Journal of Mathematics
Volume 2021 (2021), Article ID 5575128, 10 pages
https://doi.org/10.1155/2021/5575128

Research Article

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

Xiao Liang¹ and Bo Tang1,2

¹School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang, Hubei 441053, China
²School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, China

Correspondence should be addressed to Xiao Liang; xiao88160@126.com

Received 28 February 2021; Revised 1 April 2021; Accepted 13 April 2021; Published 24 April 2021

Academic Editor: Ahmet Ocak Akdemir

Copyright © 2021 Xiao Liang and Bo Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

The coupled nonlinear Schrödinger equation (CNLSE) can be employed in simulating the propagation of the optical soliton in a birefringent fiber [1–3]. A soliton is a solitary pulse which can travel at a constant speed and keep a stationary shape due to the balancing of the self-phase modulation and the group velocity dispersion effect in fiber optics [4]. According to Agrawal [5], in a fiber communication system, the input pulse may be orthogonally polarized in a birefringent fiber. The polarized components can form solitary waves, which are named as vector solitons. Because of the nonlinear coupling effect, the vector solitons can propagate undistorted even when the components have different widths or peak powers.

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 [6–8]. It is also shown in [9, 10] that the anomalous dispersion or diffusion effects are better described by the spatial fractional derivatives. The anomalous effects reflect the Lévy-type particle movement, different from Brownian motion, which depicts the classical random movement of particles. Therefore, the space-time fractional coupled nonlinear Schrödinger equation (FCNLSE) is useful in modeling solitons in fractional fiber optics.

In this article, we consider the FCNLSE given as follows [11]:

with the initial conditions

and homogeneous Dirichlet boundary conditions on , where and complex functions and represent the amplitudes of orthogonally polarized waves in a birefringent optical fiber. is the Caputo derivative of in time, is the Riesz derivative of in space, , , and the parameters and are some real constants.

Both analytical treatments and numerical methods have been investigated for the fractional Schrödinger equations and some novel types of nonlinear Schrödinger equations. In [12], an extended sinh-Gordon equation expansion method is adopted to solve the space-time fractional Schrödinger equation analytically. In [13], a modified residual power series method is implemented on the fractional Schrödinger equation. In [14], the L1 scheme together with the Fourier–Galerkin spectral method is employed to discretize the time-fractional Schrödinger model. In [15], a Fourier spectral exponential splitting scheme is constructed to solve the space-fractional initial boundary value problems. In [16], a generalized exponential rational function method is applied to a new extension of the nonlinear Schrödinger equation. In [17], a cubic-quartic nonlinear Schrödinger equation is solved analytically for the dark, singular, and bright-singular soliton solutions. In [18], a modified expansion function method and an extended sinh-Gordon method are proposed for the M-fractional paraxial nonlinear Schrödinger equation to obtain soliton solutions.

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 [19], by applying the Padé approximation. Then, we combine the modified ETD scheme with a fourth-order fractional compact scheme in space. During this procedure, the nonlinear term of the equation is computed explicitly, and the calculation of the fractional exponential time integral is undertaken more efficiently.

2. Discretization in Space

The spatial Riesz derivative is defined in [10] as

where . and are the left and right Riemann–Liouville derivatives:

in which is the gamma function.

It is stated in [20] that the approximation of the left derivative is calculated using matrix :

where

with , where is a single spatial step.

Similarly, the approximation of the right derivative is calculated using matrix :

Matrices and are transposes to each other in (5) and (6).

Furthermore, we use the centered difference method for the fractional derivative to approximate the Riesz derivative in the following way [21]:

where

Noticed that scheme (8) is second-order convergent in space, Ding et al. [22] generated a compact scheme to improve the order of convergence:

where and is the second-order approximation (8). As been proved by Theorem 11 in [22], compact scheme (10) is fourth-order convergent spatially.

3. The Exponential Time Integrator

We obtain a system of time-fractional equations after discretizing FCNLSE (1) in space:

where denotes the Caputo derivative, is the matrix in the Riesz derivative approximation, contains the nonlinear function and the boundary conditions, and with , and the initial condition is .

As been computed using the variation of constant formula in [23], system (11) has an analytical solution:

where denotes the inverse function of the Laplace transform to , and can be calculated taking advantage of the Mittag-Leffler (ML) function :

Formula (12) can be written in a discrete form after discretization on with an equal-spaced mesh-grid :

Then, the ETD scheme can be denoted as [19, 23]

where is the numerical approximation to , and the convolution weights can be computed as

Scheme (15) is called the one-step ETD scheme.

Garrappa and Popolizio proved in [19] that the one-step ETD scheme (15) has the absolute approximation error satisfying

where is the temporal step number and is a constant relating to and . This inequality tells us that ETD scheme (15) is first-order convergent temporally.

The two-step ETD scheme is also constructed in [19]:

where

Garrappa and Popolizio also proved in [19] that the two-step ETD scheme (18) has the absolute approximation error satisfying

where is the temporal step number and is a constant relating to and . This inequality tells us that ETD scheme (18) is -order convergent temporally.

