  The collective oscillations of 1D repulsive Bose gas with external harmonic
confinement in two different regimes are studied. The first regime is the mean
field regime when the density is high. The second regime is the Tonks-Girardeau
regime when the density is low. We investigate the resonances under periodic
modulations of the trap potential and the effective nonlinearity. Modulations
of the effective nonlinear coefficient result from modulations of the atomic
scattering length by the Feshbach resonance method or variations of the
transverse trap frequency. In the mean field regime we predict the bistability
in the nonlinear oscillations of the condensate. In the Tonks-Girardeau regime
the resonance has the character of a linear parametric resonance. In the case
of rapid strong modulations of the nonlinear coefficient we find analytical
expressions for the nonlinearity managed soliton width and the frequency of the
slow secondary oscillations near the fixed point.
