Structure and Collapse of a Dipolar Bose-Einstein Condensate*

A Bose-Einstein condensate formed of dipolar particles is known to be unstable against collapse if the attractive part of the dipolar interaction is strong enough. Here we explore the limits of this stability, with special attention to the possibiliity of the condensate collapsing locally rather than collapsing as a whole toward the center of the trap. We explicitly connect local collapse to the presence of low-energy ``roton-like'' modes, and we propose experiments that could probe this local collapse.