To relief the burden of computing the function , we transform it using the multiplication of eigenvectors and functions of eigenvalues [23]:

where is diagonalizable, with ’s to be its eigenvalues, is the composition of ’s eigenvectors, and . 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.

Moreover, we use the Padé approximation to compute the value of the ML function [24, 25]:

with coefficients

After simplification, formula (22) becomes

The Padé approximation (24) to the ML function can be applied to the one-step and two-step ETD schemes (15) and (18) to enhance the efficiency.

4. Approximation Error Analysis

The approximation error of formula (24) is defined as [24]

Then, we have

where

in which

Then, we compute the error of approximation as

As stated by Sarumi et al. [24], to make the approximation of reliable for , we need to have . This is why we use to approximate the Mittag-Leffler function.

5. Numerical Experiments

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. [12]:

with initial conditions

and homogeneous Dirichlet boundary conditions on , where the parameters can be chosen as , , , , , and .

The analytical solutions to FCNLSE (32) are given in [12] as

where and for valid solitons.

We plot the traces of numerical solutions to FCNLSE (32) with initial conditions (33) using the one-step ETD scheme (15) and the central difference method (8) for different and values in Figures 1–4. It can be seen from the plots that and travel in the same pace and direction. This is due to the fact that and model vector solitons in a birefringent fiber. Because of the nonlinear coupling effect, the vector solitons can propagate undistorted even when the components have different widths or peak powers.

Figure 1: The trace of the solution to of FCNLSE (32) with and .

Figure 2: The trace of the solution to of FCNLSE (32) with and .

Figure 3: The trace of the solution to of FCNLSE (32) with and .

Figure 4: The trace of the solution to of FCNLSE (32) with and .

In Tables 1 and 2, the temporal convergence rates of the two-step ETD scheme (18) are computed according to formulas (30) and (31). The spatial step size is chosen as which is relatively small. The experiments are performed for both and . It can be noticed from the convergence rates that the order of convergence for is around 1.6, and the order of convergence for is around 1.8, which means the two-step ETD scheme (18) has a temporal order of .

Table 1: Temporal convergence rates of the two-step ETD scheme with Padé approximation for FCNLSE (32) with .

for
Order
for
Order

0.1
2.3309e − 3
—
2.2265e − 3
—

0.05
7.6231e − 4
1.6124
7.2183e − 4
1.6250

0.025
2.4983e − 4
1.6094
2.3503e − 4
1.6188

0.0125
8.1755e − 5
1.6116
7.6287e − 5
1.6233

Table 2: Temporal convergence rates of the two-step ETD scheme with Padé approximation for FCNLSE (32) with .

for
Order
for
Order

0.1
1.8527e − 3
—
1.7532e − 3
—

0.05
5.3631e − 4
1.7885
4.9810e − 4
1.8155

0.025
1.5159e − 4
1.8229
1.3967e − 4
1.8344

0.0125
4.2812e − 5
1.8241
3.9562e − 5
1.8198

Secondly, we solve FCNLSE (32) with initial conditions

and homogeneous Dirichlet boundary conditions on , where the parameters are chosen as and .

In Figures 5–12, the evolution traces of solutions to FCNLSE (32) with initial conditions (35) are depicted with different values of and , using the two-step ETD scheme (18) in time and the compact scheme (10) in space. It can be observed from the mesh plots that the absolute values of and remain the same, which means the magnitudes of the pulses remain identical, while the real parts of and remain opposite to each other. It can also be seen from the evolution profiles that different and values result in different diffusion effects and time delay effects.

Figure 5: The trace of the solution to of FCNLSE (32) with and .

Figure 6: The trace of the solution to the real part of with and .

Figure 7: The trace of the solution to of FCNLSE (32) with and .

Figure 8: The trace of the solution to the real part of with and .

Figure 9: The trace of the solution to of FCNLSE (32) with and .

Figure 10: The trace of the solution to the real part of with and .

Figure 11: The trace of the solution to of FCNLSE (32) with and .

Figure 12: The trace of the solution to the real part of with and .

In Table 3, the computation time is recorded solving FCNLSE (32) using the two-step ETD scheme (18) taking advantage of the Padé approximation (24) for different and values by counting the CPU time used in MATLAB. As we took the similar experiments without using the Padé approximation, the CPU time needed for the computation is about 15 times longer. This indicates the efficiency and necessity of the Padé approximation for the ETD schemes.

Table 3: Computation time (CPU time in s) needed for solving FCNLSE (32) with initial conditions (35) via the two-step ETD scheme (18) utilizing the Padé approximation (24), taking and .

0.3
0.3
0.3
0.6
0.6
0.6

1.2
1.5
1.8
1.2
1.5
1.8

t = 0.25
2.452
2.445
2.360
2.313
2.388
2.426

t = 0.5
4.736
4.823
4.633
4.577
4.579
4.752

t = 0.75
7.305
7.537
7.345
7.292
7.334
7.436

t = 1
9.238
9.426
9.332
9.443
9.563
9.573

6. Conclusion

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 1–4 express the bright soliton solutions, and Figures 5–12 depict orthogonally polarized optical waves in a birefringent fiber. The main contribution of this paper is the modification of the exponential time-differencing methods by applying the Padé approximation, as well as obtaining the soliton solutions to the fractional coupled nonlinear Schrödinger equation, which might be applicable in the industry of fiber optics.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Both authors contributed equally.

