Model equations for water waves in the presence of surface tension

We consider the system describing nonlinear waves on the free surface of an inviscid fluid layer. In the presence of surface tension two-dimensional waves are characterized by two parameters: λ, the inverse square of the Froude number and b the Bond number. A general reduction method is applied to derive, from the full system, reduced equations in which the bounded coordinate of the domain is eliminated. Two cases are treated: b > 1/3, λ near λ 0 = 1 and (b, λ) near the singular point (1/3, 1). After suitable scalings the limiting reduced systems lead to the Korteweg-de Vries equation and the Kawahara equation respectively.