A Study of a Nonlinear Schr\"odinger Equation for Optical Fibers

Abstract

Non linear fiber optics concerns with the non linear optical phenomena occurring inside optical fibers. The propagation of light in single-mode fibers is governed by the one-dimensional nonlinear Schr\"odinger equation (NLS) in the presence of attenuation, dispersion, and non linear effects. In this NLS the role of space and time is exchanged with respect to the standard NLS as introduced in the almost entire mathematical literature. Such an exchange is far from being only formal as it enters in the interpretation of basic ideas as well-posedness and stability, as well as in the phenomenological meaning of the predictions formulated by such model. In this dissertation, the physical bases of the optical fibers are provided, and a derivation of the NLS from the Maxwell's equations is reviewed. Furthermore, problems of local nature (local existence of solutions, uniqueness) and problems of global nature (global existence) are studied by the Kato's method based on a fixed point argument and Strichartz's estimates. Moreover, tools usually employed to study global problems connected with finite-time blow up of solutions, are here used to show a closeness result between the NLS for optical fibers and an integrable NLS. Integrability is pursued by means of the Painlev\'e analysis that allows to describe quite a large class of integrable equations like the transformed equation from the standard NLS. Among these functions, one can be described as a standard NLS with an additional linear harmonic oscillator term. This equation is showed to be close in $L^2$-norm to the NLS for single-mode fibers. Finally, the dissertation focuses on two kinds of stationary solutions: the so-called time-homogenous, that is given by a stationary wave oscillating in space, and the soliton-like solution. Results are obtained about stability properties of them.