# New High-Order Energy Preserving Method of the Fractional Coupled Nonlinear Schrödinger Equations

Authors: Jianqiang Sun; Jiameng Kong; Lijuan Zhang; Jingxian Zhang
Journal: Journal of Mathematics
 (2022)

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## Abstract

The symplectic structure is given for the fractional coupled nonlinear Schrödinger equations. The Fourier spectral method and the fourth-order combination average vector field (AVF) method are applied to discretize the structure, and a new format for the fractional coupled nonlinear Schrödinger equations is obtained. The numerical experiments are showed to illustrate the property of the new format. The new scheme can maintain the energy conservation property better than the classical symplectic scheme.


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## Body

## 1. Introduction

The fractional coupled nonlinear Schrödinger (FCNLS) equations are the generalization of the classical Schrödinger equations, which is introduced by Laskin via replacing Brownian motions in Feynman path integrals by the Lévy ones [1]. It involves the fractional Laplacian with the Lévy index 1<α≤2 instead of the usual one. Constitutive equations for fractional order viscoelastic beam are constructed in the manner of Euler–Bernoulli beam theory by Yildirim and Alkan [2]. In the special case of α=2, the fractional nonlinear Schrödinger equation can be simplified as the classical Schrödinger equation, which describes the evolution of microscopic particles and has been studied extensively [3–5]. While for α=1, the equation collapses to the relativistic Hartree equation describing the dynamics of boson stars [6–8]. The fractional coupled nonlinear Schrödinger equations have been attracting great interest for scholars. Ortigueira proposed the so-called fractional-centered difference to approximate the fractional Laplacian operator [9] and this method is of the second-order accuracy. Bai et al. proposed the preconditioned modified HSS iteration method [10]. Atangana and Cloot [11] studied the Crank–Nicolson difference scheme for the space fractional variable-order Schrödinger equations [11]. Wei et al. [12] investigated the implicit fully discrete local discontinuous Galerkin methods for the fractional equations and coupled one which involves the Caputo-type time fractional derivative [12, 13]. Ran and Zhang [14] proposed an implicit conservative difference scheme for numerically solving the strongly CNLS equations [14]. Wang et al. [15] studied the difference method for the space fractional coupled nonlinear Schrödinger equations [15]. Wang and Huang [16] constructed the structure-preserving algorithms with the Fourier pseudo-spectral approximation to the spatial fractional operator [16]. Alkan et al. [17] used a sinc-collocation method to approximate the solution of fractional order boundary value problem [17].The construction of energy conservation schemes plays an important role in the numerical solution of energy conservation partial differential equations. Recently, energy preserving methods for the Hamiltonian system have received much attention [18–20]. Quispel and McLaren [21] constructed the energy preserving scheme of Hamilton system by the average vector field (AVF) method [21], and it can accurately simulate the evolution of the system over a long period of time and preserve the energy conservation of the system. Cai et al. [22] structured a fourth-order AVF method based on the composition technique [22]. At the same time, the second-order AVF method has also been proposed to solve the multisymplectic structure PDEs, which can also preserve the energy conservation [23]. Based on the composition technique and the AVF method, we proposed a fourth-order energy preserving composition scheme of the multisymplectic structure PDEs [24]. The fractional coupled nonlinear Schrödinger equations have the energy conservation property. Here, we constructed the fourth-order scheme based on the AVF method by using the composition technique, applied the scheme for the fractional coupled nonlinear Schrödinger equations, and compared the energy preserving property and accuracy of new schemes with the other classical schemes to demonstrate the performance of our scheme.This paper is organized as following. In Section2, the symplectic structure of the FCNLS equations are proposed. In Section 3, the structures are discretized by the Fourier spectrum method on the spatial direction. Then, the fourth-order combination AVF method, the second-order AVF method, and the symplectic method are used to discrete the time direction, respectively. In Section 4, the numerical experiments are reported. The accuracy and energy conservation property of the fourth-order combination AVF schemes are investigated. At last, we have conclusive remarks in Section 5.


