High Accuracy Split-Step Finite Difference Method for Schrödinger-KdV Equations

Transcription

1 Commun. Theor. Phys Vo. 70, No. 4, October, 208 High Accuracy Spit-Step Finite Difference Method for Schrödinger-KdV Equations Feng Liao 廖锋, and Lu-Ming Zhang 张鲁明 2 Schoo of Mathematics and Statistics, Changshu Institute of Technoogy, Changshu 25500, China 2 Coege of Science, Naning University of Aeronautics and Astronautics, Naning 206, China Received March, 208; revised manuscript received Juy 6, 208 Abstract In this artice, two spit-step finite difference methods for Schrödinger-KdV equations are formuated and investigated. The main features of our methods are based on: i The appications of spit-step technique for Schrödingerike equation in time. ii The utiizations of high-order finite difference method for KdV-ike equation in spatia discretization. iii Our methods are of spectra-ike accuracy in space and can be reaized by fast Fourier transform efficienty. Numerica experiments are conducted to iustrate the efficiency and accuracy of our numerica methods. DOI: 0.088/ /70/4/43 Key words: spit-step method, Schrödinger-KdV equations, finite difference method, fast Fourier transform Introduction The noninear Schrödinger-KdV equations [ 2] iϵu t u xx = 2 uv, v t + 2 v xxx + 2 v2 + u 2 x = 0, can be used to mode the noninear dynamics behavior of one-dimensiona Langmuir and ion-asoustic waves in a system of coordinates moving at the ion-acoustic speed. Here ϵ is a positive constant, u is compex function describing eectric fied of Langmuir osciations whie v is rea function describing ow-frequency density perturbation. Many works have been concentrated on the numerica studies of this probem. Bai and Zhang [3] formuated a finite eement method FEM to study Schrödinger- KdV equations. Later, Bai [4] deveoped a spit-step quadratic B-spine finite eement method SSQBS-FEM for Schrödinger-KdV equations. Appert and Vacavik [5] soved the Schrödinger-KdV equations using a finite difference method FDM. Gobabai a Safdari-Vaighani [6] empoyed a meshess technique based on radia basis function RBF coocation method. Zhang et a. did some works concerning Schrödinger-KdV equations using average vector fied AVF method and muti-sympectic Fourier pseudospectra MSFP method. [7 8] Some other numerica methods for Schrödinger-KdV equations, such as variationa iteration method, decomposition method and homotopy perturbation method, readers are reffered to Refs. [9 ] and reference therein. The main purpose of this paper is to construct high accuracy spit-step finite difference SSFD method for Schrödinger-KdV equations. Spit-step or time-spitting method has evoved as a vauabe technique for the numerica approximation of partia differentia equations PDE. Wang [2] presented a time-spitting finite difference TSFD method for various versions of noninear Schrödinger equation. To improve the accuracy of TSFD, Dehghan and Taeei [3] constructed a compact time-spitting finite difference scheme, which was proved to be unconditionay stabe and preserve some invariant properties. Wang and Zhang [4] proposed an efficient spit-step compact finite difference method for the cubicquintic compex Ginzburg-Landau equations both in one dimension and in muti-dimensions. However, a of