  Breathing solitons are nonlinear waves in which the energy concentrates in a
localized and oscillatory fashion. Similarly to stationary solitons, breathers
in dissipative systems can form stable bound states displaying molecule-like
dynamics, which are frequently called breather molecules. So far, the
experimental observation of optical breather molecules and the real-time
detection of their dynamics have been limited to diatomic molecules, that is,
bound states of only two breathers. In this work, we report on the observation
of different types of breather complexes in a mode-locked fibre laser:
multi-breather molecules, and molecular complexes originating from the binding
of two breather-pair molecules or a breather pair molecule and a single
breather. The inter-molecular temporal separation of the molecular complexes
attains several hundreds of picoseconds, which is more than an order of
magnitude larger than that of their stationary soliton counterparts and is a
signature of long-range interactions. Numerical simulations of the laser model
support our experimental findings. Moreover, non-equilibrium dynamics of
breathing solitons are also observed, including breather collisions and
annihilation. Our work opens the possibility of studying the dynamics of
many-body systems in which breathers are the elementary constituents.
