  Continuous-wave-driven Kerr nonlinear microresonators give rise to
self-organization in terms of dissipative Kerr solitons, which constitute
optical frequency combs that can be used to generate low-noise microwave
signals. Here, by applying either amplitude or phase modulation to the driving
laser we create an intracavity potential trap, to discipline the repetition
rate of the solitons. We demonstrate that this effect gives rise to a novel
spectral purification mechanism of the external microwave signal frequency,
leading to reduced phase noise of the output signal. We experimentally observe
that the microwave signal generated from disciplined solitons follows the
external drive at long time scales, but exhibits an unexpected suppression of
the fast timing jitter. Counter-intuitively, this filtering takes place for
frequencies that are substantially lower than the cavity decay rate. As a
result, while the long-time-scale stability of the Kerr frequency comb
repetition rate is improved by more than 4 orders of magnitude as a result of
locking to the external microwave signal, the soliton stream shows a reduction
of the phase noise by 30 dB at offset frequencies above 10 kHz.
