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 inter-atomic 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 $^{7}$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-vs.-norm diagram features two cuspidal points, resulting in a typical \textit{swallowtail pattern}, which is a generic feature of 3D solitons supported by low- (2D) or fully-dimensional lattice potentials. 
D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, L.-C. Crasovan, Y.   V. Kartashov, L. Torner