the BdG spectra, and the predicted oscillation frequencies, for the symmetric two soliton state for different values of the optical lattice strength: V$_{0}$ = 0, V$_{0}$ = 8 . 3 × 10 − 14 eV, V$_{0}$ = 1 . 65 × 10 − 13 eV and V$_{0}$ = 4 . 96 × 10 − 13 eV. In the case of a purely harmonic potential, V$_{0}$ = 0, one observes the constant eigenfrequency of the dipolar mode at ω$_{z}$ , as well as the eigenfrequency of one anomalous mode bifurcating from the dipolar mode in the linear limit. The eigenfrequency at 2 ω$_{z}$ is imaginary due to a degeneracy of a negative and a positive Krein sign mode. The former one corresponds to the second anomalous mode and the latter one to the quadrupole mode. This degeneracy is lifted for increasing chemical potential. The predictions of Eqs. (24)-(25) agree very well with the BdG results for the anomalous modes in the case of large chemical potentials. 

For V$_{0}$ = 8 . 3 × 10 − 14 eV the degeneracy between the anomalous modes and the dipolar and the quadrupole mode is lifted in the linear limit. Thus, there exists an energy gap between the anomalous mode and the dipolar mode and, more importantly, an energy gap between the second anomalous mode and the quadrupole mode. A particularly useful conclusion stemming from the above findings is that the two soliton state can be stabilized in a harmonic trap by adding a small optical lattice. 

For V$_{0}$ = 1 . 65 × 10 − 13 eV the system is stable as well. The anomalous modes decrease with increasing chemical potential and reach similar values for a large chemical potential. This denotes that the two solitons are decoupled for a large chemical potential. This can be understood at least in the framework of the effective potential (23): the range of the interaction potential scales with the density n$_{0}$ of the background atom cloud and thereby with the chemical potential as well. Thus, for a large chemical potential, i.e., a large density, the interaction between solitons is strong but of short range too. 

For V$_{0}$ = 4 . 96 × 10 − 13 eV the anomalous modes have similar values even in the linear limit. Thus, the two solitons are already almost decoupled in the linear limit due to their large separation. The two anomalous modes collide with the dipolar and quadrupole modes for increasing chemical potential, leading to oscillatory instabilities, and bifurcate from these modes again for a larger chemical potential. Although the predictions of Eqs. (24)-(25) agree qualitatively with the BdG results, there is no quantitative agreement due to the crossings and collisions of the soliton modes with the modes of the condensate. 

