  We consider a trapped atomic Bose--Einstein condensate penetrated by a
repulsive Gaussian potential and theoretically investigate the dynamics induced
by oscillating the Gaussian potential. Our study is based on the numerical
calculation of the two-dimensional Gross--Pitaevskii equation. Our calculation
reveals the dependence of the characteristic behavior of the condensate on the
amplitude and frequency of the oscillating potential. These dynamics are deeply
related to the nucleation and dynamics of quantized vortices and solitons. When
the potential oscillates with a large amplitude, it nucleates many vortex pairs
that move away from the potential. When the amplitude of the oscillation is
small, it nucleates solitons through annihilation of vortex pairs. We discuss
three issues concerning the nucleation of vortices. The first is the phase
diagram for the nucleation of vortices and solitons near the oscillating
potential. The second is the mechanism and critical velocity of the nucleation.
The critical velocity of the nucleation is an important issue in quantum
fluids, and we propose a new expression for the velocity containing both the
coherence length and the size of the potential. The third is the divergence of
the nucleation time, which is the time it takes for the potential to nucleate
vortices, near the critical parameters for vortex nucleation.