these methods require to sove tridiagona inear agebraic equations in impementation, and the computationa cost wi be increased aong with the increment of the spatia accuracy. Recenty, Wang et a. constructed a time-spitting compact finite difference method for Gross-Pitaevskii equation, which is reaized by discrete fast discrete Sine transform, and there is no need to sove inear agebraic equations. [5] Subsequenty, Wang [6] considered sixthorder compact time-spitting finite difference method for nonoca Gross-Pitaevskii equation, the method is of spectra-ike accuracy in space, and conserves the tota mass and energy of the system in the discretized eve. It shoud be noted that the methods in Refs. [5 6] are not fit for soving the PDE with odd-order partia derivatives. In this paper, we formuate a high accuracy and fast sover for Schrödinger-KdV equations based on discrete Fourier transform, which is of sixth or eighth-order accuracy in space and can be reaized by FFT efficienty. The ayout of the paper is as foows: In Sec. 2, we formuate a sixth-order spit-step finite difference SSFD- 6 method. Then we estabish an eighth-order spit-step Supported by the Nationa Natura Science Foundation of China under Grant No. 578 Corresponding author, E-mai: wawd3kwcom@63.com c 208 Chinese Physica Society and IOP Pubishing Ltd

2 44 Communications in Theoretica Physics Vo. 70 finite difference SSFD-8 method in Sec. 3. Numerica investigations of our numerica methods are conducted in Sec. 4, and some concusions are drawn in Sec Sixth-Order Spit-Step Finite Difference Method In this paper, we consider the genera forms of Schrödinger-KdV equations iu t + γu xx = ξuv, x, t > 0, 2 v t + αv xxx + fv x = ω u 2 x, x, t > 0, 3 with the initia vaue and periodic-boundary conditions of ux L, t = ux R, t, vx L, t = vx R, t, t 0, 4 ux, 0 = u 0 x, vx, 0 = v 0 x, x, 5 where = [x L, x R ], γ, ξ, α, ω are known constants, u 0 x and v 0 x are periodic functions with the period x R x L. It is easy to verify that probem 2 5 preserves the tota mass Qt = u 2 dx = Q0, 6 and the tota energy Et = γω u x 2 dx + α v x 2 dx ξ 2 F vdx + ω v u 2 dx = C, 7 where F v = v fsds. The proof detais of 6 and 7 0 are provided in Appendix A. 2. Sixth-Order Difference Approximation Formua Choose a mesh size h := x R x L /J with J an even positive integer, time step τ, and denote grid points with coordinates x, t n := x L + h, nτ for = 0,,..., J and n 0. Define { X J = span Φ x = e iµ x x L : x, µ = }, = J/2,..., J/2, x R x L Y J = {u = u 0, u,..., u J C J+ : u 0 = u J }. For any genera periodic function ux on and a vector u Y J, et P J : L 2 X J be the standard L 2 -proection operator onto X J, I J : C X J and I J : Y J X J be the trigonometric operator, i.e. with P J ux = I J ux = J 2/ J 2/ û = x R x L ũ = J J û Φ x, ũ Φ x, x, u Φ x. uxφ xdx, Obviousy, P J and I J are identica operators over X J. For any u, v Y J, the inner product and norm are defined as foows: J u, v = u v, u 2 = u, u, J δ x + u = h u + u 2. h Suppose that gx is an x R x L periodic function, then for the approximation of the first-order derivative g x x, we have the foowing formua, i.e. ag + g + ag + = β h g +2 g 2 + α h g + g + Oh 6, 8 where g = gx, g = g x x and a, α, β are undetermined parameters, which depend on the accuracy-order constraints. Base on Tayor s expansion, we have α 2α + 4β = 2a +, 3 + 8β 3 = a, α β 5 = a 2. 