Solitary Waves on a Two-Layer Fluid

This paper deals with weakly nonlinear long gravity waves on a stably stratified two-layer fluid. By using the reductive perturbation method, it is found that the fast mode is always governed by a Korteweg-de Vries (K-dV) equation whose coefficients depend on the thickness ratio and the density ratio. On the other hand, the slow mode is also governed, in general, by another K-dV equation except near and at the critical thickness ratio. At the critical thickness ratio, however, the slow mode is shown to be governed by a modified K-dV equation with cubic nonlinearity and near the critical thickness ratio it is governed by an equation of a combined form of the K-dV and modified K-dV equation. Steady solitary wave solutions to these equations are investigated in detail. A special solution representing a dispersive bore or hydraulic jump is also found.