Commun Theor Phys (Being, China 45 (006 pp 199 06 c International Academic Publishers Vol 45, No, February 15, 006 Further Extended Jacobi Elliptic Function Rational Expansion Method and New Families of Jacobi Elliptic Function Solutions to (+1-Dimensional Dispersive Long Wave Equation ZHANG Yuan-Yuan, WANG Qi, and ZHANG Hong-Qing Department of Applied Mathematics, Dalian University of Technology, Dalian 11604, China (Received March 31, 005; Revised June 3, 005 Abstract In this paper, a further extended Jacobi elliptic function rational expansion method is proposed for constructing new forms of exact solutions to nonlinear partial differential equations by making a more general transformation For illustration, we apply the method to (+1-dimensional dispersive long wave equation and successfully obtain many new doubly periodic solutions When the modulus m 1, these solutions degenerate as soliton solutions The method can be also applied to other nonlinear partial differential equations PACS numbers: 030Jr, 0545Yv Key words: doubly periodic solution, soliton solution, (+1-dimensional dispersive long wave equation 1 Introduction In recent years, searching for explicit exact solutions to nonlinear evolution equations has always been one central theme in mathematics and physics With the development of solitary theory, numerous work has been done on solitary wave solutions to nonlinear partial differential equations (NPDEs by using various powerful methods, such as the inverse scattering method, [1,] the truncated Painlevé expansion, [3] Hirota bilinear method, [4] the tanh method, [5 8] various extended tanh methods, [9 17] generalized hyperbolic function method, [18,19] various generalized Riccati equation expansion methods, [0 4] and various rational expansion methods, [5 8] etc Recently, Chen [9] firstly extended the Jacobi elliptic function rational expansion method by using 1 Jacobi elliptic functions and their rational combination, and obtained more forms of and more general exact doubly periodic solutions of some nonlinear partial differential equations In this paper, we would like to further extend Chen s method by using other 6 Jacobi elliptic functions and their rational combination According to the properties of these Jacobi elliptic functions, we successfully construct new forms of exact solutions that cannot be obtained by Chen s method So the method we propose can be seen as the supplement of Chen s The algorithm and its application are demonstrated later with the (+1-dimensional dispersive long wave equation Summary of Further Extended Jacobi Elliptic Function Rational Expansion Method The main steps of our further extended method are listed as follows Step 1 For a given nonlinear partial differential equation with some physical fields u i (x, y, t in three variables x, y, t, F m (u i, u i,t, u i,x, u i,y, u i,tt, u i,xt, u i,yt, u i,xx, u i,yy, u i,xy, = 0 (i = 1,,, n, (1 by using the travelling wave transformation u i (x, y, t = u i (ξ, ξ = k(x + ly + λt, ( where k, l, and λ are constants to be determined later the NPDE (1 is reduced to a nonlinear ordinary differential equation G m (u i, u i, u i, = 0 (3 Step We introduce the 6 Jacobi elliptic functions used in this paper as follows: sn ξ, ns ξ, cn ξ, nc ξ, sc ξ, and cs ξ They are the Jacobian elliptic sine function, the Jacobian elliptic cosine function, and the Jacobian elliptic functions degenerated by the two kinds of functions Other Jacobian functions can be found in Refs [30] [3], namely, [30 3] ns ξ = 1 sn ξ, nc ξ = 1 cn ξ, nd ξ = 1 dn ξ, The project partially supported by the State Key Basic Research Program of China under Grant No 004 CB 318000 E-mail: mathzhyy@yahoocomcn (4a
