Efficient Exponential Time-Differencing Methods for the Optical Soliton Solutions to the Space-Time Fractional Coupled Nonlinear Schrödinger Equation

Xiao Liang,Bo Tang,Ahmet Ocak Akdemir

doi:10.1155/2021/5575128

Abstract

The coupled nonlinear Schrödinger equation is used in simulating the propagation of the optical soliton in a birefringent fiber. Hereditary properties and memory of various materials can be depicted more precisely using the temporal fractional derivatives, and the anomalous dispersion or diffusion effects are better described by the spatial fractional derivatives. In this paper, one-step and two-step exponential time-differencing methods are proposed as time integrators to solve the space-time fractional coupled nonlinear Schrödinger equation numerically to obtain the optical soliton solutions. During this procedure, we take advantage of the global Padé approximation to evaluate the Mittag-Leffler function more efficiently. The approximation error of the Padé approximation is analyzed. A centered difference method is used for the discretization of the space-fractional derivative. Extensive numerical examples are provided to demonstrate the efficiency and effectiveness of the modified exponential time-differencing methods.

Highlights

Introduction e coupled nonlinearSchrodinger equation (CNLSE) can be employed in simulating the propagation of the optical soliton in a birefringent fiber [1,2,3]
We obtain a system of time-fractional equations after discretizing fractional coupled nonlinear Schrodinger equation (FCNLSE) (1) in space: zα ztα U(t) + AU(t) F(U(t)), (11)
E Padeapproximation (24) to the ML function can be applied to the one-step and two-step exponential time-differencing (ETD) schemes (15) and (18) to enhance the efficiency

Summary

The Exponential Time Integrator

We obtain a system of time-fractional equations after discretizing FCNLSE (1) in space: zα ztα U(t) + AU(t) F(U(t)),. E two-step ETD scheme is constructed in [19]: Un eα, tn; A􏼁U0 + W(n1)F U0􏼁 + 􏽘 W(n2,j)F􏼐Uj􏼑. Garrappa and Popolizio proved in [19] that the twostep ETD scheme (18) has the absolute approximation error Errn ‖U(tn) − Un‖ satisfying. To relief the burden of computing the function eα,β(t; A), we transform it using the multiplication of eigenvectors and functions of eigenvalues [23]:. Using this decomposition, we avoid the computation of the ML function of matrices, which is really time consuming. We only need to calculate the ML function with inputs of numbers and multiply the matrices, which reduces the time of computation significantly. E Padeapproximation (24) to the ML function can be applied to the one-step and two-step ETD schemes (15) and (18) to enhance the efficiency. As stated by Sarumi et al [24], to make the approximation is is wofhRy mαw,β,neruelsieabRle3α,,2βfotroβa≠pαp,rowxeimneaeted to have m ≥ n + 1. the Mittag-Leffler function

Numerical Experiments

Conclusion