Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to (1+1)-Dimensional Dispersive Long Wave Equation

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1 Commun. Theor. Phys. (Beijing, China) 43 (005) pp c International Academic Publishers Vol. 43, No. 6, June 15, 005 Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to (1+1)-Dimensional Dispersive Long Wave Equation WANG Qi, 1 CHEN Yong,,3, and ZHANG Hong-Qing 1 1 Department of Applied Mathematics, Dalian University of Technology, Dalian 11604, China Department of Mathematics, Ningbo University, Ningbo 31511, China 3 Department of Physics, Shanghai Jiao Tong University, Shanghai 00030, China (Received November 1, 004; Revised November 3, 004) Abstract In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases. PACS numbers: kf Key words: elliptic equation rational expansion method, rational form solitary wave solutions, rational form Jacobi and Weierstrass doubly periodic wave solutions, symbolic computation, (1+1)-dimensional dispersive long wave equation 1 Introduction Recently, in Ref. [1], Fan proposed an algebraic method which further exceeds the applicability of the tanh method [ 8] in obtaining a series of travelling wave solutions including the soliton, triangular periodic, Jacobi, and Weierstrass doubly periodic solutions. More recently, making use of a new more general ansatz, we [9] presented the generalized algebraic method to uniformly construct a series of new and general travelling wave solution for nonlinear evolution equations (NLEEs). On the other hand, in order to extend the algebraic method [1] by Fan and the generalized method [9] by us, we [10] presented the generalized method to uniformly construct a series of soliton-like solutions and double-like periodic solutions for NLEEs. In this paper, we propose a new algebraic method, named elliptic equation rational expansion (EERE) method, which further exceeds the applicability of the above methods [1,9] in obtaining a series of travelling wave solutions including rational form solitary wave solutions, triangular periodic wave solutions and rational wave solutions. For illustration, we apply the proposed method to solve the (1+1)-dimensional dispersive long wave equation (DLWE) and successfully construct various exact solutions especially including rational form kink-shaped and bell-shaped soliton solutions, and rational form Jacobi and Weierstrass doubly periodic solutions. This paper is arranged as follows. In Sec., a detailed derivation of the proposed method will be given. The application of the proposed method to DLWE is illustrated in Sec. 3. Conclusions will be presented finally. Summary of Elliptic Equation Rational Expansion Method In the following we would like to outline the main steps of our method: Step 1 For a given NLEE system