  The observation of temporal dissipative Kerr solitons in optical
microresonators provides, on the applied side, compact sources of coherent
optical frequency combs that have already been applied in coherent
communications, dual comb spectroscopy and metrology. On a fundamental level,
it enables the study of soliton physics in driven nonlinear cavities.
Microresonators are commonly multimode and, as a result, inter-mode
interactions inherently occur among mode families - a condition referred to as
"avoided mode crossings". Avoided mode crossings can cause soliton decay, but
can also modify the soliton spectrum, leading to e.g. the formation of
dispersive wave and inducing a spectral recoil. Yet, to date, the entailing
temporal soliton dynamics from inter-mode interactions has rarely been studied,
but is critical to understand regimes of soliton-stability. Here we report the
discovery of an inter-mode breather soliton. Such breathing dynamics occurs
within a laser detuning range where conventionally stationary dissipative
solitons are expected. We demonstrate experimentally the phenomenon in two
microresonator platforms (crystalline magnesium fluoride and photonic
chip-based silicon nitride microresonators), and theoretically describe the
dynamics based on a pair of coupled Lugiato-Lefever equations. We demonstrate
experimentally that the breathing is associated with a periodic energy exchange
between the soliton and another optical mode family. We further show that
inter-mode interactions can be modeled by a response function acting on
dissipative solitons. The observation of breathing dynamics in the
conventionally stable soliton regime is critical to applications, ranging from
low-noise microwave generation, frequency synthesis to spectroscopy. On a
fundamental level, our results provide new understandings of the rich
dissipative soliton dynamics in multimode nonlinear cavities.
