^{1}, A. Maluckov

^{2}, L. Salasnich

^{3}, B. A. Malomed

^{4}and Lj. Hadžievski

^{1}

### Abstract

The Bose–Einstein condensate (BEC), confined in a combination of the cigar-shaped trap and axial optical lattice, is studied in the framework of two models described by two versions of the one-dimensional (1D) discrete nonpolynomial Schrödinger equation (NPSE). Both models are derived from the three-dimensional Gross–Pitaevskii equation (3D GPE). To produce “model 1” (which was derived in recent works), the 3D GPE is first reduced to the 1D continual NPSE, which is subsequently discretized. “Model 2,” which was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction in the dimension. The two models seem very different; in particular, model 1 is represented by a single discrete equation for the 1D wave function, while model 2 includes an additional equation for the transverse width. Nevertheless, numerical analyses show similar behaviors of fundamental unstaggered solitons in both systems, as concerns their existence region and stability limits. Both models admit the collapse of the localized modes, reproducing the fundamental property of the self-attractive BEC confined in tight traps. Thus, we conclude that the fundamental properties of discrete solitons predicted for the strongly trapped self-attracting BEC are reliable, as the two distinct models produce them in a nearly identical form. However, a difference between the models is found too, as strongly pinned (very narrow) discrete solitons, which were previously found in model 1, are not generated by model 2—in fact, in agreement with the continual 1D NPSE, which does not have such solutions either. In that respect, the newly derived model provides for a more accurate approximation for the trapped BEC.

We appreciate valuable discussions with D. E. Pelinovsky. G.G., A.M., and Lj.H. acknowledge the support from the Ministry of Science, Serbia (through Project No. 141034). L.S. and B.A.M. appreciate a partial support from CARIPARO Foundation through “Progetti di Eccellenza 2006.” The work of B.A.M. was also supported in part by the German-Israel Foundation through Grant No. 149/2006. This author acknowledges hospitality of the Vinča Institute of Nuclear Research (Belgrade, Serbia).

I. INTRODUCTION II. THE DERIVATION OF THE DISCRETE ONE-DIMENSIONAL SYSTEMS A. Model 1: The reduction in the dimension followed by the discretization B. Model 2: The discretization followed by the reduction in the dimension III. FUNDAMENTAL BRIGHT SOLITONS A. The existence of fundamental solitons B. The norm and free energy of fundamental unstaggered solitons C. Dynamical considerations IV. CONCLUSION

### Key Topics

- Bose Einstein condensates
- 18.0
- Lagrangian mechanics
- 12.0
- Wave functions
- 11.0
- Numerical modeling
- 8.0
- Free energy
- 7.0

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