with some physical fields u i (x, y, t) in three variables x, y, t, F i (u i, u it, u ix, u iy, u itt, u ixt, u iyt, u ixx, u iyy, u ixy,...) = 0, (1) by using the wave transformation u i (x, y, t) = U i (ξ), ξ = k(x + ly + λ t), () where k, l, and λ are constants to be determined later, the nonlinear partial differential equation (1) is reduced to a nonlinear ordinary differential equation (ODE): G i (U i, U i, U i,...) = 0. (3) Step We introduce a new ansätz in terms of finite rational formal expansion in the following forms: m i a ij φ j (ξ) U i (ξ) = a i0 + (µφ(ξ) + 1) j, (4) j=1 where the new variable φ = φ(ξ) satisfies ( dφ ) φ = = h0 + h 1 φ + h φ + h 3 φ 3 + h 4 φ 4, (5) dξ The project supported by the National Outstanding Youth Foundation of China under Grant No and the China Postdoctoral Science Foundation under Grant No chenyong@dlut.edu.cn

2 976 WANG Qi, CHEN Yong, and ZHANG Hong-Qing Vol. 43 and h ρ, a i0, and a ij (ρ = 0, 1,..., 4; i = 1,,...; j = 1,,..., m i ) are constants to be determined later. Step 3 The underlying mechanism for a series of fundamental solutions such as polynomial, exponential, solitary wave, rational, triangular periodic, Jacobi and Weierstrass doubly periodic solutions to occur is that different effects act to change wave forms in many nonlinear equations, i.e. dispersion, dissipation and nonlinearity, either separately or various combination are able to balance out. We define the degree of U i (ξ) as D[U i (ξ)] = n i, which gives rise to the degrees of other expressions as D[U (α) i ] = n i +α, D[U β (α) i (U j ) s ] = n i β+(α+n j )s. (6) Therefore we can get the value of m i in Eq. (4). If n i is a nonnegative integer, then we first make the transformation U i = V i i. n Step 4 Substitute Eq. (4) into Eq. (3) along with Eq. (5) and then set all coefficients of ( ) q 4 φ p (ξ) h ρ φ ρ p = 1,,... ; q = 0, 1, ρ=0 of the resulting system s numerator to be zero to get an over-determined system of nonlinear algebraic equations with respect to k, µ, a i0, and a ij (i = 1,,...; j = 1,,..., m i ). Step 5 Solving the over-determined system of nonlinear algebraic equations by use of Maple, we will end up with the explicit expressions for k, µ, a i0, and a ij (i = 1,,...; j = 1,,..., m i ). Step 6 It is well known that the general solutions of Eq. (5) are Case A If h 3 = h 4 = 0, equation (5) possesses the following solutions: φ = h 0 ξ, h 1 = h = 0, h 0 > 0 ; (7) φ = h 0 h h 1ξ, h = 0, h 1 0 ; (8) φ = h 1 + exp( h ξ), h 0 = h 1, h > 0 ; (9) h 4h φ = h 1 h + h 1 h sin( h ξ), h 0 = 0, h < 0 ; (10) φ = h 1 h + h 1 h sinh( h ξ), h 0 = 0, h > 0. (11) Case B If h 1 = h 3 = 0, equation (5) possesses the following solutions: φ = h sech ( h ξ), h 0 = 0, h > 0, h 4 < 0 ; (1) h 4 φ = h ( tanh h ) h 4 ξ, h 0 = h, h < 0, h 4 > 0 ; (13) 4h 4 φ = h sec( h