| The motion of a fluid with a free surface in a container subject to resonant forcing from the bottom and side is considered. The evolution of the free surface is shown to be governed by a forced KdV equation with an additional Fredholm integral term and is named the KdV-Fredholm equation. The steady-state of the forced KdV-Fredholm is then studied asymptotically in the limit of small dispersion. Three cases are considered, the forced KdV, corresponding to the KdV-Fredholm without the Fredholm term; the forced KdV with a weak Fredholm term; and the steady-state forced KdV-Fredholm equation. A generalized multiple scales approach is reviewed to study the forced KdV and the weak KdV-Fredholm equations. This approach is then combined with boundary layer analysis for integral equations to study the forced KdV-Fredholm. The analytic work is compared with numerical results. |
