A class of model equations for bi-directional propagation of capillary-gravity waves

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter alpha = a/h(0) and wavelength parameter beta = (h(0)/l)(2), where a and l are the actual amplitude and wavelength of the surface wave, and h(0) is the height of the undisturbed water surface from the flat bottom topography. These equations are also characterized by the surface tension parameter, namely the Bond number tau = Gamma /pgh(0)(2) where Gamma is the surface tension coefficient, p is the density of water, and g is the acceleration due to gravity. The traveling solitary wave solutions are explicitly constructed for a class of lower order Boussinesq system. From the Boussinesq equation of higher order, the appropriate equations to model solitary waves are derived under appropriate scaling in two specific cases: (i) beta much less than (1/3 - tau) less than or equal to 1/3 and (ii) (1 /3 - tau) = 0 (beta). The case (i) leads to the classical Boussinesq equation whose fourth-order dispersive term vanishes for tau = 1/3. This emphasizes the significance of the case (ii) that leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation. (C) 2002 Elsevier Science Ltd. All rights reserved.