  We study stability of the first excited state of quasi-one-dimensional
Bose-Einstein condensates in a double-well potential, which is called
"$\pi$-state". The density notch in the $\pi$-state can be regarded as a
standing dark soliton. From the excitation spectrum, we determine the critical
barrier height, above which the $\pi$-state is dynamically unstable. We find
that the critical barrier height decreases monotonically as the number of
condensate atoms increases. We also simulate the dynamics of the $\pi$-state by
solving the time-dependent Gross-Pitaevskii equation. We show that due to the
dynamical instability the dark soliton starts to move away from the trap center
and exhibits a large-amplitude oscillation.
