Abstract
Using the forced Korteweg-de Vries equation as a model, we investigate the interaction of a solitary wave with an isolated moving external force of small amplitude. We use both analysis and numerical calculations. Our analysis is based on a multi-scale perturbation asymptotic analysis and is carried through to either the first- or second-order, depending on whether the first or more accurate second-order expression for the solitary wave speed is used. The perturbation theory leads to a pair of autonomous equations for the solitary wave amplitude and position and these are analyzed in detail for the two extreme cases when the lengthscale of the external force is either much greater, or much less, than that of the solitary wave. The theory predicts that the main régimes 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 either partially or totally trapped at the location of the external force. These theoretical predictions are confirmed in our numerical simulations of the forced Korteweg-de Vries equation.
Published Version
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