An Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson Equation

Commun. Theor. Phys. (Beijing, China) 50 (008) pp. 309 314 c Chinese Physical Society Vol. 50, No., August 15, 008 An Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson Equation ZHAO Xue-Qin 1, and ZHI Hong-Yan 1 Department of Mathematics, Qufu Normal University, Shandong Rizhao 7686, China Department of Applied Mathematics, Dalian University of Technology, Dalian 11604, China (Received April 9, 007; Revised May 8, 008) Abstract With the aid of computerized symbolic computation, an improved F-expansion method is presented to uniformly construct more new exact doubly periodic solutions in terms of rational formal Jacobi elliptic function of nonlinear partial differential equations (NPDEs). The coupled Drinfel d Sokolov Wilson equation is chosen to illustrate the method. As a result, we can successfully obtain abundant new doubly periodic solutions without calculating various Jacobi elliptic functions. In the limit cases, the rational solitary wave solutions and trigonometric function solutions are obtained as well. PACS numbers: 0.30.Jr, 05.45.Yv Key words: Jacobi elliptic function, doubly periodic solution, rational solitary wave solution 1 Introduction The nonlinear partial differential equations are related to nonlinear science such as physics, mechanics, biology, etc. To further explain some physical phenomena, searching for exact solutions of NPDEs is very important. Up to now, there have been many powerful methods, such as inverse scattering method, [1] Bäcklund transformation method, [] Hirota s bilinear method, [3] Darboux transformation method, [4] homogeneous balance method, [5,6] hyperbolic function expansion method, [7,8] sine-cosine method, [9] various tanh methods, [101] various generalized Riccati equation expansion methods, [1314] and so on. Recently, various Jacobin elliptic function expansion methods [1517] and F-expansion method [180] which can be thought of as a generalization of Jacobi elliptic function expansion method since F here stands for everyone of Jacobi elliptic functions, have been developed to obtain Jacobi elliptic function solutions of a wide class of nonlinear partial differential equations. It is remarkable about the Jacobin elliptic function method and F-expansion method that they allow one to find both Jacobin elliptic function solutions, triangular function solutions, and solition solutions using the same and unique procedure. In this paper, we will improve F-expansion method by means of a general ansätz and more new solutions of an auxiliary equation. For illustration, we apply the improved method to the coupled Drinfel d Sokolov Wilson