  Breathers and rogue waves of special coupled nonlinear Schr\"odinger systems
(the Manakov equations) are studied analytically. These systems model the
orthogonal polarization modes in an optical fiber with randomly varying
birefringence. Unlike the situation in a waveguide with zero birefringence,
rogue waves can occur in these Manakov systems with dispersion and nonlinearity
of opposite signs, provided that a group velocity mismatch is present. The
criterion for the existence of rogue waves correlates exactly with the onset of
modulation instability. Theoretically the Hirota bilinear transform is employed
and rogue waves are obtained as a long wave limit of breathers. In terms of
wave profiles, a black rogue wave (intensity dropping to zero) and the
transition to a four-petal configuration are identified. Sufficiently strong
modulation instabilities of the background may overwhelm or mask the
development of the rogue waves, and such thresholds are correlated to actual
physical properties of optical fibers. Numerical simulations on the evolution
of breathers are performed to verify the prediction of the analytical
formulations.
