  Rotation symmetry of microresonators supports the generation of phase-locked
counter-propagating (CP) solitons that can potentially miniaturize dual-comb
systems. Realization of these dual-comb compatible solitons in photonic
integrated circuits remains a challenge. Here, we synthesized such CP solitons
in an integrated silicon nitride microresonator and observed forced soliton
oscillation due to rhythmic, time-varying soliton interactions. The
interactions result in seconds mutual-coherence passively. Temporal motion in
the soliton streams is discerned by measuring a quadratic-scaling frequency
noise peaks and an inverse quadratic-scaling microcomb sidebands. By generating
a CP soliton trimer to have two synchronized solitons in one of the orbiting
directions, we resolve the incapability of measuring two unsynchronized CP
soliton dimer pulses by optical cross-correlation, and show CP solitons undergo
complex motion trajectory. We further prove that precise dual-comb spectroscopy
with an acquisition time as short as 0.6 $\mu$s is feasible using these
solitons, although the temporal motion limits the dynamic range. Besides
revealing soliton interactions with different group velocities, our work
propels the realization of photonic integrated dual-comb spectrometers with
high passive coherence.