ξ), h 0 = 0, h < 0, h 4 > 0 ; (14) h 4 ( ) h h φ = tan h 4 ξ, h 0 = h, h > 0, h 4 > 0 ; (15) 4h 4 φ = 1 h4 ξ, h 0 = h = 0, h 4 > 0 ; (16) φ = sn (ξ), h 0 = 1, h = (m + 1), h 4 = m ; (17) φ = cd (ξ), h 0 = 1, h = (m + 1), h 4 = m ; (18) φ = cn (ξ), h 0 = 1 m, h = m 1, h 4 = m ; (19) φ = dn (ξ), h 0 = m 1, h = m, h 4 = 1 ; (0) φ = ns (ξ), h 0 = m, h = (m + 1), h 4 = 1 ; (1) φ = dc (ξ), h 0 = m, h = (m + 1), h 4 = 1 ; () φ = nc (ξ), h 0 = m, h = m 1, h 4 = 1 m ; (3) φ = nd (ξ), h 0 = 1, h = m, h 4 = 1 m ; (4) φ = cs (ξ), h 0 = 1 m, h = m, h 4 = 1 ; (5) φ = sc (ξ), h 0 = 1, h = m, h 4 = 1 m ; (6)

3 No. 6 Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to 977 φ = sd (ξ), h 0 = 1, h = m 1, h 4 = m (m 1) ; (7) φ = ds (ξ), h 0 = m (m 1), h = m 1, h 4 = 1 ; (8) φ = ns (ξ) ± cs (ξ), h 0 = 1 4, h = 1 m, h 4 = 1 4 ; (9) φ = nc (ξ) ± sc (ξ), φ = ns (ξ) ± ds (ξ), φ = sn (ξ) ± i cn (ξ), h 0 = 1 m 4, h = 1 + m, h 4 = 1 m 4 ; (30) h 0 = m 4, h = m, h 4 = 1 4 ; (31) h 0 = m 4, h = m, h 4 = m 4, (3) where m is a modulus. The Jacobi elliptic functions are doubly periodic and possess properties of triangular functions: sn (ξ) + cn (ξ) = 1, dn (ξ) = 1 m sn ξ, sn (ξ) = cn (ξ) dn (ξ), cn (ξ) = sn (ξ) dn (ξ), dn (ξ) = m sn (ξ) cn (ξ). When m 1, the Jacobi functions degenerate to the hyperbolic functions, i.e. sn (ξ) tanh(ξ), cn (ξ) sech (ξ). When m 0, the Jacobi functions degenerate to the triangular functions, i.e. sn (ξ) sin(ξ), cn (ξ) cos(ξ). The more detailed notations for the Weierstrass and Jacobi elliptic functions can be found in Refs. [11] and [1]. Case C If h 4 = 0, equation (5) possesses the following solutions: φ = h ( ) sech h ξ, h 0 = h 1 = 0, h > 0 ; (33) h 3 φ = h h 3 sec ( h ) ξ, h 0 = h 1 = 0, h < 0 ; (34) φ = 4 h 3 ξ, h 0 = h 1 = h = 0 ; (35) φ = ( h3 ξ, g, g 3 ), h = 0, h 3 > 0, (36) where g = 4h 1 /h 3 and g 3 = 4h 0 /h 3 are called invariants of Weierstrass elliptic function. Case D If h 0 = h 1 = 0, equation (5) possesses the following solutions: h sec ( h ξ/) φ = h h 4 tan(, h < 0 ; (37) h ξ/) + h 3 φ = h sech ( h ξ/) h h 4 tanh( h ξ/) h 3, h > 0, (38) Thus according to Eqs. (), (4), (7) (38) and the conclusions in Step 5, we can obtain some rational formal travelling-wave solutions of Eq. (1). Remark If we set the parameters in Eq. (4) to different values, the Fan s method can be recovered by the EERE method. The concrete case is as follows: Setting µ = 0, we just recover the solutions obtained by Fan s method. [1] 3 Exact Solutions of the (1+1)-Dimensional Dispersive Long Wave Equation Let us consider the (1+1)-dimensional dispersive long wave equation (DLWE), i.e., v t + vv x + w x = 0, w t + (wv) x v xxx = 0, (39) where w 1 is the elevation of the water wave, v is the surface velocity of water along x-direction. The equation system (39) can be traced back to the