Collisional phase shifts of ring dark solitons in inhomogeneous Bose–Einstein condensates

The head-on collisions of two ring dark solitons in inhomogeneous Bose–Einstein condensates (BECs) with thin disk-shaped potential are studied by the extended Poincaré–Lighthill–Kuo (PLK) perturbation method. The result shows that the system admits a solution with two concentric ring solitons, one moving inwards and the other moving outwards, which in small-amplitude limit, are described by two modified cylindrical KdV equations in the respective reference frames. In particular, the analytical phase shifts induced by the head-on collisions between two ring dark solitary waves are derived, and the result shows that the phase shifts change with the radial coordinate r   according to the (1+σr2)r−1/3(1+σr2)r−1/3 law (where σ∼ωr2/ωz2), which are quite different with the homogeneous case.