FIG. 6. (Color online) Top panels: Fixed points (lines) of the system (26)-(27) and position of the solitons (circles for the symmetric two-soliton state, × for the two-soliton branch with one soliton near the center and one in one well, and stars for the state with both dark solitons in the same well) calculated by the numerical integration of Eq. (1) for µ = 3 . 8 × 10 − 12 eV. Bottom panels: BdG spectra of the two-soliton states for µ = 3 . 8 × 10 − 12 eV as a function of V$_{0}$ ; left corresponds to the symmetric two-soliton state, middle to the always unstable state with one soliton near the center and one in one well, while the right corresponds to the state with both dark solitons in the same well. Notice also the agreement of the theoretically predicted modes from Eq. (28) with the numerical ones. there exists only the symmetric solution with two nodes. However, for V$_{0}$$_{,cr}$ = 3 . 4 × 10 − 12 eV, two asymmetric states appear stemming from a saddle-node bifurcation. For all three states, the agreement between the fixed points obtained from Eqs. (26)-(27) and the position of the solitons is excellent. The bottom panels of Fig. 6 show the BdG spectra for the symmetric and the two asymmetric two soliton states as a function of V$_{0}$ for fixed µ = 3 . 8 × 10 − 12 eV. The solid 23  0  50  100  150 r  (2$^{π}$ Hz)  0  5  10  0  0.5  1 i  (2$^{π}$ Hz) V$_{0}$ (10$^{-12}$ eV)  0  50  100  150 r  (2$^{π}$ Hz)  0  50  0.5  1 i  (2$^{π}$ Hz V$_{0}$ (10$^{-12}$ eV)  0  50  100  150 r  (2$^{π}$ Hz)  0  5  0.5  1 i  (2$^{π}$ Hz) V$_{0}$ (10$^{-12}$ eV) 0 0.2 0.4 0.6 0.8 1 −4 −2 0 2 4 V$_{0}$ (10 −12  eV) z 0 1,2  (  m)