Exact soliton-like solutions of the radial Gross-Pitaevskii equation

We construct exact ring soliton-like solutions of the cylindrically symmetric (i.e. radial) Gross-Pitaevskii equation with a potential, using the similarity transformation method. Depending on the choice of the allowed free functions, the solutions can take the form of stationary dark or bright rings whose time dependence is in the phase dynamics only, or oscillating and bouncing solutions, related to the second Painleve transcendent. In each case the potential can be chosen to be time independent.