Infinite Sequence Soliton-Like Exact Solutions of (2 + 1)-Dimensional Breaking Soliton Equation

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1 Commun. Theor. Phys. 55 (0) Vol. 55, No. 6, June 5, 0 Infinite Sequence Soliton-Like Exact Solutions of ( + )-Dimensional Breaking Soliton Equation Taogetusang,, Sirendaoerji, and LI Shu-Min (Ó ) College of Mathematical Science, Inner Mongolia Normal University, Huhhot 000, China College of Mathematical Science, Bao Tou Teachers College, Baotou 04030, China (Received November, 00; revised manuscript received January 0, 0) Abstract To seek new infinite sequence soliton-like exact solutions to nonlinear evolution equations (NEE(s)), by developing two characteristics of construction and mechanization on auxiliary equation method, the second kind of elliptic equation is highly studied and new type solutions and Bäcklund transformation are obtained. Then (+)-dimensional breaking soliton equation is chosen as an example and its infinite sequence soliton-like exact solutions are constructed with the help of symbolic computation system Mathematica, which include infinite sequence smooth soliton-like solutions of Jacobi elliptic type, infinite sequence compact soliton solutions of Jacobi elliptic type and infinite sequence peak soliton solutions of exponential function type and triangular function type. PACS numbers: 0.30.Jr Key words: the second kind of elliptic equation, Bäcklund transformation, nonlinear evolution equation, infinite sequence soliton-like exact solution Introduction Solutions of nonlinear evolution equations and its algebraic and geometric properties are very important to study in soliton theory. A new method which is called the auxiliary equation method have been applied to find many new solutions of the nonlinear evolution equations in Refs. [] [9]. The exact solutions of nonlinear evolution equation are constructed by taking traveling transformation ξ = kx + ωt to change nonlinear evolution equations into nonlinear ordinary differential equations and then applying Riccati equation or other auxiliary equations. [ 3] A new application of the auxiliary equation method is the construction of soliton-like solutions of nonlinear evolution equations, that is to say, taking transformations ξ = xp(y) + q(y, t), ξ = xp(y) + q(y, z, t), ξ = p(y, t) + q(x, t), and ξ = p(y) + q(x, z, t), etc. to exchange ( + )-dimensional and constant coefficients nonlinear nonlinear evolution equations into variable coefficients nonlinear ordinary differential equations, then the soliton-like exact solutions can be found by using the auxiliary equation method. [4 6] By use of Refs. [] [6] and summing up the characteristics of construction and mechanization of auxiliary equations, the Bäcklund transformation and the formula of nonlinear superposition of the solutions to several kinds of auxiliary equations are proposed to seek infinite sequence exact solutions of nonlinear evolution equations with the help of symbolic computation