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Research article Special Issues

Some identities of degenerate higher-order Daehee polynomials based on λ-umbral calculus

  • The degenerate versions of special polynomials and numbers, initiated by Carlitz, have regained the attention of some mathematicians by replacing the usual exponential function in the generating function of special polynomials with the degenerate exponential function. To study the relations between degenerate special polynomials, λ-umbral calculus, an analogue of umbral calculus, is intensively applied to obtain related formulas for expressing one λ-Sheffer polynomial in terms of other λ-Sheffer polynomials. In this paper, we study the connection between degenerate higher-order Daehee polynomials and other degenerate type of special polynomials. We present explicit formulas for representations of the polynomials using λ-umbral calculus and confirm the presented formulas between the degenerate higher-order Daehee polynomials and the degenerate Bernoulli polynomials, for example. Additionally, we investigate the pattern of the root distribution of these polynomials.

    Citation: Dojin Kim, Sangbeom Park, Jongkyum Kwon. Some identities of degenerate higher-order Daehee polynomials based on λ-umbral calculus[J]. Electronic Research Archive, 2023, 31(6): 3064-3085. doi: 10.3934/era.2023155

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  • The degenerate versions of special polynomials and numbers, initiated by Carlitz, have regained the attention of some mathematicians by replacing the usual exponential function in the generating function of special polynomials with the degenerate exponential function. To study the relations between degenerate special polynomials, λ-umbral calculus, an analogue of umbral calculus, is intensively applied to obtain related formulas for expressing one λ-Sheffer polynomial in terms of other λ-Sheffer polynomials. In this paper, we study the connection between degenerate higher-order Daehee polynomials and other degenerate type of special polynomials. We present explicit formulas for representations of the polynomials using λ-umbral calculus and confirm the presented formulas between the degenerate higher-order Daehee polynomials and the degenerate Bernoulli polynomials, for example. Additionally, we investigate the pattern of the root distribution of these polynomials.



    In the paper, we consider fractional Klein-Gordon-Schrödinger (KGS) equations with the fractional Laplacian

    ituν(Δ)α2u+uf1(|u|2,ϕ)=0, (1.1)
    ttϕΔϕ+ϕf2(|u|2,ϕ)=0, (1.2)

    where, f1(|u|2,ϕ)=f(|u|2,ϕ)|u|2, f2(|u|2,ϕ)=f(|u|2,ϕ)ϕ,f:R+×RR, u(,t), ϕ(,t) are complex and real valued unknown functions, and the fractional Laplace operator [1,2,3,4] is defined by

    (Δ)α2u(x)=Cn,αRnu(x)u(y)|xy|n+α/2dy.

    Let ϕt=v. Then, the fractional KGS system (1.1)–(1.2) can be expressed as the following form

    iutν(Δ)α2u=uf1(|u|2,ϕ), (1.3)
    ϕt=v, (1.4)
    vtΔϕ+ϕ=f2(|u|2,ϕ). (1.5)

    The fractional KGS system (1.3)–(1.5) has the following mass and energy conserved laws

    Mass: M(t)=+|u(x,t)|2dx=M(0),Energy: E(t)=+12(v2+|ϕ|2+ϕ2)+ν|(Δ)α4u|2f(|u|2,ϕ)dx=E(0).

    When α=2, the fractional KGS system (1.1) and (1.2) reduces to the integer KGS system [5,6,7,8,9,10,11,12]. When α2, the fractional KGS system [13,14,15,16,17] describes various important physical phenomena. Recently, some attentions have been paid to fractional KGS system with fractional Laplacian, which are fractional version of classical KGS equations and consider long-range interactions. In general, the analytic solution of the fractional KGS system (1.1), (1.2) can not be derived due to nonlocality and nonlinearity. Therefore, numerical method plays an important role in the study of the fractional KGS system (1.1), (1.2). Various numerical methods [13,14,15,16,17] such as the finite difference method, pseudo-spectral method, and symplectic method have been studied the fractional KGS system with f(|u|2,ϕ)=|u|2ϕ, and the stability and convergence of the numerical methods have been discussed.

    It is not difficult to find that there are some deficiencies in the numerical schemes of the fractional KGS system. The first deficiencie is that the above finite difference methods are based on fractional center difference scheme, and approximate solutions converge to the exact solution at the rate O(τ2+h2). To the best of the authors'knowledge, there are few research on higher-order scheme for the fractional KGS system. The second deficiencie that the above numerical schemes are only based on one dimension fractional KGS system, and high dimension fractional KGS systems are rarely studied. In addition, there exist few reports on numerical scheme for general fractional KGS system in Eqs (1.1), (1.2) with f(|u|2,ϕ)|u|2ϕ.

    The main goal of this paper is to construct structure-preserving scheme [18,19,20,21,22] for solving one dimension and two dimension fractional KGS equations with fractional Laplacian operator. Some numerical schemes were proposed to approximate the fractional Laplacian operator. The fractional center difference scheme based on fractional Laplace operator was first developed in [23]. Based on this work, some high-order schemes were constructed, and the results have be applied to some fractional difference equations. In addition, finite element scheme and Fourier spectral scheme of fractional Laplace operator have been studied with some special boundary conditions. In this paper, we use high-order difference scheme for fractional Laplace operator with zero boundary condition. By using some useful lemmas, we prove that the scheme can preserve the mass and energy conservation laws.

    All structure-preserving schemes for the nonlinear fractional KGS system (f(|u|2,ϕ)|u|2ϕ) face how to efficiently solve a large nonlinear system at each time step. In this paper, we consider structure-preserving scheme for general fractional KGS equations (1.3)–(1.5). The high order central differences are used for the space direction, with the Crank-Nicolson scheme applied to the time direction. However, the numerical scheme is nonlinear scheme, and it takes too much time in the numerical simulation. Recently, scalar auxiliary variable scheme was introduce for gradient flows, and the numerical scheme only requiring solving decoupled linear systems at each time step. As far as we know, there exist few reports on scalar auxiliary variable scheme for fractional KGS system (1.3)–(1.5). In this paper, we show a new scalar auxiliary variable scheme to solve two dimension fractional KGS system (1.3)–(1.5).

    In this paper, we first show numerical scheme to solve one dimension fractional KGS system (1.3)–(1.5), and discussed conservation, existence and uniqueness, stability and convergence of the numerical scheme. Second, we show numerical scheme for two dimension fractional KGS system by high order finite difference scheme in span and Crank-Nicolson scheme in time, and discussed conservation of the numerical scheme. Third, we show scalar auxiliary variable scheme to solve two dimension fractional KGS equations. Finally, some numerical examples are given, confirm theoretical results and demonstrate the efficiency of the numerical schemes.

    The outline of the paper is as follows. In Section 2, the structure-preserving scheme is proposed for one dimension fractional KGS system, and convergence and stability of the numerical scheme is proved. In Section 3, the high conservative difference scheme is proposed for two dimension fractional KGS system. In Section 4, the numerical experiments are given, and the results verify the efficiency of the conservative difference scheme. Finally, a conclusion and some discussions are given in Section 5.

    In the section, we show structure-preserving scheme for one dimension fractional KGS system with f(|u|2,ϕ)|u|2ϕ, and prove that the scheme can preserve mass and energy conservation laws. Moreover, we show that the arising scheme is uniquely solvable, and approximate solutions converge to the exact solution at the rate O(τ2+h4).

