Numerical solutions of a variable-coefficient nonlinear Schrödinger equation for an inhomogeneous optical fiber

Abstract

This paper investigates a variable-coefficient nonlinear Schrödinger equation for an inhomogeneous optical fiber. Numerical one- and two-solitonic envelopes of the electrical field via the fourth-order split-step Runge–Kutta, split-step Fourier and Runge–Kutta methods with equal grids in the τ axis and equal grids in the ξ axis are graphically presented, respectively, where τ and ξ represent the retarded time and normalized distance along the fiber. 2-norm of the relative errors between the analytical solutions under the Painlevé integrability condition and numerical solutions are given, where the CPU time is also shown. Relative errors and CPU time of the numerical one- and two-soliton solutions with equal grids in the ξ axis are bigger than those with equal grids in the τ axis, which does not mean that the results with equal grids in the ξ axis are infeasible. Compared with the numerical solutions with equal grids in the τ axis, those with equal grids in the ξ axis are closer to the analytical solutions without the Painlevé integrability condition. With respect to the relative errors and CPU time, one could choose the split-step Fourier method to derive the numerical one-soliton solutions, while the numerical two-soliton solutions are gotten with the RK method. The attenuation coefficient makes the amplitudes of the solitons decrease, the group velocity dispersion coefficient leads to the periodic solitons, while the effect of the attenuation coefficient is more obvious than that of the nonlinearity parameter. Effects of ηR,1 and ηI,1 on the solitonic weak interaction between the two solitons are investigated: Solitonic weak interaction between the two solitons enhances with ηR,1 and ηI,1 increasing, where ηR,1 and ηI,1 are the frequencies of the two solitons.