system Mathematica. [7 9] By highly studying conclusions in Ref. [9], new type solutions to the second kind of elliptic equation are developed to construct new infinite sequence exact solutions (+)-dimensional breaking soliton equation, [5 6] which include infinite sequence smooth soliton solutions of Jacobi elliptic function type, hyperbolic function type, and triangular function type, infinite sequence compact soliton solutions of Jacobi elliptic type and triangular function type, and infinite sequence peak soliton solutions of hyperbolic function type and triangular function type. u t + αu xxy + 4αuv x + 4αu x v = 0, u y = v x, () where α is an arbitrary constant. Bäcklund Transformation and New Type Solutions of Second of Elliptic Equation. Quasi Bäcklund Transformation of Second Kind of Elliptic Equation According to Ref. [9], several type solutions and Bäcklund transformation are obtained by highly studying the second kind of elliptic equation. The achievements are of significance to seek infinite sequence solutions of nonlinear evolution equations. ) = (ψ (ξ)) = aψ(ξ) + bψ (ξ) + cψ 3 (ξ), () If ψ n (ξ) is a solution of Eq. (), the following ψ n (ξ) is also the solution of Eq. () Supported by the Natural Natural Science Foundation of China under Grant No , the Science Research Foundation of Institution of Higher Education of Inner Mongolia Autonomous Region, China under Grant No. NJZZ0703 and the Natural Science Foundation of Inner Mongolia Autonomous Region, China under Grant No. 00MS0 Corresponding author, tgts@imnu.edu.cn c 0 Chinese Physical Society and IOP Publishing Ltd

2 950 Communications in Theoretical Physics Vol. 55 a + (b ± b 4ac)ψ n (ξ) ±b +, (n =,,...), (3) b 4ac ± cψ n (ξ) a[ b + b 4ac cψ n (ξ)] c[a + (b, (n =,,...), (4) b 4ac)ψ n (ξ)] ab ± a b (b 4ac) 4abcψ n (ξ) + [ b c c b (b 4ac)]ψ n (ξ) abc + b cψ n (ξ) + bc ψ n (ξ), (n =,,...), (5) a [ 3A 9ψ n (ξ) ] 3A (b b 3ac) + 3A cψ n (ξ) ± 3(b +, (n =,,...), (6) b 3ac)ψ n (ξ) a [ 3B ± 9ψ n (ξ)] 3B (b + b 3ac) + 3B cψ n (ξ) ± 3( b +, b 3ac)ψ n (ξ) (n =,,...), (7) 3A ( b + b 3ac) 3A cψ n (ξ) ± 3(b + b 3ac)ψ n (ξ) c [ 3A 9ψ n, (ξ)] (n =,,...), (8) 3B (b + b 3ac) 3B cψ n (ξ) ± 3( b + b 3ac)ψ n (ξ) c [ 3B ± 9ψ n, (n =,,...), (9) (ξ)] where A = c [b3 9abc + (b 3ac) 3/ ], B = c [b3 9abc (b 3ac) 3/ ] ; a, b, and c are arbitrary constants to be determined by the second kind of elliptic equation.. New Solutions of Second Kind of Elliptic Equation In recent years, numerous work has been done on exact solutions to nonlinear evolution equations by using the Jacobi elliptic function method. (i) The known solutions of the second kind of elliptic equation When a = 4, b = 4(+k ), c = 4k, then the solution of Eq. () takes form ψ(ξ) = sn (ξ, k). (0) When a = 4( k ), b = 4(k ), c = 4k, Eq. () has the following solution ψ(ξ) = cn (ξ, k). () Combing the two solutions with Eqs. (3) (9), infinite sequence exact solutions of Jacobi function type of Eq. () can be obtained. (ii) The new solutions of the second kind of