^{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.

The dynamics of a dilute quantum gas which forms the Bose–Einstein condensate (BEC) is very accurately described by the three-dimensional Gross–Pitaevskii equation (3D GPE). This equation treats effects of collisions between atoms in the condensate in the mean-field approximation. In experimentally relevant settings, the BEC is always confined by a trapping potential. In many cases, the trap is designed to have the “cigar-shaped” form, allowing an effective reduction in the dimension from 3 to 1. In turn, the one-dimensional (1D) dynamics of the trapped condensate may be controlled by means of an additional periodic potential, induced by an optical lattice (OL), which acts along the axis of the “cigar.” If the OL potential is sufficiently strong, the eventual dynamical model reduces to a 1D discrete equation. In both the continual and discrete versions of the 1D description, a crucially important feature is the form of nonlinearity in the respective equations. In the limit of low density, the nonlinearity is cubic—the same as in the underlying 3D GPE. In the general case, a consistent derivation, which starts from the cubic nonlinearity in 3D, leads to 1D equations with a nonpolynomial nonlinearity, the respective model being called the “nonpolynomial Schrödinger equation” (NPSE). The discrete limit of the latter equation, corresponding to the action of the strong axial OL potential, was derived and investigated recently. An essential asset of both versions of the NPSE, continual and discrete ones, is that they predict the onset of the collapse (formation of a singularity in the condensate with attraction between atoms) in the framework of the 1D description, thus complying with the fundamental property of the BEC, which was predicted by the underlying 3D GPE and observed experimentally. However, in the case when the OL potential is very strong, an alternative way to derive the 1D discrete model may start with the discretization of the 3D GPE, followed by the reduction in the dimension in the cigar-shaped trap. In this work, we report a new discrete model (“model 2”) derived in this way, which seems very different from the previously known discrete 1D NPSE (which we call “model 1”). In particular, while model 1 amounts to a single discrete equation for the 1D complex wave function,model 2 incorporates an additional equation for the transverse width. Nevertheless, numerical analysis performed in the present work shows a remarkably similar behavior of fundamental localized modes in the form of unstaggered discrete solitons in both systems. The similarity pertains to the existence region for the solitons and their stability limits. Importantly, both models admit the collapse and produce similar predictions for the collapse threshold. Thus, basic properties of discrete solitons found in the model of the strongly trapped BEC are trustworthy, as they are reproduced independently by the two very different models. Nevertheless, a difference between the two models is also found: very narrow discrete solitons, which exist in model 1, are absent in model 2. In fact, the continual 1D NPSE does not give rise to such extremely narrow solutions either, thus indicating that the newly derived model, although having a more complex mathematical form, eventually provides a more accurate approximation.

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

## Figures

Values of the transverse width as functions of the chemical potential for fundamental on-site solitons in model 1 (the ordinary model) are shown by curves, which are marked by triangles for the central site [in panel (a), the full and empty circles almost coincide]. full circles for the first neighbors, and empty circles for the second neighbors to the central site. In model 2 (the new system) the values of for fundamental on-site solitons are shown by dashed lines for the central site, dotted lines for the first neighbors, and dashed-dotted lines for the second neighbors to the central site. The intersite coupling constants are (a) and (b) .

Values of the transverse width as functions of the chemical potential for fundamental on-site solitons in model 1 (the ordinary model) are shown by curves, which are marked by triangles for the central site [in panel (a), the full and empty circles almost coincide]. full circles for the first neighbors, and empty circles for the second neighbors to the central site. In model 2 (the new system) the values of for fundamental on-site solitons are shown by dashed lines for the central site, dotted lines for the first neighbors, and dashed-dotted lines for the second neighbors to the central site. The intersite coupling constants are (a) and (b) .

The same as in Fig. 1 but for families of intersite solitons found in both models 1 and 2.

The same as in Fig. 1 but for families of intersite solitons found in both models 1 and 2.

The cw solutions are unstable to modulational perturbations in regions below curves , which are determined by condition , see Eq. (29). The solid and dashed curves correspond to models 1 and 2, respectively. The figure pertains to and .

The cw solutions are unstable to modulational perturbations in regions below curves , which are determined by condition , see Eq. (29). The solid and dashed curves correspond to models 1 and 2, respectively. The figure pertains to and .

The norm (power) vs chemical potential for on-site fundamental solitons in both models 1 and 2. The coupling constants are (a) and (b) , which correspond to strongly discrete systems and quasicontinual systems, respectively.

The norm (power) vs chemical potential for on-site fundamental solitons in both models 1 and 2. The coupling constants are (a) and (b) , which correspond to strongly discrete systems and quasicontinual systems, respectively.

The norm (power) vs chemical potential for intersite fundamental solitons in models 1 and 2. The coupling constants are (a) and (b) .

The norm (power) vs chemical potential for intersite fundamental solitons in models 1 and 2. The coupling constants are (a) and (b) .

Free energy vs norm for on-site and intersite fundamental solitons in both models 1 and 2 for (a) and (b) . Solid and dashed lines correspond to the on-site solitons in models 2 and 1, respectively. Dotted and dashed-dotted lines represent, respectively, intersite solitons in models 2 and 1.

Free energy vs norm for on-site and intersite fundamental solitons in both models 1 and 2 for (a) and (b) . Solid and dashed lines correspond to the on-site solitons in models 2 and 1, respectively. Dotted and dashed-dotted lines represent, respectively, intersite solitons in models 2 and 1.

Pure real unstable eigenvalues (“ev”) for intersite unstaggered solitons in models 1 and 2 are shown by black and white circles, respectively, for . Note that in this case, pure real eigenvalues for on-site solitons have not been found.

Pure real unstable eigenvalues (“ev”) for intersite unstaggered solitons in models 1 and 2 are shown by black and white circles, respectively, for . Note that in this case, pure real eigenvalues for on-site solitons have not been found.

The same as in Fig. 7 but at . In this case, plots (a) and (b) display the eigenvalues for the on-site solitons and intersite solitons, respectively.

The same as in Fig. 7 but at . In this case, plots (a) and (b) display the eigenvalues for the on-site solitons and intersite solitons, respectively.

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