    The fractional KGS system (1.1)–(1.2) contains a fractional Schrödinger equation and a classic Klein-Gordon equation, and we consider boundary condition

    u(x,t)=0,xR/Ω;ϕ(x,t)=0,xΩ; Ω=(a,b).

    Let M,N. Then, choose time-step τ=T/N and mesh size h=(ba)/M. Denote

    xj=a+jh, tn=nτ, j=0, 1, 2,, M, n=0, 1, 2, 

    Then

    unj=u(xj,tn), Unju(xj,tn), ϕnj=ϕ(xj,tn), Φnjϕ(xj,tn).

    Define

    Ωh={xj|1jM1}, Ωτ={tn|1nN1},ˉΩh={xj|0jM}, ˉΩτ={tn|0nN}.

    Suppose w={wnj; j=0, 1, 2 M, n=0, 1, 2 N} be a discrete function in ˉΩh×ˉΩτ, and

    Z0h={w=wj|w0=wM=0, j=0, 1, 2,  M}.

    For convenience, we define the finite difference operators

    (wnj)t=wn+1jwnjτ, (wnj)ˆt=wn+1jwn1j2τ, wn+12j=wn+1j+wnj2, ˜wn+12j=wn+1j+wn1j2.

    For any grid function u={uj}, v={vj}, define the discrete inner product, L2-norm and Lp-norm as

    u,v=hM1j=1uj¯vj, u2=u,u,uplph=hM1j=1|uj|p, 1p<+, ulh=sup0<j<M1|uj|.

    For 0δ1, we also define the fractional Sobolev norm uHδ and semi-norm |u|Hδ as

    u2Hδ=π/hπ/h(1+|k|2δ)|ˆu(k)|2dk, |u|2Hδ=π/hπ/h|k|2δ|ˆu(k)|2dk,

    where

    ˆu(k)=12πhjzujeikxj,uj=12ππ/hπ/hˆu(k)eikxjdk.

    Lemma 1. [24] Let f(x)C6[xL,xR], 2<j<M2. Then

    f(xj)=43f(xj+h)2f(xj)+f(xjh)h213f(xj+2h)2f(xj)+f(xj2h)h2+O(h4).

    When j=1,M1, then

    f(x1)=76f(x2)2f(x1)+f(x0)h216f(x3)2f(x2)+f(x1)h2112f(x0)1144f(4)(x0)+O(h4),f(xM1)=76f(xM)2f(xM1)+f(xM2)h216f(xM1)2f(xM2)+f(xM3)h2112f(xM)1144f(4)(xM)+O(h4).

    Lemma 2. [20] Suppose that uL1(R) and

    uL4+α(R):={u|+(1+|ξ|)4+α|ˆu(ξ)|dξ<}.

    Then for a fixed h, we can obtain high order scheme

    43Δαhu(x)13Δα2hu(x)=(Δ)α2u(x)+O(h4),

    where

    Δαhu(x)=hαk=g(α)ku(xkh), g(α)k=(1)kΓ(α+1)Γ(α2k+1)Γ(α2+k+1).

    Lemma 3. For any grid functions unZ0h, we can obtain

    Im43Δαhun+1213Δα2hun+12,un+12=0,Re43Δαhun+1213Δα2hun+12,unt=12τ(43ΔαhUn+1213Δα2hUn+1243ΔαhUn2+13Δα2hUn2),

    where Im(s),Re(s) mean taking the imaginary part and the real part of a complex number s, respectively.

    Lemma 4. [25] (Diserete Sobolev inequality) For every 12δ1, there exist a constant C=C(δ)>0 independent of h>0 such that

    ulhCσuHδh,

    for all ul2h.

    Lemma 5. [25] (Gagliardo-Nirenberg inequality) For any 14<δ01, there exist a constant Cδ0=C(δ0)>0 independent of h>0 such that

    ul4hCδ0uδ0/δHδu1δ0/δ.

    Applying Crank-Nicolson scheme in time and higher order difference scheme in space to the fractional KGS system (1.3)–(1.5), we can obtain the following numerical scheme

    i(Unj)tνˆΔαhUn+12j=12[f(|Un+1j|2,Φnj)f(|Unj|2,Φnj)|Un+1j|2|Unj|2+f(|Un+1j|2,Φn+1j)f(|Unj|2,Φn+1j)|Un+1j|2|Unj|2]Un+12j, (2.1)
    (Φnj)t=Vn+12j, (2.2)
    (Vnj)t˜Δ2hΦn+12j+Φn+12j=12[f(|Unj|2,Φn+1j)f(|Unj|2,Φnj)Φn+1jΦnj+f(|Un+1j|2,Φn+1j)f(|Un+1j|2,Φnj)Φn+1jΦnj], (2.3)
    U0j=u0(xj),Φ0j=ϕ0(xj),V0j=ϕ1(xj), (2.4)
    Un0=UnM,Φn0=ΦnM, (2.5)

    where

    ˆΔαhUn+12j=43ΔαhUn+12j13Δα2hUn+12j,
    ˜Δ2hΦn+12j={76Δ2hΦn+121112Δ2hΦn+122,j=1,43Δ2hΦn+12j13Δ22hΦn+12j,2<j<M2,76Δ2hΦn+12M1112Δ2hΦn+12M2,j=M1.

    Theorem 1. The scheme (2.1)(2.5) is conservative in the sense

    Mass: Mn=Mn1=M0,Energy: En=En1=E0,

    where

    Mn=Un2,En=ν(43ΔαhUn+1213Δα2hUn+12)+12[43Δ2hΦn+1213Δ22hΦn+12+h6(|Δ2hΦn+112|2+|Δ2hΦn+1M12|2)+Φn+12+Vn+12]hM1j=1|f(|Un+1j|2,Φn+1j).

    Proof: Multiplying h(Un+1j+Unj) to Eq (2.1) and summing it up for 1jM1 can yield

    iUnt,2Un+12νˆΔαhUn+12,2Un+12=12[f(|Un+1|2,Φn)f(|Un|2,Φn)|Un+1|2|Un|2+f(|Un+1|2,Φn+1)f(|Un|2,Φn+1)|Un+1|2|Un|2]Un+12,2Un+12. (2.6)

    Taking the imaginary part of Eq (2.6) and noting that Lemma 3 yields Un+12=Un2. When n=0,1,2,, we can obtain Mn=Mn1=M0.

    Multiplying 2h(Un+1jUnj)/τ to Eq (2.1) and summing it up for 1jM1 can yield

    iUnt,2UntνˆΔαhUn+12,2Unt=12[f(|Un+1|2,Φn)f(|Un|2,Φn)|Un+1|2|Un|2+f(|Un+1|2,Φn+1)f(|Un|2,Φn+1)|Un+1|2|Un|2]Un+12,2Unt. (2.7)

    It follows from Lemma 3 that

    ˆΔαhUn+12,2Unt=1τ(43ΔαhUn+1213Δα2hUn+1243ΔαhUn2+13Δα2hUn2).

