Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire

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 schrödinger 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.L'étude de la propagation de la lumière dans les fibres optiques monomodes requiert la prise en compte de plusieurs phénomènes compliqués tels que la dispersion modale de polarisation et l'effet Kerr. Il s'est avéré que l'évolution de l'enveloppe lentement variable du champ électrique est bien décrite par un système couplé d'équations de Schrödinger non linéaires à coefficients aléatoires : l'équation de Manakov PMD. Cette équation fait intervenir différentes échelles dont le ratio est donné par un petit paramètre. La première partie de ce travail consiste à étudier le comportement asymptotique de la solution de l'équation de Manakov PMD lorsque ce petit paramètre tend vers zéro. En généralisant la théorie de l'Approximation-Diffusion au cadre de la dimension infinie, on a montré que la dynamique asymptotique est donnée par une équation aux dérivées partielles stochastiques dirigée par un mouvement brownien de dimension trois. Dans une seconde partie, nous proposons un schéma de différences finies de type Crank Nicolson pour cette équation pour lequel nous obtenons un ordre de convergence en probabilité d'ordre 1/2. La discrétisation du bruit doit être implicite afin d'obtenir un schéma conservatif et stable. Enfin la dernière partie est relative à la simulation numérique de la dispersion modale de polarisation et à ses effets sur la propagation et la collision de solitons de Manakov. Dans ce cadre, on propose une méthode de réduction de variance valable pour les équations aux dérivées partielles stochastiques

Strong order of convergence of a semidiscrete scheme for the stochastic Manakov equation

It is well accepted by physicists that the Manakov PMD equation is a good model to describe the evolution of nonlinear electric fields in optical fibers with randomly varying birefringence. In the regime of the diffusion approximation theory, an effective asymptotic dynamics has recently been obtained to describe this evolution. This equation is called the stochastic Manakov equation. In this article, we propose a semidiscrete version of a Crank Nicolson scheme for this limit equation and we analyze the strong error. Allowing sufficient regularity of the initial data, we prove that the numerical scheme has strong order 1/2

Role of nonlinearity in non-Hermitian quantum mechanics: Description of linear quantum electrodynamics from the nonlinear Schrödinger-Poisson equation

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

In this article, we propose a generalization of the theory of diffusion approximation for random ODE to a nonlinear system of random Schrödinger 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-Marty, where a linear electric field is considered, and we get an asymptotic dynamic for the nonlinear electric field

Bloch theory and spectral gaps for linearized water waves

International audienceThe system of equations for water waves, when linearized about equilibrium of a fluid body with a varying bottom boundary, is described by a spectral problem for the Dirichlet -- Neumann operator of the unperturbed free surface. This spectral problem is fundamental in questions of stability, as well as to the perturbation theory of evolution of the free surface in such settings. In addition, the Dirichlet -- Neumann operator is self-adjoint when given an appropriate definition and domain, and it is a novel but very natural spectral problem for a nonlocal operator. In the case in which the bottom boundary varies periodically, $\{y = -h + b(x)\}$ where $b(x+\gamma) = b(x)$, $\gamma \in \Gamma$ a lattice, this spectral problem admits a Bloch decomposition in terms of spectral band functions and their associated band-parametrized eigenfunctions. In this article we describe this analytic construction in the case of a spatially periodic bottom variation from constant depth in two space dimensional water waves problem, giving a construction of the Bloch eigenfunctions and eigenvalues as a function of the band parameters and a description of the Dirichlet -- Neumann operator in terms of the bathymetry $b(x)$. One of the consequences of this description is that the spectrum consists of a series of bands separated by spectral gaps which are zones of forbidden energies. For a given generic periodic bottom profile $b(x)=\varepsilon \beta(x)$, every gap opens for a sufficiently small value of the perturbation parameter $\varepsilon$

Analysis and simulation of rare events for SPDEs

International audienceIn this work, we consider the numerical estimation of the probability for a stochastic process to hit a set B before reaching another set A. This event is assumed to be rare. We consider reactive trajectories of the stochastic Allen-Cahn partial differential evolution equation (with double well potential) in dimension 1. Reactive trajectories are defined as the probability distribution of the trajectories of a stochastic process, conditioned by the event of hitting B before A. We investigate the use of the so-called Adaptive Multilevel Splitting algorithm in order to estimate the rare event and simulate reactive trajectories. This algorithm uses a reaction coordinate (a real valued function of state space defining level sets), and is based on (i) the selection, among several replicas of the system having hit A before B, of those with maximal reaction coordinate; (ii) iteration of the latter step. We choose for the reaction coordinate the average magnetization, and for B the minimum of the well opposite to the initial condition. We discuss the context, prove that the algorithm has a sense in the usual functional setting, and numerically test the method (estimation of rare event, and transition state sampling)