9 The inear equations 9 is unique sovabe, i.e. a = /3, α = 7/9, β = /36. Then we obtain the sixth-order difference approximation for the first-order partia derivative 3 g + g + 3 g + = 36h g +2 g h g + g. 0 Next, we approximate the third-order derivative f xxx x via the foowing formua, i.e. bg ag3 + g3 + ag bg3 +2 = β h 3 g +2 g 2 + α h 3 g + g + Oh 6. It foows from Tayor s expansion, we obtain α α + 2β = 0, 3 + 8β = 2a + 2b +, 3 α β 5 = a + 4b, α β 35 = a 2 + 4b 3, 2 which is unique sovabe, i.e. a = 4/9, b = /26, α = 40/2, β = 20/2. Thus, the sixth-order difference approximation for the third-order partia derivative is given as foows 26 g g3 + g g g3 +2

3 No. 4 Communications in Theoretica Physics 45 = 20 2h 3 g +2 g h 3 g + g. 3 Finay, we approximate the second-order derivative f xx x as foows ag 2 + g2 + ag 2 + = β h 2 g +2 2g + g 2 + α h 2 g + 2g + g + Oh 6. 4 From Tayor s expansion, we have 2a + = α + 4β, a = α 2 + 4β 3, a 2 = α β 45, 5 which is unique sovabe and a = 2/, α = 2/, β = 3/44. Then the sixth-order difference approximation for the second-order partia derivative is given as foows 2 g2 + g2 + 2 g2 + = 3 44h 2 g +2 2g + g h 2 g + 2g + g Spit-Step Finite Difference Method The discrete Fourier transform for {g } and its inverse are provided as foows, i.e. g = J g Φ x, g = J J/2 g Φ x. 7 Simiary, wa can define discrete Fourier transform for g g 2, and g 3 and their inverse. From Eqs. 0 and 7, we have, 3 J/2 x g e 2πi/J + = J/2 36h + 7 J/2 9h J/2 g e 2πi/J+2 g e 2πi/J+ x g e 2πi/J + 3 J/2 J/2 J/2 g e 2πi/J 2 g e 2πi/J x g e 2πi/J+, 8 which gives x g = 4i 9h i 4π 8h sin sin J + J 2 3 cos J + g x T 6 g. 9 Simiary, from Eqs. 3, 6, and 7, we obtain xxx g = 4π 40i sin 80i sin J 2h π J J cos J + g xxx T 6 g, h cos 4π 24 xx g = 2 J + h cos 2 J 4 cos J + g xx T 6 g, 2 where { x g }, { xx g }, and { xxx g } represent the Fourier transform of {g }, {g 2 }, and {g 3 }, respectivey. In the rest of this section, we formuate a spitstep finite difference SSFD method for probem 2 5. Firsty, we discrete KdV-ike equation 3 in tempora direction as foows v n+ v n + α 2 xxxv n+ + xxx v n + x fv n = ω x u n 2, 22 for = 0,,..., J. Acting the discrete Fourier transform on Eq. 22 and considering the orthogonaity of the Fourier basis functions, we obtain ṽ n+ ṽ n + α 2 xxxṽ n+ + xxx ṽ n n + x F = ω x Gn, 23 for = J/2, J/2 +,..., J/2, where { xxx ṽ n}, n { x F }, and { x Gn } represent the discrete Fourier transform of { xxx v n}, { xfv n}, and { x u n 2 }, respectivey. From Eqs. 9, 20 and 23, we obtain ṽ n+ ṽ n + α 2 xxx T 6 ṽ n+ + ṽ n + x T 6 F n ω G n = 0, 24 for n and = J/2, J/2 +,..., J/2, where { F n} and { G n } express the discrete Fourier transform of {fv n} and { un 2 }, respectivey. Secondy, we utiize spit-step method to discrete Schrödinger-ike equation 2 in time, we obtain iu t = ξ 2 uv, iu t + γu xx = 0, iu t = ξ 2 uv, t [t n, t n+ ], 25 Directy from the noninear subprobem in Eq. 25, we have i du u = ξ 2 vdt, t [t n, t n+ ]. 26 Integrating Eq. 26 from t n to t n+, and then approximating the integra on [t n, t n+ ] via trapezoida rue, we obtain ux, t n+ = ux, t n e 0.25iξτvx,t n+vx,t n+. 27 For the inear subprobem in Eq. 25, we discretize it in