    Taking the real part of Eq (2.7) and noting that Lemma 3 yields

    ντ(43ΔαhUn+1213Δα2hUn+1243ΔαhUn2+13Δα2hUn2)=h2τM1j=1[f(|Un+1j|2,Φn+1j)f(|Unj|2,Φn+1j)+f(|Un+1j|2,Φnj)f(|Unj|2,Φnj)]. (2.8)

    Multiplying 2h(Vn+1jVnj)/τ to Eq (2.2) and summing it up for 1jM1 can yield

    Φnt,2Vnt=Vn+12,2Vnt. (2.9)

    Multiplying 2h(Φn+1jΦnj)/τ to Eq (2.3) and summing it up for 1jM1 can yield

    Vnt,2Φnt˜Δ2hΦn+12,2Φnt+Φn+12,2Φnt=12[f(|Un|2,Φn+1)f(|Un|2,Φn)Φn+1Φn+f(|Un+1|2,Φn+1)f(|Un+1|2,Φn)Φn+1Φn],2Φnt. (2.10)

    It follows from literature [24] that

    ˜Δ2hΦn+12,2Φnt=1τ[(43Δ2hΦn+1213Δ22hΦn+1243Δ2hΦn2+13Δ22hΦn2)+h6(|Δ2hΦn+112|2|Δ2hΦn12|2)+(|Δ2hΦn+1M12|2|Δ2hΦnM12|2)].

    Noting that equation Φnt,Vnt=Vnt,Φnt, we obtain

    1τ[(43Δ2hΦn+1213Δ22hΦn+1243Δ2hΦn2+13Δ22hΦn2)+h6(|Δ2hΦn+112|2|Δ2hΦn12|2)+(|Δ2hΦn+1M12|2|Δ2hΦnM12|2)]+1τ(Φn+12Φn2)+1τ(Vn+12Vn2)=hτM1j=1[f(|Un+1j|2,Φn+1j)f(|Un+1j|2,Φnj)+f(|Unj|2,Φn+1j)f(|Unj|2,Φnj)]. (2.11)

    Combining Eqs (2.8) and (2.11), we obtain

    ν(43ΔαhUn+1213Δα2hUn+12)+12[43Δ2hΦn+1213Δ22hΦn+12+h6(|Δ2hΦn+112|2+|Δ2hΦn+1M12|2)+Φn+12+Vn+12]hM1j=1|f(|Un+1j|2,Φn+1j)=ν(43ΔαhUn213Δα2hUn2)+12[43Δ2hΦn213Δ22hΦn2+h6(|Δ2hΦn12|2+|Δ2hΦnM12|2)+Φn+12+Vn2]hM1j=1f(|Unj|2,Φnj).

    When n=0,1,2,, we can obtain En=En1=E0.

    In [15], we show the theoretical analysis for f(|u|2,ϕ)=|u|2ϕ+|u|4ϕ. In similar analysis of literature [15], we can obtain the theoretical analysis for f(|u|2,ϕ)=|u|2lϕm when l,m satisfy certain conditions. For simplicity of notation, we only consider f(|u|2,ϕ)=|u|2ϕ in next theoretical analysis.

    Theorem 2. The scheme (2.1)(2.5) is bounded in the discrete lh.

    Proof: Using Theorem 1 can obtain Un=˜C and

    En=ν(43ΔαhUn+1213Δα2hUn+12)+12[43Δ2hΦn+1213Δ22hΦn+12+h6(|Δ2hΦn+112|2+|Δ2hΦn+1M12|2)+Φn+12+Vn+12]hM1j=1|Un+1j|2Φn+1j. (2.12)

    Noting that Young's inequality ab14a2+b2, we can obtain

    hM1j=1|Un+1j|2Φn+1jhM1j=1|Un+1j|2|Φn+1j|hM1j=1(|Un+1j|4+14|Φn+1j|2)=Un+14l4h+14Φn+12. (2.13)

    Using Gagliardo-Nirenberg inequality and noting that 14<σ0<α4 can yield

    Un+14l4hˆCδ0Un+18δ0αHα2Un+148δ0αˆCδ0(εUn+12Hα2+C(ε)). (2.14)

    It follows from [22] that there exists a constant 1Cα(2π)α such that

    43ΔαhUn+1213Δα2hUn+12=Cα|Un+1|2Hα2, (2.15)
    43Δ2hΦn+1213Δ22hΦn+12=C|Φn+1|2H1. (2.16)

    Substituting Eqs (2.15), (2.16) into Eq (2.12) can yield

    Cαν|Un+1|2Hα2+12[C|Φn+1|2H1+h6(|Δ2hΦn+112|2+|Δ2hΦn+1M12|2)+Φn+12+Vn+12]En+ˆCδ0(εUn+12Hα2+C(ε))+14Φn+12.

    Noting that Un2Hα2=|Un|2Hα2+Un2,En=E0, we have

    (νCαˆCδ0ε)|Un+1|2Hα2+12[C|Φn+1|2H1+h6(|Δ2hΦn+112|2+|Δ2hΦn+1M12|2)+12Φn+12+Vn+12]ˆCδ0C(ε)+E0.

    When ε<νCαˆCδ0, there exist a constant C, such that

    |Φn+1|2H1C,Φn2C,Un2Hα2C.

    It follows from Lemma 4 that UnlhC,ΦnlhC.

    Theorem 3. The solution of the numerical scheme (2.1)(2.5) exists and unique.

    Proof: Let Pn=UnˆUn,Rn=VnˆVn,Qn=ΦnˆΦn. Then, it follows from Eqs (2.1)–(2.15) that

    i(Pnj)tνˆΔαhPn+12j=(Un+12jΦn+12jˆUn+12jˆΦn+12j),1<j<M1,1<n<N, (2.17)
    (Qnj)t=Rn+12j,1<j<M1,1<n<N, (2.18)
    (Rnj)t˜Δ2hQj+Qn+12j=12(|Unj|2+|Un+1j|2)12(|ˆUnj|2+|ˆUn+1j|2),1<j<M1,1<n<N. (2.19)

    Multiplying h(Pn+1j+Pnj) to Eq (2.17) and summing it up for 1<j<M1 can yield

    iPntνˆΔαhPn+12,Pn+12=Un+12Φn+12ˆUn+12ˆΦn+12,Pn+12. (2.20)

    Taking the imaginary part of Eq (2.20) can obtain

    12τ(Pn+12Pn2)=Im(Un+12Φn+12ˆUn+12ˆΦn+12),Pn+12. (2.21)

    Noting that Theorem 2, then we can obtain

    12τ(Pn+12Pn2)C(Pn+12+Pn2+Qn+12+Qn2). (2.22)

    Multiplying (Rn+1jRnj)/τ to Eq (2.18) and summing it up for 1<j<M1 can yield

    Qnt,Rnt=Rn+12,Rnt. (2.23)

    Multiplying (Qn+1jQnj)/τ to Eq (2.19) and summing it up for 1<j<M1 can yield

    Rnt,Qnt+[(43Δ2hQn+1213Δ22hQn+1243Δ2hQn2+13Δ22hQn2)+h6(|Δ2hQn+112|2|Δ2hQn12|2)+h6(|Δ2hQn+1M12|2|Δ2hQnM12|2)]+Qn+1,Qnt=12(|Un|2+|Un+1|2)12(|ˆUn|2+|ˆUn+1|2),Qnt. (2.24)

    It follows from Theorem 2 that the Eq (2.24) can be expressed as the following form

    Rnt,Qnt+1τ[(43Δ2hQn+1213Δ22hQn+1243Δ2hQn2+13Δ22hQn2)+h6(|Δ2hQn+112|2|Δ2hQn12|2)+h6(|Δ2hQn+1M12|2|Δ2hQnM12|2)]+1τ(Qn+1Qn)C2(Pn2+Rn+12+Rn).

