CAPILLARY GRAVITY-WAVES OF SOLITARY TYPE AND ENVELOPE SOLITONS ON DEEP-WATER

The existence of steady solitary waves on deep water was suggested on physical grounds by Longuet-Higgins (1988) and later confirmed by numerical computation (Longuet-Higgins 1989; Vanden-Broeck & Dias 1992). Their numerical methods are accurate only for waves of finite amplitude. In this paper we show that solitary capillary-gravity waves of small amplitude are in fact a special case of envelope solitons, namely those having a carrier wave of length 2pi(T/rhog)1/2 (g = gravity, T = surface tension, rho = density). The dispersion relation c2 = 2(1-11/32 alpha(max)2) between the speed c and the maximum surface slope alpha(max) is derived from the nonlinear Schrodinger equation for deep-water solitons (Djordjevik & Redekopp 1977) and is found to provide a good asymptote for the numerical calculations.