content_image/337186.jpg content_image/337185.jpg
can conclude that the bifurcation is of the supercritical pitchfork type. Two modes look similar to the negative energy modes in panel five describing a two-soliton state in one well and the third mode looks similar to a mode of one-soliton in a harmonic trap. 

From the above analysis, we can readily derive some general conclusions concerning dark soliton states in a double well potential. The sum of negative energy modes and imaginary eigenfrequency modes is equal to the number of nodes in the wavefunction profile which, for large values of the chemical potential, represent dark solitons. In the linear limit, the BdG eigenfrequencies of a state correspond to the energy differences between that and other states of the linear system. For a negative energy difference the corresponding mode has a negative energy. Dark solitons located at the center of the barrier are known to be linearly unstable (for sufficiently high chemical potential) associated with a purely imaginary eigenfrequency [37, 54]. Thus, symmetric states with an odd number of nodes always have a mode which, above a critical value of the chemical potential, becomes purely imaginary. At this critical point additional stable asymmetric dark-soliton states emerge through supercritical pitchfork bifurcations. 

## IV. STATICS VS. DYNAMICS OF DARK SOLITON STATES 

A. The one-soliton state 

Having investigated in detail the statics of matter-wave dark solitons in double-well poten- tials, we now proceed to compare these results to the soliton dynamics, using the theoretical background exposed in Sec. III.B. Particularly, considering the case of a single dark soliton, we can readily obtain from Eqs. (12) and (14) the following approximate equation of motion for the dark soliton center: 

<img src='content_image/337186.jpg'>

It is straightforward to find the fixed points z$_{o}$, p$_{o}$ ≡ dz$_{o}$/dt of the above dynamical system, which are given by 

<img src='content_image/337185.jpg'>

It is clear that the fixed points correspond to intersections of the straight line f$_{1}$ ( z$_{o}$ ) = z$_{o}$ and the sinusoidal curve f$_{2}$ ( z$_{o}$ ) = kV 0 2 ω 2 ds sin(2 kz$_{o}$ ). The fixed point at ( z$_{o}$, p$_{o}$ ) = (0 , 0) exists 

