We consider the three-dimensional (3D) mean-field model for the Bose-Einstein
condensate (BEC), with a 1D nonlinear lattice (NL), which periodically changes
the sign of the nonlinearity along the axial direction, and the
harmonic-oscillator trapping potential applied in the transverse plane. The
lattice can be created as an optical or magnetic one, by means of available
experimental techniques. The objective is to identify stable 3D solitons
supported by the setting. Two methods are developed for this purpose: The
variational approximation, formulated in the framework of the 3D
Gross-Pitaevskii equation, and the 1D nonpolynomial Schr\"{o}dinger equation
(NPSE) in the axial direction, which allows one to predict the collapse in the
framework of the 1D description. Results are summarized in the form of a
stability region for the solitons in the plane of the NL strength and
wavenumber. Both methods produce a similar form of the stability region. Unlike
their counterparts supported by the NL in the 1D model with the cubic
nonlinearity, kicked solitons of the NPSE cannot be set in motion, but the kick
may help to stabilize them against the collapse, by causing the solitons to
shed excess norm. A dynamical effect specific to the NL is found in the form of
freely propagating small-amplitude wave packets emitted by perturbed solitons.Comment: 14 pages, 8 figures. To be published in J. Phys. B: At. Mol. Opt.
Phy