equation and successfully construct more types of doubly periodic solutions than those obtained by F-expansion method. The rest of this paper is organized as follows. In Sec., we simply summarize the mathematical framework of the improved F-expansion method. In Sec. 3, we apply the approach to the coupled Drinfel d Sokolov Wilson equation and bring out many solutions. Conclusions will be presented finally. Summary of Improved F-expansion Method In the following we would like to outline the main steps of our improved method: Step 1 For a given NPDEs system with some physical fields u i (x, t) in two variables x, t, Θ i (u i, u it, u ix, u ixt,...) = 0, (1) by using the wave transformation u i (x, t) = U i (ξ), ξ = lx + λt, () where λ and l are constants to be determined later. Then the nonlinear partial differential equation (1) is reduced to a nonlinear ordinary differential equation (ODE): i (U i, U i, U i,...) = 0. (3) Step We search for solutions of Eq. (3) in the following form n i U i (ξ) = a i0 + a ij F j b ij + (F + µ) j j=1 + c ij F j1 a + bf + d ij a + bf (F + µ) j, (4) where µ, a i0, a ij, b ij, c ij, and d ij (j = 1,...,n i ; i = 1,,...) are constants to be determined later and the new variable F satisfies F = a + bf, (5) where a, b, and c are constants. Step 3 Determine n i by balancing the nonlinear terms with the highest-order derivative terms in Eq. (1) or Eq. (3). Step 4 With the aid of Maple, substituting Eq. (4) into Eq. (3) along with Eq. (5), and collecting the coefficients of the same power F i ( a + bf ) j (i = 0, 1,,...; j = 0, 1). Set each of the obtained coefficients to be zero to get a set of over-determined algebraic equations. Step 5 Solving the over-determined algebraic equations by use of Maple, we would end up with the explicit expressions for λ, l, µ, a i0, a ij, b ij, c ij, and d ij (i = 1,,...; j = 1...,n i ). Step 6 According to Eqs. () and (4), the conclusions obtained in Step 5 and solutions of Eq. (5), we can obtain The project supported by National Natural Science Foundation of China under Grant No. 10771118 E-mail: xqzhao197@16.com

310 ZHAO Xue-Qin and ZHI Hong-Yan Vol. 50 many rational formal Jacobi elliptic function solutions of Eq. (1). 3 Exact Solutions of Coupled Drinfel d Sokolov Wilson Equation The coupled Drinfel d Sokolov Wilson equation reads: u t + 3vv x = 0, (6a) v t + v xxx + uv x + u x v = 0, (6b) which was introduced as a model of water waves. [1,] There is a mount of papers devoted to this equation. [35] Here our proposed method gives a series of doubly periodic solutions to the system as follows. According to the above method, to seek travelling wave solutions of Eqs. (6), we make the