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 α=a/h₀ and wavelength parameter β=(h₀/l)², where a and l are the actual amplitude and wavelength of the surface wave, and h₀ 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 τ=Γ/ρgh₀², where Γ is the surface tension coefficient, ρ 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) β?(1/3−τ)?1/3 and (ii) (1/3−τ)=O(β). The case (i) leads to the classical Boussinesq equation whose fourth-order dispersive term vanishes for τ=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.