    # Optimal error estimate of a conservative Fourier pseudo-spectral method for the space fractional nonlinear Schrödinger equation

In this paper, we consider the error analysis of a conservative Fourier pseudo-spectral method that conserves mass and energy for the space fractional nonlinear Schrödinger equation. We give a new fractional Sobolev norm that can construct the discrete fractional Sobolev space, and we also can prove some important lemmas for the new fractional Sobolev norm. Based on these lemmas and energy method, a priori error estimate for the method can be established. Then, we are able to prove that the Fourier pseudo-spectral method is unconditionally convergent with order O(τ^2+N^α/2-r) in the discrete L^∞ norm, where τ is the time step and N is the number of collocation points used in the spectral method. Numerical examples are presented to verify the theoretical analysis.

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## 1 Introduction

The nonlinear fractional Schrödinger equation is a generalization of the classical Schrödinger equation. It has found several applicaitions in physics, such as nonlinear optics , propagation dynamics  and water wave dynamics . In this paper, we consider the following space fractional nonlinear Schrödinger (FNLS) equation

 iut−(−Δ)α2u+β|u|2u=0,  x∈Ω,  0

with the periodic boundary condition

 u(x,t)=u(x+L,t),x∈Ω, 0

and the initial condition

 u(x,0)=φ(x),  x∈Ω, (1.3)

where , and is a complex-valued wave function, parameter is a real constant, and is a complex-value initial data. The fractional Laplacian acting on periodic function defined by 

 (−△)α2u=∑k∈Z|μk|αˆukeiμkx,  μ=2πL, (1.4)

where

 u=∑k∈Zˆukeiμkx,  ˆuk=1L∫Ωu(x)e−iμkxdx. (1.5)

When , the equation (1.1) reduces to the classical nonlinear Schrödinger (NLS) equation. Due to self-adjoint property of the fractional Laplacian, the solution of satisfies the following mass and energy conservation laws:

 Mass:  M(t)=∫Ω|u(x,t)|2dx=M(0), (1.6) Energy:  E(t)=∫Ω|(−Δ)α4u(x,t)|2−β2|u(x,t)|4dx=E(0). (1.7)

Various numerical methods have been developed in the literatures for the space FNLS equation, including finite difference methods [29, 28, 36, 33, 30], finite element methods [20, 19], spectral methods [34, 2]. In the past few decades, structure-preserving methods which can inherit the intrinsic geometric properties of the given dynamical system have attracted a lot of interest due to the superior properties in long time numerical simulation over traditional methods. For more details, readers can refer to [8, 11, 18]. Recently, structure-preserving numerical methods have been extended to solve the space FNLS equation. For example, in , Wang et al. first constructed a mass conservative Crank-Nicolson difference scheme, and they further proposed a linearly implicit difference scheme that conserves mass and energy in . In [30, 33], the modified mass and energy conservative Crank-Nicolson difference schemes were presented. Other works related to the conservative method can be founded in [24, 20].

Spectral and pseudo-spectral methods have been proved to be an efficient and high order numerical method in solving smooth problems . Over the past few years, though structure-preserving spectral methods have been widely used to solve Hamiltonian PDEs [3, 6, 17, 9]. Only in very recently years, structure-preserving Fourier pseudo-spectral methods are extended to solve the space FNLS equation. For instance, in , Wang and Huang proposed the symplectic and multi-symplectic Fourier pseudo-spectral methods for the space FNLS equation. In , a mass and energy conservative Fourier pseudo-spectral method was constructed. The numerical results show that the conservative Fourier pseudo-spectral method is efficient and stable for long-term numerical simulation. However the unconditionally convergent results on the conservative Fourier pseudo-spectral method for the space fractional PDEs have not been obtained. Actually, with the help of the defined fractional Sobolev norm and the discrete uniform Gagliardo-Nirenberg inequality in , we can easily prove that the conservative Fourier pseudo-spectral method for the space FNLS equation is unconditionally convergent in the discrete norm, but the challange problem is error estimate in norm. For the classical NLS equations, in , Gong first established the semi-norm equivalence between the finite difference method and the Fourier pseudo-spectral method and thus obtained the unconditionally convergent results on the Fourier pseudo-spectral in the discrete norm. Then based on this equivalence, the error estimates of the Fourier pseudo-spectral method in the discrete norm were obtained in . However, this error analysis technique for establishing semi-norm equivalence can not extend to the FNLS equations. By reading the finite difference methods for the fractional PDEs with fractional Laplacian [36, 31, 12], we know that under the homogeneous Dirichlet boundary condition, the Riesz derivative is discreted instead of fractional Laplacian due to the equivalence between Riesz derivative and fractional Laplacian. But this equivalence does not hold in the case of periodic boundary condition. That’s why even now these is no corresponding finite difference method has been used to solve the space fractional PDEs under the periodic boundary condition. Therefore, for the FNLS equation, it is impossible to establish semi-norm equivalence between the finite difference method and the Fourier pseudo-spectral method in the error analysis.

