Stabilizations of Two-Dimensional Trapped and Untrapped Matter Waves via a Feshbach Resonance Technique

We have studied the dynamics of two-dimensional (2D) trapped and untrapped Bose--Einstein condensates (BECs) with a rapid periodic modulation of the scattering length via a Feshbach resonance technique, a → a₀+a₁sin(Ωt) with an attractive (negative) mean value and the large constants a₀, a₁ and Ω. Applying a variation approximation (VA), the critical threshold for the collapse of the 2D trapped vortex BEC is predicted and the collapse is prevented by causing the scattering length oscillating rapidly. On the other hand, with analytical calculation, we prove that the stabilization of a bright soliton in a 2D untrapped BEC is impossible for enough large interaction intensity and the upper limit of the intensity for the soliton stabilization is derived.