Stability Properties of Light Propagating in Fiber Optics

The study of the propagation of light is a fascinating fields of research that is interesting in itself because it gives rise to beautiful and complex mathematical structures. Furthermore, understanding the propagation of light is crucial in applications such as lasers and optical fibers. In this proposal I will focus on a model that has been proven to be quite successful in describing the propagation of light in a certain type of fiber optics: the Manakov system. This system consists of two differential equations, that is two equations involving derivatives. The solutions to these equations represent possible behaviors for the light. The goal of the project is to study solutions to the Manakov system which are called solitons. This type of solutions has been proven to be very useful in several fields of science. In the context of fiber optics, the hope is that solitons can provide a very efficient way to send information on long distances through fiber optics. More precisely, the project deals with the study of the stability properties of soliton solutions. Stability is a fundamental concept in physics. An illustration of this concept is given by trying to make a pencil stand on its lead. In theory, it is possible but, in practice, because it is such an unstable state, it cannot be done. The same concept applies to solutions of differential equations: some are stable and some are not. In the case of differential equations though, it is often necessary to develop some sophisticated mathematical methods to study stability. However, it is crucial to study the stability of the solutions of the Manakov system because only stable solutions can be observed experimentally. Additionally, I intend to study generalizations of the Manakov system which take into account additional physical properties of the fiber optics. I plan to use the results obtained for the Manakov system and use them to study (by a perturbative approach) these more general systems. The research of this proposal can be summarized as followed: • Develop a method to study the stability of certain solutions in the context of optics. • Study the physical conditions under which stable solutions of the Manakov system propagate. • Extend the study to systems that include more physical effects than the Manakov system.