## A variational reduction and the existence of a fully-localised solitary wave for the three-dimensional water-wave problem with weak surface tension. (arXiv:1603.09189v2 [math.AP] UPDATED)

Fully localised solitary waves are travelling-wave solutions of the
three-dimensional gravity-capillary water wave problem which decay to zero in
every horizontal spatial direction. Their existence has been predicted on the
basis of numerical simulations and model equations (in which context they are
usually referred to as `lumps'), and a mathematically rigorous existence theory
for strong surface tension (Bond number $\beta$ greater than $\frac{1}{3}$) has
recently been given. In this article we present an existence theory for the
physically more realistic case $0<\beta<\frac{1}{3}$. A classical variational
principle for fully localised solitary waves is reduced to a locally equivalent
variational principle featuring a perturbation of the functional associated
with the Davey-Stewartson equation. A nontrivial critical point of the reduced
functional is found by minimising it over its natural constraint set.查看全文