  We study numerically the outcome of the phase separation instability of a
dual-species Bose-Einstein condensate in an optical lattice. When only one
excitation mode is unstable a bound pair of a bright and dark soliton-like
structures periodically appears and disappears, whereas for more than one
unstable mode a persistent soliton-antisoliton pair develops. The oscillating
soliton represents a regime where the two-species condensate neither remains
phase-separated nor is dynamically stable.
