Matter-wave dark solitons in a double-well potential

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.