4 46 Communications in Theoretica Physics Vo. 70 time as foows i u n + γ 2 xxu n+ + xx u n = τ un+ Foowed by discrete Fourier transform and Eq. 2, we have ũ n+ = 2 + iτγ xxt 6 ũ n 2 iτγ xx T From Eqs. 27 and 29, the sequentia subprobems 25 can be soved as foows: u = u n e 0.25iξτvn +vn+, 30 = 2 + iτγ xxt 6 ũ 2 iτγ xx T 6, 3 ũ u n+ = u e 0.25iξτvn +vn+, 32 where {ũ } and {ũ } express the discrete Fourier transform of {u } and {u }, respectivey. From above discussion, Eqs. 24 and comprise the detais of sixth-order spit-step finite difference SSFD-6 method. However, SSFD-6 is a three-eve scheme, which requires a two-eve scheme to cacuate u and v. In this artice, we compute u and v via the foowing two-eve noninear impicit scheme, i.e. ṽ s+ τ ṽ x T 6 + α 2 xxx T 6 ṽ s+ + ṽ 0 s F + F 0 s ω G + G 0 = 0, 33 u = u 0 e 0.25iξτv0 +vs+, 34 = 2 + iτγ xxt 6 ũ 2 iτγ xx T 6, 35 ũ u s+ for s 0, where { transform of {F s F s = u e 0.25iξτv0 +vs+, 36 s F } and { } and {G s s G } being the Fourier }, respectivey, with = fv s, G s = u s 2. Obviousy, above twoeve impicit scheme requires iteration computation, the iteration initia vaues are given by u 0 = u 0, v0 = v 0 and the iteration scheme continues unti foowing condition is satisfied, i.e. max 0 J vs+ v s 0 2. Above a, the detais of SSFD-6 are provided as foows: Compute {u 0 }J and {v0 }J via initia condition 5, and cacuate {ũ 0 }J/2 and {ṽ0 }J/2 via FFT; Initiaize {ũ 0 {ṽ 0 } J/2 : = {ṽ0 }J/2 s := 0; Cacuate {ṽ Compute {u } J/2 : = {ũ 0 }J/2 ; } J/2 } J via 33; and by virtu of and define WHILE max 0 J vs+ DO v s > 0 2 a. Evauate {ṽ s+ } J/2 via 24; b. Compute {ũ s+ and update s : = s + ; END WHILE {ṽ s+ } J Let {ũ }J/2 : = {ũs+ } J/2, {u }J {v s+ } J WHILE n < N DO : = {us+ and define n : = ; by virtu of } J/2, {ṽ }J/2 : = } J, {v }J : = c. Cacuate {ṽ n+ } J/2 by means of 23; d. Compute {u n+ } J via and update n := n +. END WHILE Theorem The discretizations Eqs for Schödinger-ike equation posses the foowing property: where Q n = u n 2. Q n+ = Q n, n 0, Proof Noticing the Parserva s identity J/2 thus from Eq. 3, we have J h ũ n 2 = J u n 2, J J u 2 = h Directy from Eqs. 30 and 32, we have J h J h J u n 2 = h u n+ u u 2, J 2 = h u From Eqs. 37 and 38, we can see that the concusion of this theorem hods. Remark Theorem impies that SSFD-6 method preserves the tota mass in discrete eve. We do not expect that SSFD-6 conserves the tota energy, but the energy can be discretized as E n = γω ξ δ+ x u n 2 + α 2 δ+ x v n 2 J J h F v n + ωh v n u n Theorem aso demonstrates that the compex component u n is convergent in the sense of L 2 -norm. Simiar to the methods which have been done in Ref. [7], we can obtain the L 2 -error estimates of u n, i.e. u, t n u n C 0 τ 2 + h 6,

5 No. 4 Communications in Theoretica Physics 47 where C 0 is a constant independent of h, τ. For simpicity of this paper, we omit the proof detais. Next, we wi prove the convergence of v n via mathematica induction argument method. Theorem 2 Let u n, v n X J be the numerica approximations of SSFD-6. If ux, t, vx, t, and f are sufficienty smooth, there exist two constants h 0 > 0 and C independent of τ or n and h, such for any h h 0 and τ = oh, vx, t n v n Cτ 2 + h 6, n 0. Proof See Appendix B. 3 