Abstract
Using the forced Korteweg-de Vries equation as a canonical model, we investigate the interaction of a solitary wave with a moving external force of small amplitude. The analysis is a multi-scale asymptotic procedure, which leads to a pair of autonomous ordinary differential equations for the solitary wave amplitude and position. These are analysed in detail for the case when the lengthscale of the external force is much greater than that of the solitary wave. For the case of a single isolated force this theory predicts that the main regimes are passage, when the solitary wave passes through the external force with some amplitude modulation, repulsion, when the solitary wave is reflected by the external force with a significant amplitude change, and trapping, when the solitary wave remains in the vicinity of the external force location for an extended period of time. This theory is then extended to the case when the external force has two forcing centres, and we show that there is then a similar set of regimes but with rather more complicated dynamical behaviour.
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