## 2. Symplectic Structure of the FCNLS Equations

We consider the fractional coupled nonlinear Schrödinger equations1<α≤2(1)iut−−Δα/2u+ρu2+v2u=0,t>0,ivt−−Δα/2v+ρv2+u2v=0,t>0,with the initial conditions ux,0=u0x,vx,0=v0x, where i=−1, the parameter ρ>0 is a real constant. When x∈ℝ, the fractional Laplacian can be defined by the Fourier transform as(2)−−Δα/2ux,t=−∫ℝξαu^ξ,teiξxdξ,where the Fourier transform is defined by(3)u^ξ,t=12π∫ℝux,te−iξxdx.Letux,t=px,t+iqx,t, vx,t=wx,t+iϕx,t, equation equation (1) can be written as(4)pt−−Δα/2q+ρp2+q2+w2+ϕ2q=0,qt+−Δα/2p−ρp2+q2+w2+ϕ2p=0,wt−−Δα/2ϕ+ρw2+ϕ2+p2+q2ϕ=0,ϕt+−Δα/2w−ρw2+ϕ2+p2+q2w=0.Equation (4) can be transformed as the following symplectic structure(5)zt=JδHzδz,where z=p,q,w,ϕT, and the Hamiltonian function reads H=1/2∫Ω−−Δα/4p2−−Δα/4q2−−Δα/4w2−−Δα/4ϕ22dx+ρ/4∫Ωp2+q2+w2+ϕ22dx, and(6)J=0100−1000000100−10.


## 3. Discretization for the FCNLS Equations

### 3.1. Spatial Discretization for the FCNLS Equations

The fractional Laplacian operatorℒ is defined by its symbols in the Fourier space, so we use the Fourier pseudo-spectral method to discretize them. We first construct a discrete approximation of the solution through interpolating trigonometric polynomial of the solution at collocation points and then approximate its fractional derivative in the frequency space based on the symbol of the operator [24–27].Give a positive even integerN, set the mesh size h=b−a/N, and let xj=a+jh,0≤j≤N−1 be the grid points in [a, b], which are referred to as the Fourier collocation points. Then the interpolation approximation INpx denoted by pNx of the function px has the form(7)INpx=pNx=∑k=−N/2N/2p˜keikμx−a,where p˜k=1/Nck∑j=0N−1pxje−ikμxj−a and μ=2π/b−a, ck=1 for k<N/2 and ck=2 for k=±N/2. Then we approximate the fractional Laplacian of px by(8)−−Δα/2pNxj=−∑k=−N/2N/2kμαp˜keikμxj−a,and similarly, approximate the operator ℒ by(9)ℒpNxj=∑k=−N/2N/2ikμkμα−2/2p˜keikμxj−a.In order to facilitate the analysis below, we rewrite the above approximation in a matrix form in the physical space. To this end, denotingpj=pxj and plugging p˜k into (8) yield(10)−−Δα/2pNxj=−∑k=−N/2N/2kμα1Nck∑d=0N−1pde−ikμxd−aeikμxj−a=∑d=0N−1pd−∑k=−N/2N/21Nckkμαeikμxj−xd=D2αpj,where D2α is an N×N matrix with elements(11)D2αj,d=−∑k=−N/2N/21Nckkμαeikμxj−xd,and p=p0,…,pN−1T. Plugging p˜k into (9) yields(12)ℒpNxj=∑k=−N/2N/2ikμkμα−2/21Nck∑d=0N−1pde−ikμxdeikμxj,=∑d=0N−1pd∑k=−N/2N/21Nckikμkμα−2/2eikμxj−xd=D1αpj,where D1α is an N×N matrix with elements(13)D1αj,d=∑k=−N/2N/21Nckikμkμα−2/2eikμxj−xd.Applying the Fourier pseudo-spectral method to equation (4) in spatial direction, we can obtain(14)dpjdt=−D2αqj−ρpj2+qj2+wj2+ϕj2qj,dqjdt=D2αpj+ρpj2+qj2+wj2+ϕj2pj,dwjdt=−D2αϕj−ρpj2+qj2+wj2+ϕj2ϕj,dϕjdt=D2αwj+ρpj2+qj2+wj2+ϕj2wj,where j=0,…,N−1. The equation (14) can be expressed as a finite dimensional canonical Hamiltonian form as(15)dZdt=JN∇ZHZ,where Z=p0,…,pN−1,q0,…,qN−1,w0,…,wN−1,ϕ0,…,ϕN−1T.(16)JN=OIOO−IOOOOOOIOO−IO,where I is an N-th identity matrix, and O is an N-th zero matrix. The corresponding Hamiltonian function is(17)HZ=−12pTD2αp+qTD2αq+wTD2αw+ϕTD2αϕ+ρ4∑j=0N−1pj2+qj2+wj2+ϕj22.


