Tunneling of a dipolar bose-einstein condensate in an optical lattice

We have studied the tunneling and fluctuations of a dipolar Bose-Einstein condensate in an optical lattice, it is found that there exist the tunneling and fluctuations between lattices l and l+1, l and l-1, respectively. In particular, when the optical lattice is infinitely long and the spin excitations are in the long-wavelength limit, tunneling effects disappear between lattices l and l+1, and that l and l-1, in this case the fluctuations are a constant, and the magnetic soliton appears.