V. DARK SOLITON DYNAMICS: NUMERICAL RESULTS So far, we have examined the existence, stability and bifurcations of the branches of dark-solitonic states in the double-well potential setting. We now proceed to investigate the dynamics of these states by direct numerical integration of Eq. (1), using as initial condi- tions stationary soliton states, perturbed along the direction of eigenvectors associated with particular eigenfrequencies (especially ones associated with instabilities). To better explain the above, we should recall that anomalous modes are connected to the positions and oscil- lation frequencies of the solitons; therefore, perturbations of stationary dark soliton states along the directions of the eigenvectors of the anomalous modes result in the displacement of the positions of the solitons, thus giving rise to interesting soliton dynamics. Before proceeding with the dynamics of the (excited) soliton states, let us make a remark concerning the ground state of the system. As mentioned above, the value of the lowest BdG mode of the ground state is given, in the linear limit, by the energy difference between the ground state and the first excited state. However, this energy gap determines the oscillation frequency of atoms between the wells of the double-well potential, which suggests that the lowest BdG mode is connected to an oscillation between the wells. Let us define the atom numbers N$_{L}$ and N$_{R}$ in the left and right well, respectively, as N$_{L}$ = ∫ 0 −∞ | ψ | $^{2}$dz and N$_{R}$ = ∫ + ∞ 0 | ψ | $^{2}$dz (the total number of atoms is N = N$_{L}$ + N$_{R}$ ). In Fig 8, we show the time evolution of N$_{L}$ (solid line) and N$_{R}$ (dashed line) for the ground state, when perturbed by the eigenvector corresponding to the lowest BdG mode (parameter values are µ = 3 × 10 − 12 eV and V$_{0}$ = 1 . 12 × 10 − 12 eV). One can clearly observe the oscillation of the atom numbers in the different wells, with a characteristic frequency known as Josephson oscillation frequency [43], determined by the magnitude of the respective BdG eigenmode. Let us now proceed with the one-soliton states. Fig. 9 shows the time evolution of the symmetric one soliton state perturbed by the eigenvector corresponding to the unstable mode, for µ = 3 × 10 − 12 eV and V$_{0}$ = 4 . 96 × 10 − 13 eV. The instability manifests itself as a drift of the soliton from the trap center to the rims of the background atom cloud, where the soliton is reflected back and forth. One clearly observes that the soliton is located at an unstable position, since a small perturbation leads to a large amplitude oscillation of the soliton. On the other hand, Fig. 10 shows the time evolution of the asymmetric one soliton state perturbed by the eigenvector corresponding to the unstable mode for µ = 3 × 10 − 12 26