works of Broer, [13] Kaup, [14] Jaulent-Miodek, [15] Martinez, [16] Kupershmidt, etc. [17] A good understanding of all solutions of Eq. (39) is very helpful for coastal and civil engineers to apply the nonlinear water wave model in a harbor and coastal design. Therefore, finding more types of exact solutions of Eq. (39) is of fundamental interest in fluid dynamics. There are an amount of papers devoted to this equations. [18 1] By considering the wave transformations v(x, t) = V (ξ), w(x, t) = W (ξ), and ξ = k(x + λt), we change Eq. (39) to the form λv +V V +W = 0, λw +(W V ) k V = 0. (40) According to the proposed method, we expand the solution of Eq. (40) in the form, m v a j φ j (ξ) V (ξ) = a 0 + (µφ(ξ) + 1) j, j=1 m w A j φ j (ξ) W (ξ) = A 0 + (µφ(ξ) + 1) j, j=1 where φ(ξ) satisfies Eq. (5). Balancing the term V with term (W V ) and the term W with term V V in Eq. (40)

4 978 WANG Qi, CHEN Yong, and ZHANG Hong-Qing Vol. 43 gives m v = 1 and m w =. So we have V (ξ) = a 0 + W (ξ) = A 0 + a 1φ(ξ) µφ(ξ) + 1, A 1φ(ξ) µφ(ξ) A φ (ξ) (µφ(ξ) + 1), (41) where φ(ξ) satisfies Eq. (5). With the aid of Maple, substituting Eq. (4) along with Eq. (3) into Eq. (1), yields a set of algebraic equations 4 4 for φ p (ξ)( ρ=0 h ρφ ρ ) q, (p = 0, 1,... ; q = 0, 1). Setting the coefficients of these terms φ p (ξ)( ρ=0 h ρφ ρ ) q to zero yields a set of over-determined algebraic equations with respect to a 0, a 1, A 0, A 1, A, µ, and k. By use of the Maple software package Charsets by Dongming Wang, which is based on the Wu-elimination method, [] solving the over-determined algebraic equations, we get the following results: k = k, a 1 = ± 3 3 h4 3 h 3 µ + 3h µ 3µ 3 h µ 4 h 0 k, a 0 = ±( kh 3 + kh µ 3kµ h kh 0 µ 3 ) 3 h 4 3 h 3 µ + 3 µ 3 µ 3 h 1 + 3µ 4 h 0 λ 3h 4 3h 3 µ + 3h µ 3µ 3 h µ 4 h 0, A 0 = k ( 1 h 1 µh 4 + 4µ h 0 h 4 + 4h h 4 + 3µ 4 h 1 + 8h 0 µ 6 + 6h 3 µ h 1 ) 1(h 4 h 3 µ + h µ µ 3 h 1 + µ 4 h 0 ) k ( 16 h 3 h 0 µ 3 4 h µ 3 h h µ 4 h 0 1 µ 5 h 1 h 0 h 3 ) 1(h 4 h 3 µ + h µ µ 3 h 1 + µ 4 h 0 ) A 1 = 1 3 k ( h 3 + h µ 3 µ h 1 + 4h 0 µ 3 ), A = 3 k (h 4 h 3 µ + h µ µ 3 h 1 + µ 4 h 0 ). (4) Then according to Eqs. (4), we obtain the following solutions of DLWE (Note Since tan- and cot-type solutions appear in pairs with tanh- and coth-type solutions, respectively, we omit them in this paper. In addition, some rational solutions are also omitted). Family 1 When h 3 = h 4 = 0 and h 0 = h 1 /4h, we obtain the following solutions for the v 1 = a 0 ± 3 h µ 3µ 3 h 1 + 3µ 4 h 1 /4h kφ, (43a) 3(µ φ + 1) w 1 = A 0 + k ( h µ 3µ h 1 + h 1 µ 3 /h )φ 3(µ φ + 1), k (h µ µ 3 h 1 + µ 4 h 1 /4h )φ 3(µ φ + 1), (43b) where φ(ξ) = h 1 /h + exp( h ξ), ξ = k(x + λt), k, a 0, and A 0 are determined by Eq. (4), µ, h > 0, h 1, a 1, and λ are arbitrary constants. Family When h 0 = h 3 = h 4 = 0, we obtain the following