Eighth-Order Spit-Step Finite Difference Method To construct eighth-order spit-step finite difference SSFD-8 method for Schrödinger-KdV equations, we provide the eighth-order finite difference formuas for the first, second and third-order derivatives as foows: 36 g +2 + g g + + g + g = 25 26h g +2 g h g + g, g g g2 + + g2 + g2 = h 2 g +2 2g + g h 2 g + 2g + g, 4 66 g g g3 + + g3 + g3 = 5 328h 3 g +3 g h 3 g +2 g h 3 g + g. 42 This together with Eq. 7, we obtain the reationship between x g, xx g, xxx g, and g as foows 25i 08h 8 55 x g = 4π cos J xx g = 393h cos 4π 2 J 4π sin J + 40i 27h sin cos J J + g x T 8 g, h cos 2 J 23 4π 79 cos J cos J + g xx T 8 xxx g = 5i 664h sin 6π 3 J + 60i 83h sin 4π 3 J 2545i 664h sin 3 J 4π 83 cos J Simiar to the anaysis in Sec. 2, the detais of SSFD-8 are provided as foows ṽ n+ ṽ n + α 2 xxx T 8 ṽ n+ + ṽ n cos J + xxx T 8 + x T 8 g, 44 g. 45 F n ω G n = 0, 46 u = u n e 0.25iξτvn +vn+, 47 = 2 + iτγ xxt 8 ũ 2 iτγ xx T 8, 48 ũ u n+ = u e 0.25iξτvn +vn+, 49 for n, and u, v can be cacuated iterativey as we have done in Eqs Comparing with SSFD-6, we can see that the computationa compexity of SSFD-8 is identica to SSFD-6, but SSFD-8 is more accurate than SSFD-6 in spatia direction. Thus the computationa compexity of our methods wi not increase aong with the increment of spacia accuracy. Based on this, we can design more higher accurate SSFD method, which can achieve spectra-ike accuracy in space when more higher-order finite difference method is investigated. 4 Numerica Resuts In this section, we wi provide some numerica exampes to test the performance of SSFD method for Schrödinger- KdV equations. Based on the works of Refs. [2, 8], we provide two kinds of soitary-wave soutions for probem 2 3 with fv = θv 2. Case If γmξ + 2δ + M 2 /4γ3γθ + αξ = 0, 3αξ γθ and 4αδ + M 2 /4γ Mγ, ux, t = ± 6 δ + M 2 ξ 4γ vx, t = 6 δ + M 2 ξ 4γ γθ + αξ sechµζtanhµζ e im/2γx+δt, γω sech 2 µζ, x R, t 0. 50

6 48 Communications in Theoretica Physics Vo. 70 Case 2 If 3αξ = γθ and 4αδ + M 2 /4γ Mγ, ux, t = 6α δ γ 2 + M 2 θω 4γ vx, t = 2 δ + M 2 ξ 4γ Mγ 4α δ + M 2 sechµζ e im/2γx+δt, 4γ sech 2 µζ, x R, t 0. 5 Here µ = /γδ + M 2 /4γ, ζ = x Mt, and M, δ are free parameters. In this paper, we provide foowing two types of soitary-wave soutions reated to Case and Case 2, respectivey. Exampe Let γ = 3/2, ξ = /2, α = /2, θ = /2, ω = /2, and M = 9/20, δ = 27/800. Exampe 2 Let γ =, ξ =, α = /3, θ =, ω = /2, and M =, δ = /4. We cacuate the L 2 and L norm errors using the formuas J J Eh, τ = h u n ux, t n 2 + h v n vx, t n 2 E u h, τ + E v h, τ, E h, τ = max 0 J un ux, t n + max 0 J vn vx, t n E h, u τ + E h, v τ. Tabe The errors and convergence ratio of SSFD-6 for Exampe at t = with = [ 28, 28]. Eh,τ E h τ Eh, τ og 2 E h, τ og h,τ 2 Eh/2,τ/8 E h/2,τ/8 / /2 / /4 / /8 / Tabe 2 The errors and convergence ratio of SSFD-6 for Exampe 2 at t = with = [ 28, 28]. Eh,τ E h τ Eh, τ og 2 E h, τ og h,τ 2 Eh/2,τ/8 E h/2,τ/8 / /2 / /4 / /8 / Fig. Numerica soutions of Exampe for t [0, 20] with = [ 28, 28]. The errors and convergence ratio of SSFD-6 are examined in Tabes 2, which demonstrate that SSFD-6 has sixthorder accuracy in space. Figures and 2 simuate the numerica soutions of SSFD-6 for Exampe and Exampe 2, respectivey, with h = /2 and τ = /80.

