Position-dependent fusion and non-fusion of colliding dark solitons in harmonically trapped Bose–Einstein condensates

Dark-soliton collisions exhibit two distinct regimes, fusion and non-fusion, characterized by different density profiles at the collision instant. For symmetric soliton pairs on a homogeneous background, the two regimes are separated by the critical velocity vc=cs/2. In a harmonically trapped Bose–Einstein condensate, however, both the background density and the soliton velocities vary along the trajectories, making the incoming local conditions position dependent. Combining the exact homogeneous two-soliton solution with a local-density description of the trapped background, we obtain ρpred=n(xcpred)[2Aavg,local(xcpred)−1]2, where xcpred is the encounter position predicted from the isolated-soliton trajectories and Aavg,local=|v1−v2|/(2cs) is the normalized local relative speed evaluated at that position. The expression predicts the minimum density of the merged dip in the fusion regime and the height of the central bump in the non-fusion regime, with the local threshold at Aavg,local=1/2. Gross–Pitaevskii simulations of symmetric and asymmetric soliton pairs in both head-on and catch-up geometries provide a direct determination of the collision outcome and of the collision-center density ρmeas. After normalization by n(xcpred), the measured densities collapse onto the curve (2Aavg,local−1)2 over both collision regimes.