TY - UNPB

T1 - Do true elevation gravity-capillary solitary waves exist? a numerical investigation

AU - Champneys, AR

AU - Vanden-Broek, J-M

AU - Lord, GJ

N1 - Additional information: Preprint of a paper later published by Cambridge University Press (2002), Journal of Fluid Mechanics, 454, pp.403-417, ISSN 0022-1120

PY - 2000

Y1 - 2000

N2 - 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 of `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 greater than 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 tau between 9/50 and 1/3.

AB - 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 of `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 greater than 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 tau between 9/50 and 1/3.

M3 - Working paper and Preprints

BT - Do true elevation gravity-capillary solitary waves exist? a numerical investigation

ER -