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We derive exact solitonic solutions of a class of Gross-Pitaevskii equationswith time-dependent harmonic trapping potential and interatomic interaction. Wefind families of exact single-solitonic, multi-solitonic, and solitary wavesolutions. We show that, with the special case of an oscillating trappingpotential and interatomic interaction, a soliton can be localized indefinitelyat an arbitrary position. The localization is shown to be experimentallypossible for sufficiently long time even with only an oscillating trappingpotential and a constant interatomic interaction.

Soliton localization in Bose-Einstein condensates with time-dependent harmonic potential and scattering length