### 3.2. Time Discretization for the Symplectic FCNLS Equations

The combination AVF method is applied to discretize equation (15), then we get(18)Zn,1−Znτ=c1JN∫01∇H1−ξZn+ξZn,1dξ,Zn,2−Zn,1τ=c2JN∫01∇H1−ξZn,1+ξZn,2dξ,Zn+1−Zn,2τ=c3JN∫01∇H1−ξZn,2+ξZn+1dξ,where τ is the time step, c1=c3=1/2−23,c2=−23/2−23. It also can be written as(19)pjn,1−pjnτ=−c1D2αqn+qn,12j+ρ12qjn3pjn2+2pjnpjn,1+pjn,12+3qjn2+2qjnqjn,1+qjn,12+3wjn2+2wjnwjn,1+wjn,12+3ϕjn2+2ϕjnϕjn,1+ϕjn,12+qjn,1pjn2+2pjnpjn,1+3pjn,12+qjn2+2qjnqjn,1+3qjn,12+wjn2+2wjnwjn,1+3wjn,12+ϕjn2+2ϕjnϕjn,1+3ϕjn,12,qjn,1−qjnτ=c1D2αpn+pn,12j+ρ12pjn3pjn2+2pjnpjn,1+pjn,12+3qjn2+2qjnqjn,1+qjn,12+3wjn2+2wjnwjn,1+wjn,12+3ϕjn2+2ϕjnϕjn,1+ϕjn,12+pjn,1pjn2+2pjnpjn,1+3pjn,12+qjn2+2qjnqjn,1+3qjn,12+wjn2+2wjnwjn,1+3wjn,12+ϕjn2+2ϕjnϕjn,1+3ϕjn,12,wjn,1−wjnτ=−c1D2αϕn+ϕn,12j+ρ12ϕjn3pjn2+2pjnpjn,1+pjn,12+3qjn2+2qjnqjn,1+qjn,12+3wjn2+2wjnwjn,1+wjn,12+3ϕjn2+2ϕjnϕjn,1+ϕjn,12+ϕjn,1pjn2+2pjnpjn,1+3pjn,12+qjn2+2qjnqjn,1+3qjn,12+wjn2+2wjnwjn,1+3wjn,12+ϕjn2+2ϕjnϕjn,1+3ϕjn,12,ϕjn,1−ϕjnτ=c1D2αwn+wn,12j+ρ12wjn3pjn2+2pjnpjn,1+pjn,12+3qjn2+2qjnqjn,1+qjn,12+3wjn2+2wjnwjn,1+wjn,12+3ϕjn2+2ϕjnϕjn,1+ϕjn,12+wjn,1pjn2+2pjnpjn,1+3pjn,12+qjn2+2qjnqjn,1+3qjn,12+wjn2+2wjnwjn,1+3wjn,12+ϕjn2+2ϕjnϕjn,1+3ϕjn,12,pjn,2−pjn,1τ=−c2D2αqn,1+qn,22j+ρ12qjn,13pjn,12+2pjn,1pjn,2+pjn,22+3qjn,12+2qjn,1qjn,2+qjn,22+3wjn,12+2wjn,1wjn,2+wjn,22+3ϕjn,12+2ϕjn,1ϕjn,2+ϕjn,22+qjn,2pjn,12+2pjn,1pjn,2+3pjn,22+qjn,12+2qjn,1qjn,2+3qjn,22+wjn,12+2wjn,1wjn,2+3wjn,22+ϕjn,12+2ϕjn,1ϕjn,2+3ϕjn,22,qjn,2−qjn,1τ=c2D2αpn,1+pn,22j+ρ12pjn,13pjn,12+2pjn,1pjn,2+pjn,22+3qjn,12+2qjn,1qjn,2+qjn,22+3wjn,12+2wjn,1wjn,2+wjn,22+3ϕjn,12+2ϕjn,1ϕjn,2+ϕjn,22+pjn,2pjn,12+2pjn,1pjn,2+3pjn,22+qjn,12+2qjn,1qjn,2+3qjn,22+wjn,12+2wjn,1wjn,2+3wjn,22+ϕjn,12+2ϕjn,1ϕjn,2+3ϕjn,22,wjn,2−wjn,1τ=−c2D2αϕn,1+ϕn,22j+ρ12ϕjn,13pjn,12+2pjn,1pjn,2+pjn,22+3qjn,12+2qjn,1qjn,2+qjn,22+3wjn,12+2wjn,1wjn,2+wjn,22+3ϕjn,12+2ϕjn,1ϕjn,2+ϕjn,22+ϕjn,2pjn,12+2pjn,1pjn,2+3pjn,22+qjn,12+2qjn,1qjn,2+3qjn,22+wjn,12+2wjn,1wjn,2+3wjn,22+ϕjn,12+2ϕjn,1ϕjn,2+3ϕjn,22,ϕjn,2−ϕjn,1τ=c2D2αwn,1+wn,22j+ρ12wjn,13pjn,12+2pjn,1pjn,2+pjn,22+3qjn,12+2qjn,1qjn,2+qjn,22+3wjn,12+2wjn,1wjn,2+wjn,22+3ϕjn,12+2ϕjn,1ϕjn,2+ϕjn,22+wjn,2pjn,12+2pjn,1pjn,2+3pjn,22+qjn,12+2qjn,1qjn,2+3qjn,22+wjn,12+2wjn,1wjn,2+3wjn,22+ϕjn,12+2ϕjn,1ϕjn,2+3ϕjn,22,pjn+1−pjn,2τ=−c3D2αqn,2+qn+12j+ρ12qjn,23pjn,22+2pjn,2pjn+1+pjn+12+3qjn,22+2qjn,2qjn+1+qjn+12+3wjn,22+2wjn,2wjn+1+wjn+12+3ϕjn,22+2ϕjn,2ϕjn+1+ϕjn+12+qjn+1pjn,22+2pjn,2pjn+1+3pjn+12+qjn,22+2qjn,2qjn+1+3qjn+12+wjn,22+2wjn,2wjn+1+3wjn+12+ϕjn,22+2ϕjn,2ϕjn+1+3ϕjn+12,qjn+1−qjn,2τ=c3D2αpn,2+pn+12j+ρ12pjn,23pjn,22+2pjn,2pjn+1+pjn+12+3qjn,22+2qjn,2qjn+1+qjn+12+3wjn,22+2wjn,2wjn+1+wjn+12+3ϕjn,22+2ϕjn,2ϕjn+1+ϕjn+12+pjn+1pjn,22+2pjn,2pjn+1+3pjn+12+qjn,22+2qjn,2qjn+1+3qjn+12+wjn,22+2wjn,2wjn+1+3wjn,12+ϕjn2+2ϕjnϕjn,1+3ϕjn,12,wjn+1−wjn,2τ=−c3D2αϕn,2+ϕn+12j+ρ12ϕjn,23pjn,22+2pjn,2pjn+1+pjn+12+3qjn,22+2qjn,2qjn+1+qjn+12+3wjn,22+2wjn,2wjn+1+wjn+12+3ϕjn,22+2ϕjn,2ϕjn+1+ϕjn+12+ϕjn+1pjn,22+2pjn,2pjn+1+3pjn+12+qjn,22+2qjn,2qjn+1+3qjn+12+wjn,22+2wjn,2wjn+1+3wjn+12+ϕjn,22+2ϕjn,2ϕjn+1+3ϕjn+12,ϕjn+1−ϕjn,2τ=c3D2αwn,2+wn+12j+ρ12wjn,23pjn,22+2pjn,2pjn+1+pjn+12+3qjn,22+2qjn,2qjn+1+qjn+12+3wjn,22+2wjn,2wjn+1+wjn+12+3ϕjn,22+2ϕjn,2ϕjn+1+ϕjn+12+wjn+1pjn,22+2pjn,2pjn+1+3pjn+12+qjn,22+2qjn,2qjn+1+3qjn+12+wjn,22+2wjn,2wjn+1+3wjn+12+ϕjn,22+2ϕjn,2ϕjn+1+3ϕjn+12.Theorem 1.
System (18) can preserve the energy conservation property.Proof.
To the first equation in scheme (18), taking the scaling product of both sides of the equation with respect to ∫01∇H1−ξZ0+ξZ1dξ, we can get(20)c1h∫01Z1−Z0∇H1−ξZ0+ξZ1dξ=0,c1h∫01ddξH1−ξZ0+ξZ1dξ=0.
From the fundamental theorem of calculus, we can obtain(21)c1hHZ1−HZ0=0.
The second and third equations in scheme (18) are the same as the first equation. Therefore, the new fourth composition scheme (18) can preserve the energy conservation.
We also apply the following second-order AVF method to discretize equation (15):(22)Zn+1−Znτ=JN∫01∇H1−ξZn+ξZn+1dξ,where τ is the time step. The following symplectic scheme of the FCNLS equations is also given to compare with the AVF method of symplectic FCNLS equations [28–30]:(23)Zn+1−Znτ=JN∇HZn+Zn+12.


