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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 

