Mathematical analysis of adaptive step-size techniques when solving the nonlinear Schrödinger equation for simulating light-wave propagation in optical fibers

In optics the nonlinear Schrödinger equation (NLSE) which modelizes light-wave propagation in an optical fibre is the most widely solved by the Symmetric Split-Step method. The practical efficiency of the Symmetric Split-Step method is highly dependent on the computational grid points distribution along the fiber and therefore an efficient adaptive step-size control strategy is mandatory. A lot of adaptive step-size methods designed to be used in conjunction with the Symmetric Split-Step method for solving the various forms taken by the NLSE can be found in the literature dedicated to optics. These methods can be gathered together into 2 groups. Broadly speaking, a first group of methods is based on the observation along the propagation length of the behavior of a given optical quantity (e.g. the photons number) and the step-size at each computational step is set so as to guarantee that the known properties of the quantity are preserved. Most of the time these approaches are derived under specific assumptions and the step-size selection criterion depends on the fiber parameters. The second group of methods makes use of some mathematical concepts to estimate the local error at each computational grid point and the step-size is set so as to maintain it lower than a prescribed tolerance. This approach should be preferred due to its generality of use but suffers of a lack of understanding in the mathematical concepts of numerical analysis it involves. The aim of this paper is to present an analysis of local error estimate and adaptive step-size control techniques for solving the NSLE by the Symmetric Split-Step method with all the unavoidable mathematical rigor required for a comprehensive understanding of the topic.