at which the symmetric mode gets destabilized, and simultaneously a stable asymmetric one emerges. In accordance with what is said above, the bifurcation is of the supercritical type [@Joseph], i.e., the emerging branches of the asymmetric states immediately go forward". Conclusions about the stability and instability of the solution branches displayed in Fig. \[bifurcation diagrams\] were produced by means of the well-known method [@VK2] based on numerical computation of (in)stability eigenvalues (imaginary parts of eigenfrequencies) for small perturbations added to the stationary solutions, using the respective linearized (Bogoliubov - de Gennes) equations. In particular, the instability of those symmetric states which coexist with asymmetric ones is always represented by a single pair of purely imaginary eigenfrequencies.

The fact that the asymmetric modes, when they exists, have smaller values of $E$ for given $N$, hence they realize the system’s GS \[see Figs. [bifurcation diagrams]{}(d-f)\], can be easily understood: having their center shifted from the layer occupied by the positive potential ($|x|<0.5$) to a region where $H=0$ \[$0.5<|x|<3$, see Figs. \[potential-fig\] and [symmetric and asymmetric modes]{}\], they obviously reduce the integral value of $E$. The same argument explains why, for given $k$, the asymmetric modes feature smaller $N$: having smaller overlap with 