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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.

The coupled nonlinear Schrödinger equation (CNLSE) can be employed in simulating the propagation of the optical soliton in a birefringent fiber [

During the last few decades, researchers have found that hereditary properties and memory of various materials can be depicted more precisely using the temporal fractional derivatives [

In this article, we consider the FCNLSE given as follows [

Both analytical treatments and numerical methods have been investigated for the fractional Schrödinger equations and some novel types of nonlinear Schrödinger equations. In [

However, to the best of our current knowledge, numerical methods for the coupled space-time fractional Schrödinger equations are rarely considered. In this paper, we modify the exponential time-differencing (ETD) method for the time-fractional nonlinear PDEs, introduced in [

The spatial Riesz derivative is defined in [

It is stated in [

Similarly, the approximation of the right derivative

Matrices

Furthermore, we use the centered difference method for the fractional derivative to approximate the Riesz derivative in the following way [

Noticed that scheme (

We obtain a system of time-fractional equations after discretizing FCNLSE (

As been computed using the variation of constant formula in [

Formula (

Then, the ETD scheme can be denoted as [

Scheme (

Garrappa and Popolizio proved in [

The two-step ETD scheme is also constructed in [

Garrappa and Popolizio also proved in [

To relief the burden of computing the function

Moreover, we use the Padé approximation

After simplification, formula (

The Padé approximation (

The approximation error of formula (

Then, we have

Then, we compute the error of approximation as

As stated by Sarumi et al. [

We tested the ETD schemes with Padé approximation on an initial boundary value problem with analytical solutions. The numerical errors in this section are computed as

The rate of convergence in time is computed as

The experiments were compiled on an Intel Core i5-6200U 2.30 GHz workstation, and MATLAB R2016b was chosen as computation software.

Firstly, we consider the following FCNLSE as suggested by Esen et al. [

The analytical solutions to FCNLSE (

We plot the traces of numerical solutions to FCNLSE (

The trace of the solution to

The trace of the solution to

The trace of the solution to

The trace of the solution to

In Tables

Temporal convergence rates of the two-step ETD scheme with Padé approximation for FCNLSE (

Order | Order | |||
---|---|---|---|---|

0.1 | 2.3309 | — | 2.2265 | — |

0.05 | 7.6231 | 1.6124 | 7.2183 | 1.6250 |

0.025 | 2.4983 | 1.6094 | 2.3503 | 1.6188 |

0.0125 | 8.1755 | 1.6116 | 7.6287 | 1.6233 |

Temporal convergence rates of the two-step ETD scheme with Padé approximation for FCNLSE (

Order | Order | |||
---|---|---|---|---|

0.1 | 1.8527 | — | 1.7532 | — |

0.05 | 5.3631 | 1.7885 | 4.9810 | 1.8155 |

0.025 | 1.5159 | 1.8229 | 1.3967 | 1.8344 |

0.0125 | 4.2812 | 1.8241 | 3.9562 | 1.8198 |

Secondly, we solve FCNLSE (

In Figures

The trace of the solution to

The trace of the solution to the real part of

The trace of the solution to

The trace of the solution to the real part of

The trace of the solution to

The trace of the solution to the real part of

The trace of the solution to

The trace of the solution to the real part of

In Table

Computation time (CPU time in s) needed for solving FCNLSE (

0.3 | 0.3 | 0.3 | 0.6 | 0.6 | 0.6 | |
---|---|---|---|---|---|---|

1.2 | 1.5 | 1.8 | 1.2 | 1.5 | 1.8 | |

2.452 | 2.445 | 2.360 | 2.313 | 2.388 | 2.426 | |

4.736 | 4.823 | 4.633 | 4.577 | 4.579 | 4.752 | |

7.305 | 7.537 | 7.345 | 7.292 | 7.334 | 7.436 | |

9.238 | 9.426 | 9.332 | 9.443 | 9.563 | 9.573 |

To solve the space-time fractional coupled nonlinear Schrödinger equation efficiently, we employed exponential time-differencing schemes for the fractional derivative in time. During this process, the Mittag-Leffler function is computed using the Padé approximation. It has been shown in the numerical experiments that the Padé approximation reduces the computational time markedly compared to the original exponential time-differencing scheme. The error of the Padé approximation to the Mittag-Leffler function has been analyzed, and the convergence rates of the schemes have been computed and demonstrated in the Numerical Experiments section. Figures

No data were used to support this study.

The authors declare that they have no conflicts of interest.

Both authors contributed equally.

This work was supported by the Natural Science Foundation of Hubei Province, China (Grant no. 2019CFB243) and the National Natural Science Foundation of China (Grant no. 12026263).