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Lieb's soliton-like excitations in harmonic trap

We study the solitonic Lieb II branch of excitations in one-dimensional Bose-gas in homogeneous and trapped geometry. Using Bethe-ansatz Lieb's equations we calculate the "effective number of atoms" and the "effective mass" of the excitation. The equations of motion of the excitation are defined by the ratio of these quantities. The frequency of oscillations of the excitation in a harmonic trap is calculated. It changes continuously from its "soliton-like" value ω_h/\sqrt{2} in the high density mean field regime to ω_h in the low density Tonks-Girardeau regime with ω_h the frequency of the harmonic trapping. Particular attention is paid to the effective mass of a soliton with velocity near the speed of sound.