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The dynamics of fluxons in a long Josephson junction driven by time-varying non-uniform bias currents are described by a generalization of the sine-Gordon equation. This equation has solitary wave solutions which correspond to current vortices or quantized packets of magnetic flux in the junction. As with the sine-Gordon equation, multi fluxon solutions may be demonstrated for the long Josephson junction. Our numerical calculations show that several fluxons may be launched or annihilated at the end of a junction. We also show multiple steady-state conditions which correspond to one or more flux quanta trapped in the junction.