  Dissipative solitons are self-localized structures resulting from a double
balance between dispersion and nonlinearity as well as dissipation and a
driving force. They occur in a wide variety of fields ranging from optics,
hydrodynamics to chemistry and biology. Recently, significant interest has
focused on their temporal realization in driven optical microresonators, known
as dissipative Kerr solitons. They provide access to coherent, chip-scale
optical frequency combs, which have already been employed in optical metrology,
data communication and spectroscopy. Such Kerr resonator systems can exhibit
numerous localized intracavity patterns and provide rich insights into
nonlinear dynamics. A particular class of solutions consists of breathing
dissipative solitons, representing pulses with oscillating amplitude and
duration, for which no comprehensive understanding has been presented to date.
Here, we observe and study single and multiple breathing dissipative solitons
in two different microresonator platforms: crystalline $\mathrm{MgF_2}$
resonator and $\mathrm{Si_3N_4}$ integrated microring. We report a
deterministic route to access the breathing state, which allowed for a detailed
exploration of the breathing dynamics. In particular, we establish the link
between the breathing frequency and two system control parameters - effective
pump laser detuning and pump power. Using a fast detection, we present a direct
observation of the spatiotemporal dynamics of individual solitons, revealing
irregular oscillations and switching. An understanding of breathing solitons is
not only of fundamental interest concerning nonlinear systems close to critical
transition, but also relevant for applications to prevent breather-induced
instabilities in soliton-based frequency combs.