## 3.1. Spatial Discretization for the FCNLS Equations

The fractional Laplacian operatorℒ is defined by its symbols in the Fourier space, so we use the Fourier pseudo-spectral method to discretize them. We first construct a discrete approximation of the solution through interpolating trigonometric polynomial of the solution at collocation points and then approximate its fractional derivative in the frequency space based on the symbol of the operator [24–27].Give a positive even integerN, set the mesh size h=b−a/N, and let xj=a+jh,0≤j≤N−1 be the grid points in [a, b], which are referred to as the Fourier collocation points. Then the interpolation approximation INpx denoted by pNx of the function px has the form(7)INpx=pNx=∑k=−N/2N/2p˜keikμx−a,where p˜k=1/Nck∑j=0N−1pxje−ikμxj−a and μ=2π/b−a, ck=1 for k<N/2 and ck=2 for k=±N/2. Then we approximate the fractional Laplacian of px by(8)−−Δα/2pNxj=−∑k=−N/2N/2kμαp˜keikμxj−a,and similarly, approximate the operator ℒ by(9)ℒpNxj=∑k=−N/2N/2ikμkμα−2/2p˜keikμxj−a.In order to facilitate the analysis below, we rewrite the above approximation in a matrix form in the physical space. To this end, denotingpj=pxj and plugging p˜k into (8) yield(10)−−Δα/2pNxj=−∑k=−N/2N/2kμα1Nck∑d=0N−1pde−ikμxd−aeikμxj−a=∑d=0N−1pd−∑k=−N/2N/21Nckkμαeikμxj−xd=D2αpj,where D2α is an N×N matrix with elements(11)D2αj,d=−∑k=−N/2N/21Nckkμαeikμxj−xd,and p=p0,…,pN−1T. Plugging p˜k into (9) yields(12)ℒpNxj=∑k=−N/2N/2ikμkμα−2/21Nck∑d=0N−1pde−ikμxdeikμxj,=∑d=0N−1pd∑k=−N/2N/21Nckikμkμα−2/2eikμxj−xd=D1αpj,where D1α is an N×N matrix with elements(13)D1αj,d=∑k=−N/2N/21Nckikμkμα−2/2eikμxj−xd.Applying the Fourier pseudo-spectral method to equation (4) in spatial direction, we can obtain(14)dpjdt=−D2αqj−ρpj2+qj2+wj2+ϕj2qj,dqjdt=D2αpj+ρpj2+qj2+wj2+ϕj2pj,dwjdt=−D2αϕj−ρpj2+qj2+wj2+ϕj2ϕj,dϕjdt=D2αwj+ρpj2+qj2+wj2+ϕj2wj,where j=0,…,N−1. The equation (14) can be expressed as a finite dimensional canonical Hamiltonian form as(15)dZdt=JN∇ZHZ,where Z=p0,…,pN−1,q0,…,qN−1,w0,…,wN−1,ϕ0,…,ϕN−1T.(16)JN=OIOO−IOOOOOOIOO−IO,where I is an N-th identity matrix, and O is an N-th zero matrix. The corresponding Hamiltonian function is(17)HZ=−12pTD2αp+qTD2αq+wTD2αw+ϕTD2αϕ+ρ4∑j=0N−1pj2+qj2+wj2+ϕj22.