7 No. 4 Communications in Theoretica Physics 49 Fig. 2 Numerica soutions of Exampe 2 for t [0, 20] with = [ 28, 28]. Fig. 3 The discretization errors of the conservative quantities for Exampe with h = /2, τ = /80 and = [ 28, 28]. Fig. 4 The discretization errors of the conservative quantities for Exampe 2 with h = /2, τ = /80 and = [ 28, 28]. From Tabe 3, we can see that SSFD-8 is of eighth-order accuracy in space. We have compared the accuracy of SSFD-6 and SSFD-8 in Tabe 4, which indicates that SSFD-8 is more accurate than SSFD-6. Tabe 3 The errors and convergence ratio of SSFD-8 for Exampe 2 at t = with = [ 28, 28]. Eh,τ E h τ Eh, τ og 2 E h, τ og h,τ 2 Eh/2,τ/6 / E h/2,τ/6 /2 / /4 / /8 /

8 420 Communications in Theoretica Physics Vo. 70 Fig. 5 Numerica soutions of genera noninearity for t [0, 20] with h = /2, τ = /80 and = [ 28, 28]. Fig. 6 The discretization errors of the conservative quantities for genera noninearity with h = /2, τ = /80 and = [ 28, 28]. Tabe 4 Comparison of L 2 and L error norms for numerica soutions of Exampe at t = with = [ 28, 28]. SSFD-6 SSFD-8 h, τ Eh, τ E h, τ Eh, τ E h, τ, / /2, / /4, / /8, / To vaidate the conservation properties, we have computed the tota mass and energy of Exampe and Exampe 2, the discretization errors of the conservative quantities are potted in Fig. 3 and Fig. 4, which demonstrate that SSFD-6 preserves the tota mass and energy very we. A comparative study has been conducted with some existing methods and the resuts are reported in Tabe 5. We choose quadratic spine functions as basis functions of FEM [3] and SSQBS- FEM [4] for spacia discretization. From Tabe 5, we can see that SSFD-6 and SSFD-8 are more efficient and accurate than other three methods. As can be seen from Tabe 5 that SSFD-8 is more accurate than SSFD-6, but the compexity of SSFD-8 is identica to SSFD-6. It shoud be pointed that the standard fourth order Runge-Kutta method is used to sove the continuous time system of FEM, [3] hence FEM [3] is expected to spent more CPU time than SSQBS-FEM. [4] Since the coefficient matrix of FDM [5] is time-varying when we evauate the compex component u n, hence FDM [5] requires more computationa time than SSFD-6 and SSFD-8.

9 No. 4 Communications in Theoretica Physics 42 Tabe 5 Comparison of L error norms for numerica soutions of Exampe at t = 0. with h =, τ = 0.00 and = [ 64, 64]. SSFD-6 SSFD-8 FEM [3] SSQBS-FEM [4] FDM [5] E u h, τ E v h, τ E h, τ CPU time 0.29/s 0.28/s.504/s 0.488/s 0.372/s To examine SSFD-6 sti works for the genera noninearity, we take in the Schrödinger-KdV Eqs. 2 3 with fv = sinv. Choosing the parameters γ, ξ, α, ω and the initia data same as Exampe, the corresponding numerica resuts are shown in Fig. 5 and the discretization errors of the conservative quantities are potted in Fig Concusion In this paper, two spit-step finite difference methods are presented for soving Schrödinger-KdV numericay. The merit of our methods are of spectra-ike accuracy in space and can be reaized by fast Fourier transform. The computationa compexity of our methods wi not increase aong with the increment of spacia accuracy. Numerica resuts demonstrate the precision and conservation properties of our methods. Appendix A Proof Making the compex conugate inner product of Eq. 2 with u, then taking the imaginary part, we get d u 2 dt = 0, A dt which impies that Eq. 6 hods. Computing the the compex conugate inner product of Eq. 2 with u t, then taking the rea part, we have γ d 2 dt u x 2 dx = ξre vuū t dx. A2 Foowed by Re vuū t dx = d v u 2 dx v t u 2 dx, A3 2 dt 2 then we obtain γ d u x 2 dx + ξ d dt dt v u 2 dx = ξ v t u 2 dx. A4 Making the inner product of Eq. 3 with u 2, fv and v xx, respectivey, we obtain v t u 2 dx α u 2 x v xx dx + u 2 f x vdx = 0, A5 d F vdx α v xx f x vdx dt = ω u 2 f x vdx, A6 d v x 2 dx + f x vv xx dx 2 dt = ω u 2 x v xx dx. A7 It foows from A5 A7 that 7 is satisfied. Appendix B Denote the trigonometric interpoations of the numerica soutions as u n I x = I J u n x, v n I x = I J v n x, n 0, where u n, v n Y J. Define the error functions e n ux = ux, t n u n I x, e n v x = vx, t n v n I x, n 0. Denote the L 2 -proected soutions as u J x, t = P J ux, t n = v J x, t = P J vx, t n = J/2 J/2 û n Φ x, ˆv n Φ x. Acting L 2 -proected operator on Eq. 3, using Tayor s expansion and considering the orthogonaity of the basis functions, we have ˆv n+ ˆv n + α 2 x T 6 ˆv n+ ˆv n + x T 6 ˆF n ωĝn = ˆR n, A8 where R n = Rx, t n and Rx, t n C 0 τ 2 + h 6. Subtracting Eq. 24 form A8, we obtain the error equation: e v n+ e v n Mutipying A9 with e n+ v J/2 + α 2 xxx T 6 e v n+ + e v n + x T 6 n ˆF F n ωĝn G n = ˆR n. n + e v, summing up for = J/2,..., J/2 and taking the rea part, we have { e v n+ 2 J/2 e v n J/2 2 + Re xt 6 ˆF n F n ωĝn G n n+ ev + e n } v A9

10 422 Communications in Theoretica Physics Vo. 70 = Re { J/2 ˆR n ev n+ + e v n }. A0 Noticing xt 6 29, = J/2,..., J/2. A 6h With the hep of mathematica induction argument, we assume that e m u C 0 τ 2 +h 6, e m v C 0 τ 2 +h 6, m n. A2 This together with the inverse inequaity, triange inequaity and Soboev inequaity, we have u m C 0, v m C 0, m n. A3 Considering the conditions of Theorem 2 and A3, we obtain fv, t n fv n C 0 e v, u, t n 2 u n 2 C 0 e u, m n. A4 It foows from Parserva s identity and A0 A3, we have v 2 e n v 2 en+ C2 0 h en v 2 + e n u 2 + e n+ v 2 + e n u 2 +C0 R 2 n 2 + e n+ v 2 + e n u 2. A5 Directy from A5 and induction argument A2, we have e n+ v Cτ 2 + h 6, A6 where C is a constant independent of τ and h. This competes the proof. References [] B. L. Guo and F. X. Chen, Noninear Ana. Theor. Methods App [2] N. N. Rao, Pramana J. Phys [3] D. M. Bai and L. M. Zhang, Phys. Lett. A [4] D. M. Bai and L. M. Zhang, Commun. Noninear Sci. Numer. Simuat [5] K. Appert and J. Vacavik, Phys. Fuids [6] A. Gobabai and A. Safdari-Vaighani, Comput [7] H. Zhang, S. H. Song, X. D. Chen, and W. E. Zhou, Chin. Phys. B [8] H. Zhang, S. H. Song, W. E. Zhou, and X. D. Chen, Chin. Phys. B [9] M. A. Abdou and A. A. Soiman, Phys. D Noninear Phenomena [0] D. Kaya and M. E-Sayed, Phys. Lett. A [] S. Kucukarsan, Noninear Ana. Rea Word App [2] H. Q. Wang, App. Math. Comput [3] M. Dehghan and A. Taeei, Comput. Phys. Commun [4] S. S. Wang and L. M. Zhang, Copmut. Phys. Commun [5] H. Q. Wang, X. Ma, J. L. Lu, and W. Gao, App. Math. Comput [6] H. Q. Wang, Int. J. Comput. Math [7] N. Wang and C. M. Huang, Comput. Math. App [8] N. N. Rao, J. Phys. A: Math. Gen