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We develop a theory of the generation of new spectral components in optical fibers pumped with a solitonic pulse and a weak continuous wave (cw). We derive the wave number matching conditions for the above process and present an analytical method of finding the amplitudes of the generated waves. We discuss related effects of the depletion of the cw pump and spectral recoil on the soliton. We also point out examples of the generation of supercontinuum spectra in fibers, where mixing between solitons and dispersive waves plays an important role.