  We study fingering instabilities and pattern formation at the interface of an
oppositely polarized two-component Bose-Einstein condensate with strong
dipole-dipole interactions in three dimensions. It is shown that the rotational
symmetry is spontaneously broken by fingering instability when the
dipole-dipole interactions are strengthened. Frog-shaped and mushroom-shaped
patterns emerge during the dynamics due to the dipolar interactions. We also
demonstrate the spontaneous density modulation and domain growth of a
two-component dipolar BEC in the dynamics. Bogoliubov analyses in the
two-dimensional approximation are performed, and the characteristic lengths of
the domains are estimated analytically. Patterns resembling those in magnetic
classical fluids are modulated when the number ratio of atoms, the trap ratio
of the external potential, or tilted polarization with respect to the z
direction is varied.
