  Motivated by recent experimental results, we study beating dark-dark solitons
as a prototypical coherent structure that emerges in two-component
Bose-Einstein condensates. We showcase their connection to dark- bright
solitons via SO(2) rotation, and infer from it both their intrinsic beating
frequency and their frequency of oscillation inside a parabolic trap. We
identify them as exact periodic orbits in the Manakov limit of equal inter- and
intra-species nonlinearity strengths with and without the trap and showcase the
persistence of such states upon weak deviations from this limit. We also
consider large deviations from the Manakov limit illustrating that this
breathing state may be broken apart into dark-antidark soliton states. Finally,
we consider the dynamics and interactions of two beating dark-dark solitons in
the absence and in the presence of the trap, inferring their typically
repulsive interaction.
