  In this article we propose a generalization of the theory of diffusion
approximation for random ODE to a nonlinear system of random Schr\"{o}dinger
equations. This system arises in the study of pulse propagation in randomly
birefringent optical fibers. We first show existence and uniqueness of
solutions for the random PDE and the limiting equation. We follow the work of
Garnier and Marty [Wave Motion 43 (2006) 544-560], Marty [Probl\`{e}mes
d'\'{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. Paul
Sabatier], where a linear electric field is considered, and we get an
asymptotic dynamic for the nonlinear electric field.