following transformation, u = U(ξ), v = V (ξ), ξ = lx + λt, (7) where l and λ are constants to be determined later, and thus equation (6) becomes λu + 3lV V = 0, (8a) λv + l 3 V + luv + lu V = 0. (8b) Balancing the highest derivative terms with nonlinear terms in Eqs. (8), we suppose that equations (8) have the following formal solutions U = a 10 + a 11 F + a 1 F + b 11 F + µ + b 1 (F + µ) + c 11 a + bf + c 1 F a + bf + d 11 a + bf + d 1 a + bf F + µ (F + µ), (9a) V = a 0 + a 1 F + b 1 F + µ + c 1 a + bf + d 1 a + bf, (9b) F + µ where F satisfies Eq. (5). a 10, a 11, a 1, b 11, b 1, c 11, c 1, d 11, d 1, a 0, a 1, b 1, c 1, and d 1 are constants to be determined later. With the aid of Maple, substituting Eqs. (9) into Eqs. (8) along with Eq. (5), collecting the coefficients of F i ( a + bf ) j (j = 0, 1; i = 0, 1,,...), and setting them to be zero, we get a set of over-determined algebraic equations with respect to a 10, a 11, a 1, b 11, b 1, c 11, c 1, d 11, d 1, a 0, a 1, b 1, c 1, d 1, l, and λ. By use of Maple, solving the over-determined algebraic equations, we get the following results. Case 1 Case c 1 = c 11 = d 11 = d 1 = c 1 = d 1 = a 11 = a 1 = a 1 = 0, µ = µ, l = l, b 1 = l ( bµ 4 c b µ + 1 µ ac + ba ), b 1 = 3 l ( bµ + a + µ 4 c ), b µ 1µ ac ba bµ 4 c a 10 = 3l µ ( 4 bµ c + 4 µ 4 c + b ) ( 4 (bµ + a + µ 4, λ = l3 b µ bµ 4 c 1 µ ac ba ) c) bµ + a + µ 4, c b 11 = 3 ( b + µ c ) l b µ 1µ ac ba bµ 4 c ( b + µ c ) µ l µ, a 0 = (bµ + a + µ 4. c) b 11 = a 0 = µ = c 1 = d 11 = c 1 = a 11 = 0, c 11 = 3l c (b + 6 ac) b + 1, ac λ = ( b + 6 ac ) l 3, b 1 = l (6 ac + b ac) c (b + 6 ac), a 1 = l c (b + 6 ac), b 1 = 3 ( l a, a 10 = 3l b + 8 b ac + 1 ac ) 4 (b + 6, d 1 = 1 b + 1 acl, ac) l = l, a 1 = 3 l c, d 1 = 3l (6 ac + b ac) c(b + 6 ac). For simplicity, we set ( N = (b µ bµ 4 c 1 µ ac ba), Q = 3l µ 4 bµ c + 4 µ 4 c + b ) 4(bµ + a + µ 4, c) M = 1 b µ 1 µ ac ba bµ 4 c. Equation (5) has many kinds of solutions, next we will show some solutions of rational forms expressed in terms of different Jacobi elliptic functions (i) If a = A, b = 4 k, and c = 1/A, then the new solution to Eq. (5) is A cn (ξ, k) (ξ, k)dn (ξ, k), sn where sn (ξ, k) and cn (ξ, k) are Jacobi elliptic sine and cosine functions, dn (ξ, k) is Jacobi elliptic function of the third kind, k (0 < k < 1) denotes the modulus of the Jacobi elliptic functions. This solution to Eq. (5) is new which has not

