  We investigate out-of-equilibrium stationary processes emerging in a Discrete
Nonlinear Schroedinger chain in contact with a heat reservoir (a bath) at
temperature $T_L$ and a pure dissipator (a sink) acting on opposite edges. We
observe two different regimes. For small heat-bath temperatures $T_L$ and
chemical-potentials, temperature profiles across the chain display a
non-monotonous shape, remain remarkably smooth and even enter the region of
negative absolute temperatures. For larger temperatures $T_L$, the transport of
energy is strongly inhibited by the spontaneous emergence of discrete
breathers, which act as a thermal wall. A strongly intermittent energy flux is
also observed, due to the irregular birth and death events of the breathers.
The corresponding statistics exhibits the typical signature of rare events of
processes with large deviations. In particular, the breather lifetime is found
to be ruled by a stretched-exponential law.