## 3.2. Time Discretization for the Symplectic FCNLS Equations

The combination AVF method is applied to discretize equation (15), then we get(18)Zn,1−Znτ=c1JN∫01∇H1−ξZn+ξZn,1dξ,Zn,2−Zn,1τ=c2JN∫01∇H1−ξZn,1+ξZn,2dξ,Zn+1−Zn,2τ=c3JN∫01∇H1−ξZn,2+ξZn+1dξ,where τ is the time step, c1=c3=1/2−23,c2=−23/2−23. It also can be written as(19)pjn,1−pjnτ=−c1D2αqn+qn,12j+ρ12qjn3pjn2+2pjnpjn,1+pjn,12+3qjn2+2qjnqjn,1+qjn,12+3wjn2+2wjnwjn,1+wjn,12+3ϕjn2+2ϕjnϕjn,1+ϕjn,12+qjn,1pjn2+2pjnpjn,1+3pjn,12+qjn2+2qjnqjn,1+3qjn,12+wjn2+2wjnwjn,1+3wjn,12+ϕjn2+2ϕjnϕjn,1+3ϕjn,12,qjn,1−qjnτ=c1D2αpn+pn,12j+ρ12pjn3pjn2+2pjnpjn,1+pjn,12+3qjn2+2qjnqjn,1+qjn,12+3wjn2+2wjnwjn,1+wjn,12+3ϕjn2+2ϕjnϕjn,1+ϕjn,12+pjn,1pjn2+2pjnpjn,1+3pjn,12+qjn2+2qjnqjn,1+3qjn,12+wjn2+2wjnwjn,1+3wjn,12+ϕjn2+2ϕjnϕjn,1+3ϕjn,12,wjn,1−wjnτ=−c1D2αϕn+ϕn,12j+ρ12ϕjn3pjn2+2pjnpjn,1+pjn,12+3qjn2+2qjnqjn,1+qjn,12+3wjn2+2wjnwjn,1+wjn,12+3ϕjn2+2ϕjnϕjn,1+ϕjn,12+ϕjn,1pjn2+2pjnpjn,1+3pjn,12+qjn2+2qjnqjn,1+3qjn,12+wjn2+2wjnwjn,1+3wjn,12+ϕjn2+2ϕjnϕjn,1+3ϕjn,12,ϕjn,1−ϕjnτ=c1D2αwn+wn,12j+ρ12wjn3pjn2+2pjnpjn,1+pjn,12+3qjn2+2qjnqjn,1+qjn,12+3wjn2+2wjnwjn,1+wjn,12+3ϕjn2+2ϕjnϕjn,1+ϕjn,12+wjn,1pjn2+2pjnpjn,1+3pjn,12+qjn2+2qjnqjn,1+3qjn,12+wjn2+2wjnwjn,1+3wjn,12+ϕjn2+2ϕjnϕjn,1+3ϕjn,12,pjn,2−pjn,1τ=−c2D2αqn,1+qn,22j+ρ12qjn,13pjn,12+2pjn,1pjn,2+pjn,22+3qjn,12+2qjn,1qjn,2+qjn,22+3wjn,12+2wjn,1wjn,2+wjn,22+3ϕjn,12+2ϕjn,1ϕjn,2+ϕjn,22+qjn,2pjn,12+2pjn,1pjn,2+3pjn,22+qjn,12+2qjn,1qjn,2+3qjn,22+wjn,12+2wjn,1wjn,2+3wjn,22+ϕjn,12+2ϕjn,1ϕjn,2+3ϕjn,22,qjn,2−qjn,1τ=c2D2αpn,1+pn,22j+ρ12pjn,13pjn,12+2pjn,1pjn,2+pjn,22+3qjn,12+2qjn,1qjn,2+qjn,22+3wjn,12+2wjn,1wjn,2+wjn,22+3ϕjn,12+2ϕjn,1ϕjn,2+ϕjn,22+pjn,2pjn,12+2pjn,1pjn,2+3pjn,22+qjn,12+2qjn,1qjn,2+3qjn,22+wjn,12+2wjn,1wjn,2+3wjn,22+ϕjn,12+2ϕjn,1ϕjn,2+3ϕjn,22,wjn,2−wjn,1τ=−c2D2αϕn,1+ϕn,22j+ρ12ϕjn,13pjn,12+2pjn,1pjn,2+pjn,22+3qjn,12+2qjn,1qjn,2+qjn,22+3wjn,12+2wjn,1wjn,2+wjn,22+3ϕjn,12+2ϕjn,1ϕjn,2+ϕjn,22+ϕjn,2pjn,12+2pjn,1pjn,2+3pjn,22+qjn,12+2qjn,1qjn,2+3qjn,22+wjn,12+2wjn,1wjn,2+3wjn,22+ϕjn,12+2ϕjn,1ϕjn,2+3ϕjn,22,ϕjn,2−ϕjn,1τ=c2D2αwn,1+wn,22j+ρ12wjn,13pjn,12+2pjn,1pjn,2+pjn,22+3qjn,12+2qjn,1qjn,2+qjn,22+3wjn,12+2wjn,1wjn,2+wjn,22+3ϕjn,12+2ϕjn,1ϕjn,2+ϕjn,22+wjn,2pjn,12+2pjn,1pjn,2+3pjn,22+qjn,12+2qjn,1qjn,2+3qjn,22+wjn,12+2wjn,1wjn,2+3wjn,22+ϕjn,12+2ϕjn,1ϕjn,2+3ϕjn,22,pjn+1−pjn,2τ=−c3D2αqn,2+qn+12j+ρ12qjn,23pjn,22+2pjn,2pjn+1+pjn+12+3qjn,22+2qjn,2qjn+1+qjn+12+3wjn,22+2wjn,2wjn+1+wjn+12+3ϕjn,22+2ϕjn,2ϕjn+1+ϕjn+12+qjn+1pjn,22+2pjn,2pjn+1+3pjn+12+qjn,22+2qjn,2qjn+1+3qjn+12+wjn,22+2wjn,2wjn+1+3wjn+12+ϕjn,22+2ϕjn,2ϕjn+1+3ϕjn+12,qjn+1−qjn,2τ=c3D2αpn,2+pn+12j+ρ12pjn,23pjn,22+2pjn,2pjn+1+pjn+12+3qjn,22+2qjn,2qjn+1+qjn+12+3wjn,22+2wjn,2wjn+1+wjn+12+3ϕjn,22+2ϕjn,2ϕjn+1+ϕjn+12+pjn+1pjn,22+2pjn,2pjn+1+3pjn+12+qjn,22+2qjn,2qjn+1+3qjn+12+wjn,22+2wjn,2wjn+1+3wjn,12+ϕjn2+2ϕjnϕjn,1+3ϕjn,12,wjn+1−wjn,2τ=−c3D2αϕn,2+ϕn+12j+ρ12ϕjn,23pjn,22+2pjn,2pjn+1+pjn+12+3qjn,22+2qjn,2qjn+1+qjn+12+3wjn,22+2wjn,2wjn+1+wjn+12+3ϕjn,22+2ϕjn,2ϕjn+1+ϕjn+12+ϕjn+1pjn,22+2pjn,2pjn+1+3pjn+12+qjn,22+2qjn,2qjn+1+3qjn+12+wjn,22+2wjn,2wjn+1+3wjn+12+ϕjn,22+2ϕjn,2ϕjn+1+3ϕjn+12,ϕjn+1−ϕjn,2τ=c3D2αwn,2+wn+12j+ρ12wjn,23pjn,22+2pjn,2pjn+1+pjn+12+3qjn,22+2qjn,2qjn+1+qjn+12+3wjn,22+2wjn,2wjn+1+wjn+12+3ϕjn,22+2ϕjn,2ϕjn+1+ϕjn+12+wjn+1pjn,22+2pjn,2pjn+1+3pjn+12+qjn,22+2qjn,2qjn+1+3qjn+12+wjn,22+2wjn,2wjn+1+3wjn+12+ϕjn,22+2ϕjn,2ϕjn+1+3ϕjn+12.Theorem 1.
System (18) can preserve the energy conservation property.Proof.
To the first equation in scheme (18), taking the scaling product of both sides of the equation with respect to ∫01∇H1−ξZ0+ξZ1dξ, we can get(20)c1h∫01Z1−Z0∇H1−ξZ0+ξZ1dξ=0,c1h∫01ddξH1−ξZ0+ξZ1dξ=0.
From the fundamental theorem of calculus, we can obtain(21)c1hHZ1−HZ0=0.
The second and third equations in scheme (18) are the same as the first equation. Therefore, the new fourth composition scheme (18) can preserve the energy conservation.
We also apply the following second-order AVF method to discretize equation (15):(22)Zn+1−Znτ=JN∫01∇H1−ξZn+ξZn+1dξ,where τ is the time step. The following symplectic scheme of the FCNLS equations is also given to compare with the AVF method of symplectic FCNLS equations [28–30]:(23)Zn+1−Znτ=JN∇HZn+Zn+12.