No. An Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson Equation 311 been shown in literature. So based on the new solution, we can derive new solutions to Eqs. (6), u 1 = Q + 3l µ ( b + µ c ) sn (ξ)dn (ξ) ( 3l bµ + a + µ 4 c ) sn (ξ)dn (ξ) Acn (ξ) + µ sn (ξ)dn (ξ) [A cn(ξ) + µ sn (ξ)dn (ξ)], v 1 = M ( b + µ c ) µ l Nl sn (ξ)dn (ξ) c M [A cn(ξ) + µ sn (ξ)dn (ξ)]. (ii) If a = A k 4, b = k 4, and c = 1/A, then the solution to Eq. (5) is A dn (ξ, k) sn (ξ, k)cn (ξ, k), u = Q + 3l µ ( b + µ c ) sn (ξ)cn(ξ) ( 3l bµ + a + µ 4 c ) sn (ξ)cn (ξ) A dn(ξ) + µ sn (ξ)cn (ξ) [A dn (ξ) + µ sn (ξ)cn (ξ)], v = M ( b + µ c ) µ l Nl sn (ξ)cn (ξ) c M [A dn(ξ) + µ sn (ξ)cn (ξ)]. (iii) If a = ( k + k 1 )/4, b = k / + 3k 1 1 and c = ( k + k 1 )/4, then the solution to Eq. (5) is k cn(ξ, k)sn (ξ, k) k sn (ξ, k) + (k 1 1) dn(ξ, k) + k 1 1 3l µ ( b + µ c ) ( k sn (ξ, k) + (k 1 1) dn (ξ, k) + k 1 1 ) u 3 = Q k cn (ξ, k)sn (ξ, k) + µ (k sn (ξ, k) + (k 1 1) dn(ξ, k) + k 1 1) ( 3l bµ + a + µ 4 c )( k sn (ξ, k) + (k 1 1) dn(ξ, k) + k 1 1 ) [k cn (ξ, k)sn (ξ, k) + µ (k sn (ξ, k) + (k 1 1) dn (ξ, k) + k 1 1)], v 3 = M ( b + µ c ) µ l Nl [ k sn (ξ, k) + (k 1 1) dn (ξ, k) + k 1 1 ] c M [k cn (ξ, k)sn (ξ, k) + µ (k sn (ξ, k) + (k 1 1) dn(ξ, k) + k 1 1)]. (iv) If a = k /4 ( B C ), b = k / 1 and c = k /4 ( B C ), then the solution to Eq. (5) is (B B k C )/(B C ) sn (ξ, k) + cn (ξ, k) B dn (ξ, k) + C 3 ( b + µ c ) l µ B C (B dn (ξ) + C) u 4 = Q + B C [cn (ξ) + µ B dn (ξ) + µ C] C B + B k sn (ξ) 3 ( bµ + a + cµ 4) l ( B C ) (B dn (ξ) + C) [ B C [cn(ξ) + µ B dn (ξ) + µ C] C B + B k sn (ξ) ], v 4 = M ( b + µ c ) µ l Nl µ B C (B dn (ξ) + C) c M B C [cn (ξ) + µ B dn (ξ) + µ C] C B + B k sn (ξ). (v) If a = 1/4 ( B C ), b = 1/ k and c = (B C )/4, then the solution to Eq. (5) is (B + C k k B )/(B C ) sn (ξ, k) + dn (ξ, k) B cn (ξ, k) + C 3 l µ ( b + µ c ) B C (B cn (ξ) + C) u 5 = Q + B C [dn (ξ) + µ B cn (ξ) + µ C] C k B k + B sn (ξ) 3 l ( bµ + a + cµ 4) B C (B cn (ξ) + C) [ B C [dn (ξ) + µ B cn(ξ) + µ C] C k B k + B sn (ξ) ], v 5 = M ( b + µ c ) µ l Nl (B cn(ξ) + C) c M B C [dn (ξ) + µ B cn (ξ) + µ C] C k B k + B sn (ξ). (vi) If a = (k 4 k + 1)/4 ( B C ), b = (k + 1)/ and c = (B C )/4, then the solution to Eq. (5) is (B C k )/(B C )cn (ξ, k) + dn (ξ, k) B sn (ξ, k) + C and the new solutions to Eqs. (6) is 3 l µ ( b + µ c ) B C (B sn (ξ) + C) u 6 = Q + B C [dn (ξ) + µ B sn (ξ) + µ C] C k B k + B cn (ξ)

31 ZHAO Xue-Qin and ZHI Hong-Yan Vol. 50 3 l ( bµ + a + cµ 4) B C (B sn (ξ) + C) [ B C [dn (ξ) + µ B sn (ξ) + µ C] C k B k + B cn(ξ) ], v 6 = M ( b + µ c ) µ l Nl (B sn (ξ) + C) c M B C [dn (ξ) + µ B sn (ξ) + µ C] C k B k + B cn (ξ). (vii) If a = A, b = + k, and c = (k 4 + 1 k )/A, then the solution to Eq. (5) is A sn(ξ, k) cn (ξ, k)dn (ξ, k) is u 7 = Q + 3l µ ( b + µ c ) cn (ξ)dn (ξ) A sn (ξ) + µ cn (ξ)dn (ξ) 3l ( bµ + a + µ 4 c ) cn (ξ)dn (ξ) [A sn(ξ) + µ cn (ξ)dn (ξ)] v 7 = M ( b + µ c ) µ l Nl cn (ξ)dn (ξ) c M [A sn(ξ) + µ cn (ξ)dn (ξ)]. (viii) If a = (k + k + 1)/4A, b = (k 