To obtain the norm error estimates of the Fourier pseudo-spectral method for the space FNLS equation, in this paper, we introduce the discrete fractional Sobolev space with a new discrete fractional Sobolev norm. We establish several lemmas for the new discrete fractional norm, based on these important lemmas and the energy method, a prior estimate for the method is estimated. Then we can prove that the conservative Fourier pseudo-spectral method is unconditionally convergent with order of in the disctete norm.

The rest of the paper is organized as following. In section 2, we construct the discrete fractional Sobolev space by introducing a new fractional Sobolev norm and we also prove some important lemmas for the new fractional Sobolev norm. In section 3, a conservative Fourier pseudo-spectral scheme for the FNLS equation is given, we show the numerical scheme satisfies discrete conservation laws and obtain a priori estimate. In section 4, the convergence property of the scheme is analyzed. Subsequently in section 5, we carry out some numerical experiments to confirm our theoretical results and show the efficiency of the scheme. Finally, we give a conclusion in section 6.

## 2 Fourier pseudo-spectral method

Let be an even integer, we define step size in space: . Then, the spatial grid points are defined as follows:. For any positive integer , we define the time-step: . Then grid points in space and time are given by , where . For a grid function , we introduce the following notations:

 δ+xunj=unj+1−unjh,un+12j=un+1j+unj2,δ+tunj=un+1j−unjτ.

Let be the space of grid functions defined on . For any grid function , we define the discrete inner product and associated norm

 (u,v)h=1NN−1∑j=0uj¯¯¯vj,  ∥u∥2h=(u,u)h. (2.1)

We also define the discrete norm as

 ∥u∥plph=1NN−1∑j=0|uj|p,   1≤p<+∞, (2.2)

and the discrete norm as

 ∥u∥l∞h=max0≤j≤N−1|uj|. (2.3)

### 2.1 Disctete Fractional Sobolev norm

We define a function space by

 SN=span{gj(x),  j=0,1,⋯,N−1},

where is a trigonometric polynomial defined by

 gj(x)=1NN/2∑k=−N/21cke% ikμ(x−xj), (2.4)

where

 ck={1, |k|

Then, we define the interpolation operator

by

 INu(x)=N−1∑j=0ujgj(x)=N/2∑k=−N/2ˆukeikμx, (2.5)

where

 ˆuk=1NckN−1∑j=0uje−ikμxj,  −N/2≤k≤N/2, (2.6)

and for . Therefore we have the inverse transformation

 uj=(INu)(xj)=N/2−1∑k=−N/2ˆukeikμxj. (2.7)

For any , we have and the Parseval’s theorem gives

 (u,v)h=N/2−1∑k=−N/2ˆuk¯¯¯ˆvk. (2.8)

Given a constant , we define the discrete fractional Sobolev norm and semi-norm as

 |u|2Hσh=N/2−1∑k=−N/2|μk|2σ|ˆuk|2,              ∥u∥2Hσh=N/2−1∑k=−N/2(1+|μk|2σ)|ˆuk|2. (2.9)

Clearly, We can easily prove that the discrete Sobolev spaces is the normed linear spaces according to the norm defined in . Next, we introduce the following lemmas, which are important for unconditional convergence analysis of the conservative Fourier pseudo-spectral method.

###### Lemma 2.1

(Discrete uniform Sobolev inequality). For any there exists a constant independent of such that

 ∥u∥l∞h≤C∥u∥Hσh. (2.10)

Proof. From the inverse transformation and the Cauchy-Schwarz inequality, we obtain

 ∥u∥l∞h ≤N/2−1∑k=−N/2|ˆuk| (2.11) =N/2−1∑k=−N/21(1+|μk|2σ)12(1+|μk|2σ)12|ˆuk| ≤(N/2−1∑k=−N/211+|μk|2σ)12(N/2−1∑k=−N/2(1+|μk|2σ)|ˆuk|2)12 ≤(N/2−1∑k=−N/211+|μk|2σ)12∥u∥Hσh.

For this implies and thus the proof is completed.

###### Lemma 2.2

For there exist a constant such that

 ∥u∥Hσ0h≤C∥u∥σ0σHσh∥u∥1−σ0σh. (2.12)

Proof. From the definition of and the Hölder’s inequality, we have

 ∥u∥2Hσ0h =N/2−1∑k=−N/2(1+|μk|2σ0)|ˆuk|2 (2.13) =N/2−1∑k=−N/2((1+|μk|2σ)|ˆuk|2)σ0σ(|ˆuk|2)1−σ0σ(1+|μk|2σ0(1+|μk|2σ)σ0σ) ≤C(N/2−1∑k=−N/2(1+|μk|2σ)|ˆuk|2)σ0σ(N/2−1∑k=−N/2|ˆuk|2)1−σ0σ =C(N/2−1∑k=−N/2(1+|μk|2σ)|ˆuk|2)σ0σ(N/2−1∑k=−N/2|ˆuk|2)1−σ0σ =C(∥u∥σ0σHσh∥u∥1−σ0σh)2,

where the inequality holds due to the fact for . Thus the proof is completed.

###### Lemma 2.3

( Hausdorff-Young inequality). If then

 (hN−1∑j=0|uj|p)1p≤(N/2−1∑k=−N/2|ˆuk|q)1q. (2.14)

Proof. From the inverse transformation , we have

 sup0≤j≤N−1|uj|≤N/2−1∑k=−N/2|ˆuk|, (2.15)

the Parseval’s identity gives

 hN−1∑j=0|uj|2=N/2−1∑k=−N/2|ˆuk|2. (2.16)

Then using the Riesz-Thorin Interpolation theorem (see Theorem 8.6 in [5, page 316]), we can obtain the conclusion.

###### Lemma 2.4

For any there exists a constant independent of such that

 ∥u∥lph≤Cσ0∥u∥σ0σHσh∥u∥1−σ0σh,    2≤p≤+∞,  σ0≤σ≤1. (2.17)

Proof. By Lemma 2.3 and Hölder’s inequality, for such that we have

 (hN−1∑j=0|uj|p)1p ≤(N/2−1∑k=−N/2|ˆuk|q)1q (2.18) =(N/2−1∑k=−N/21(1+|μk|2σ0)q2(1+|μk|2σ0)q2|ˆuk|q)1q ≤(N/2−1∑k=−N/2(1+|μk|2σ0)|ˆuk|2)12(N/2−1∑k=−N/21(1+|μk|2σ0)q2−q)2−q2q ≤∥u∥Hσ0h(N/2−1∑k=−N/21(1+|μk|2σ0)q2−q)2−q2q.

Then for we have

 ∥u∥lph≤~Cσ0∥u∥Hσ0h, (2.19)

where is independent of . Combining the above inequality with gives and thus completes the proof.

### 2.2 Discrete fractional Laplacian

Applying the fractional Laplacian to the interpolated function yields

 (−Δ)α2INu(x)=1NN−1∑j=0ujN/2∑p=−N/21cp|μp|αeipμ(x−xj), (2.20)

and thus

 (−Δ)α2INu(xk) =N/2−1∑p=−N/2dp(1NN−1∑j=0uje−2πijpN)e2πipkN, (2.21)

where

 dp=|μp|α,−N/2≤p≤N/2−1. (2.22)

For ,  we define a discrete fractional Laplacian by

 ((−Δ)α2dU)k =N/2−1∑p=−N/2dp(1NN−1∑j=0Uje−2πijpN)e2πipkN, (2.23)

By using the notation of the discrete Fourier transform and its inverse:

 (FdU)k=1NN−1∑j=0Uje−2πijkN,(F−1dˆU)j=N/2−1∑k=−N/2ˆUke2πijkN, (2.24)

the discrete fractional Laplacian can be expressed as

 (−Δ)α2dU=F−1dΛαFdU, (2.25)

where Next, we give several lemmas that show the relationship between discrete fractional Soboolv semi-norm and fractional Laplacian.

###### Lemma 2.5

For any grid function , we have

 (Dαu,u)h=|u|2Hα/2h,1<α≤2. (2.26)

Proof. Using the Parseval’s identity , we have

 (Dαu,u)h =(F−1dΛαFdu,u)h (2.27) =N/2−1∑k=−N/2(ΛαFdu)k(Fd¯¯¯u)k=N/2−1∑k=−N/2dkˆuk¯¯¯ˆuk=|u|2Hα/2h.
###### Lemma 2.6

For any two grid functions , we have

 (Dαu,v)h=(Dα/2u,Dα/2v)h,1<α≤2. (2.28)

Proof. Using the Parseval’s identity , we have

 (Dαu,v)h =(F−1dΛαFdu,v)h (2.29) =N/2−1∑k=−N/2(ΛαFdu)k(Fd¯¯¯v)k =N/2−1∑k=−N/2(Λα2Fdu)k(Λα2Fd¯¯¯v)k=(Dα/2u,Dα/2v)h.
###### Lemma 2.7

For any two grid functions , we have

 (Dαu,v)h≤|u|Hα/2h|v|Hα/2h,1<α≤2. (2.30)

Proof. Using the Parseval’s identity , we have

 (Dαu,v)h =(F−1dΛαFdu,v)h (2.31) =N/2−1∑k=−N/2(ΛαFdu)k(Fd¯¯¯v)k=N/2−1∑k=−N/2dkˆuk¯¯¯ˆvk.