    Combining Eq (2.23) can yield

    1τ(Rn+12Rn)+1τ[(43Δ2hQn+1213Δ22hQn+1243Δ2hQn2+13Δ22hQn2)+h6(|Δ2hQn+112|2|Δ2hQn12|2)+h6(|Δ2hQn+1M12|2|Δ2hQnM12|2)]+1τ(Qn+1Qn)C2(Pn2+Rn+12+Rn). (2.25)

    Adding Eqs (2.22) and (2.25) can yield

    1τ(Rn+12Rn)+1τ(Pn+12Pn)+1τ(Qn+1Qn)+1τ[(43Δ2hQn+1213Δ22hQn+1243Δ2hQn2+13Δ22hQn2)+h6(|Δ2hQn+112|2|Δ2hQn12|2)+h6(|Δ2hQn+1M12|2|Δ2hQnM12|2)]C2(Pn2+Pn+12+Qn2+Qn+12+Rn+12+Rn).

    Let

    Bn=Rn+12+Pn+12+Qn+1+43ΔαhQn+1213Δα2hQn+12+h6|Δ2hQn+112|2h6|Δ2hQn+1M12|2.

    Then, we can obtain

    BnBn1Cτ(Bn+Bn1).

    It follows from Gronwall's inequality that

    max1nN(Bn2)e4CTB02=0.

    Noting that Pn+12+Qn+12Bn, we can obtain Pn+12=0,Qn+12=0.

    Let unj=u(xj,tn), vnj=v(xj,tn), ϕnj=ϕ(xj,tn). Then, we define the local truncation error as

    Rn1=i(unj)tνˆΔαhun+12j+un+12jϕn+12j,1<j<M1,1<n<N, (2.26)
    Rn2=(ϕnj)tvn+12j,1<j<M1,1<n<N, (2.27)
    Rn3=(vnj)t˜Δ2hϕj+ϕn+12j12(|unj|2+|un+1j|2),1<j<M1,1<n<N. (2.28)

    According to Taylor expansion, we obtain the following result.

    Theorem 4. |Rnj|ˆC(τ2+h4) holds as τ,h0.

    Theorem 5. Suppose that the problem (1.1),(1.2) has a smooth solution, then the solution Un,Φn of difference scheme (2.1)(2.5) converges to the true solution u,ϕ with order O(τ2+h4) by thelh norm.

    Proof: Let enj=Unjunj,ηnj=Φnjϕnj, ξj=Vnjvnj. Then, we obtain

    Rn1=i(enj)tνˆΔαhen+12j+Fn+12j,1<j<M1,1<n<N, (2.29)
    Rn2=(ηnj)tξn+12j,1<j<M1,1<n<N, (2.30)
    Rn3=(ξnj)t˜Δ2hηj+ηn+12jGnj,1<j<M1,1<n<N, (2.31)

    where

    Fn+12=Un+12Φn+12un+12ϕn+12,Gn=12(|Un|2+|Un+1|2)12(|un|2+|un+1|2)=12(Un¯Un+Un+1¯Un+1)12(un¯un+un+1¯un+1)=12(en¯Un+un¯en+en+1¯Un+1+un+1¯en+1).

    Multiplying h(en+1j+enj) to Eq (2.29) and summing it up for 1<j<M1 can yield

    Rn1,2en+12=ient,2en+12νˆΔαhen+12,2en+12+Fn+12,2en+12. (2.32)

    Taking the imaginary part of Eq (2.32) can yield

    1τ(en+12en2)Rn12+C(en+12+en2+ηn+12+ηn2). (2.33)

    Multiplying h(ξn+1jξnj)/τ to Eq (2.30) and summing it up for 1<j<M1 can yield

    Rn2,ξnt=ηntξn+12j,ξnt. (2.34)

    Multiplying h(ηn+1jηnj)/τ to Eq (2.31) and summing it up for 1<j<M1 can yield

    Rn3,ηnt=ξnt,ηnt+1τ[(43Δ2hηn+1213Δ22hηn+1243Δ2hηn2+13Δ22hηn2)+h6(|Δ2hηn+112|2|Δ2hηn12|2)+(|Δ2hηn+1M12|2|Δ2hηnM12|2)]+1τ(ηn+1ηn)+Gn,ηnt. (2.35)

    It follows from Cauchy-Schwarz inequality that

    G,ηnt=G,Rn2ηn+12C(ηn+12+ηn2+en+12+en2), (2.36)
    Rn3,ηnt=Rn3,Rn2ηn+12C(Rn32+Rn22+ηn+12+ηn2). (2.37)

    Substituting Eqs. (2.36) and (2.37) into Eq (2.35) and noting that Eqs (2.33) and (2.34) yields

    1τ(en+12en2)+1τ(ξn+12ηn2)+1τ[(43Δαhηn+1213Δα2hηn+1243Δαhηn2+13Δα2hηn2)+h6(|Δαhηn+112|2|Δαhηn12|2)+(|Δαhηn+1M12|2|ΔαhηnM12|2)]+1τ(ηn+12ηn2)C(Rn12+Rn22+Rn32+ηn+12+ηn2+en+12+en2).

    Let

    Bn=en+12+ξn+12+43Δαhηn+1213Δα2hηn+12+h6(|Δαhηn+112|2+|Δαhηn+1M12|2)+ηn+12.

    Then, we can obtain

    BnBn1τ(|Rn12+|Rn22+|Rn32)+Cτ(Bn+Bn1).

    It follows from Gronwall's inequality that

    max1nNBn(B0+τNl=1(|Rl12+|Rl22))e8CT(B0+ˆCT(τ2+h4)2)e8CT.

    Noting that B0=O(τ2+h4), we can obtain

    enC(τ2+h4),ηnH1C(τ2+h4),ηnC(τ2+h4). (2.38)

    Multiplying h(en+1jenj)/τ to Eq (2.29) and summing it up for 1<j<M1 can yield

    Rn1,en+1en=ient,en+1enνˆΔαhen+12,en+1en+Fn+12,en+1en. (2.39)

    It follows from Eq (2.32) that can obtain

    ReFn+12,en+1en=τReFn+12,iνˆΔαhen+12+iFn+12iRn1=τImFn+12,νˆΔαhen+12τFn+12,Rn1,

    and

    ˆΔαhen+12,en+1en=ˆCατ(|en+1|2Hα/2|en|2Hα/2).

    Noting that

    |ImFn+12,νˆΔαhen+12|C(|Fn+12|2+|e|nHα2+|e|n+1Hα2),

    we can yield

    |en+1|2Hα/2|en|2Hα/2τ(|en+1|2Hα/2+|en|2Hα/2+|Fn+12|2Hα/2+ReRn1,en+1en). (2.40)

    It follows from [15] that can obtain

    |Fn+12|2Hα/2C(|en+1|2Hα/2+|en|2Hα/2+(τ2+h4)2), Fn+122C(τ2+h4)2.

    Thus, Eq (2.40) can be expressed as

    |en+1|2Hα/2|en|2Hα/2Cτ(|en+1|2Hα/2+|en|2Hα/2+(τ2+h2)2)+ReRn1,en+1en. (2.41)

    Summing up the superscript n to N and then replacing N by n, we get

    |en+1|2Hα/2Cnl=0|el|2Hα/2+CT(τ2+h4)2.

    It follows from Gronwall Inequality that

    |en+1|Hα/2C(τ2+h4).

    Noting that Eq (2.38) can yield elhC(τ2+h4), ηlhC(τ2+h4).

