  We numerically investigate azimuthal modulation instability in an optical
fiber supporting orbital angular momentum modes only, i.e. a vortex fiber, by
means of the scalar multimode unidirectional pulse propagation equation. We
demonstrate that the nonlinear stage of azimuthal modulation instability taking
place in such a ring-core fiber, with anomalous rotation group-velocity
dispersion, can be simply described by analytical breather solutions of the
corresponding nonlinear Schr\"odinger equation. Azimuthal soliton dynamics and
nonlinear compression are also unveiled as well as specific spatial rotating
features as a function of the topological charge involved. Our results open a
new route for studying transverse nonlinear waves in optical fibers and for
manipulating orbital angular momentum states.
