Head-on collision of ring dark solitons in Bose--Einstein condensates

The ring dark solitons and their head-on collisions in aBose--Einstein condensates with thin disc-shaped potentialare studied. It is shown that the system admits a solution withtwo concentric ring solitons, one moving inwards and the othermoving outwards, which in small-amplitude limit, are described bythe two cylindrical KdV equations in the respective referenceframes. By using the extended Poincar'e--Lighthill--Kuo perturbation method, the analytical phase shifts followingthe head-on collisions between two ring dark solitary waves arederived. It is shown that the phase shifts decrease with theradial coordinate $r$ according to the $r^{-1/3}$ law and dependon the initial soliton amplitude and radius.