solutions of where v = ±(kh µ 3kµ h 1 ) 3h µ 3µ 3 h 1 λ 3h µ 3µ 3 h 1 ± w = A 0 + k (h µ 3µ h 1 )φ 3(µφ + 1) 3h µ 3µ 3 h 1 kφ, (44a) 3(µφ + 1) k (h µ µ 3 h 1 )φ 3(µφ + 1), (44b) φ(ξ) = h 1 + h 1 sinh( h ξ), ξ = k(x + λt), h h k and A 0 are determined by Eq. (4), h > 0, h 1, µ, a 1, and λ are arbitrary constants. Family 3 When h 0 = h 1 = h 3 = 0, we obtain the following solutions of v 3 = a 0 ± 3 h h µ k h /h 4 sech ( h ξ) 3(µ h /h 4 sech ( h ξ) + 1), (45a) w 3 = A 0 + k h µ h /h 4 sech ( h ξ) 3(µ h /h 4 sech ( h ξ) + 1) + k (h 4 + h µ )h sech ( h ξ) 3h 4 (µ h /h 4 sech ( h ξ) + 1), where ξ = k(x + λt), k is determined by Eq. (4), h > 0, h 4 < 0, µ, a 1, and λ are arbitrary constants. (45b)

5 No. 6 Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to 979 Family 4 When h 1 = h 3 = 0 and h 0 = h /4h 4, we obtain the following solutions of DLWE, 1 h h µ + 3µ 4 h /h 4 k h /h 4 tanh( h ξ/) v 4 = a 0 ± µ h /h 4 tanh(, (46a) h ξ/) + w 4 = A 0 k ( h µ h µ 3 /h 4 ) h /h 4 tanh( h ξ/) 3(µ h /h 4 tanh( h ξ/) + ) k ( 4h 4 4h µ µ 4 h /h 4 )h tanh ( h ξ/) 3h 4 ( µ h /h 4 tanh( h ξ/) + ), (46b) where ξ = k(x + λt), k, a 0, A 0 are determined by Eq. (4), h < 0, h 4 > 0, µ, a 1, and λ are arbitrary constants. Family 5 When h 0 = h 1 = 0, we obtain the following solutions of v 5 = a 0 ± 3 h 4 3h 3 µ + 3h µ kh sech ( h ξ/) 3(h µ sech ( h ξ/) + h h 4 tanh( h ξ/) h 3 ), (47a) k ( h 3 + h µ)h sech ( h ξ/) w 5 = A 0 + 3(h µ sech ( h ξ/) + h h 4 tanh( h ξ/) h 3 ) k (h 4 h 3 µ + h µ )h sech 4 ( h ξ/) 3(h µ sech ( h ξ/) + h h 4 tanh( h ξ/) h 3 ), (47b) where ξ = k(x + λt), k, a 0, and A 0 are determined by Eq. (4), h > 0, h 3, h 4, µ, a 1, and λ are arbitrary constants. Family 6 When h = h 4 = 0, we obtain the following solutions for the where v 6 = a 0 ± 3h 3 µ 3µ 3 h 1 + 3µ 4 h 0 kφ 3(µ φ + 1) w 6 = A 0 + k ( h 3 3µ h 1 + 4h 0 µ 3 )φ 3(µφ + 1), (48a) k ( h 3 µ µ 3 h 1 + µ 4 h 0 )φ 3(µφ + 1), (48b) ( h 3 ξ ) φ(ξ) =, g, g 3, ξ = k(x + λt), g = 4 h 1, g 3 = 4 h 0, h 3 h 3 and k, A 0 are determined by Eq. (4), h 3 > 0, h 0, h 1, µ, a 1, and λ are arbitrary constants. Family 7 When h 1 = h 3 = 0, h 0 = 1, h = (m + 1), and h 4 = m, we obtain the following solutions for the v 7 = a 0 ± 3m + 3( m 1)µ + 3µ 4 k sn (ξ) 3(µ sn (ξ) + 1) w 7 = A 0 + k (( m 1)µ + 4µ 3 ) sn (ξ) 3(µ sn (ξ) + 1), (49a) k (m + ( m 1)µ + µ 4 ) sn (ξ) 3(µ sn (ξ) + 1), (49b) where ξ = k(x + λt), k, and A 0 are determined by Eq. (4), µ, a 1, and λ are arbitrary constants. Family 8 When h 1 = h 3 = 0, h 0 = 1, h = (m + 1), and h 4 = m, we obtain the following solutions for the DLWE, v 8 = a 0 ± 3m + 3( m 1)µ + 3µ 4 k cd (ξ) 3(µ cd (ξ) + 1) w 8 = A 0 k ( µ( m 1) 4 µ 3 ) cd (ξ) 3(µ cd (ξ) + 1), (50a) + k ( m ( m 1)µ µ 4 ) cd (ξ) 3(µ cd (ξ) + 1), (50b) where ξ = k(x + λt), k, and A 0 are determined by Eq. (4), µ, a 1, and λ are arbitrary constants. Family 9 When h 1 = h 3 = 0, h 0 = 1 m, h = m 1, and h 4 = m, we obtain the following solutions for the v 9 = a 0 ± 3m + 3(m 1)µ + 3µ 4 (1 m ) k cn (ξ) 3(µ cn (ξ) + 1) w 9 = A 0 k ( (m 1)µ 4(1 m )µ 3 ) cn (ξ) 3(µ cn (ξ) + 1), (51a) + k (m (m 1)µ µ 4 (1 m )) cn (ξ) 3(µ cn (ξ) + 1), (51b)

