Two routes to the one-dimensional discrete nonpolynomial Schr\"odinger equation

TL;DR

This paper compares two derivations of the 1D discrete nonpolynomial Schr"odinger equation for Bose-Einstein condensates in traps, showing they produce similar soliton behaviors but differ in some solutions, with the new model being more accurate.

Contribution

The paper introduces a new derivation of the 1D discrete NPSE from the 3D GPE and compares it with the existing model, highlighting their similarities and differences.

Findings

Both models predict similar soliton existence and stability regions.

Collapse of localized modes is observed in both models.

The new model does not produce strongly pinned solitons, aligning with the continual NPSE.

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 Schr\"odinger 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", that was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction of 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.