A Conservative Crank-Nicolson Fourier Spectral Method for the Space Fractional Schrödinger Equation with Wave Operators

In this paper, the Crank-Nicolson Fourier spectral method is proposed for solving the space fractional Schrödinger equation with wave operators. The equation is treated with the conserved Crank-Nicolson Fourier Galerkin method and the conserved CrankNicolson Fourier collocation method, respectively. In addition, the ability of the constructed numerical method to maintain the conservation of mass and energy is studied in detail. Meanwhile, the convergence with spectral accuracy in space and secondorder accuracy in time is verified for both Galerkin and collocation approximations. Finally, the numerical experiments verify the properties of the conservative difference scheme and demonstrate the correctness of theoretical results.


Introduction
The Schrödinger equation is one of the most basic equations in quantum mechanics, which was proposed by Austrian physicist Schrödinger in 1926. The equation can correctly describe the quantum behaviors of wave function, which has made great contributions to the study of quantum mechanics. Since then, the Schrödinger system has attracted a large number of mathematicians and physicists to explore the characteristics of its solution and physical applications. The study of conservative methods for the Schrödinger equation is one of the most popular research fields.
Over the past decades, most of the researches on the conservative method of the Schrödinger equation focus on the integerorder Schrödinger equation (e.g., see Refs. [1][2][3][4][5][6][7]). As models of science and engineering are needed to be more realistic, the fractional-order Schrödinger equation becomes one of the most important models in the fields of Bose-Einstein condensation, plasma, nonlinear optics, fluid dynamics [8,9], etc. However, few studies have been investigated on conservative methods for the fractional Schrödinger equation. Besides that, most of the existing fractional-order conservative methods are finite element and finite difference methods [10,11].
From the viewpoint of mathematics, the solution of the Schrödinger system has important geometric structures such as energy conservation and multisymplectic structure. Therefore, these properties should be maintained as much as possible in the construction of numerical methods. In this paper, we consider the following nonlinear fractional Schrödinger equation: subject to the boundary condition and the initial conditions ϕ y, 0 ð Þ= ϕ 0 y ð Þ, ϕ t y, 0 where β and κ are positive real constants, 1 < α ≤ 2, and i 2 = −1. ϕ 0 ðyÞ and ϕ 1 ðyÞ are given real functions. The fractional Laplacian operator ð−ΔÞ α/2 can be defined as a pseudodifferential operator with the symbol −jξj α : where F is the Fourier transform andû is the Fourier transform of u.
The spectral method is a generalization of a standard separation variable method, for which Chebyshev polynomials and Legendre polynomials are generally used as the basic functions of approximate expansions. And the Fourier series is convenient to deal with the periodic boundary conditions. Bridges and Reich [12] first put forward the Hamiltonian system using the Fourier spectrum discrete method in 2001. Based on their theoretical ideas, Chen and Qin [13] in the same year proposed the Fourier pseudo-spectral method for the Hamiltonian partial differential equation and used it to integrate the nonlinear Schrödinger equation with periodic boundary conditions. For more comprehensive work on the different conservative Fourier pseudo-spectral methods, refer to [2,[14][15][16] and their references.
In fact, the nonlinear fractional Schrödinger equation ((6)-(8)) has two conserved quantities: where with The outline of the remainder of this paper is as follows. In Section 2, a conserved Crank-Nicolson Fourier Galerkin method and a conserved Crank-Nicolson Fourier collocation method are constructed to discrete time variables and spatial variables. Energy-preserving and mass-preserving properties of the new method are investigated, and the error estimate is derived in Section 3. In Section 4, numerical experiments are presented to illustrate the theoretical results. Finally, the conclusions are given in Section 5.

Crank-Nicolson Fourier Spectral Method and Conservation Laws
Let C ∞ per ðΩÞ be the set of all complex-valued and 2π-periodic C ∞ -functions on Ω. Denote ð·, · Þ as the inner product on the space L 2 per ðΩÞ with the L 2 norm k·k L 2 per ðΩÞ (abbreviated as k·k): For μ as a nonnegative real number, let H μ per ðΩÞ be the closure of C ∞ per ðΩÞ. Note that H 0 per ðΩÞ = L 2 per ðΩÞ. For any function uðxÞ ∈ L 2 per ðΩÞ, the following equations [17] can be developed easily: where the Fourier coefficients are arranged aŝ For the Fourier transform of fractional Laplacian − ð−ΔÞ α/2 , we have In order to discretize the equation in the temporal direction, the time step is defined by τ = T/N t . Denote difference operator where n is a positive integer (0 ≤ n ≤ N t ). Therefore, the Crank-Nicolson method was used to discretize equation (6) in the time axis.
2 Journal of Function Spaces where R n = Oðτ 2 Þ: 2.1. Crank-Nicolson Fourier Galerkin Method. For positive even number N, the basis function space can be constructed as where the norm and seminorm of H α per ðΩÞ are characterized by Let The orthogonal operators P N : L 2 per ðΩÞ → S N are defined as follows: Lemma 1 [18,19]. Suppose that u ∈ H s per ðΩÞ for all 0 ≤ μ ≤ s; it holds that Denote The time variables of equation (6) are discretized by the Crank-Nicolson method. And the discrete Fourier Galerkin approximation for equation (6) has a modified scheme as follows: where u n+1 N ∈ S N , ∀v ∈ S N . 2.2. Crank-Nicolson Fourier Collocation Method. For positive even number N, consider the points x j = 2πj/N, j = 0, 1, ⋯, N − 1, as collocation nodes. The discrete Fourier coefficients [18] of a function u on ½0, 2π with respect to the collocation points are the following form: Using the inversion formula, we have Define the interpolation operator I N [18] at the collocation points: According to (27), Lemma 2 [18,19]. For any u ∈ H s per ðΩÞ, s ≥ 1, the estimate in the sense of the Sobolev norm.
Combining Lemma 2 and the triangle inequality, Corollary 3 is drawn.