6 980 WANG Qi, CHEN Yong, and ZHANG Hong-Qing Vol. 43 Family 10 When h 1 = h 3 = 0, h 0 = m 1, h = m, and h 4 = 1, we obtain the following solutions for the v 10 = a 0 ± 3 + 3( m )µ + 3µ 4 (m 1) k dn (ξ) 3(µ dn (ξ) + 1) w 10 = A 0 k ( ( m )µ 4 (m 1)µ 3 ) dn (ξ) 3(µ dn (ξ) + 1), (5a) + k (1 ( m )µ µ 4 (m 1)) dn (ξ) 3(µ dn (ξ) + 1), (5b) Family 11 When h 1 = h 3 = 0, h 0 = m, h = (1 + m ), and h 4 = 1, we obtain the following solutions for the v 11 = a 0 ± 3 + 3( m 1)µ + 3µ 4 m k ns (ξ) 3(µ ns (ξ) + 1) w 11 = A 0 k ( µ ( m 1) 4m µ 3 ) ns (ξ) 3(µ ns (ξ) + 1), (53a) + k ( 1 ( m 1)µ µ 4 m ) ns (ξ) 3(µ ns (ξ) + 1), (53b) where ξ = k(x + λt), k, and A 0 are determined by Eq. (4), µ, a 1, and λ are arbitrary constants. Family 1 When h 1 = h 3 = 0, h 0 = m, h = (1 + m ), and h 4 = 1, we obtain the following solutions for the v 1 = a 0 ± 3 + 3( m 1)µ + 3µ 4 m k dc (ξ) 3(µ dc (ξ) + 1) w 1 = A 0 k ( ( m 1)µ 4m µ 3 ) dc (ξ) 3(µ dc (ξ) + 1), (54a) + k ( 1 ( m 1)µ µ 4 m ) dc (ξ) 3(µ dc (ξ) + 1), (54b) Family 13 When h 1 = h 3 = 0, h 0 = m, h = m 1, and h 4 = 1 m, we obtain the following solutions for the v 13 = a 0 ± 3 3m + 3(m 1)µ 3µ 4 m k nc (ξ) 3(µ nc (ξ) + 1) w 13 = A 0 + k ((m 1)µ 4m µ 3 ) nc (ξ) 3(µ nc (ξ) + 1), (55a) k (1 m + (m 1)µ µ 4 m ) nc (ξ) 3(µ nc (ξ) + 1), (55b) Family 14 When h 1 = h 3 = 0, h 0 = 1, h = m, and h 4 = m 1, we obtain the following solutions for the v 14 = a 0 ± 3m 3 + 3( m )µ 3µ 4 k nd (ξ) 3(µ nd (ξ) + 1) w 14 = A 0 k ( ( m )µ + 4µ 3 ) nd (ξ) 3(µ nd (ξ) + 1), (56a) + k ( m + 1 ( m )µ + µ 4 ) nd (ξ) 3(µ nd (ξ, m) + 1), (56b) where ξ = k(x + λt), k, and A 0 are determined by Eq. (4), µ, a 1, and λ are arbitrary constants. Family 15 When h 1 = h 3 = 0, h 0 = 1 m, h = m, and h 4 = 1, we obtain the following solutions for the v 15 = a 0 ± 3 + 3( m )µ + 3µ 4 (1 m ) kcs (ξ) 3(µcs (ξ) + 1) w 15 = A 0 + k (( m )µ + 4(1 m )µ 3 )cs (ξ) 3(µcs (ξ) + 1), (57a) k (1 + ( m )µ + µ 4 (1 m ))cs (ξ) 3(µcs (ξ) + 1), (57b) Family 16 When h 1 = h 3 = 0, h 0 = 1, h = m, and h 4 = 1 m, we obtain the following solutions for the v 16 = a 0 ± 3 3m + 3( m )µ + 3µ 4 k sc (ξ) 3(µ sc (ξ) + 1), (58a)