elliptic equation In the virtue of Jacobi elliptic function, we get sn(ξ + 4K(k)) = sn(ξ), cn(ξ + 4K(k)) = cn(ξ), () ψ(ξ) = where K(k) = = π/ 0 0 k sin ϕ dϕ ( x )( k x ) dx, 0 k. According to Eqs. (0) and (), the new type solutions of the second kind of elliptic equation are introduced as follows to seek infinite sequence compact soliton solutions. When a = 4, b = 4( + k ), c = 4k, Eq. () has the following solution { sn (ξ, k), 0 ξ K(k), ψ(ξ) = (3) 0, others. When a = 4( k ), b = 4(k ), c = 4k, Eq. () has the following solution ψ(ξ) = { cn (ξ, k), K(k) ξ K(k) 0, others., (4).3 Bäcklund Transformation and New Solutions to Second Kind of Elliptic Equation under Special Conditions (i) If c = 0, Eq. () becomes the following equation ) = (ψ (ξ)) = bψ (ξ) + aψ(ξ), (5) By computation, the following solutions of Eq. (5) are acquired { a b a b sin( bξ) (b < 0), kπ π bξ kπ + 3π 0, others; (k Z), ψ(ξ) = b a b sin( bξ) (b < 0), kπ + π bξ kπ + 5π (k Z), a (7) b, others; { a ψ(ξ) = b + a b sin( bξ) (b < 0), kπ + π bξ kπ + 5π (k Z), (8) 0, others; (6)

3 No. 6 Communications in Theoretical Physics 95 ψ(ξ) = b + a b sin( bξ) (b < 0), kπ π bξ kπ + 3π (k Z), a (9) b, others; ψ(ξ) = b a b cos( bξ) (b < 0), kπ π bξ kπ + π (k Z), (0) 0, others; ψ(ξ) = b a b cos( bξ) (b < 0), kπ bξ kπ + π (k Z), a b, others; () ψ(ξ) = b + a b cos( bξ) (b < 0), kπ π bξ kπ + π (k Z), a b, others; () ψ(ξ) = b + a b cos( bξ) (b < 0), kπ bξ kπ + π (k Z), (3) 0, others. (ii) Bäcklund transformation of Eq. (5) If ψ(ξ) is the solution of Eq. (5), the following ψ(ξ) is also the solution of Eq. (5) ψ(ξ) = [ a(p ± bd + p bp ) b bd + p ψ(ξ) + bdψ (ξ) ], (4) where d and p are arbitrary constants, and p 0. (iii) If b 4ac = 0, Eq. () becomes the following equation ) ( = ψ (ξ) ) = cψ 3 (ξ) + bψ (ξ) + b ψ(ξ). (5) 4c By computation, we get new type solutions of Eq. (5) as follows [ b [ + exp ( b/ ξ )] ] ψ(ξ)= [ ( )] (c > 0, b < 0), (6) c exp b/ ξ ψ(ξ) = b ( b ) c tan 4 ξ (c > 0, b > 0). (7) When b 4ac = 0, based on Eqs. (3) (9), yields Bäcklund transformation of Eq. (5), then combing the Bäcklund transformation with Eqs. (6) (7), we can construct infinite sequence exact solutions of Eq. (5). (iv) If a = c = 0, Eq. () becomes the following equation ( ψ(ξ) ) = bψ (ξ). (8) By computation, we gain the following solutions of Eq. (8) ψ(ξ) = exp( bξ ) (b > 0), (9) ψ(ξ) = exp(i bξ ) (b < 0). (30) (v) Bäcklund transformation of Eq. (8) If ψ(ξ) is the solution of Eq. (8), the following ψ(ξ) is also the solution of Eq. (8) ψ(ξ) = 4( n ± bd)ψ(ξ) + 4nψ (ξ) + 4dψ (ξ) 4 ( bmψ(ξ) ± lψ (ξ) ± mψ (ξ) ), (3) ψ(ξ) = d( bψ(ξ) + ψ (ξ) ) qψ(ξ) + lψ (ξ) + mψ (ξ), (3) n(+( n bd)ψ(ξ)+nψ ψ(ξ) (ξ)+dψ (ξ)) = ( bnm ± l)ψ(ξ) + l nψ (ξ) + m nψ (ξ), (33) where d, m, l, q, b, and n are arbitrary constants, d, m, l, q, and b are not all to be zero and b > 0, n > 0. (vi) If a = 0, Eq. () becomes the following equation ) ( = ψ (ξ) ) = bψ (ξ) + cψ 3 (ξ), (34) By computation, we search for the following solutions of Eq. (34) ψ(ξ) = b c + b[ + exp( bξ ) ] c [ exp( bξ ) ] (b > 0), (35) ψ(ξ) = b c b ( tan ( b/)ξ ) (b < 0). (36) c When a = 0, Eqs. (3) (9) become Bäcklund transformation of Eq. (34), then combing the Bäcklund transformation with Eqs. (35) and (36), we can get infinite sequence exact solutions of Eq. (34). 