6k + 1)/, and c = (A k + A + ka )/4, then the solution to Eq. (5) (1 + sn (ξ, k))(1 k sn (ξ, k)) A cn(ξ, k)dn (ξ, k) and the new solutions to Eqs. (6) is 3l µ ( b + µ c ) A cn(ξ)dn (ξ) u 8 = Q + 1 k sn (ξ) + sn (ξ) ksn (ξ) + µ A cn(ξ)dn (ξ) is 3A l ( bµ + a + µ 4 c ) cn (ξ)dn (ξ) (1 k sn (ξ) + sn (ξ) ksn (ξ) + µ A cn(ξ)dn (ξ)), v 8 = M ( b + µ c ) µ l bµ + a + µ 4 c M Nl A cn(ξ)dn (ξ) [ 1 k sn (ξ) + sn (ξ) k (sn (ξ)) + µ A cn(ξ)dn (ξ) (ix) If a = (k + k + 1)/4A, b = 9k 6k + 10/, and c = (A k + A + ka )/4, then the solution to Eq. (5) (1 + sn (ξ, k))(1 + k sn (ξ, k)) A cn(ξ, k)dn (ξ, k) 3l µa ( b + µ c ) cn (ξ)dn (ξ) u 9 = Q + 1 + k sn (ξ) + sn (ξ) + ksn (ξ) + µ A cn(ξ)dn (ξ) 3l A ( bµ + a + µ 4 c ) cn (ξ)dn (ξ) [1 + k sn (ξ) + sn (ξ) + ksn (ξ) + µ A cn(ξ)dn (ξ)], v 9 = M ( b + µ c ) µ l Nl A cn(ξ)dn (ξ) c M [1 + k sn (ξ) + sn (ξ) + ksn (ξ) + µ Acn (ξ)dn (ξ)]. (x) If a = 4k/A, b = k 1 6 k, and c = A k + A + ka, then the solution to Eq. (5) is 1 ksn (ξ, k) A cn(ξ, k)dn (ξ, k), u 10 = Q + 3l µa ( b + µ c ) ( cn (ξ)dn (ξ) 1 ksn (ξ) + µ A cn(ξ)dn (ξ) 3l A bµ + a + µ 4 c ) cn (ξ)dn (ξ) [1 ksn (ξ) + µ A cn(ξ)dn (ξ)], v 10 = M ( b + µ c ) µ l Nl A cn(ξ)dn (ξ) c M [1 ksn (ξ) + µ A cn(ξ)dn (ξ)]. (xi) If a = 4k/A, b = 6k k 1, and c = A k + A ka, then the solution to Eq. (5) is 1 + ksn (ξ, k) A cn(ξ, k)dn (ξ, k) u 11 = Q + 3l µa ( b + µ c ) ( cn (ξ)dn (ξ) ksn (ξ) + 1 + µ A cn(ξ)dn (ξ) 3l A bµ + a + µ 4 c ) cn (ξ)dn (ξ) [ksn (ξ) + 1 + µ A cn(ξ)dn (ξ)], v 11 = M ( b + µ c ) µ l bµ + a + µ 4 c Nl A cn(ξ)dn (ξ) M [ksn (ξ) + 1 + µ A cn(ξ)dn (ξ)]. ].

No. An Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson Equation 313 (xii) If a = k 4 /4 + k (k 1 1) + k 1 k, b = (k 6k 1 )/, and c = 1/4, then the solution to Eq. (5) is (k 1 1)sn (ξ, k) + dn (ξ, k) + 1 3l µ ( b + µ c ) sn (ξ)cn (ξ) u 1 = Q + (k 1 1)sn (ξ) + dn (ξ) + 1 + µ sn (ξ)cn(ξ) 3l ( bµ + a + µ 4 c ) sn (ξ)cn (ξ) [(k 1 1)sn (ξ) + dn (ξ) + 1 + µ sn (ξ)cn (ξ)], v 1 = M ( b + µ c ) µ l Nl sn (ξ)cn(ξ) c M [(k 1 1)sn (ξ) + dn (ξ) + 1 + µ sn (ξ)cn (ξ)]. (viii) If a = k 4 /4 k (k 1 + 1) + + k 1 k, b = (k + 6k 1 )/, and c = 1/4, then the solution to Eq. (5) is 1 (k 1 + 1)sn (ξ, k) dn (ξ, k) and the new solutions to Eqs. (6) is 3l µ ( b + µ c ) sn (ξ)cn (ξ) u 13 = Q + 1 (k 1 + 1)sn (ξ) dn (ξ) + µ sn (ξ)cn(ξ) 3l ( bµ + a + µ 4 c ) sn (ξ)cn (ξ) [1 (k 1 + 1)sn (ξ) dn (ξ) + µ sn (ξ)cn (ξ)], v 13 = M ( b + µ c ) µ l Nl sn (ξ)cn(ξ) c M [1 (k 1 + 1)sn (ξ) dn (ξ) + µ sn (ξ)cn (ξ)]. (xiv) If a = 8k 1 4k 1 k 8 k + 8, b = + 6k 1 k, and c = 1, then the solution to Eq. (5) is cn (ξ, k) k 1 sn (ξ, k) 3l µ ( b + µ c ) ( sn (ξ)cn (ξ) u 14 = Q k 1 sn (ξ) cn (ξ) µ sn (ξ)cn (ξ) 3l bµ + a + µ 4 c ) sn (ξ)cn (ξ) [k 1 sn (ξ) cn (ξ) µ sn (ξ)cn(ξ)], v 14 = M ( b + µ c ) µ l Nl sn (ξ)cn (ξ) c M [k 1 sn (ξ) cn (ξ) µ sn (ξ)cn(ξ)]. (xv) If a = 4k 1 k 8k 1 8 k + 8, b = 6k 1 k, and c = 1, then the solution to Eq. (5) is cn (ξ, k) + k 1 sn (ξ, k) u 15 = Q + 3l µ ( b + µ c ) sn (ξ)cn (ξ) cn (ξ) + k 1 sn (ξ) + µ sn (ξ)cn (ξ) ( 3l bµ + a + µ 4 c ) sn (ξ)cn (ξ) [cn (ξ) + k 1 sn (ξ) + µ sn (ξ)cn(ξ)], v 15 = M ( b + µ c ) µ l Nl sn (ξ)cn (ξ) c M [cn (ξ) + k 1 sn (ξ) + µ sn (ξ)cn(ξ)]. (xvi) If a = 4k ( 4 k + k 1 k k 1 ) /( k + k 1 ), b = (k 4 + 8 k + 4 k k 1 8k 1 8)/( k + k 1 ), and c = 1/( k + k 1 ), then the solution to Eq. (5) is dn (ξ, k) + k 1 sn (ξ, k) cn(ξ, k), u 16 = Q + 3l µ ( b + µ c ) ( sn (ξ)cn(ξ) dn 3l bµ + a + µ 4 c ) sn (ξ)cn (ξ) [ (ξ) + µ sn (ξ)cn(ξ) + k 1 dn ], (ξ) + µ sn (ξ)cn (ξ) + k 1 v 16 = M ( b + µ c ) µ l bµ + a + µ 4 c Nl sn (ξ)cn(ξ) M [ dn (ξ) + µ sn (ξ)cn (ξ) + k 1 ], where ξ = lx [l ( 3 bµ 4 c b µ + 1 µ ac + ba ) /(bµ + a + µ 4 c)]t, k 1 = 1 k, µ, l, A, B, and C are arbitrary constants. It is known that when k 0, sn ξ sin ξ, cn ξ cos ξ, dn ξ 1, and when k 1, sn ξ tanhξ, cn ξ sech ξ, dn ξ sech ξ. So we can also obtain solutions of Eqs. (6) expressed in terms of hyperbolic functions and trigonometric functions. For example, if the modulus k 1, (u 1, v 1 ) degenerates as rational solitary solution of Eqs. (6), namely 3l µ ( b + µ c ) tanh (ξ)sech (ξ) u 17 = Q + sech (ξ) tanh (ξ) + µ tanh (ξ)sech (ξ) + 1

314 ZHAO Xue-Qin and ZHI Hong-Yan Vol. 50 3l ( bµ + a + µ 4 c ) tanh (ξ)sech (ξ) [ sech (ξ) tanh (ξ) + µ tanh(ξ)sech (ξ) + 1 ], v 17 = M ( b + µ c ) µ l bµ + a + µ 4 c Nl tanh(ξ)sech (ξ) M [ sech (ξ) tanh (ξ) + µ tanh(ξ)sech (ξ) + 1 ]. Remark 1 These solutions are obtained by substituting the relevant results in Case 1 into Eqs. (9), so they are only some solutions of Eqs. (6). Other solutions are omitted so as not to be too verbose. Remark All the above solutions of the coupled Drinfel d Sokolov Wilson equation are quite new and different from those reported in Refs. [4] and [5]. Remark 3 Compared with the various Jacobi elliptic function expansion methods, [1517] our improved method can not only obtain abundant doubly periodic solutions without calculating various Jacobi elliptic functions, but also get different types of doubly periodic solutions. 4 Conclusions In this paper, the improved F-expansion method is presented for finding exact solutions of nonlinear partial differential equations. The coupled Drinfel d Sokolov Wilson equation is chosen to illustrate the method such that sixteen families of Jacobi elliptic function solutions are obtained. When the modulus k 1, these obtained solutions degenerate as rational solitary wave solutions. The algorithm can also