7 No. 6 Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to 981 w 16 = A 0 + k (( m )µ + 4µ 3 ) sc (ξ) 3(µ sc (ξ) + 1) k (1 m + ( m )µ + µ 4 ) sc (ξ) 3(µ sc (ξ) + 1), (58b) where ξ = k(x + λt), k, and A 0 are determined by Eq. (4), µ, a 1, and λ are arbitrary constants. Family 17 When h 1 = h 3 = 0, h 0 = 1, h = m 1, and h 4 = m (m 1), we obtain the following solutions for the v 17 = a 0 ± 3m (m 1) + 3(m 1)µ + 3µ 4 k sd (ξ) 3(µ sd (ξ) + 1) w 17 = A 0 + k ((m 1)µ + 4µ 3 ) sd (ξ) 3(µ sd (ξ) + 1), (59a) k (m (m 1) + (m 1)µ + µ 4 ) sd (ξ) 3(µ sd (ξ) + 1), (59b) Family 18 When h 1 = h 3 = 0, h 0 = m (m 1), h = m 1, and h 4 = 1, we obtain the following solutions for the v 18 = a 0 ± 3 + 3(m 1)µ + 3µ 4 m (m 1) k ds (ξ) 3(µ ds (ξ) + 1) w 18 = A 0 k ( (m 1)µ 4m (m 1)µ 3 ) ds (ξ) 3(µ ds (ξ) + 1), (60a) + k ( 1 (m 1)µ µ 4 m (m 1)) ds (ξ) 3(µ ds (ξ) + 1), (60b) Family 19 When h 1 = h 3 = 0, h 0 = 1/4, h = (1 m )/, and h 4 = 1/4, we obtain the following solutions for the 3 + 6(1 m )µ v 19 = a 0 ± + 3µ 4 k( ns (ξ) ± cs (ξ)), (61a) 3(µ( ns (ξ) ± cs (ξ)) + 1) w 19 = A 0 + k ((1 m )µ + µ 3 )( ns (ξ) ± cs (ξ)) 3(µ( ns (ξ) ± cs (ξ)) + 1) k (1 + (1 m )µ + µ 4 )( ns (ξ) ± cs (ξ)) 6(µ( ns (ξ) ± cs (ξ)) + 1), (61b) Family 0 When h 1 = h 3 = 0, h 0 = (1 m )/4, h = (1 + m )/, and h 4 = (1 m )/4, we obtain the following solutions for the 3 3m + 6(1 + m v 0 = a 0 ± )µ + 3µ 4 (1 m )k( nc (ξ) ± sc (ξ)), (6a) 3(µ( nc (ξ) ± sc (ξ)) + 1) w 0 = A 0 k ( (1 + m )µ (1 m )µ 3 )( nc (ξ) ± sc (ξ)) 3(µ ( nc (ξ) ± sc (ξ)) + 1) + k ( 1 + m (1 + m )µ µ 4 (1 m ))( nc (ξ) ± sc (ξ)) 6(µ( nc (ξ) ± sc (ξ)) + 1), (6b) Family 1 When h 1 = h 3 = 0, h 0 = m 4 /4, h = (m )/, and h 4 = 1/4, we obtain the following solutions for the 3 + 6(m )µ v 1 = a 0 ± + 3µ 4 m 4 k( ns (ξ) ± ds (ξ)), (63a) 3(µ( ns (ξ) ± ds (ξ)) + 1) w 1 = A 0 k ( (m )µ m 4 µ 3 )( ns (ξ) ± ds (ξ)) 3(µ( ns (ξ) ± ds (ξ)) + 1) + k ( 1 (m )µ µ 4 m 4 )( ns (ξ) ± ds (ξ)) 6(µ( ns (ξ) ± ds (ξ)) + 1), (63b) Family When h 1 = h 3 = 0, h 0 = m /4, h = m /, and h 4 = m /4, we obtain the following solutions for the 3m + 6(m v = a 0 ± )µ + 3µ 4 m k( sn (ξ) ± i cn (ξ)), (64a) 3(µ( sn (ξ) ± i cn (ξ)) + 1)

8 98 WANG Qi, CHEN Yong, and ZHANG Hong-Qing Vol. 43 w = A 0 k ( (m )µ µ 3 m )( sn (ξ) ± i cn (ξ)) 3(µ( sn (ξ) ± i cn (ξ)) + 1) k (m + (m )µ + µ 4 m )( sn (ξ) ± i cn (ξ)) 6(µ( sn (ξ) ± i cn (ξ)) + 1), (64b) 4 Summary and Conclusion Based on the EERE method and symbolic computation, we obtain many types of solutions including rational form solitary wave solutions, rational form Jacobi, and Weierstrass doubly periodic solutions for the (1+1)-DLWE. The success of the EERE method lies in the following fact: We introduce a new ansätz in terms of finite rational formal expansion (4), which is more general than known ansätz in tanh method [1 3,5 9] for seeking travelling wave solutions of NLEEs. Writing the soliton solutions of a nonlinear equation as the polynomials of auxiliary variables of the elliptic equation, the equation can be changed into a nonlinear system of algebraic equations. The system can be solved with the help of symbolic computation system such as Maple. The algorithm can be also applied to a wide class of NLEEs in mathematical physics. The