The second panel shows the BdG spectra along the second branch, solutions of which represent states with a density minimum at the center of the barrier. In the linear limit one can assign to each mode an energy difference of the linear problem similar as in the BdG spectrum of the ground state. However in this case the frequency difference of the first excited state to the ground state is negative (according to our definition) leading to a negative energy mode. As one can see in the inset, the lowest mode has negative energy close to the linear limit. However, increasing the chemical potential the eigenfrequency of this mode is moving rapidly towards the origin of the spectrum and, at µ = 2 . 1437 × 10 − 12 eV, becomes and remains purely imaginary, thus carrying zero energy (red + symbols). This happens exactly at the point where the third branch bifurcates as one can see in the inset of Fig. 1. For the above mentioned potential parameters, this bifurcation happens in the weakly-interacting regime, where Eq. (3) reduces to the cubic NLS equation. Thus, one can apply the Galerkin-type approach of Ref. [50] and find that the critical value of the norm N at which the bifurcation occurs is given by where ω$_{0}$$_{,}$$_{1}$ are the first two lowest eigenvalues of the operator 1 + ˆ H , A$_{1}$ = ∫ ψ 4 $_{1}$dz , B = ∫ ψ 2 $_{0}$ψ 2 $_{1}$dz , while ψ$_{0}$$_{,}$$_{1}$ denote the ground state and the first excited state of the operator 1 + ˆ H , respectively. One can also determine the critical chemical potential at which the symmetry breaking is expected to occur, namely µ$_{cr}$ = 1+ ω$_{0}$ +3 BN$_{cr}$ [50] leading for (these parameter values) to µ$_{cr}$ = 2 . 1436 × 10 − 12 eV, which is in very good agreement with our numerical findings. The third panel shows the BdG spectra along the third branch which bifurcates at µ = 2 . 1437 × 10 − 12 eV from the second one. All the eigenfrequencies of the arising asymmetric state are real close to the linear limit, thus it is concluded that this state is linearly stable and that the bifurcation is a supercritical pitchfork. This state has one negative energy mode as well, since it is a one-soliton state. The eigenfrequency of the negative energy mode increases with increasing chemical potential, passes through the mode corresponding to the second excited state of the linear limiting case, and finally collides at some critical value of the chemical potential, µ = 2 . 76 × 10 − 12 eV, with the mode corresponding to the third excited state; this results in the generation of a quartet of unstable eigenfrequencies and, at this point, the state becomes oscillatory unstable. However, the magnitude of the instability 13 N$_{cr}$ = ω$_{1}$ − ω$_{0}$ 3 B A$_{1}$ , (17)