the gaps between the ground state and the first excited state. Therefore, these modes with positive energy occur in the linear limit at lower frequencies. One of the negative energy modes collides at some critical value of the chemical potential with the mode stemming from the fourth excited state and forms a quartet of complex frequencies, thus leading to an oscillatory instability. The seventh panel shows the BdG spectra along the seventh branch, the solutions of which corresponds to the symmetric three-soliton state exhibiting a linear counterpart. The negative energy modes occur at the energy gaps to the ground state, the first and the second excited states. The eigenfrequency of the latter becomes, for increasing µ , purely imaginary at the point where the eighth branch bifurcates. The other two anomalous modes are almost degenerate, reflecting again the fact that the solitons in the different wells are almost decoupled. One of the negative energy modes collides with a mode with positive energy and forms a quartet of complex frequencies. However, the magnitude of the imaginary part is much smaller than the magnitude of the purely imaginary mode. The behavior of the purely imaginary mode is similar to the behavior of the mode in panel two (describing a single soliton at the center of the barrier). The behavior of the other two modes is similar to the modes in panel six describing one soliton in each well. Finally, the eighth panel shows the BdG spectra along the eighth branch, the solutions of which corresponds to the asymmetric three-soliton state. This state, which has two solitons in one well and one soliton in the other well, emerges at µ ∼ 2 . 74 × 10 − 12 eV, where the seventh state (i.e., the one corresponding to the nonlinear continuation of the third excited state), becomes unstable. The critical value of the number of atoms for which the bifurcation occurs can be predicted in the same way as for the bifurcation of the one soliton branch; the result is: where ω$_{2}$$_{,}$$_{3}$ are the third and fourth lowest eigenvalues of operator 1 + ˆ H , C$_{1}$ = ∫ ψ 4 $_{3}$dz , B = ∫ ψ 2 $_{2}$ψ 2 $_{3}$dz , while ψ$_{2}$$_{,}$$_{3}$ denote the second and third excited state of the operator 1 + ˆ H , respectively. The corresponding critical chemical potential is then given by µ (3) cr = 1 + ω$_{2}$ + 3 BN (3) cr leading (for these parameters) to µ (3) cr = 2 . 7420 × 10 − 12 eV, which is in very good agreement with the numerical critical point given above. Right after the bifurcation, the asymmetric three soliton states are stable and, thus, we N (3) cr = ω$_{3}$ − ω$_{2}$ 3 D C$_{1}$ , (18) 15