Higher order corrections to the KdV approximation for water waves

In order to investigate corrections to the common KdV approximation to long waves, we derive modulation equations for the evolution of long wavelength initial data for the water wave and Boussinesq equations. The equations governing the corrections to the KdV approximation are identical for both systems and are explicitly solvable. We prove estimates showing that they do indeed give a significantly better approximation than the KdV equation alone. We also present the results of numerical experiments which show that the error estimates we derive for the correction to the Boussinesq equation are essentially optimal.