Long internal waves of finite amplitude

We derive new nonlinear evolution equations for long internal waves in a two-fluid system, when the thickness of the lower layer is effectively infinite, by making the only assumption that the thickness of the upper layer is small compared with the characteristic wavelength. The resulting equations have the full nonlinearity of the original problem and retain the leading-order dispersive effects. For large amplitude internal solitary waves, we show that our model captures the scaling relation between the amplitude and the characteristic wavelength observed experimentally by Koop and Butler.