Long Nonlinear Waves in Resonance with Topography

The evolution of periodic long surface waves over a periodic bottom topography resonant with the waves is studied. Coupled Korteweg–de Vries equations are derived and describe the evolution in terms of interaction between right- and left-traveling waves. The coupling arises from the cumulative effect of wave scattering. We discuss the various conserved quantities of the system and compute solutions for the initial value problem and for the time-periodic problem of fluid "sloshing" in a tank. Some three-dimensional extensions are discussed.