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The dynamics of a bright matter wave soliton in a quasi 1D Bose-Einsteincondensate with periodically rapidly varying trap is considered. The governingequation is derived based on averaging over fast modulations of theGross-Pitaevskii (GP) equation. This equation has the form of GP equation witheffective potential of more complicated structure than unperturbed trap. Forthe case of inverted (expulsive) quadratic trap corresponding to unstable GPequation, the effective potential can be stable. For the bounded in space trappotential it is showed that the bifurcation exists, i.e.,the single wellpotential bifurcates to the triple well effective potential. Stabilization ofBEC cloud on-site state in the temporary modulated optical lattice is found.(analogous to the Kapitza stabilization of the pendulum). The predictions ofthe averaged GP equation are confirmed by the numerical simulations of GPequation with rapid perturbations.Comment: 15 pages, 4 figure

The dynamics of bright matter wave solitons in a quasi one-dimensional Bose Einstein condensate with a rapidly varying trap