content_image/371990.jpg content_image/371991.jpg content_image/371993.jpg content_image/371994.jpg content_image/371997.jpg content_image/371999.jpg content_image/372000.jpg
Using d $^{2}$z$_{i}$ /dt 2 = − (1 / 2) dV$_{eff}$ ( z$_{1}$, z$_{2}$ ) /dz$_{i}$ (with i = 1 , 2) as per Eq. (12), it is straightforward to derive the following system of equations of motion, 

<img src='content_image/371990.jpg'>

<img src='content_image/371991.jpg'>

which, accordingly, leads to the following system for the fixed points ( z (1) o , z (2) o ): 

<img src='content_image/371993.jpg'>

<img src='content_image/371994.jpg'>

Thus, the fixed points for the two soliton states depend on the density of the background cloud and thereby on the chemical potential. Due to the presence of the external potentials, the density of the background is inhomogeneous. In our analysis, we replace the inhomo- geneous density, with the density of the background at z = 0 using the TF approximation; namely, n$_{0}$ → n$_{0}$ ( z = 0) = (( µ − V$_{0}$ ) 2 − 1) / 4. ( i ) 

Considering small deviations from the fixed points, i.e., z$_{i}$ = z o + ϵη$_{i}$ exp( iωt ) (with i = 1 , 2), leads to the eigenvalue problem for the normal modes: 

<img src='content_image/371997.jpg'>

with 

<img src='content_image/371999.jpg'>

<img src='content_image/372000.jpg'>

Diagonalizing the above matrix, we obtain the eigenfrequencies of the two anomalous modes of the two-soliton states. Notice that since different two soliton states lead to different fixed points, the oscillation frequencies for the two solitons will differ as well. Negative eigenvalues of this 2 × 2 system lead to imaginary frequencies describing unstable states; however, as mentioned in the previous section, one cannot describe oscillatory instabilities. 

In the top panel of Fig. 6, we show the fixed points of the system (26)-(27) and the zero points of the solutions with two nodes of Eq. (1) corresponding to the positions of the two solitons as a function of V$_{0}$ (the chemical potential is fixed: µ = 3 . 6 × 10 − 13 eV). For small V$_{0}$ 

