  In this work, we explore systematically various SO(2)-rotation-induced
multiple dark-dark soliton breathing patterns obtained from stationary and
spectrally stable multiple dark-bright and dark-dark waveforms in trapped
one-dimensional, two-component atomic Bose-Einstein condensates (BECs). The
stationary states stem from the associated linear limits (as the eigenfunctions
of the quantum harmonic oscillator problem) and are parametrically continued to
the nonlinear regimes by varying the respective chemical potentials, i.e., from
the low-density linear limits to the high-density Thomas-Fermi regimes. We
perform a Bogolyubov-de Gennes (BdG) spectral stability analysis to identify
stable parametric regimes of these states. Upon SO(2)-rotation, the stable
steady-states, one-, two-, three-, four-, and many dark-dark soliton breathing
patterns are observed in the numerical simulations. Furthermore, analytic
solutions up to three dark-bright solitons in the homogeneous setting, and
three-component systems are also investigated.
