Nonautonomous matter-wave solitons near the Feshbach resonance

By means of analytical and numerical methods, we reveal the main features of nonautonomous matter-wave solitons near the Feshbach resonance in a one-dimensional Bose-Einstein condensate confined by a harmonic potential with a varying-in-time longitudinal trapping frequency. Based on the generalized nonautonomous Gross-Pitaevskii model, we show that solitons in nonautonomous physical systems exist only under certain conditions so that varying-in-time nonlinearity and confining harmonic potential cannot be chosen independently; they satisfy the exact integrability scenarios and complement each other. We focus on the most physically important examples where the applied magnetic field is either a linearly or a periodically varying-in-time function. In the case of periodically varying scattering length, variations of confining harmonic potential are found to be sign-reversible (periodic attractive and repulsive) to support the soliton-management regime. We investigate the losses of validity of one-dimensional (1D) approximation in the cases when, by the joint action of varying-in-time nonlinearity and confining potential, the atom cloud can be compressed from an initially elongated quasi-1D cigar-shaped geometry to a final ball-shaped three-dimensional geometry and the induced soliton collapse may occur.