  Capillary instability and the resulting dynamics in an immiscible
two-component Bose-Einstein condensate are investigated using the mean-field
and Bogoliubov analyses. A long, cylindrical condensate surrounded by the other
component is dynamically unstable against breakup into droplets due to the
interfacial tension arising from the quantum pressure and interactions. A
heteronuclear system confined in a cigar-shaped trap is proposed for realizing
this phenomenon experimentally.
