A mathematical justification of the forced Korteweg-de Vries equation for capillary-gravity waves

The forced Korteweg-de Vries (KdV) equation is the KdV equation with a forcing term and arises as a model for several physical situations. In this paper, we study the validity of this modeling for capillary-gravity waves in an infinitely long canal over an uneven bottom. An underlying background flow of the water together with an uneven bottom causes a deriving force in the KdV equation in some scaling limit. We will show that the solutions of the full problem for capillary-gravity waves split up into two waves moving with different propagation speeds and that the shape of each wave is governed by a forced KdV equation in a slow time scale.