The Rigorous Approximation of Long-Wavelength Capillary-Gravity Waves

In a previous paper we proved that long-wavelength solutions of the water-wave problem in the case of zero surface tension split up into two wave packets, one moving to the right and one to the left, where each of these wave packets evolves independently as a solution of a Korteweg-de Vries (KdV) equation. In this paper we examine the effect of surface tension on this scenario. We find that we obtain three different physical regimes depending on the strength of the surface tension. For weak surface tension, the propagation of the wave packets is very similar to that in the zero surface tension case. For strong surface tension, the evolution is again governed by a pair of KdV equations, but the coefficients in these equations have changed in such a way that the KdV soliton now represents a wave of depression on the fluid surface. Finally, at a special, intermediate value of the surface tension (where the Bond number equals ⅓) the KdV description breaks down and it is necessary to introduce a new approximating equation, the Kawahara equation, which is a fifth order, nonlinear partial differential equation. In each of these regimes we give rigorous estimates of the difference between the solution of the appropriate modulation equation and the solution of the true water-wave problem.