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

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


Introduction
e 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 simpli ed as the classical Schrödinger equation, which describes the evolution of microscopic particles and has been studied extensively [3][4][5]. While for α 1, the equation collapses to the relativistic Hartree equation describing the dynamics of boson stars [6][7][8].
e fractional coupled nonlinear Schrödinger equations have been attracting great interest for scholars. Ortigueira proposed the so-called fractional-centered di erence to approximate the fractional Laplacian operator [9] and this method is of the second-order accuracy. Bai et al. proposed the preconditioned modi ed HSS iteration method [10]. Atangana and Cloot [11] studied the Crank-Nicolson di erence 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 di erence scheme for numerically solving the strongly CNLS equations [14]. Wang et al. [15] studied the di erence 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]. e construction of energy conservation schemes plays an important role in the numerical solution of energy conservation partial di erential equations. Recently, energy preserving methods for the Hamiltonian system have received much attention [18][19][20]. Quispel and McLaren [21] constructed the energy preserving scheme of Hamilton system by the average vector eld (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]. e 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.
is paper is organized as following. In Section 2, 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. en, the fourthorder 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. e accuracy and energy conservation property of the fourth-order combination AVF schemes are investigated. At last, we have conclusive remarks in Section 5.

Symplectic Structure of the FCNLS Equations
We consider the fractional coupled nonlinear Schrödinger with the initial conditions u( , the parameter ρ > 0 is a real constant. When x ∈ R, the fractional Laplacian can be defined by the Fourier transform as where the Fourier transform is defined by Let (1) can be written as Equation (4) can be transformed as the following symplectic structure where z � (p, q, w, ϕ) T , and the Hamiltonian function reads

Discretization for the FCNLS Equations
3.1. Spatial Discretization for the FCNLS Equations. e fractional Laplacian operator L 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][25][26][27].
Give a positive even integer N, set the mesh size , which are referred to as the Fourier collocation points.
en the interpolation approximation where en we approximate the fractional Laplacian of p(x) by and similarly, approximate the operator L by In order to facilitate the analysis below, we rewrite the above approximation in a matrix form in the physical space. To this end, denoting p j � p(x j ) and plugging p k into (8) yield where D α 2 is an N × N matrix with elements 2 Journal of Mathematics and p � (p 0 , . . . , p N− 1 ) T . Plugging p k into (9) yields where D α 1 is an N × N matrix with elements Applying the Fourier pseudo-spectral method to equation (4) in spatial direction, we can obtain where j � 0, . . . , N − 1. e equation (14) can be expressed as a finite dimensional canonical Hamiltonian form as where I is an N-th identity matrix, and O is an N-th zero matrix. e corresponding Hamiltonian function is

Time Discretization for the Symplectic FCNLS Equations.
e combination AVF method is applied to discretize equation (15), then we get where τ is the time step, . It also can be written as Journal of Mathematics 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 From the fundamental theorem of calculus, we can obtain e second and third equations in scheme (18) are the same as the first equation. erefore, the new fourth composition scheme (18) can preserve the energy conservation.
We also apply the following second-order AVF method to discretize equation (15): where τ is the time step. e following symplectic scheme of the FCNLS equations is also given to compare with the AVF method of symplectic FCNLS equations [28][29][30]:

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. e energy error is defined as where I(Z 0 ) is the initial energy, and I(Z n ) is the energy value at t n � nτ. e maximal module error of numerical solution and exact solution is defined as e order of convergence is defined as We consider the FCNLS equations. Equation (1) has the initial conditions such as 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 where α � 2. e exact solution can be given by First, we consider the symplectic structure of FCNLS equations. e discrete energy corresponding to (15) can be expressed as       Journal of Mathematics 7 and the discrete mass corresponding to (15) can be written as (31) Table 1 shows the error of numerical solution and exact solution and the order of convergence of the di erent schemes with di erent 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. e new fourth energy preserving combination scheme is more accurate than other two schemes. Table 2 shows the mass values of three numerical schemes with di erent 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 nally the AVF scheme. ese data can be obtained by the Matlab programs based on the symplectic scheme and the combination AVF scheme of the paper.
From Figures 1 and 2, we can get that the numerical solution obtained from the fourth-order combination AVF scheme can well simulate the waves. e energy errors of the fourth-order combination AVF method with di erent α values are plotted in Figure 3. e error is up to 10 − 13 . Figure 4 shows the energy error obtained from the secondorder AVF method with two di erent α values. e error is up to 10 − 14 . Figure 5 shows the energy error obtained from the second-order symplectic method with di erent α values. e 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.

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.
e new high order scheme can accurately simulate evolution of the equations and maintain the energy conservation well.

Data Availability
ese data can be obtained by the Matlab programs based on symplectic scheme and the combination AVF scheme of the paper.

Conflicts of Interest
e authors declare that they have no con icts of interest.