3 Application of Method Given nonlinear evolution equation ((+)-dimensional NEEs) H(u, u x, u y, u t, u xx, u xt, u tt, u xy, u yy, u yt,...) = 0. (37) Assume the solution of Eq. (37) as follows u(x, y, t) = u(ξ) = g 0 (y, t) + g (y, t)ψ(ξ) (ξ = px + q(y, t)), (38) where g 0 (y, t), g (y, t), and q(y, t) are functions to be determined with respect to (y, t), p is a constant to be determined and ψ(ξ) is determined by Eq. (). Substituting Eqs. () and (38) with ξ = px+q(y, t) into Eq. (37), setting coefficients of ψ j (ξ)(ψ (ξ)) k (k = 0, ; j = 0,,,...) to zero, yields a set of nonlinear differential equations about g 0 (y, t), g (y, t), and q(y, t). Solving the set with the help of symbolic computation system Mathematica, and taking each solution with infinite sequence solutions determined by the solutions of Eq. () (Including the special conditions) and Bäcklund transformation into Eq. (38), we can get infinite sequence peak solitary wave

4 95 Communications in Theoretical Physics Vol. 55 solutions, infinite sequence compact soliton solutions, and infinite sequence smooth soliton solutions. 4 Infinite Sequence Soliton-Like Exact Solutions of (+)-Dimensional Breaking Soliton Equation In this section, we would like to apply our method to obtain infinite sequence soliton-like exact solutions of the (+)-dimensional breaking soliton equation. Set the solutions of (+)-dimensional breaking soliton equation as follows u(x, y, t) = u(ξ) = g 0 (y, t) + g (y, t)ψ(ξ) (ξ = px + q(y, t)), (39) v(x, y, t) = v(ξ) = f 0 (y, t) + f (y, t)ψ(ξ) (ξ = px + q(y, t)). (40) Taking Eqs. (), (39), and (40) with ξ = px + q(y, t) into Eqs. (), equating coefficients of ψ j (ξ) ( ψ (ξ) ) k (k = 0, ; j = 0,,, 3, 4) to zero, yields a set of nonlinear differential equations about g 0 (y, t), g (y, t), f 0 (y, t), f (y, t), and q(y, t). Solving the set by use of Mathematica, we get g 0 (y, t)=g 0, g (y, t)= 3cp 8, f (y, t)= 3cp 8p q y(y, t), f 0 (y, t) = [ qt (y, t) + [4g 0 α + bp α]q y (y, t) ], (4) 4pα g 0 (y, t) = g 0 (y), g (y, t) = g, f (y, t) = 3cp 8 q y(y, t), a = 0, f 0 (y, t) = [ g q t (y, t) 8g pα + p α[ bg + 3cg 0 (y)]q y (y, t) ] ; (4) g 0 (y, t) = g 0 (y), g (y, t) = g, f (y, t) = 3cp 8 q y(y, t), b 4ac = 0, f 0 (y, t) = [ g q t (y, t) 8g pα + p α[ bg + 3cg 0 (y)]q y (y, t) ] ; (43) where g 0, g, and p are arbitrary constants and not zero. Substituting Eqs. (4) (43) into Eqs. (39) and (40), we win the following exact solutions of ( + )-dimensional breaking soliton equation u(x, y, t) = [ qt (y, t) + α[8g 0 + bp + 3cp ψ(px + q(y, t))]q y (y, t) ], v(x, y, t) = g cp ψ(px + q(y, t)); (44) u(x, y, t) = [ g q t (y, t) + p α[ bg 8g pα + 3cg 0 (y) 3cg ψ(px + q(y, t))]q y (y, t) ], v(x, y, t) = g 0 (y) + g ψ(px + q(y, t)); (45) u(x, y, t) = [ g q t (y, t) + p α[ bg + 3b 8g pα 4a g 0(y) 3b 4a g ψ(px + q(y, t))]q y (y, t) ], v(x, y, t) = g 0 (y) + g ψ(px + q(y, t)). (46) According to the known solutions and Bäcklund transformation of the second kind of elliptic equation, we can construct infinite sequence exact solutions of Eqs. (), then combing the obtained solutions with Eqs. (44) (46), new infinite sequence exact solutions of Eqs. () are obtained as follows (i) Infinite sequence smooth soliton-like exact solutions (a) Infinite sequence smooth soliton-like exact solutions of Jacobi elliptic function type u n (x, y, t) = [ qt (y, t) + α[8g 0 + bp + 3cp ψ n (px + q(y, t))]q y (y, t) ], v n (x, y, t) = g cp ψ n (px + q(y, t)), ab ± a b (b 4ac) 4abcψ n (ξ) + [ b c c b (b 4ac)]ψn (ξ) abc + b cψ n (ξ) + bc ψn (ξ), [A + Bcn (ξ, k)] B ψ 0 (ξ) = C [dn (ξ, k) Nsn(ξ, k)], N = A + B k, a = A + B C, b = ( k C ), c = (A B (n =,,...); (47) ) (b) Infinite sequence smooth soliton-like exact solutions of hyperbolic function type. When k =, the solutions (47) will degenerate to the following infinite sequence smooth exact solutions of hyperbolic function type u n (x, y, t) = [ qt (y, t) + α[8g 0 + bp + 3cp ψ n (px + q(y, t))]q y (y, t) ], v n (x, y, t) = g cp ψ n (px + q(y, t)), ab + 4acψ n (ξ) + bcψ n (ξ) c[a + bψ n (ξ) + cψ n (ξ)], ψ 0(ξ) = C [A + Bsech(ξ)] A C [sech(ξ) N tanh(ξ)], N = A + B, a = A + B C, b =, c = (A B (n =,,...); (48) ) (c) Infinite sequence smooth soliton-like exact solutions of triangular function type

5 No. 6 Communications in Theoretical Physics 953 When k = 0, the obtained solutions (47) will degenerate as the following infinite sequence smooth exact solutions of triangular function type u n (x, y, t) = [ qt (y, t) + α[8g 0 + bp + 3cp ψ n (px + q(y, t))]q y (y, t) ], v n (x, y, t) = g cp ψ n (px + q(y, t)), ab + 4acψ n (ξ) + bcψn (ξ) c[a + bψ n (ξ) + cψn (ξ)] (b 4ac = 0), [A + B cos(ξ)] B ψ 0 (ξ) = C [ N sin(ξ)], N = A + B, a = A + B C C, b =, c = (A B ). (49) (ii) Infinite sequence compact soliton exact solutions of Jacobi elliptic function type Now we will show two kinds of Bäcklund transformation of infinite sequence compact soliton exact solutions of Jacobi elliptic function type to Eqs. () as follows u n (x, y, t) = [ qt (y, t) + α[8g 0 + bp + 3cp ψ n (px + q(y, t))]q y (y, t) ], v n (x, y, t) = g cp ψ n (px + q(y, t)) (n =,,...) 3A ( b + b 3ac) 3A cψ n (ξ) ± 3(b + b 3ac)ψ n (ξ) c [ 3A 9ψ n (ξ)], { sn a = 4, b = 4( + k ), c = 4k (ξ, k), 0 ξ K(k),, ψ 0 (ξ) = 0, others, u n (x, y, t) = [ qt (y, t) + α[8g 0 + bp + 3cp ψ n (px + q(y, t))]q y (y, t) ], (50) v n (x, y, t) = g cp ψ n (px + q(y, t)) (n =,,...), 3A ( b + b 3ac) 3A cψ n (ξ) ± 3(b + b 3ac)ψ n (ξ) c [ 3A 9ψ n (ξ)], { a = 4( k ), b = 4(k ), c = 4k cn, ψ 0 (ξ) = (ξ, k), K(k) ξ K(k) 0, others,, (5) where A = (/c )[b 3 9abc + (b 3ac) 3/ ]. (iii) Infinite sequence peak soliton exact solutions (a) Infinite sequence peak soliton exact solutions of exponential function type When a = 0, b > 0, according to Eq. (8) and solution (35), we can get infinite sequence exact solutions of Eq. (34). Then substituting the solutions into Eq. (45), infinite sequence peak soliton exact solutions of exponential function type of ( + )-dimensional breaking soliton equation are gained (given Bäcklund transformation of solutions as follows) u n (x, y, t) = [ g q t (y, t) + p α[ bg + 3cg 0 (y) 8g pα 3cg ψ n (px + q(y, t))]q y (y, t) ], v n (x, y, t) = g 0 (y) + g ψ n (px + q(y, t)) (n =,, 3,...), 6A 0 cψ n (ξ) + 6bψ n (ξ) c[ 6A 0 9ψ n (ξ)], b 3 ) (A 0 = 4 c, ψ 0 (ξ) = b c + b[ + exp( bξ ) ] c [ exp( bξ ) ]. (5) When b 4ac = 0, according to the following Eq. (53), we can obtain infinite sequence peak soliton exact solutions of exponential function type of Eqs. () u n (x, y, t) = [ g q t (y, t)+p α [ bg + 3b 8g pα 4a g 0(y) ] ] 3b 4a g ψ n (px + q(y, t)) q y (y, t), v n (x, y, t) = g 0 (y) + g ψ n (px + q(y, t)) (n =,, 3,...), ab + 4acψ n (ξ) + bcψn (ξ) c[a + bψ n (ξ) + cψn (ξ)] (b 4ac = 0), ψ 0 (ξ) = [ b [ + exp ( ( b/) ξ )] [ ( )] ] c exp ( b/) ξ (c > 0, b < 0). (53) (b) Infinite sequence peak soliton exact solutions of triangular function type When a = 0, b < 0, according to the following Eq. (54), we can obtain infinite sequence peak soliton exact solutions of triangular function type of Eqs. () u n (x, y, t) = 8g pα [ g q t (y, t) + p α[ bg + 3cg 0 (y) 3cg ψ n (px + q(y, t))]q y (y, t)], v n (x, y, t) = g 0 (y) + g ψ n (px + q(y, t)) (n =,, 3,...), bψ n (ξ) b + cψ n (ξ),

6 954 Communications in Theoretical Physics Vol. 55 ψ 0 (ξ) = b c b ( tan ( b/)ξ ). (54) c When b 4ac = 0, according to the following Eq. (55), we can obtain infinite sequence peak soliton exact solutions of triangular function type of Eqs. () u n (x, y, t) = [ [ g q t (y, t)+p α bg + 3b 8g pα 4a g 0(y) ] ] 3b 4a g ψ n (px + q(y, t)) q y (y, t), v n (x, y, t) = g 0 (y) + g ψ n (px + q(y, t)) (n =,, 3,...), ab + 4acψ n (ξ) + bcψn (ξ) c[a + bψ n (ξ) + cψn (ξ)] (b 4ac = 0), ψ 0 (ξ) = b ( b ) c tan 4 ξ (c > 0, b > 0). (55) In all the Bäcklund transformations, p, g 0, and g are arbitrary constants; g 0 (y), q(y, t) are arbitrary continuous functions with respect to (y, t). 5 Conclusion In the past decades, In the past decades, the auxiliary equation method has been successfully applied to find smooth soliton solutions to nonlinear evolution equations in Refs. [] [6]. However, to our knowledge, only a little work was done to search for peak and compact soliton solutions to nonlinear evolution equations. For example, in 993, Camassa and Holm found peak solitary wave solutions in shallow water wave CH equation with small amplitude, Rosenau and Hyman found compact soliton solutions in K(m, n) equation. The peak solitary wave solutions and compact soliton solutions to nonlinear evolution equations have been drawn considerable attention by mathematicians and physicists. And many solving methods have been presented to seek two kinds of solutions. But the methods have only obtained finite peak solitary wave solutions and compact soliton solutions. According