TY - JOUR

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

AU - Champneys, A. R.

AU - Vanden-Broeck, J. M.

AU - Lord, G. J.

PY - 2002/3/10

Y1 - 2002/3/10

N2 - This paper extends the numerical results of Hunter and 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 t 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 t 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 zerotail-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 t 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 < t < 1/3. This finding confirms and explains previous asymptotic results by Yang and Akylas.

AB - This paper extends the numerical results of Hunter and 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 t 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 t 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 zerotail-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 t 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 < t < 1/3. This finding confirms and explains previous asymptotic results by Yang and Akylas.

UR - http://www.scopus.com/inward/record.url?scp=0037051411&partnerID=8YFLogxK

M3 - Article

SN - 0022-1120

VL - 454

SP - 403

EP - 417

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -