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We investigate theoretically nonlinear transmission in space-division multiplexed (SDM) systems using multimode fibers exhibiting rapidly varying birefringence. A primary objective is to generalize the Manakov equations, well known in the case of single-mode fibers. We first investigate the case where linear coupling among spatial modes of the fiber is weak and derive new Manakov equations after averaging over random birefringence fluctuations. Such an averaging reduces the number of intermodal nonlinear terms drastically since all four-wave-mixing terms vanish. Cross-phase modulation terms still affect multimode transmission but their effectiveness is reduced. We verify the accuracy of new Manakov equations by simulating the transmission of multiple 114-Gb/s bit streams in the PDM-QPSK format over different modes of a multimode fiber and comparing the numerical results with those obtained by solving the full stochastic equations. The agreement is excellent in all cases studied. A major benefit of the new Manakov equations is that they typically reduce the computation time by more than a factor of 10. Our results show that birefringence fluctuations improve system performance by reducing the impact of fiber nonlinearities. The extent of improvement depends on the fiber design and how many spatial modes are used for SDM transmission. We also consider the case where all spatial modes experience strong random linear coupling modeled using a random matrix. We derive new Manakov equations in this regime and show that the impact of some nonlinear effects can be reduced relatively to single-modes fibers. Finally, we extend our analysis to multicore fibers and show that the Manakov equations obtained in the strong- and weak-coupling regimes can still be used depending on the extent of coupling among fiber cores.