Heriot-Watt Mathematics Report Series
HWM00-28, 15 Dec 2000
Do true elevation gravity-capillary solitary waves exist? a numerical investigation
A R Champneys, J M Vanden-Broek and G J Lord
Abstract
This paper extends the numerical results of Hunter and Vanden-Broek
(1983) and Vanden-Broek (1991) which were concerned with studies
of solitary waves on the surface
of fluids of finite depth under the action of gravity and surface tension.
The aim of this paper is to answer the question
of whether small-amplitude elevation solitary waves exist.
Several analytical results have proved that bifurcating from
Froude number $F=1$, for Bond number $\tau$
between 0 and 1/3, there are families `generalised' solitary waves
with periodic tails whose minimum amplitude is an exponentially
small function of $F-1$. An open problem (which, for $\tau$ sufficiently
close to $1/3$, was recently proved by S.-M. Sun to be false) is
whether this amplitude can ever be zero, which would give
a truly localised solitary wave.
The problem is first addressed in terms of model equations taking the
form of generalised 5th-order KdV equations, where it is demonstrated
that if such a zero-tail amplitude solution occurs, it does so along
codimension-one lines in the parameter plane. Moreover, along solution
paths of generalised solitary waves a topological distinction is
found between cases where the tail does vanish and those where it does
not. This motivates a new set of numerical results for the full
problem, formulated using a boundary integral method, namely to probe
the size of the tail amplitude as $\tau$ varies for fixed $F>1$.
The strong conclusion from the numerical results is that true
solitary waves of elevation do not} exist for the steady
gravity-capillary waver wave problem at least for $9/50 < \tau < 1/3$.
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Publication
A R Champneys, J M Vanden-Broeck, G J Lord
, Do true elevation gravity-capillary solitary waves exist? A numerical investigation, Journal of Fluid Mechanics, 454, 403-417, (2002).
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