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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 largerdoi:10.3390/e19090445 fatcat:bhalslbeargo3iswiaq4zl2lgi