method can be extended to find rational form soliton-like solutions and more types of rational form double periodic solutions of Eq. (1). Only will the restriction on ξ(x, y, t) in Eq. (4) as merely a linear function x, y, t, and the restrictions on the coefficients h ρ, a i0 and a ij (ρ = 0, 1,..., 4; i = 1,,...; j = 1,,..., m i ) as constants be removed. Acknowledgments We are grateful to Prof. S.Y. Lou for his enthusiastic guidance and help. References [1] E. Fan, Comput. Phys. Commun. 153 (003) 17; E. Fan, Chaos, Solitons and Fractals 15 (003) 559; E. Fan, Chaos, Solitons and Fractals 16 (003) 819; E. Fan, Chaos, Solitons and Fractals 19 (004) 1141; E. Fan, Phys. Lett. A300 (00) 43. [] E.J. Parkes and B.R. Duffy, Comput. Phys. Commun. 98 (1996) 88; E.J. Parkes and B.R. Duffy, Phys. Lett. A9 (1997) 17. [3] M.L. Wang, Phys. Lett. A16 (1996) 67. [4] Y.T. Gao and B. Tian, Comput. Phys. Commun. 133 (001) 158; Y.T. Gao and B. Tian, Int. J. Mod. Phys. C1 (001) 1431; B. Tian and Y.T. Gao, Int. J. Mod. Phys. C1 (001) 361. [5] S.K. Liu, et al., Phys. Lett. A90 (001) 7. [6] E. Fan, Phys. Lett. A77 (000) 1; E. Fan, Comput. Math. Appl. 43 (00) 671. [7] Z.Y. Yan, Comput. Phys. Commun. 153 (003) 145. Z.Y. Yan and H.Q. Zhang, Phys. Lett. A85 (001) 355. [8] Y. Chen, B. Li, and H.Q. Zhang, Int. J. Mod. Phys. C13 (003) 99; Y. Chen, Z.Y. Yan, and H.Q. Zhang, Phys. Lett. A307 (003) 107; Y. Chen and Y. Zheng, Int. J. Mod. Phys. C14(5) (003) 601; Y. Chen and B. Li, Chaos, Solitons and Fractals 19(4) (004) 977; Q. Wang, Y. Chen, and H.Q. Zhang, Commun. Theor. Phys. (Beijing, China) 41 (004) 81. [9] Y. Chen, Q. Wang, and B. Li, Commun. Theor. Phys. (Beijing, China) 4 (004) 39; Y. Chen, Q. Wang, and B. Li, Chaos, Solitons and Fractals (3) (004) 675; Y. Chen and Q. Wang, Int. J. Mod. Phys. C15 (004) 595. [10] Y. Chen, Q. Wang, and B. Li, Commun. Theor. Phys. (Beijing, China) 4 (004) 655; Y. Chen and Q. Wang, Chaos, Solitons and Fractals 3 (005) 801; Y. Chen, Q. Wang, and B Li, Chaos, Solitons and Fractals 3 (005) [11] K. Chandrasekharan, Elliptic Function, Springer, Berlin (1978). [1] Patrick Du Val, Elliptic Function and Elliptic Curves, Cambridge University Press, Cambridge (1973). [13] L.J.F. Broer, Appl. Sci. Res. 31 (1975) 377. [14] D.J. Kaup, Prog. Theor. Phys. 54 (1975) 7. [15] M. Jaulent and J. Miodek, Lett. Math. Phys. 1 (1976) 43. [16] L. Martinez, J. Math. Phys. 1 (1980) 34. [17] B.A. Kupershmidt, Commun. Math. Phys. 99 (1985) 51. [18] C.L. Chen and S.Y. Lou, Chaos, Solitons and Fractals 16 (003) 7. [19] M.L. Wang, Phys. Lett. A199 (1995) 169. [0] X.D. Zheng, Y. Chen, and H.Q. Zheng, Phys. Lett. A311 (003) 145. [1] Q. Wang and Y. Chen, and H.Q. Zhang, Chaos, Solitons and Fractals 3 (005) 477. [] W. Wu, in Algorithms and Computation, eds. D.Z. Du, et al., Springer, Berlin (1994) p. 1.