The KdV Equation on the Half-Line: Time-Periodicity and Mass Transport

The work presented here emanates from questions arising from experimental observations of the propagation of surface water waves. The experiments in question featured a periodically moving wavemaker located at one end of a flume that generated unidirectional waves of relatively small amplitude and long wavelength when compared with the undisturbed depth. It was observed that the wave profile at any point down the channel very quickly became periodic in time with the same period as that of the wavemaker. One of the questions dealt with here is whether or not such a property holds for model equations for such waves. In the present discussion, this is examined in the context of the Korteweg-de Vries equation using the recently developed version of the inverse scattering theory for boundary-value problems put forward by Fokas and his collaborators. It turns out that solutions of the Korteweg-de Vries equation generated by periodic forcing at the boundary do exhibit asymptotic temporal periodicity at any fixed point down the channel. However, a more subtle issue to do with conservation of mass fails to hold at the second order in a small parameter, which is the typical wave amplitude divided by the undisturbed depth.