  We investigate the structure of trapped Bose-Einstein condensates (BECs) with
long-range anisotropic dipolar interactions. We find that a small perturbation
in the trapping potential can lead to dramatic changes in the condensate's
density profile for sufficiently large dipolar interaction strengths and trap
aspect ratios. By employing perturbation theory, we relate these oscillations
to a previously-identified "roton-like" mode in dipolar BECs. The same physics
is responsible for radial density oscillations in vortex states of dipolar BECs
that have been predicted previously.
