Role of stationary invariant manifolds in the spatiotemporal dynamics of a nonlinear-wave system of finite extension: Application to polarization attraction in optical fibers

The study of the spatiotemporal dynamics of two counterpropagating beams in optical fibers has recently been the subject of a growing renewed interest. This system has been shown to exhibit a phenomenon of polarization attraction which can be used to achieve a complete polarization of an initially unpolarized beam, almost without any loss of energy. In previous works, a theoretical description of this phenomenon has been developed in the particular case where the underlying stationary system exhibits the important property of integrability. Our aim here is to provide a generalization of the theoretical description to nonintegrable stationary systems. The analysis reveals that the spatiotemporal dynamics of the system relaxes towards a stationary trajectory whose geometric structure is revealed by the stable and unstable manifolds of some singular fixed points of the stationary system. We illustrate the theory by considering the representative and concrete example of a weakly birefringent optical fiber system.