Collective oscillations of one-dimensional Bose-Einstein gas in a time-varying trap potential and atomic scattering length

The collective oscillations of one-dimensional (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 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. We confirm the analytical predictions by direct numerical simulations of the 1D Gross-Pitaevskii equation and the effective nonlinear Schrödinger equation with quintic nonlinearity and trap potential.