Corollary 3.
For any u ∈ H s per ðΩÞ, s ≥ 1, there exists a constant C independent of u and N, such that Using the Fourier collocation method to discrete the spatial variables of the equation, we get the fully discrete scheme for equations (6)-(8) as the following forms: 3 Journal of Function Spaces Applying the Fourier transformation to (24), we get the following form: where

Theory Analysis of Conservation
Theorem 4. The Crank-Nicolson Fourier Galerkin method (24) of solving equations (6)-(8) preserves the discrete mass and discrete energy: where Proof. We derive the full discrete Fourier Galerkin method: Let v =ũ n N in equation (37); it holds that Taking the imaginary part of equation (38), due to Therefore, thus, The above equality indicates that the method (24) maintains the conservation of the discrete mass. The following items consider the conservation of the discrete energy.
Let v = δ̂tu n N ; according to equation (37), we also get Taking the real part of (42), due to Re δ t 2 u n N , δ̂tu n Journal of Function Spaces Re therefore, using (43)-(46), we obtain thus, Based on the above analysis, the method (24) also maintains the conservation of the discrete energy. ☐

Theory Analysis of Convergence
In order to simplify the notation, we always assume that C is a positive constant in this article, which might be different in every formula.
Lemma 5 [20]. For any discrete function u n N , it holds that u n+1 Proof. Using Theorem 4, it yields thus, Because of β > 0, it satisfies Sum the inequalities of Lemma 5 from 0 to n yields Adding (53) and (54), we can obtain the following items: For τ is sufficiently small (τ < 1), this implies

Journal of Function Spaces
According to the discrete Gronwall's inequality, there is Therefore, ☐ Theorem 7. If s ≥ 1, assume that u ∈ C 2 ðI ; H α per ðΩÞ ∩ H s ðΩÞÞ is the exact solution of (6)-(8), and u n N is the numerical solution of (24). It possesses the following conclusion: Proof. Let u * = P N u, e = u − u N , ξ = u − u * , and η = u * − u N ; then, e n = ξ n + η n . From triangle inequality and Lemma 1, it yields According to the orthogonality of the projection operator P N , we get The authors derive the full discrete Fourier Galerkin method: Subtracting equation (62) from equation (61), due to thus, According to the orthogonality of operator P N , i.e., ðP N u − u, vÞ = 0, ∀v ∈ S N . Therefore, Let v = δ̂tη n in (64), and taking the real part, due to Re δ t 2 e n , δ̂tη n À Á = Re δ t 2 η n , δ̂tη n À Á Re therefore, using (66)-(68), this implies where G n j j= β 2 u n−1 2 + u n+1 2 u n − u n−1 Journal of Function Spaces Thus, according to Lemma 6, we can get Note Lemma 1; it gives that Then, Thus, (69) becomes Because of and from Lemma 5, it gives that Then, combining (74) and (76) leads to Summing above inequalities (77) from 1 to n yields Hence, using the discrete Gronwall's inequality gives thus, Substituting (80) into (60) can yield which immediately gives conclusion. ☐ Similar to the proof of Theorem 7, we can obtain the following theorem. Theorem 8. Let s ≥ 1; assume that u ∈ C 2 ðI ; H α per ðΩÞ ∩ H s ðΩÞÞ is the exact solution of (6)-(8), and u n N is the numerical solution of (32). It possesses the following conclusion:

Numerical Example
Numerical examples will be proposed in this section to verify the correctness of the theoretical analysis, that is, the convergence of the numerical method and its ability to maintain discrete mass and discrete energy.
There is no exact solution of (83) known for 1 < α < 2. Therefore, numerical solution calculated by the method (24) with N = 1024 and τ = 2 −10 is taken as the reference solution. Let Φ be the numerical solution, and calculate the error at t = t n in the sense of the discrete L 2 norm: The convergence rates in the direction of time and space are calculated as Let T = 1. Tables 1 and 2 show that the numerical method is proven to have spectral accuracy in space and second-order accuracy in time for solving equation (83) with α = 1:4 and α = 1:6.   Table 2: Errors and convergence rates in space for τ = 2 −10 and T = 1.  Journal of Function Spaces Figures 3-6 show the ability of the numerical method (24) to maintain the discrete mass and discrete energy when α = 1:4 and α = 1:6. It can be seen from the figure that the numerical method (24) maintains the discrete mass and discrete energy well.

Conclusion
For the fractional Schrödinger equation with wave operators, we successfully constructed the effective conservative Crank-Nicolson Fourier spectral method for solving this equation, based on the relative theory of a fractional-order derivative and its property. We give the strict theoretical derivation for the convergence rate of the numerical method, i.e., Oðτ 2 + N −s kuk s Þ. Finally, numerical examples are introduced to verify the correctness of the theoretical results and the validity of our numerical methods. Both theoretical derivation and numerical experiment verify that the numerical method can keep the energy conservation and mass conservation of the original fractional Schrödinger equation. Both environmental noise and regime switching are important factors [21][22][23][24][25][26][27][28][29]; we will introduce them in the model ((6)-(8)) in the future.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.