Mathematical analysis of light propagation in optical fibers with randomly varying birefringence

The study of light propagation in monomode optical fibers requires to take care of various complex phenomena such as the polarization mode dispersion (PMD) and the Kerr effect. It has been proved that the slowly varying envelope of the electric field is well described by a coupled non linear schrodinger equation with random coefficients called the Manakov PMD equation. The particularity of this equation is the presence of various length scales whose ratio is given by a small parameter. The first part of this thesis is concerned with the theoretical study of the asymptotic dynamic of the solution of the Manakov PMD equation as this parameter goes to zero. Generalizing the theory of the Diffusion Approximation in the infinite dimensional setting, we were able to prove that the asymptotic dynamic is given by a stochastic partial differential equation driven by three Brownian motions. In a second part, we propose a Crank Nicolson scheme for this equation and we prove that the order of convergence is 1/2. The discretization of the noise term is taken implicit so that the scheme is conservative and stable. Finally the last part is concerned with numerical simulations of the PMD and propagation and collision of Manakov solitons. The above scheme is implemented and we propose a variance reduction method valid in the context of stochastic partial differential equations.