  We study the spin dynamics in a spin-1 ferromagnetic Bose-Einstein condensate
with magnetic dipole-dipole interaction (MDDI) based on the Gross-Pitaevskii
and Bogoliubov theories. We find that various magnetic structures such as
checkerboards and stripes emerge in the course of the dynamics due to the
combined effects of spin-exchange interaction, MDDI, quadratic Zeeman and
finite-size effects, and non-stationary initial conditions. However, the
short-range magnetic order observed by the Berkeley group [Phys. Rev. Lett.
{\bf 100}, 170403 (2008)] is not fully reproduced in our calculations; the
periodicity of the order differs by a factor of three and the checkerboard
pattern eventually dissolves in our numerical simulations. Possible reasons for
the discrepancy are discussed.

Quantum Gases