An Embedded Split-Step method for solving the nonlinear Schrödinger equation in optics

Abstract

In optics the nonlinear Schrödinger equation (NLSE) which modelizes wave propagation in an optical fiber is mostly 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, therefore an efficient adaptive step-size control strategy is mandatory. The most common approach for step-size control is the "step-doubling" approach. It provides an estimation of the local error for an extra computational cost of around 50 %. Alternatively there exist in optics literature other approaches based on the observation along the propagation length of the behavior of a given optical quantity. The step-size at each computational step is set so as to guarantee that the known properties of the quantity are preserved. These approaches derived under specific physical assumptions are low cost but suffer from a lack of generality. In this paper we present a new method for estimating the local error in the Symmetric Split-Step method when solving the NLSE. It conciliates the advantages of the step-doubling approach in terms of generality without the drawback of requiring a significant extra computational cost. The method is related to Embedded Split-Step methods for nonlinear evolution problems.

Simulation of white light generation and near light bullets using a novel numerical technique

“…Per propagation slice the operators are applied in an iterative fashion. However, the operators do not commute and a Strang symmetrisation scheme must be implemented to reduce this error [9]. The symmetrisation relies on how fast operators vary with the propagation coordinate.…”

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

“…We have shown that when solving the NLSE (where the nonlinear problem (8) admits an explicit solution), the Step-Doubling method (also referred as the Local Error method [25]) is the method of choice for adaptive step-size control. One alternative approach would be to use an Embedded Split-Step (ESS) scheme, see [4] for a presentation of the ESS method applied to the NLSE and a comparison of the 2 methods. We have also highlighted the difficulty in designing an adaptive step-size control strategy for solving the GNLSE by the S3F method.…”

“…N = N 0 as defined in (6)) the nonlinear ODE problem (8) admits an analytical solution and the local error in the S3F method only amounts to the splitting error as given by (10). In such a case, although an approach similar to the one performed to estimate the local error when solving a nonlinear ODE by an ERK method could be considered (involving 2 embedded Split-Step schemes of different orders [4,17]), the usual method for estimating the splitting local error is based on the step-doubling technique. Actually, it is the adaptive step-size control used in conjunction with the S3F method for solving the NLSE presented in [25].…”

A three operator split-step method covering a larger set of non-linear partial differential equations

“…The Split-Step Fourier Method has been extensively explored in one dimensional systems such as the one-dimensional cubic Non-Linear Schrodinger Equation (NLSE) that models the substantive spectral broadening of light (accordingly named supercontinuum generation) as it propagates in fiber. Past studies focused primarily on estimating step-size dependent error and deriving adaptive step-size algorithms for the implementation of the method [12][13][14]. Other studies explored the stability of the method [15] or its application around certain bound solutions of a NLSE such as soliton formation [16].…”