the occurrence of additional states, for sufficiently large nonlinearity (large atom numbers), which do not have a linear counterpart. Next, let us discuss the dynamics of dark solitons in the considered setup with an external potential in the weakly-interacting limit (4 | ψ | 2 ≪ 1). Therefore, we factorize the total wavefunction ψ = n$_{0}$u into the background density n$_{0}$ and the soliton wavefunction u . We assume that the background density is given by the ground state wavefunction of the condensate and that we can use the TF approximation to describe the latter. Then Eq. (3) may be expressed as a perturbed NLS equation for the soliton wavefunction u , namely, where the effective perturbation P ( u ) is given by: Here, V ( z ) = (1 / 2)Ω $^{2}$z2 + V$_{0}$ cos $^{2}$( kz ) is the trapping potential, and all terms of P ( u ) are assumed to be of the same order. Furthermore, assuming that the length scale of the trap is much larger than the width of the soliton, one can apply the adiabatic perturbation theory for dark solitons devised in Ref. [25, 69] to derive the following equation of motion for the slowly-varying dark soliton center z$_{0}$ ( t ) [40]: where the effective potential felt by the soliton is given by: At this point we should make the following remarks. First, for the derivation of Eq. (12), it was assumed that the background density of the condensate can be described via the TF approximation; this actually means that Eq. (12) is valid only for sufficiently large number of atoms. At the same time, however, the derivation of Eq. (12) was done in the framework of the cubic NLS model, i.e., the weakly-interacting limit of Eq. (3), which is valid for sufficiently small number of atoms. Nevertheless, Eq. (12) may still be relevant to provide some estimates. It is known that the soliton oscillation frequency (for a BEC confined in a harmonic trap) is up-shifted for large number of atoms due to the dimensionality of the system (i.e., the effect of transverse dimensions, which are taken into regard in the derivation 8 V$_{eff}$ ( z ) = 1 2 Ω $^{2}$z2 + 1 2 V$_{0}$ [ 1 $_{−}$( π 2 3 − 2 ) k 2 6 µ ] cos(2 kz ) . (13) d $^{2}$z$_{0}$ dt 2 = − 1 2 dV$_{eff}$ dz$_{0}$ , (12) P ( u ) = (1 − | u | $^{2}$) V u + 1 2 1 µ − V ( z ) dV dz ∂u ∂z . (11) i∂$_{t}$u + 1 2 ∂ 2 $_{z}$u − µ ( | u | 2 − 1) u = P ( u ) , (10)