is much smaller than in the previous case of the purely imaginary eigenfrequency. The fourth and fifth panel show the spectra along the fourth and fifth branch, the solutions of which correspond to asymmetric two-soliton states having no linear counter-part. The previous one has got one imaginary and one negative energy mode and the latter one has two negative energy modes. These modes result from the presence of two dark solitons. The state considered in the fourth panel has two dark solitons, one soliton located approximately at the barrier and one soliton in one well. The eigenfrequency corresponding to the former one is purely imaginary and looks similar to the mode of the symmetric one soliton state of the second panel. The eigenfrequency of the other anomalous mode decreases with increasing chemical potential and collides with a mode with positive energy, thus generating a quartet of complex eigenfrequencies. The magnitude of the imaginary part is again much smaller than the magnitude of the purely imaginary eigenfrequency. The generation of dynamic instabilities is illustrated e.g. in panel five. The collision of one of the anomalous modes with a mode of positive Krein sign could lead to the emergence of a quartet of complex eigenfrequencies, resulting in the concurrent presence of an apparent level crossing in the real part of the eigenfrequency along with a nonzero imaginary part of the relevant eigenfrequency mode. At µ ≃ 2 . 828 × 10 − 12 eV, the states of the fourth and fifth branch collide and disappear as can be seen in Fig. 1. Just before the collision (for slightly larger µ ), the state of the fourth branch is unstable due to the presence of a mode with imaginary eigenfrequency in the BdG spectrum, while the state of the fifth branch is linearly stable since all the BdG eigenfrequencies have only real parts. Thus, it can be concluded that, at this critical point, a saddle-node bifurcation occurs, readily destroying these two states. The sixth panel shows the BdG spectra along the sixth branch, the solutions of which correspond to symmetric two-soliton states. This branch has a linear counterpart, and we can assign to each mode in the linear limit an energy difference of the linear problem. The negative energy modes occur for negative energy differences, e.g., at the energy difference to the ground state and the first excited state. The anomalous modes here are close to being degenerate. Physically speaking, this reflects the fact that the dark solitons are essentially decoupled at the linear limit due to the presence of the optical lattice. The negative energy modes are not the modes with the lowest eigenfrequency. The energy gaps between the second excited state and the third and fourth excited state, respectively, are smaller than 14