Comparative study of finite difference methods and pseudo-spectral methods for solving the nonlinear Schrödinger equation in optical fiber

Abstract

This work evaluates the suitability of the finite difference methods and the pseudo-spectral methods for validating the pulse propagation problem in an optical fiber, which is modeled by the nonlinear Schrödinger equation (NLSE) represented in a classical electromagnetic version. In particular, the finite difference methods have been reported as excellent schemes for solving the nonlinear Schrödinger type-equations represented in multiple fields of study. However, a rigorous analysis of the finite difference methods for solving specifically the NLSE in fiber has not been reported yet. On the other hand, the pseudo-spectral methods are reported as optimal schemes to integrate the NLSE in fiber. Thus, four schemes of finite difference methods and three schemes of pseudo-spectral methods are analyzed by the validation of the propagation of a fundamental soliton, which demands a high level of convergence and stability to reproduce the complex behavior involved in this pulse propagation problem. As a result, we observe that the multiple phenomenologies modeled by the NLSE in fiber are reproduced numerically with the best degree of convergence and stability by the pseudo-spectral methods, whereas the finite difference methods are not suitable to validate this pulse propagation problem due to a loss of convergence and a high computational cost.