This paper extends the numerical results of Hunter & Vanden-Broeck (1983) and
Vanden-Broeck (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 τ between 0 and 1/3, there are families
of ‘generalized’ solitary waves with periodic tails whose minimum amplitude is an
exponentially small function of F−1. An open problem (which, for τ sufficiently close
to 1/3, was recently proved by Sun 1999 to be false) is whether this amplitude can
ever be zero, which would give a truly localized solitary wave.The problem is first addressed in terms of model equations taking the form of
generalized fifth-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 generalized 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
τ 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 water wave
problem, at least for 9/50 < τ < 1=3. This finding confirms and explains previous
asymptotic results by Yang & Akylas.
Internal gravity-capillary solitary waves in finite depth
