Wave dynamics in the extended forced Korteweg-de Vries equation

The extended forced Korteweg-de Vries equation (efKdV) is a mathematical model for simulating the interaction of a shallow layer of fluid with external forcing agents. Herein we consider the dynamics of the efKdV when the forcing is assumed to be small and spatially periodic with a time-periodic variation of the phase. We show that a good heuristic understanding of the dynamics for a certain class of initial data can be found by studying a one-degree-of-freedom Hamiltonian system. One feature associated with this Hamiltonian system is that if the phase varies slowly with time, then to leading order the dynamics for the resonant solutions are governed by the forced nonlinear pendulum equation. Furthermore, we show that resonant solutions can correspond to waves which are trapped, i.e., waves which do not travel but instead oscillate. The theory is illustrated by numerical simulations.