to the references on auxiliary equation methods, summing up the characteristics of construction and mechanization of the methods and developing the characteristics, we in this paper have proposed new type solutions and Bäcklund transformation of the second kind of elliptic equation. The method is of significance to search for infinite sequence smooth soliton-like solutions of Jacobi elliptic function type, infinite sequence compact soliton solutions of Jacobi elliptic type, and infinite sequence peak soliton solutions of exponential function type and triangular function type to nonlinear evolution equations. To illustrate the method, ( + )-dimensional breaking soliton equation is chosen as an example and its new infinite sequence soliton-like exact solutions are constructed with the help of symbolic computation system Mathematica, such as smooth soliton-like exact solutions of Jacobi elliptic type, infinite sequence compact soliton solutions of Jacobi elliptic type, infinite sequence peak soliton solutions of exponential function type and triangular function type, etc. References [] E.G. Fan, Phys. Lett. A 77 (000). [] Sirendaoreji and J. Sun, Phys. Lett. A 309 (003) 69. [3] D.S. Li and H.Q. Zhang, Chin. Phys. 3 (004) 377. [4] D.S. Li and H.Q. Zhang, Acta Phys. Sin. 5 (003) 373 (in Chinese). [5] Y. Chen, B. Li, and H.Q. Zhang, Chin. Phys. (003) 940. [6] Y. Chen, Z.Y. Yan, B. Li, and H.Q. Zhang, Chin. Phys. (003). [7] H.T. Chen and H.Q. Zhang, Commun. Theor. Phys. 4 (004) 497. [8] F.D. Xie, J. Chen, and Z.S. Lü, Commun. Theor. Phys. 43 (005) 585. [9] Z.H. Pan, S.H. Ma, and J.P. Fang, Chin. Phys. B 9 (00) [0] X.D. Zhen, Y. Chen, B. Li, and H.Q. Zhang, Commun. Theor. Phys. 39 (003) 647. [] F.D. Xie and X.S. Gao, Commun. Theor. Phys. 4 (004) 353. [] Y. Chen and E.G. Fan, Chin. Phys. 6 (007) 6. [3] D.S. Li and H.Q. Zhang, Acta. Phys. Sin. 5 (003) 569 (in Chinese). [4] Y. Chen, B. Li and H.Q. Zhang, Chin. Phys. (003) 940. [5] D.S. Li and H.Q. Zhang, Commun. Theor. Phys. 40 (003) 43. [6] D.S. Li and H.Q. Zhang, Chin. Phys. 3 (004) 984. [7] D.S. Li and H.Q. Zhang, Chin. Phys. 3 (004) 377. [8] Z.S. LÜ and H.Q. Zhang, Commun. Theor. Phys. 39 (003) 405. [9] S.H. Ma, J.P. Fang, and H.P. Zhu, Acta Phys. Sin. 56 (007) 439 (in Chinese). [0] S.H. Ma, X.H. Wu, J.P. Fang, and C.L. Zheng, Acta Phys. Sin. 57 (008) (in Chinese). [] B. Li and H.Q. Zhang, Chin. Phys. B 7 (008) [] X. Zeng and H.Q. Zhang, Acta Phys. Sin. 54 (005) 504 (in Chinese). [3] Z.H. Pan, S.H. Ma, and J.P. Fang, Chin. Phys. B 9 (00) 0030(). [4] J.Y. Qiang, S.H. Ma, and J.P. Fang, Chin. Phys. B 9 (00) (). [5] Y. Chen, B. Li, and H.Q. Zhang, Commun. Theor. Phys. 40 (003) 37. [6] C.L. Zheng, J.F. Zhang, W.H. Huang, and L.Q. Chen, Chin. Phys. Lett. 0 (003) 783. [7] Taogetusang, Sirendaoerji, and Q.P. Wang, Acta Sci. J. Nat. Univ. NeiMongol 38 (009) 387 (in Chinese). [8] Taogetusang, Sirendaoerji, and S.M. Li, Chin. Phys. B 9 (00) (). [9] Taogetusang and Sirendaoerji, Acta Phys. Sin. 59 (00) 443 (in Chinese).