  Dissipative temporal Kerr solitons in optical microresonators enable to
convert a continuous wave laser into a train of femtosecond pulses. Of
particular interest are single soliton states, whose $\mathrm{sech}^{2}$
spectral envelope provides a spectrally smooth and low noise optical frequency
comb, and that recently have been generated in crystalline, silica, and
silicon-nitride resonators. Here, we study the dynamics of multiple soliton
states containing ${N}$ solitons and report the discovery of a novel, yet
simple mechanism which makes it possible to reduce deterministically the number
of solitons, one by one, i.e. ${N\! \to\! N\!-\!1\! \to\! \dots \!\to\! 1}$. By
applying weak phase modulation, we directly characterize the soliton state via
a double-resonance response. The dynamical probing demonstrates that
transitions occur in a predictable way, and thereby enables us to map
experimentally the underlying multi-stability diagram of dissipative Kerr
solitons. These measurements reveal the "lifted" degeneracy of soliton states
as a result of the power-dependent thermal shift of the cavity resonance (i.e.
the thermal nonlinearity). The experimental results are in agreement with
theoretical and numerical analysis that incorporate the thermal nonlinearity.
By studying two different microresonator platforms (integrated
$\mathrm{Si_{3}N_{4}}$ microresonators and crystalline $\mathrm{MgF_{2}}$
resonators) we confirm that these effects have a universal nature. Beyond
elucidating the fundamental dynamical properties of dissipative Kerr solitons
the observed phenomena are also of practical relevance, providing a
manipulation toolbox which enables to sequentially reduce, monitor and
stabilize the number ${N}$ of solitons, preventing it from decay. Achieving
reliable single soliton operation and stabilization in this manner in optical
resonators is imperative to applications.
