  This paper starts by an investigation of nonlinear transmission in
space-division multiplexed (SDM) systems using multimode fibers exhibiting a
rapidly varying birefringence. A primary objective is to generalize the Manakov
equations, well known in the case of single-mode fibers. We first investigate a
reference case where linear coupling among the spatial modes of the fiber is
weak and after averaging over birefringence fluctuations, we obtain new Manakov
equations for multimode fibers. Such an averaging reduces the number of
intermodal nonlinear terms drastically since all four-wave-mixing terms average
out. Cross-phase modulation terms still affect multimode transmission but their
effectiveness is reduced. We then verify the accuracy of our new Manakov
equations by transmitting multiple PDM-QPSK signals over different modes of a
multimode fiber and comparing the numerical results with those obtained by
solving the full stochastic equation. The agreement is excellent in all cases
studied. A great benefit of the new equations is to reduce the computation time
by a factor of 10 or more. Another important feature observed is that
birefringence fluctuations improve system performance by reducing the impact of
fiber nonlinearities. Finally multimode fibers with strong random coupling
among all spatial modes are considered. Linear coupling is modeled using the
random matrix theory approach. We derive new Manakov equations for multimode
fibers in that regime and show that such fibers can perform better than
single-modes fiber for large number of propagating spatial modes.