Therefore

 (Dαu,v)h (2.32)

For simplicity, we denote and as the exact value of and its numerical approximation at respectively.

## 3 Solution existence and conservation of the scheme

We discretize the FNLS equation using the Fourier pseudo-spectral method in space and the Crank-Nicolson method in time to arrive at a fully discrete system:

 iδ+tUnj−(DαUn+1/2)j+β2(|Unj|2+|Un+1j|2)Un+1/2j=0,Un∈Vh, (3.1)

where For convenience, scheme can be written in an equivalent form

 iδ+tUn−DαUn+1/2+F(Un,Un+1)=0,Un∈Vh, (3.2)

where

###### Lemma 3.1

For the approximation there exist identities:

 \rm{Im}(DαUn+1/2,Un+1/2)h=0, (3.3) \rm{Re}(DαUn+1/2,δ+tUn)h=12τ(|Un+1|2Hα/2h−|Un|2Hα/2h), (3.4)

According to Lemma 2.5 and Lemma 2.6, we can get the results immediately. Here “”and “”mean taking the imaginary part and real part of a complex number , respectively.

### 3.1 Conservation

###### Theorem 3.1

The scheme is conservative in the sense that

 Mn=M0,0≤n≤N, (3.5) En=E0,0≤n≤N, (3.6)

where

 Mn:=∥Un∥2h,En=|U|2Hα/2h−β2∥Un∥4l4h (3.7)

Proof. Computing the discrete inner product of with , then taking the imaginary part, we obtain

 12τ(∥Un+1∥2h−∥Un∥2h)=0,tn∈Ωτ, (3.8)

where Lemma 3.1 is used. This gives .

Computing the discrete inner product of with , then taking the real part, we obtain

 −12τ[(|Un+1|2Hα/2h−β2∥Un+1∥4l4h)−(|Un|2Hα/2h−β2∥Un∥4l4h)]=0,tn∈Ωτ, (3.9)

where Lemma 3.1 is used. This yields .

### 3.2 A priori estimate

###### Theorem 3.2

Then numerical solution of scheme is bounded in the following sense

 ∥Un∥h≤C1,|Un|Hα/2h≤C2,∥Un∥l∞h≤C3,0≤n≤N, (3.10)

where are some positive constants.

Proof. The proof is similar to that in [33, Theorem 3.2]. The mass conservation implies the first inequality in immediately if we choose
Next, we prove the second inequality by the energy conservation . If , according to the second term of and energy conservation , we can get the result straightforwardly. If , In the view of Lemma 2.4 with , Young’s inequality and the first inequality in , we obtain

 ∥Un∥4l4h≤Cσ0∥Un∥8σ0αHα/2h∥Un∥4−8σ0αh ≤Cσ0(ε|Un|2Hα/2h+ε∥Un∥2h+C(ε)), (3.11)

where is any arbitrary positive constant. Combing the second term of with , the energy conservation imply

 |Un|2Hα/2h =β2∥Un∥4l4h+E0 (3.12) ≤β2Cσ0(ε|Un|2Hα/2h+ε∥Un∥2h+C(ε))+E0.

Taking , we have

 |Un|2Hα/2h≤∥U0∥2h+βCσ0C(ε)+2E0:=C22,0≤n≤Nt. (3.13)

This implies the second inequality of
Finally, combining the first two inequality in with Lemma 2.1, we get the third inequality in , that is,

 ∥Un∥2l∞h≤C2σ(∥Un∥2h+|Un|2Hα/2h)≤C2σ(C21+C22):=C23,0≤n≤Nt. (3.14)

Thus the proof is completed.

### 3.3 Existence

###### Theorem 3.3

The nonlinear equation system in scheme is solvable.

Proof. The argument of the existence for the solution relies on the Browder fixed point theorem (see [1, 11]). Here we omit the proof for brevity.

## 4 Convergence of the scheme

In this section, we will establish error estimate of in the discrete norm. For simplicity, we let and assume that is a set of infinitely differentiable functions with -period defined on . is the closure of in . The semi-norm and the norm of are denoted by and , respectively.
For the given even , we introduce the projection space

 SN={u|u(x)=∑|l|≤N/2ˆuleilx},

and the interpolation space

 S′′N={u|u(x)=∑|l|≤N/2ˆulclˆuleilx,ˆu−N2=ˆuN2,}

where

It is clear that