Acknowledgments

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).

References

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Physical Review A, vol. 27, no. 6, pp. 3135–3145, 1983.

A. Aydin, “Multisymplectic integration of N-coupled nonlinear Schrödinger equation with destabilized periodic wave solutions,” Chaos Solitons and Fractals, vol. 41, pp. 735–751, 2009.

H. Bulut, T. A. Sulaiman, and B. Demirdag, “Dynamics of soliton solutions in the chiral nonlinear Schrödinger equations,” Nonlinear Dynamics, vol. 91, no. 3, pp. 1985–1991, 2018.

M. Eslami, “New exact solutions for higher order nonlinear Schrödinger equation in optical fibers,” Optical and Quantum Electronics, vol. 50, Article ID 47, 2018.

G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics Series, Academic Press, San Diego, CA, USA, 2013.

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 1999.

X. Liang, “Efficient exponential time differencing methods with Padé approximations for the semilinear space-time-fractional Schrödinger equation,” Modern Physics Letters B, vol. 34, Article ID 2050428, 2020.

S. Longhi, “Fractional Schrödinger equation in optics,” Optics Letters, vol. 40, no. 6, pp. 1117–1120, 2015.

R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000.

Q. Yang, F. Liu, and I. Turner, “Numerical methods for fractional partial differential equations with Riesz space fractional derivatives,” Applied Mathematical Modelling, vol. 34, no. 1, pp. 200–218, 2010.

M. Lakestani and J. Manafian, “Analytical treatments of the space-time fractional coupled nonlinear Schrödinger equations,” Optical and Quantum Electronics, vol. 50, Article ID 396, 2018.

A. Esen, T. A. Sulaiman, H. Bulut, and H. M. Baskonus, “Optical solitons to the space-time fractional (1 + 1)-dimensional coupled nonlinear Schrödinger equation,” Optik, vol. 167, pp. 150–156, 2018.

A. Demir, M. A. Bayrak, and E. Ozbilge, “New approaches for the solution of space-time fractional Schrödinger equation,” Advances in Difference Equations, vol. 23, Article ID 133, 2020.

J. Zhang, J. Wang, and Y. Zhou, “Numerical analysis for time-fractional Schrödinger equation on two space dimensions,” Advances in Difference Equations, vol. 23, Article ID 53, 2020.

X. Liang and A. Q. M. Khaliq, “An efficient Fourier spectral exponential time differencing method for the space-fractional nonlinear Schrödinger equations,” Computers & Mathematics with Applications, vol. 75, no. 12, pp. 4438–4457, 2018.

B. Ghanbari, H. Günerhan, O. A. Ilhan, and H. M. Baskonus, “Some new families of exact solutions to a new extension of nonlinear Schrödinger equation,” Physica Scripta, vol. 95, no. 7, pp. 1–12, 2020.

W. Gao, H. F. Ismael, A. M. Husien, H. Bulut, and H. M. Baskonus, “Optical soliton solutions of the cubic-quartic nonlinear schrödinger and resonant nonlinear schrödinger equation with the parabolic law,” Applied Sciences, vol. 10, no. 1, pp. 1–20, 2020.

W. Gao, H. F. Ismael, S. A. Mohammed, H. M. Baskonus, and H. Bulut, “Complex and real optical soliton properties of the paraxial nonlinear Schrödinger equation in Kerr media with M-fractional,” Frontiers in Physics, vol. 7, no. 197, pp. 1–8, 2019.

R. Garrappa and M. Popolizio, “Generalized exponential time differencing methods for fractional order problems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 876–890, 2011.

I. Podlubny, A. Chechkin, T. Skovranek, Y. Chen, and B. M. Vinagre Jara, “Matrix approach to discrete fractional calculus II: partial fractional differential equations,” Journal of Computational Physics, vol. 228, no. 8, pp. 3137–3153, 2009.

M. D. Ortigueira, “Riesz potential operators and inverses via fractional centred derivatives,” International Journal of Mathematics and Mathematical Sciences, vol. 2006, pp. 1–12, 2006.

H. F. Ding, C. P. Li, and Y. Q. Chen, “High-order algorithms for riesz derivative and their applications,” Abstract and Applied Analysis, vol. 23, Article ID 653797, 2014.

R. Garrappa, “Exponential integrators for time-fractional partial differential equations,” The European Physical Journal Special Topics, vol. 222, no. 8, pp. 1915–1927, 2013.

I. O. Sarumi, K. M. Furati, and A. Q. M. Khaliq, “Highly accurate global Padé approximations of generalized MittagCLeffler function and its inverse,” Journal of Scientific Computing, vol. 82, Article ID 46, 2020.

C. Zeng and Y. Chen, “Global Padé approximations of the generalized Mittag-Leffler function and its inverse,” Fractional Calculus and Applied Analysis, vol. 18, pp. 1492–1506, 2015.