    Let U=[U1,U2,,UM1], V=[V1,V2,,VM1], Φ=[Φ1,Φ2,,ΦM1]. Then, we rewrite the numerical scheme (2.1)–(2.5) as the following vector form

    iUntν˜AUn+12=12[f(|Un+1|2,Φn)f(|Un|2,Φn)|Un+1|2|Un|2+f(|Un+1|2,Φn+1)f(|Un|2,Φn+1)|Un+1|2|Un|2]Un+12,Φnt=Vn+12,Vnt˜BΦn+12+Φn+12=12[f(|Un|2,Φn+1)f(|Un|2,Φn)Φn+1Φn+f(|Un+1j|2,Φn+1)f(|Un+1|2,Φn)Φn+1Φn],U0=u0,Φ0=ϕ0,V0=ϕ1,

    where, matrices ˜A, ˜B represent differential matrix of fractional Laplacian operator and classical Laplacian operator, respectively. In order to solve above nonlinear numerical scheme, we will use the following iterative algorithm

    iUn+1(s+1)Unτν˜AUn+1(s+1)+Un2=12[f(|Un+1(s)|2,Φn)f(|Un|2,Φn)|Un+1(s)|2|Un|2+f(|Un+1(s)|2,Φn+1(s))f(|Un|2,Φn+1(s))|Un+1(s)|2|Un|2]Un+1(s)+Un2,Φn+1(s+1)Φnτ=Vn+1(s)+Vn2,Vn+1(s+1)Vnτ˜BΦn+1(s+1)+Φn2+Φn+1(s+1)+Φn2=12[f(|Un|2,Φn+1(s+1))f(|Un|2,Φn)Φn+1(s+1)Φn+f(|Un+1(s+1)|2,Φn+1(s+1))f(|Un+1(s+1)|2,Φn)Φn+1(s+1)Φn].

    In this section, we first show Crank-Nicolson scheme in time and high central difference scheme in space, and the obtained scheme preserves mass and energy conservation laws. However, the obtained discrete system is nonlinear system, and it takes too much time in the numerical simulation for two dimension case. Then, we show a equivalent form of two dimension fractional KGS system by introducing some new auxiliary variables. The new system is discretized by the scalar auxiliary variable scheme, and a linear discrete system is obtained, which can preserve energy conservation law.

    Now, we consider boundary condition

    u(x,y,t)=0, (x,y)R2/Ω; ϕ(x,y,t)=0, (x,y)Ω, Ω=(xL,xR)×(yL,yR).

    Let

    hx=(xRxL)/M, hy=(yRyL)/M, τ=T/N,

    where M, N be positive integers. Then,

    xj=xL+jhx, yk=yL+khy, tn=nτ.

    Denote

    Ωhx,hy={(xj,yk)|1jM1, 1kM1}, Ωτ={tn|1nN1},ˉΩhx,hy={(xj,yk)|0jM, 0kM}, ˉΩτ={tn|0nN}.

    Then, the grid function can be defined by Unj,ku(xj,yk,tn), Φnj,kϕ(xj,yk,tn), where unj,k=u(xj,yk,tn), ϕnj,k=ϕ(xj,yk,tn). For any grid u={uj,k}, ϕ={ϕj,k}, we can define

    u,v=hxhyM1j=1M1k=1uj,k¯vj,k, u2=u,u.

    Lemma 6. [26,27] Suppose that uL1(R2) and

    uL4+α(R2):={u|+(1+|ξ|)4+α|ˆu(ξ)|dξ<}.

    Then, for a fixed h=hx=hy, we can obtain high order scheme

    43Δαhu(x,y)13Δα2hu(x,y)=Δα2u(x,y)+O(h4), (3.1)

    where

    Δαhu(x,y)=h2αk=l=g(α)k,lu(xkh,ylh),

    and g(α)k,l are Fourier expansion coefficients of generation function

    ρ(x,y)=[4sin2(x2)+4sin2(y2)]α2,

    which can be calculated as

    g(α)k,l=1(2π)2[π,π]2ρ(x,y)ei(kx+ly)dxdy.

    Lemma 7. Let

    ˜Δ2hx,hyΦ(xj,yk)=˜Δ2hxΦ(xj,yk)+˜Δ2hyΦ(xj,yk),

    where

    ˜Δ2hxΦ(xj,yk)={76Δ2hΦ(x1,yk)n+12112Δ2hΦ(x2,yk)n+12,j=1,43Δ2hΦ(xj,yk)n+1213Δ22hΦ(xj,yk)n+12,2<j<M2,76Δ2hΦ(xM1,yk)n+12112Δ2hΦ(xM2,yk)n+12,j=M1,
    ˜Δ2hyΦ(xj,yk)={76Δ2hΦ(xj,y1)n+12112Δ2hΦ(xj,y2)n+12,k=1,43Δ2hΦ(xj,yk)n+1213Δ22hΦ(xj,yk)n+12,2<k<M2,76Δ2hΦ(xj,yM1)n+12112Δ2hΦ(xj,yM2)n+12,k=M1.

    Then, we obtain

    ΔΦ(xj,yk)=˜Δ2hx,hyΦ(xj,yk)+O(h4x+h4y).

    Applying Crank-Nicolson scheme in time and higher order difference scheme in space to two dimension fractional KGS system (1.3)–(1.5), we can obtain the following numerical scheme

    i(Unj,k)tνˆΔαhUn+12j,k=12[f(|Un+1j,k|2,Φnj,k)f(|Unj,k|2,Φnj,k)|Un+1j,k|2|Unj,k|2+f(|Un+1j,k|2,Φn+1j,k)f(|Unj,k|2,Φn+1j,k)|Un+1j,k|2|Unj,k|2]Un+12j,k, (3.2)
    j,k=1,2,,M1,n=0,1,,(Φnj,k)t=Vn+12j,k,j,k=1,2,,M1,n=0,1,, (3.3)
    (Vnj,k)t˜Δ2hx,hyΦn+12j,k+Φn+12j,k=12[f(|Unj,k|2,Φn+1j,k)f(|Unj,k|2,Φnj,k)Φn+1j,kΦnj,k+f(|Un+1j,k|2,Φn+1j,k)f(|Un+1j,k|2,Φnj,k)Φn+1j,kΦnj,k], (3.4)
    j,k=1,,M1,n=0,1,,U0j,k=u0(xj,yk),Φ0j,k=ϕ0(xj,yk),V0j,k=ϕ1(xj,yk),j,k=1,2,,M1, (3.5)
    Un0,k=UnM,k,Φn0,k=ΦnM,k, Unj,0=Unj,M,Φnj,M=Φnj,M, n=0,1,, (3.6)

    where

    ˆΔαhUn+12j,k=43ΔαhUn+12j,k13Δα2hUn+12j,k.

    Let

    ˜Δ2hx,hyΦj,k=(DxI+IDy)Φ, 43ΔαhUj,k13Δα2hUj,k=AU.

    Then, we can obtain

    iUtνAUn+12=12[f(|Un+1|2,Φn)f(|Un|2,Φn)|Un+1|2|Un|2+f(|Un+1|2,Φn+1)f(|Un|2,Φn+1)|Un+1|2|Un|2]Un+12,n=0,1,, (3.7)
    Φnt=Vn+12,n=0,1,, (3.8)
    Vnt(DxI+IDy)Φn+12+Φn+12=12[f(|Un|2,Φn+1)f(|Un|2,Φn)Φn+1Φn+f(|Un+1|2,Φn+1)f(|Un+1|2,Φn)Φn+1Φn],n=0,1,, (3.9)

    where, represents kronecker product of matrices, Dx, Dy are differential matrix of x, y direction, and I is identity matrix.

