  We derive exact solutions to the sine--Gordon equation describing localized
structures on the background of librational and rotational travelling waves. In
the case of librational waves, the exact solution represents a localized spike
in space-time coordinates (a rogue wave) which decays to the periodic
background algebraically fast. In the case of rotational waves, the exact
solution represents a kink propagating on the periodic background and decaying
algebraically in the transverse direction to its propagation. These solutions
model the universal patterns in the dynamics of fluxon condensates in the
semi-classical limit. The different dynamics is related to different outcomes
of modulational stability of the librational and rotational waves.
