  We theoretically examine the vortex states of a gas of trapped
quasi-two-dimensional ultracold bosons subject to a density-dependent gauge
potential, realizing an effective nonlinear rotation of the atomic condensate,
which we also show is within the reach of current experimental techniques with
ultracold atom experiments. The nonlinear rotation has a two-fold effect; as
well as distorting the shape of the condensate it also leads to an
inhomogeneous vorticity resulting in novel morphological and topological
states, including ring vortex arrangements that do not follow the standard
Abrikosov result. The dynamics of trapped vortices are also explored, which
differs from the case of rigid-body rotation due to the absence of a global
laboratory reference frame.