    Lemma 8. [19] Let ARn×n have eigenvalues {λj}nj=1, and let BRm×m have eigenvalues{μj}mj=1. Then the m×n eigenvalues of AB are

    λ1μ1,,λ1μm,λ2μ1,,λ2μm,,λnμ1,,λnμm.

    Lemma 9. [19] For matrixs A and B, (AB)T=ATBT.

    Let Λh=DxI+IDy. Then, it follows from Lemmas 8 and 9 that the matrix Λh and A are symmetric positive matrixs. Moreover, there exists fractional symmetric positive difference quotient operator denoted by Λ12h and A12 such that

    Λhv,v=Λ12hv,Λ12hv, Av,v=A12v,A12v.

    It follows from Lemmas 8 and 9 that the numerical scheme (3.7)–(3.9) preserves mass and energy conservation laws.

    For f(|u|2,ϕ)=|u|2ϕ, it follows from Eqs (3.2)–(3.6) that we can obtain the following numerical scheme

    i(Unj,k)tνˆΔαhUn+12j,k=Un+12j,kΦn+12j,k,j,k=1,2,,M1,n=0,1,, (3.10)
    (Φnj,k)t=Vn+12j,k,j,k=1,2,,M1,n=0,1,, (3.11)
    (Vnj,k)t˜Δ2hx,hyΦn+12j,k+Φn+12j,k=12(|Unj,k|2+|Un+1j,k|2),j,k=1,,M1,n=0,1,, (3.12)
    U0j,k=u0(xj,yk),Φ0j,k=ϕ0(xj,yk),V0j,k=ϕ1(xj,yk),j,k=1,2,,M1, (3.13)
    Un0,k=UnM,k,Φn0,k=ΦnM,k, Unj,0=Unj,M,Φnj,M=Φnj,M, n=0,1,. (3.14)

    We can prove that the resulting scheme (3.10)–(3.14) can preserve the mass and energy conservation laws. However, the above numerical scheme is nonlinear scheme. In order to construct linear scheme, we consider also the following finite difference scheme for fractional KGS system (1.3)–(1.5) with f(|u|2,ϕ)=|u|2ϕ

    i(Unj,k)tνˆΔαhUn+12j,k=Un+12j,kΦn+12j,k,j,k=1,2,,M1,n=0,1,, (3.15)
    (Φnj,k)ˆt=˜Vnj,k,j,k=1,2,,M1,n=0,1,, (3.16)
    (Vnj,k)ˆt˜Δ2hx,hy˜Φnj,k+˜Φnj,k=|Unj,k|2,j,k=2,,M2,n=0,1,, (3.17)
    U0j,k=u0(xj,yk),Φ0j,k=ϕ0(xj,yk),V0j,k=ϕ1(xj,yk),j,k=1,2,,M1, (3.18)
    Un0,k=UnM,k,Φn0,k=ΦnM,k, Unj,0=Unj,M,Φnj,M=Φnj,M, n=0,1,. (3.19)

    We can also prove that the resulting scheme (3.15)–(3.19) can preserve the mass and energy conservation laws. However, the scheme is only conservation for f(|u|2,ϕ)=|u|2ϕ. For f(|u|2,ϕ)|u|2ϕ, we use scalar auxiliary variable scheme to obtain linearly implicit scheme. Let q=f(|u|2,ϕ),1+C0. Then,

    iutν(Δα2)u=uf(|u|2,ϕ)|u|2qf(|u|2,ϕ),1+C0, (3.20)
    ϕt=v, (3.21)
    vtΔϕ+ϕ=f(|u|2,ϕ)ϕqf(|u|2,ϕ),1+C0, (3.22)
    qt=f(|u|2,ϕ)|u|2,2Re(uut)+f(|u|2,ϕ)ϕ,ϕt2f(|u|2,ϕ),1+C0. (3.23)

    Lemma 10. The fractional KGS system (3.20)(3.23) has the following energy conserved laws

    E(t)=+v2+|ϕ|2+ϕ2+2|(Δ)α4u|2+q2dxdy=E(0).

    Applying scalar auxiliary variable scheme in time and higher order difference scheme in space to the fractional KGS system (1.3)–(1.5), we can obtain the following numerical scheme

    i(Unj,k)tνˆΔαhUn+12j,k=~Uj,kn+12f(|~Uj,kn+12|2,~Φj,kn+12)|~Uj,kn+12|2Qn+12j,kf(|~Uj,kn+12|2,~Φj,kn+12),1+C0, (3.24)
    j,k=1,2,,M1,n=0,1,,(Φnj,k)t=Vn+12j,k,j=1,2,,M1,n=0,1,, (3.25)
    (Vnj,k)t˜Δ2hx,hyΦn+12j,k+Φn+12j,k=f(|~Uj,kn+12|2,~Φj,kn+12)~Φj,kn+12Qn+12j,kf(|~Uj,kn+12|2,~Φj,kn+12),1+C0, (3.26)
    j,k=1,2,,M1,n=0,1,,(Qnj,k)t=f(|~Uj,kn+12|2,~Φj,kn+12)|~Un+12j,kn+12|2,2Re(~Uj,kn+12(Unj,k)t)+f(|~Uj,k|2,~Φijn+12)~Φj,kn+12,(Φnj,k)t2f(|~Uj,kn+12|2,~Φj,kn+12),1+C0,j,k=1,,M1,n=0,1,, (3.27)

    where ~Uj,kn+12=3Unj,kUn1j,k2,~Φj,kn+12=3Φnj,kΦn1j,k2.

    Theorem 6. The scheme (3.24)(3.27) is conservative in the sense

    Energy: En=En1=E0,

    where

    En=43ΔαhUn+1213Δα2hUn+12+43Δ2hΦn+1213Δ22hΦn+12+h6(|Δ2hΦn+112|2+|Δ2hΦn+1M12|2)+Φn+12+Vn+12Qn+12.

    Proof: Multiplying 2h(Un+1j,kUnj,k)/τ to Eq (3.24) and summing it up for 1j,kM1 can yield

    iUntνˆΔαhUn+12,2Un+12t=˜Un+12f(|˜Un+12|2,˜Φn+12)|˜Un+12|2Qn+12f(|˜Un+12|2,˜Φn+12),1+C0,2Un+12t. (3.28)

    Taking the real part of Eq (3.28) yields

    1τ(43ΔαhUn+1213Δα2hUn+1243ΔαhUn2+13Δα2hUn2)=Re˜Un+12f(|˜Un+12|2,˜Φn+12)|˜Un+12|2Qn+12f(|˜Un+12|2,˜Φn+12),1+C0,2Un+12t. (3.29)

    Multiplying 2h(Vn+1jVnj)/τ to Eq (3.25) and summing it up for 1j,kM1 can yield

    Φnt,2Vnt=Vn+12,2Vnt. (3.30)

    Multiplying 2h(Φn+1jΦnj)/τ to Eq (3.26) and summing it up for 1j,kM1 can yield

    Vnt˜Δ2hx,hyΦn+12+Φn+12,2Φnt=f(|˜Un+12|2,˜Φn+12)˜Φn+12Qn+12f(|˜Un+12|2,˜Φn+12),1+C0,2Φnt. (3.31)

    It is easy to check that

    Φn+12,Φnt=1τ(Φn+12Φn2).

