  Temporal-dissipative Kerr solitons are self-localized light pulses sustained
in driven nonlinear optical resonators. Their realization in microresonators
has enabled compact sources of coherent optical frequency combs as well as the
study of dissipative solitons. A key parameter of their dynamics is the
effective-detuning of the pump laser to the thermally- and Kerr-shifted cavity
resonance. Together with the free spectral range and dispersion, it governs the
soliton-pulse duration, as predicted by an approximate analytical solution of
the Lugiato-Lefever equation. Yet, a precise experimental verification of this
relation was lacking so far. Here, by measuring and controlling the
effective-detuning, we establish a new way of stabilizing solitons in
microresonators and demonstrate that the measured relation linking soliton
width and detuning deviates by less than 1 % from the approximate expression,
validating its excellent predictive power. Furthermore, a detuning-dependent
enhancement of specific comb lines is revealed, due to linear couplings between
mode-families. They cause deviations from the predicted comb power evolution,
and induce a detuning-dependent soliton recoil that modifies the pulse
repetition-rate, explaining its unexpected dependence on laser-detuning.
Finally, we observe that detuning-dependent mode-crossings can destabilize the
soliton, leading to an unpredicted soliton breathing regime (oscillations of
the pulse) that occurs in a normally-stable regime. Our results test the
approximate analytical solutions with an unprecedented degree of accuracy and
provide new insights into dissipative-soliton dynamics.
