Stability analysis and attractor dynamics of three-dimensional dark solitons with localized dissipation

We study the stability and the attractor dynamics of an elongated Bose-Einstein condensate (BEC) with dark or gray kink solitons in the presence of localized dissipation. To this end, the three-dimensional 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 gray 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.