    Noting that equation Φnt,Vnt=Vnt,Φnt, we obtain

    1τ[(43Δ2hΦn+1213Δ22hΦn+1243Δ2hΦn2+13Δ22hΦn2)+h6(|Δ2hΦn+112|2|Δ2hΦn12|2)+(|Δ2hΦn+1M12|2|Δ2hΦnM12|2)]+1τ(Φn+12Φn2)+1τ(Vn+12Vn2)=f(|˜Un+12|2,˜Φn+12)˜Φn+12Qn+12f(|˜Un+12|2,˜Φn+12),1+C0,Φnt. (3.32)

    Multiplying 2hQn+12j,k to Eq (3.27) and summing it up for 1j,kM1 can yield

    Qnt,2Qn+12=f(|˜Un+12|2,˜Φn+12)|~Un+12n+12|2,2Re(˜Un+12Unt)+f(|˜U|2,˜Φn+12)˜Φn+12,Φnt2f(|˜Un+12|2,˜Φn+12),1+C0,2Qn+12. (3.33)

    Combining Eqs (3.29), (3.32), (3.33), we obtain

    43ΔαhUn+1213Δα2hUn+12+43Δ2hΦn+1213Δ22hΦn+12+h6(|Δ2hΦn+112|2+|Δ2hΦn+1M12|2)+Φn+12+Vn+12Qn+12=43ΔαhUn213Δα2hUn2+43Δ2hΦn213Δ22hΦn2+h6(|Δ2hΦn12|2+|Δ2hΦnM12|2)+Φn2+Vn2Qn2.

    When n=0,1,2,,n, we can obtain En=En1=E0.

    In above subsection, we construct some structure-preserving schemes to solve two dimension fractional KGS equations. In similar method of one dimension case, we can obtain the iterative algorithms of numerical scheme (3.2)–(3.6) and numerical scheme (3.10)–(3.14). Now, we consider the iterative algorithm of numerical scheme (3.15)–(3.19) and numerical scheme (3.24)–(3.27). Let U=[U11,U12,,UM1M1], V=[V11,V12,,VM1M1], Φ=[Φ11,Φ12,,ΦM1M1]. Then, we rewrite the numerical scheme (3.15)–(3.19) and numerical scheme (3.24)–(3.27) as the following vector form

    iUn+1UnτνAUn+1+Un2=Un+1+Un2Φn+1+Φn2, (3.34)
    Φn+1Φn12τ=Vn+1+Vn12, (3.35)
    Vn+1Vn12τBΦn+1+Φn12=|Un|2, (3.36)

    and

    iUntνAUn+12=˜Un+12f(|˜Un+12|2,˜Φn+12)|˜Un+12|2Qn+12f(|˜Un+12|2,˜Φn+12),1+C0, (3.37)
    Φnt=Vn+12, (3.38)
    VntBΦn+12+Φn+12=f(|˜Un+12|2,˜Φn+12)˜Φn+12Qn+12f(|˜Un+12|2,˜Φn+12),1+C0, (3.39)
    Qnt=f(|˜Un+12|2,˜Φn+12)|~Un+12n+12|2,2Re(˜Un+12(Un)t)+f(|˜U|2,˜Φn+12)˜Φn+12,(Φn)t2f(|˜Un+12|2,˜Φn+12),1+C0, (3.40)

    where, matrice B represents differential matrix of two dimension Laplacian operator.

    Consider numerical scheme (3.34)–(3.36), if (Un, Φn, Vn), n=0,1,2, are known, then Φn+1, Vn+1 of numerical scheme (3.34)–(3.36) is solve by the following linear equations

    Φn+1τVn+1=Φn1+τVn1,Vn+1τBΦn+1=τBΦn1+Vn1+2τ|Un|2

    Then, we can obtain Un+1 of numerical scheme (3.34)–(3.36) by solve linear equations

    iUn+1ντ2AUn+1+τ4Un+1(Φn+1+Φn)=iUn+ντ2AUnτ4Un(Φn+1+Φn).

    Consider numerical scheme (3.37)–(3.40), the numerical solution Un+1, Φn+1, Vn+1 of numerical scheme (3.37)–(3.40) is solve by the following linear equations

    iUn+1ντ2AUn+1τ2˜Un+12f(|˜Un+12|2,˜Φn+12)|˜Un+12|2Qn+1f(|˜Un+12|2,˜Φn+12),1+C0=iUn+ντ2AUnτ2˜Un+12f(|˜Un+12|2,˜Φn+12)|˜Un+12|2Qnf(|˜Un+12|2,˜Φn+12),1+C0,Φn+1τVn+1=ΦnτVn,Vn+1τ2BΦn+1+τ2Φn+1τ2f(|˜Un+12|2,˜Φn+12)˜Φn+12Qn+1f(|˜Un+12|2,˜Φn+12),1+C0=Vn+τ2BΦnτ2Φnτ2f(|˜Un+12|2,˜Φn+12)˜Φn+12Qnf(|˜Un+12|2,˜Φn+12),1+C0,Qn+1f(|˜Un+12|2,˜Φn+12)|~Un+12n+12|2,2Re(˜Un+12(Un)t)+f(|˜U|2,˜Φn+12)˜Φn+12,(Φn+1)2f(|˜Un+12|2,˜Φn+12),1+C0=Qn+f(|˜Un+12|2,˜Φn+12)|~Un+12n+12|2,2Re(˜Un+12(Un)t)+f(|˜U|2,˜Φn+12)˜Φn+12,(Φn)2f(|˜Un+12|2,˜Φn+12),1+C0.

    In above sections, we study some numerical schemes to solve following one dimension and two dimension fractional KGS systems:

    iutν(Δ)α2u=uf(|u|2,ϕ)|u|2, (4.1)
    ϕttΔϕ+ϕ=f(|u|2,ϕ)ϕ, (4.2)

    and

    iutν(Δ)α2u=uϕ, (4.3)
    ϕttΔϕ+ϕ=|u|2. (4.4)

    Recently, some structure-preserving schemes such as linearly implicit conservative scheme, symplectic scheme and multi-symplectic scheme have been designed and investigated for solving classical and fractional KGS system. However, in these works main system (4.3), (4.4) have been considered. As far as we know, there exist few studies on system (4.1), (4.2). In this paper, we consider not noly structure-preserving scheme of fractional KGS system (4.3), (4.4) but also fractional KGS system (4.1), (4.2). For two dimension case, we show linearly implicit conservative scheme (3.15)–(3.19) and fully implicit conservative scheme (3.10)–(3.14) to solve fractional KGS system (4.3), (4.4), and the fully implicit conservative scheme (3.10)–(3.14) is can obtain numerical result by above iterative algorithm. Moreover, we show also linearly implicit conservative scheme (3.24)–(3.27) and fully implicit conservative scheme (3.2)–(3.6) to solve fractional KGS system (4.1), (4.2). For one dimension case, we only give a fully implicit conservative scheme (2.1)–(2.5), in fact, linearly implicit(3.15)–(3.19) and (3.24)–(3.27) are still applicable to one dimensional case.

    In this section, we give some numerical experiments to show the efficiency of the structure-preserving schemes. The first numerical example shows the numerical errors and convergence rates of the structure-preserving scheme, and check conservation property of the schemes. The second numerical example shows the numerical result for the general fractional KGS system. The third numerical example shows the numerical result for two dimension fractional KGS system.

