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trapping potential. As demonstrated in Refs. [59], Eq. (1) provides accurate results in the dimensionality crossover regime and the TF limit, thus describing the axial dynamics of cigar-shaped BECs in a very good approximation to the 3D GP equation. It is worth mentioning that in the weakly-interacting limit, 4 a | ψ | 2 ≪ 1, Eq. (1) is reduced to the usual 1D GP equation with a cubic nonlinearity, characterized by an effective 1D coupling constant g$_{1}$$_{D}$ = 2 a ℏ ω ⊥ (see, e.g., discussion and references in Ref. [3]). 

Let us now assume that the trapping potential is a superposition of a harmonic trap and an optical lattice, characterized by an amplitude V$_{0}$ and a wavenumber k , namely, 

<img src='content_image/351981.jpg'>

It is clear that a proper choice of the potential parameters (and the number of atoms) may readily lead to a trapping potential that has the form of an effective double well potential; in such a setting, the height of the barrier can be tuned by the amplitude of the optical lattice. In our considerations below, we consider the case of a $^{87}$Rb condensate, confined in a harmonic trap with frequencies ω ⊥ = 10 ω$_{z}$ = 2 π × 400Hz (these values are relevant to the experiments of Refs. [13, 15]); furthermore, we will vary the optical lattice strength in the range of V$_{0}$ = 0 to V$_{0}$ = 1 . 16 × 10 − 12 eV, thereby changing the trapping potential from a purely harmonic form to a double-well potential. Finally, we will assume a fixed value of the optical lattice wavenumber, namely k = π/ 5 . 37 µ m − $^{1}$. 

Next, assuming that the density, length, time and energy are measured, respectively, in units of a , α ⊥ ≡ √ ℏ /mω ⊥ (transverse harmonic oscillator length), ω − 1 ⊥ , and ℏ ω ⊥ , we express Eq. (1) in the following dimensionless form, 

<img src='content_image/351980.jpg'>

where ˆ H = − (1 / 2) ∂ 2 z + (1 / 2)Ω $^{2}$z2 + V$_{0}$ cos $^{2}$( kz ), is the usual single-particle Hamiltonian and Ω ≡ ω$_{z}$/ω ⊥ is the normalized harmonic trap strength (note that the normalized optical lattice strength V$_{0}$ is measured in the units of energy ℏ ω ⊥ ). 

Below, we will analyze the stability of the nonlinear modes of Eq. (3) by means of the Bogoliubov-de Gennes (BdG) equations (see, e.g., Ref. [1]). In particular, once a numerically exact — up to a prescribed tolerance — stationary state, ψ$_{0}$ ( z ), is found (e.g., by a Newton- Raphson method), we consider small perturbations of this state of the form, 

<img src='content_image/351979.jpg'>

