  For a dissipative variant of the two-dimensional Gross-Pitaevskii equation
with a parabolic trap under rotation, we study a symmetry breaking process that
leads to the formation of vortices. The first symmetry breaking leads to the
formation of many small vortices distributed uniformly near the Thomas-Fermi
radius. The instability occurs as a result of a linear instability of a
vortex-free steady state as the rotation is increased above a critical
threshold. We focus on the second subsequent symmetry breaking, which occurs in
the weakly nonlinear regime. At slightly above threshold, we derive a
one-dimensional amplitude equation that describes the slow evolution of the
envelope of the initial instability. We show that the mechanism responsible for
initiating vortex formation is a modulational instability of the amplitude
equation. We also illustrate the role of dissipation in the symmetry breaking
process. All analyses are confirmed by detailed numerical computations.