    When α=2, the KGS system (4.3), (4.4) has the following solitary wave solutions [5]

    u(x,t,v)=3241v2sech2121v2(xvtx0)exp(i(vx+1v2+v42(1v2)t)),ϕ(x,t,v)=34(1v2)sech2121v2(xvtx0),

    where v is the propagating velocity of the wave and x0 is the initial phase. We consider initial-value (v=0.8,x0=10)

    u0=3241v2sech2121v2(xx0)exp(i(vx)),ϕ0=34(1v2)sech2121v2(xx0).

    In this example, we text errors, convergence orders and conservation of mass and energy of one dimension KGS system by numerical scheme (2.1)–(2.5). First, we show errors and convergence orders of numerical scheme (2.1)–(2.5) at time t=1. For α2, the numerical exact solutions are obtained by a very fine mesh and a small time step. Then, we fix the space mesh h=0.00001 and time step τ=0.00001 to test time convergence orders and space convergence orders by numerical scheme (2.1)–(2.5). The Figures 12 show time convergence orders and space convergence orders for different α, and it is found that the scheme is of order 2 in time, order 4 in space. From Figures 12, we can draw the observations: the approximate solution converge to the exact solution at the rate O(τ2+h4), and consistent with the theoretical estimates of Theorems 5–6. Second, we examine the conservation of mass and energy with x[20,20],t[0,100], τ=0.01,h=0.1. Figure 3 shows relative residuals on the mass and energy errors for different values of α by numerical scheme (2.1)–(2.5). It is found that the numerical scheme preserves mass and energy conservation very well although energy varies with α.

    Figure 1.  Errors and convergence orders in time for h=0.00001, t=1.
    Figure 2.  Errors and convergence orders in time for τ=0.00001, t=1.
    Figure 3.  Conservation of mass (left) energy (right) of fractional KGS system by numerical scheme (2.1)–(2.5).

    Consider following one dimension fractional KGS system

    itu12(Δ)α2u=uϕ+γ|u|2u, (4.5)
    ttϕ+ϕxx+ϕ=|u|2, (4.6)
    u0=3241v2sech2121v2(xx0)exp(i(vx)), (4.7)
    ϕ0=34(1v2)sech2121v2(xx0). (4.8)

    In this example, we simulate solitary wave and collisions of two solitary waves of fractional KGS system (4.5)–(4.8) by numerical scheme (2.1)–(2.5). First, we simulate solitary wave of numerical solutions for different orders α and difference parameter γ by the numerical scheme (2.1)–(2.5).

    Figures 46 display solitary wave of the numerical solutions for difference value of γ=0.8, 1.5, 2 and same value of α=2. It is found that the parameter γ affects the propagation velocity of the solitary wave, and larger γ, the propagation of the soliton got slower.

    Figure 4.  The wave forms of the numerical solution, γ=0.8,α=2.
    Figure 5.  The wave forms of the numerical solution, γ=1.5,α=2.
    Figure 6.  The wave forms of the numerical solution, γ=2,α=2.

    Figures 79 display solitary wave of the numerical solutions for difference value of α=2, 1.8, 1.5 and same value of γ=1. It is found that the parameter α affects also the propagation velocity of the solitary wave, and smaller α, the propagation of the soliton got slower.

    Figure 7.  The wave forms of the numerical solution, γ=1,α=2.
    Figure 8.  The wave forms of the numerical solution, γ=1,α=1.8.
    Figure 9.  The wave forms of the numerical solution, γ=1,α=1.5.

    Second, we consider collisions of two solitary waves with x[20,20], t[0,30], τ=0.01,h=0.1, and the initial data are chosen as (p1=10,p2=10,v1=0.8,v2=0.8)

    u0=u(xp1,0,v1)+u(xp2,0,v2),ϕ0=ϕ(xp1,0,v1)+ϕ(xp2,0,v2).

    Figures 1012 display solitary wave of the numerical solutions for difference value of γ=0.8, 1.5, 2 and same value of α=2. Figures 1315 display solitary wave of the numerical solutions for difference value of α=2, 1.8, 1.5 and same value of γ=1. It is found that the parameters α, γ affect also the propagation velocity of the solitary wave. When smaller α and larger γ, the propagation of the soliton got slower, the soliton changes faster and even a high oscillation appears.

    Figure 10.  The wave forms of the numerical solution, γ=0.8,α=2.
    Figure 11.  The wave forms of the numerical solution, γ=1.5,α=2.
    Figure 12.  The wave forms of the numerical solution, γ=2,α=2.
    Figure 13.  The wave forms of the numerical solution, γ=1,α=2.
    Figure 14.  The wave forms of the numerical solution, γ=1,α=1.8.
    Figure 15.  The wave forms of the numerical solution, γ=1,α=1.5.

    In the two example, we show some numerical results of fractional KGS system (4.5)–(4.8) for solitary wave case and collisions of two solitary waves case. In the process of time evolution, the solitary wave moves towards the boundary gradually, and produces some small waves around the solitary wave. In this paper, we only consider spatial range [20,20]. If we want to get a better numerical result, we can expand the space a bit, but it will take more calculation time.

    Consider two dimension fractional KGS system

    itu12(Δ)α2u=uϕ, (4.9)
    ttϕ+Δϕ+ϕ=|u|2, (4.10)
    u(x,y,0)=2ex2+2y2+e(x2+2y2)ei5/cosh(4x2+y2), (4.11)
    ϕ(x,y,0)=e(x2+y2),ϕt(x,0)=e(x2+y2)/2. (4.12)

    In the example, we simulate solitary wave numerical solutions for different orders α by the scalar auxiliary variable scheme (3.24)–(3.27). Figures 1618 show the surface plots of the nucleon density |u|2 and meson field ϕ for different time t=3,5,8 and same value of α=2, respectively.

    Figure 16.  Surface plots of the nucleon density |u|2 (left column) and meson field ϕ (right column).
    Figure 17.  Surface plots of the nucleon density |u|2 (left column) and meson field ϕ (right column).
    Figure 18.  Surface plots of the nucleon density |u|2 (left column) and meson field ϕ (right column).

    Figures 1921 show the surface plots of the nucleon density |u|2 and meson field ϕ for different time t=3,5,8 and same value of α=1.8, respectively. From Figures 1621, we fine that the meson field change periodically, and the order α affects the shape of nucleon field. They also show that α affects the propagation velocity of the solitary wave.

    Figure 19.  Surface plots of the nucleon density |u|2 (left column) and meson field ϕ (right column).
    Figure 20.  Surface plots of the nucleon density |u|2 (left column) and meson field ϕ (right column).
    Figure 21.  Surface plots of the nucleon density |u|2 (left column) and meson field ϕ (right column).

    In the paper, we study structure-preserving scheme to solve one dimension and two dimension space fractional KGS equations. First, we use the high central differences scheme in space and Crank-Nicolson scheme in time to discrete one dimension fractional KGS equations, which preserve mass and energy conservation laws of the fractional system. Then, we show that the arising scheme is uniquely solvable and approximate solutions converge to the exact solution at the rate O(τ2+h4). Second, we give the high central differences scheme in space, Crank-Nicolson scheme and scalar auxiliary variable scheme in time for two dimension fractional KGS equations, which preserve one or more analytical properties of the fractional system. Finally, the numerical experiments including some one dimensional and two dimensional fractional KGS systems are given to verify the correctness of theoretical results.

    The authors declare there is no conflict of interest.

    This work is supported by National Natural Science Foundation of China (No.12161070).



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