Stable three-dimensional solitons in attractive Bose-Einstein condensates loaded in an optical lattice

The existence and stability of solitons in Bose-Einstein condensates with attractive interatomic interactions, described by the Gross-Pitaevskii equation with a three-dimensional (3D) periodic potential, are investigated in a systematic form. We find a one-parameter family of stable 3D solitons in a certain interval of values of their norm, provided that the strength of the potential exceeds a threshold value. The minimum number of ${}^7\mathrm{Li}$ atoms in the stable solitons is 60, and the energy of the soliton at the stability threshold is $\approx 6$ recoil energies in the lattice. The respective energy versus norm diagram features two cuspidal points, resulting in a typical swallowtail pattern, which is a generic feature of 3D solitons supported by quasi-two-dimensional or fully dimensional lattice potentials.