establishing the existence of a low-dimensional
invariant manifold containing homoclinic orbits. The periodically modulated solitary
waves are created when a line solitary wave undergoes a dimension-breaking
bifurcation in which it spontaneously loses its spatial homogeneity in the transverse
direction; an infinite-dimensional version of the Lyapunov centre theorem
is the main ingredient in the existence theorem. The fully localised solitary
waves are obtained by finding critical points of a variational functional.

#### Video

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