be applied to many nonlinear partial differential equations in mathematical physics. Further work about various extensions and improvement of F- expansion method needs us to find more general ansätzes or more new solutions of the auxiliary equation. Acknowledgments We would express our sincere thanks to referee for his enthusiastic help and valuable suggestions. References [1] C.S. Gardner, et al., Phys. Rev. Lett. 19 (1967) 1095. [] M.R. Miura, Bäcklund Transformation, Springer-Verlag, Berlin (1978). [3] R. Hirota, Phys. Rev. Lett. 7 (1971) 119. [4] C.H. Gu, et al. Darboux Transformation in Soliton Theory and Its Geometry Applications, Shanghai Science & Technology Press, Shanghai (1999). [5] Z.Y. Yan and H.Q. Zhang, Appl. Math. Mech. 1 (000) 38. [6] M.L. Wang, Phys. Lett. A. 199 (1995) 169. [7] Y.T. Gao and B. Tian, Comput. Phys. Commun. 133 (001) 158; B. Tian and Y.T. Gao, Z. Naturscrift. A 57 (00) 39; Y.R. Shi, et al., Acta Phys. Sin. 5 ( 003) 67 (in Chinese). [8] Y.T. Gao and B. Tian, Int. J. Mod. Phys. C 1 (003) 1431. [9] C.T. Yan, Phys. Lett. A. 4 (1996) 77. [10] Y. Chen and Y. Zhen, Int. J. Mod. Phys. C 14 (003) 601; B. Li and Y. Chen, Chaos, Solitons and Fractals 1 (004) 41. [11] E.G. Fan, Phys. Lett. A 77 (000) 1; E.G. Fan, Phys. Lett. A 94 (00) 6. [1] Z.Y. Yan, Phys. Lett. A 85 (001) 355; Z.Y. Yan, Phys. Lett. A 9 (00) 100. [13] Y. Chen and B. Li, Commun. Theor. Phys. (Beijing, China) 40 (003) 137; Y. Chen and B. Li, Chaos, Solitons and Fractals 19 (004) 977; B. Li and Y. Chen, Chaos, Solitons and Fractals 1 (004) 41. [14] Q. Wang, Y. Chen, and B. Li, Appl. Math. Comput. 160 (005) 77; Q. Wang, Y. Chen, B. Li, and H.Q. Zhang, Commun. Theor. Phys. (Beijing, China) 41 (004) 81. [15] S.K. Liu, et al., Acta. Phys. Sin. 50 (001) 068 (in Chinese). [16] Y. Chen and Q. Wang, Chaos, Solitons and Fractals 4 (005) 745; Y. Chen, Q. Wang, and B. Li, Chaos, Solitons and Fractals 6 (005) 31. [17] Q. Wang, Y. Chen, and H.Q. Zhang, Chaos, Solitons and Fractals 3 (005) 477. [18] Y.B. Zhou, M.L. Wang and Y.M. Wang, Phys. Lett. A 308 (003) 31; Y.B. Zhou, M.L. Wang, and T.D. Miao, Phys. Lett. A 33 (004) 77. [19] M.L. Wang and Y.B. Zhou, Phys. Lett. A 318 (003) 84; M.L.Wang, Y.M. Wang and J.L. Zhang, Chin. Phys. 1 (003) 1341. [0] J.L. Zhang, M.L. Wang, and D.M. Cheng, et al., Commun. Theor. Phys. (Beijing, China) 40 (003) 19; J.L. Zhang, M.L. Wang, and Z.D. Fang, J. At. Mol. Phys. 1 (004) 78. [1] V.G. Drinfeld and V.V. Sokolov, Sov. Math. Dokl. 3 (1981) 457. [] G. Wilson, Phys. Lett. A 89 (198) 33. [3] Y.Q. Yao and Z.B. Li, Phys. Lett. A 97 (00) 196. [4] B. A. Kupershmidt, Commun. Math. Phys. 99 (1985) 51. [5] E.G. Fan, J. Phys. A: Math. Gen. 36 ( 003) 7009.