## 4. Numerical Experiment

In this section, numerical experiments for the fractional coupled nonlinear Schrödinger equations with periodic boundary conditions are presented to investigate the relative energy error and accuracy of convergence. The energy error is defined as(24)Et=I¯Zn−I¯Z0I¯Z0,where I¯Z0 is the initial energy, and I¯Zn is the energy value at tn=nτ. The maximal module error of numerical solution and exact solution is defined as(25)L∞τ=max1≤j≤Nujn−uxj,tn.The order of convergence is defined as(26)Ratio=logL∞τ/L∞τ/2log2.We consider the FCNLS equations. Equation (1) has the initial conditions such as(27)ux,0=sechx+d0eiV0x,vx,0=sechx−d0e−iV0x,where d0=1, V0=2, ρ=2, x∈−20,20, N=200, and τ=0.01.Since it is difficult to find the exact solution of fractional coupled Schrödinger equation and calculate the convergence order of the system, we use the following fractional Schrödinger equation to calculate the convergence order of the system(28)iut−−Δα/2u+2u2u=0,where α=2. The exact solution can be given by(29)ux,t=sechx−4tei2x−3t.First, we consider the symplectic structure of FCNLS equations. The discrete energy corresponding to (15) can be expressed as (30)I¯Z=−12pTD2αp+qTD2αq+wTD2αw+ϕTD2αϕ+ρ4∑j=0N−1pj2+qj2+wj2+ϕj22,and the discrete mass corresponding to (15) can be written as (31)M¯Z=ρ4∑j=0N−1pj2+qj2+wj2+ϕj22.Table1 shows the error of numerical solution and exact solution and the order of convergence of the different schemes with different time steps at t=1.6. It is easy to see that the orders of convergence of the fourth-order energy preserving combination scheme are almost equal to 4, and the orders of convergence of AVF scheme are almost equal to 2. The new fourth energy preserving combination scheme is more accurate than other two schemes. Table 2 shows the mass values of three numerical schemes with different α. One can observe that three numerical schemes all can preserve the conservation of mass approximately, and the symplectic scheme has the best mass preserving property, followed by the combination AVF scheme, and finally the AVF scheme. These data can be obtained by the Matlab programs based on the symplectic scheme and the combination AVF scheme of the paper.Table 1
TheL∞ errors and the order of convergence of three different schemes with h=0.02 and different time steps at t=1.6.
t=1.6AVF schemeOrderSymplectic schemeOrderCombination AVF schemeOrderτ=0.081.2743e − 01—1.2876e − 01—3.2165e − 02—τ=0.042.9558e − 022.10822.9890e − 022.10712.600e − 033.6292τ=0.027.2589e − 032.02597.3525e − 032.02351.9171e − 043.7618τ=0.011.8076e − 032.00581.8304e − 032.00641.2805e − 053.9044Table 2
The mass values of three different schemes withτ=0.02 at t=20.
t=20Initial valueAVF schemeSymplectic schemeCombination AVF schemeα=1.2109.9956110.000039.99971α=1.4109.9959810.000009.99958α=1.6109.9982510.000179.99996α=1.8109.9991710.000009.99997From Figures1 and 2, we can get that the numerical solution obtained from the fourth-order combination AVF scheme can well simulate the waves. The energy errors of the fourth-order combination AVF method with different α values are plotted in Figure 3. The error is up to 10−13. Figure 4 shows the energy error obtained from the second-order AVF method with two different α values. The error is up to 10−14. Figure 5 shows the energy error obtained from the second-order symplectic method with different α values. The error is only up to 10−4. We can get that the fourth-order combination AVF method and second-order AVF method preserve energy conservation of the system more accurately.Figure 1
Numerical solutions ofu+v in t∈0,12 at α=1.8.Figure 2
Numerical solutions ofu+v in t∈0,12 at α=1.4.Figure 3
Energy errors for the fourth-order combination AVF method int∈0,20 with different values of α.Figure 4
Energy errors for the second-order AVF method int∈0,20 with different values of α.Figure 5
Energy errors for the second-order symplectic method int∈0,20 with different values of α.


## 5. Concluding Remarks

We construct symplectic fourth-order combination AVF formats of the fractional coupled nonlinear Schrödinger equations and compare the new high order scheme with the second-order AVF formats and symplectic formats. Numerical results show that the new fourth-order combination AVF formats have fourth-order accuracy. The new high order scheme can accurately simulate evolution of the equations and maintain the energy conservation well.



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