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In this article we propose a generalization of the theory of diffusionapproximation for random ODE to a nonlinear system of random Schr\"{o}dingerequations. This system arises in the study of pulse propagation in randomlybirefringent optical fibers. We first show existence and uniqueness ofsolutions for the random PDE and the limiting equation. We follow the work ofGarnier and Marty [Wave Motion 43 (2006) 544-560], Marty [Probl\`{e}mesd'\'{e}volution en milieux al\'{e}atoires: Th\'{e}or\`{e}mes limites,sch\'{e}mas num\'{e}riques et applications en optique (2005) Univ. PaulSabatier], where a linear electric field is considered, and we get anasymptotic dynamic for the nonlinear electric field.

A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers