  We study the stability and the attractor dynamics of an elongated
Bose-Einstein condensate with dark or grey kink solitons in the presence of
localized dissipation. To this end, the 3D Gross-Pitaevskii equation with an
additional imaginary potential is solved numerically. We analyze the
suppression of the snaking instability in dependence of the dissipation
strength and extract the threshold value for the stabilization of the dark
soliton for experimentally realistic parameters. Below the threshold value, we
observe the decay into a solitonic vortex. Above the stabilization threshold,
we observe the attractor dynamics towards the dark soliton when initially
starting from a grey soliton. We find that for all initial conditions the dark
soliton is the unique steady-state of the system - even when starting from the
BEC ground state.

Quantum Gases