00 ZHANG Yuan-Yuan, WANG Qi, and ZHANG Hong-Qing Vol 45 sd ξ = sn ξ dn ξ, sn ξ sc ξ = cn ξ, cn ξ cs ξ = sn ξ, dn ξ ds ξ = sn ξ Now we introduce some new ansätze in the following forms: (i sn ξ and ns ξ rational expansion sn j (ξ u i (ξ = a i0 + a (µ 1 sn (ξ + µ ns (ξ + 1 j + b ns j (ξ (µ 1 sn (ξ + µ ns (ξ + 1 j = a i0 + a (4b sn j (ξ (µ 1 sn (ξ + µ + sn (ξ j + b 1 (µ 1 sn (ξ + µ + sn (ξ j ; (5a (ii cn ξ and nc ξ rational expansion cn j (ξ u i (ξ = a i0 + a (µ 1 cn (ξ + µ nc (ξ + 1 j + b nc j (ξ (µ 1 cn (ξ + µ nc (ξ + 1 j = a i0 + a (iii sc ξ and cs ξ rational expansion u i (ξ = a i0 + = a i0 + a a cn j (ξ (µ 1 cn (ξ + µ + cn (ξ j + b 1 (µ 1 cn (ξ + µ + cn (ξ j ; (5b sc j (ξ (µ 1 sc (ξ + µ cs (ξ + 1 j + b cs j (ξ (µ 1 sc (ξ + µ cs (ξ + 1 j sc j (ξ (µ 1 sc (ξ + µ + sc (ξ j + b 1 (µ 1 sc (ξ + µ + sc (ξ j, (5c where a i0, a, b, (i = 1,, ; j = 1,,, m i are constants to be determined later The Jacobi elliptic functions are double periodic and possess the following properties: (i Properties of triangular function (ii Derivatives of the Jacobi elliptic functions cn ξ + sn ξ = dn ξ + m sn ξ = 1, (6a ns ξ = cs ξ + 1, ns ξ = m + ds ξ, (6b sc ξ + 1 = nc ξ, m sd ξ + 1 = nd ξ (6c ξ = cn ξ dn ξ, cn ξ = sn ξ dn ξ, dn ξ = m sn ξ cn ξ, (7a ns ξ = ds ξ cs ξ, ds ξ = cs ξ ns ξ, cs ξ = ns ξ ds ξ, (7b sc ξ = nc ξ dc ξ, nc ξ = sc ξ dc ξ, cd ξ = cd ξ nd ξ, nd ξ = m sd ξ cd ξ, (7c where m is a modulus The Jacobi Glaisher functions for elliptic function can be found in Refs [30] and [31] Step 3 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 (q i ] = n i + q, D[u p i (u(q j s ] = n i p + (q + n j s (8 Balancing the highest derivative terms with nonlinear terms in Eq (3, we can get the value of m i in Eq (5 Step 4 Substitute Eq (5 into Eq (3 along with Eqs (6 and (7 and then set all the coefficients of sn i (ξ, cn i (ξ, and sc i (ξ (i = 1,, to be zero to get an over-determined system of nonlinear algebraic equations with respect to λ, l, k, a i0, a, b, µ 1, and µ (i = 1,, ; j = 1,,, m i Step 5 Solving the over-determined system of nonlinear algebraic equations by means of the Maple software Charsets by Dongming Wang, we would end up with the explicit expressions for λ, l, k, a i0, a, b, µ 1, and µ (i = 1,, ; j = 1,,, m i Step 6 By using the results obtained in the above steps, we can derive a series of Jacobi doubly periodic solutions Since lim sn ξ = tanh ξ, lim lim ns ξ = coth ξ, lim cn ξ = sech ξ, lim cs ξ = csch ξ, lim dn ξ = sech ξ, (9a ds ξ = csch ξ, (9b
No Further Extended Jacobi Elliptic Function Rational Expansion Method and New Families of 01 lim sn ξ = sin ξ, lim m 0 cn ξ = cos ξ, lim m 0 lim ns ξ = csc ξ, lim cs ξ = cot ξ, lim m 0 m 0 u i degenerate respectively as the following forms: (i Solitary wave solutions: u i (ξ = a i0 + u i (ξ = a i0 + u i (ξ = a i0 + a a a dn ξ = 1, m 0 ds ξ = csc ξ, (9d m 0 (9c tanh j (ξ (µ 1 tanh(ξ + µ /tanh(ξ + 1 j + b 1/tanh j (ξ (µ 1 tanh(ξ + µ /tanh(ξ + 1 j, (10a sech j (ξ (µ 1 sech(ξ + µ /sech(ξ + 1 j + b 1/sech j (ξ (µ 1 sech(ξ + µ 1/sech(ξ + 1 j, (10b sinh j (ξ (µ 1 sinh(ξ + µ /sinh(ξ + 1 j + b 1/sinh j (ξ (µ 1 sinh(ξ + µ /sinh(ξ + 1 j (10c (ii Triangular function formal solution: sin j (ξ u i (ξ = a i0 + a (µ 1 sin(ξ + µ /sin(ξ + 1 j + b 1/sin j (ξ (µ 1 sin(ξ + µ /sin(ξ + 1 j, (11a u i (ξ = a i0 + u i (ξ = a i0 + a a cos j (ξ (µ 1 cos(ξ + µ /cos(ξ + 1 j + b 1/cos j (ξ (µ 1 cos(ξ + µ /cos(ξ + 1 j, (11b tan j (ξ (µ 1 tan(ξ + µ /tan(ξ + 1 j + b 1/tan j (ξ (µ 1 tan(ξ + µ /tan(ξ + 1 j (11c Remark It is necessary to point out that our ansätze are different compared with the method proposed by Chen, [9] and according to the relationship between these Jacobi elliptic functions we can obtain the new solutions containing solitary wave solutions, singular solitary solutions and triangular function formal solutions that cannot be obtained by Chen s method 3 New Families of Jacobi Elliptic Function Solutions to (+1-Dimensional Dispersive Long Wave Equation In this section, we apply the method developed in Sec to the (+1-dimensional dispersive long wave equation (DLWE, ie u yt + v xx + (uu x y = 0, v t + u x + (uv x + u xxy = 0 (1 According to step 1 in Sec, we make the following transformation: u(x, t = U(ξ, v(x, t = V (ξ, ξ = k(x + ly + λt, (13 where k, l, and λ are constants to be determined later, and equation (1 becomes λlu + V + lu + luu = 0, λv + U + (UV + k lu = 0 (14 Now we consider the system (14 in the above three cases, ie Eqs (5a (5c 31 sn ξ and ns ξ Rational Expansion Now we consider the ansätz (5a Balancing the highest derivative terms with nonlinear terms in Eq (14, we suppose that equation (14 has the following formal solutions: a 1 sn (ξ + b 1 U(ξ = a 0 + µ 1 sn (ξ + µ + sn(ξ, V (ξ = A A 1 sn (ξ + B 1 0 + µ 1 sn (ξ + µ + sn(ξ + A sn 4 (ξ + B (µ 1 sn (ξ + µ + sn(ξ, (15 where a 0, a 1, b 1, A 0, A 1, A, B 1, B, µ 1, µ, are constants to be determined later With the aid of Maple, substituting Eq (15 along with Eqs (6 and (7 into Eq (14 yields a set of over-determined algebraic equations for sn i (ξ (i = 0, 1, Setting the coefficients of these terms sn i (ξ to zero yields a set of overdetermined algebraic equations with respect to a 0, a 1, b 1, A 0, A 1, B 1, A, B, µ 1, µ, k, l, and λ By use of the Maple soft package Charsets by Dongming Wang, solving the over-determined algebraic equations results in the following results:
0 ZHANG Yuan-Yuan, WANG Qi, and ZHANG Hong-Qing Vol 45 where Family 1 u 1 = a 0 + v 1 = A 0 + (1 + i 3b 1 sn (k(x + ly + λt + b 1 (sn(k(x + ly + λt + sn(k(x + ly + λt + 1, b 1 (9m +5im 3+15 5i 3l 6(1 i 3+m lb 1 (3 i 3+3m +im 3 (1 i 3+m sn (k(x + ly + λt + lb1 (5im 3+6+i 3 3(1 i 3+m sn (k(x + ly + λt + sn(k(x + ly + λt + 1 (16a sn 4 (k(x + ly + λt + lb1 (3+i 3+im 3 (1 i 3+m (sn (k(x + ly + λt + sn(k(x + ly + λt + 1, (16b λ = 6b 1m + ib 1 m 3 + 6a 0 m + 3a 0 3ia 0 3 + 6b1 ib 1 3 3(1 i, 3 + m k = 1 3i 3 + 9 6 1 i 3 + m b 1, a 0, b 1, l are arbitrary constants, ( 9ilb1 3 lb 1 + 4 m 6 + ( 36 48lb 1 + 36i 3 + 4ilb 1 3 m 4 Family A 0 = 1 1 m 6 + ( 3i 3 + 3 m 4 + ( 3i 3 3 m 1 ( 18lb1 36 6ilb 1 3 + 36i 3 m 4 + 13lb 1 5ilb 1 3 1 m 6 + ( 3i 3 + 3 m 4 + ( 3i 3 3 m u = a 0 + v = A 0 + a 1 sn(k(x + ly + λt µ 1 sn(k(x + ly + λt + 1, (17a la 1 µ 1(1+m µ 1 µ 1 +µ 1 m µ 4 1 m sn(k(x + ly + λt la 1 sn (k(x + ly + λt (µ 1 sn(k(x + ly + λt + 1 (µ 1 sn(k(x + ly + λt + 1, (17b where k = (4 µ 1 + 4 µ 1 m 4 µ 14 4 m 1 a 1, µ 1, a 0, a 1, l are arbitrary constants, Family 3 u 3 = a 0 λ = 1 a 1 µ 1 + a 1 m µ 1 a 1 µ 3 1 + a 0 µ 1 + a 0 µ 1 m a 0 µ 4 1 a 0 m µ 1 + µ 1 m µ 14 m, ( la1 4 4µ 4 1 + 8µ 1 m 4 + 3la 1 µ 4 1 4µ 4 1 4µ 8 1 + 8µ 6 1 la 6 1 µ 1 A 0 = 4((µ 1 1m + µ 1 µ 14 + ( 16µ 1 4 + 3la 1 µ 1 4 + 8µ 1 + 8µ 1 6 + la 1 6la 1 µ 1 m (4(µ 1 1m + µ 1 µ 14 b 1 µ 1 sn (k(x + ly + λt b 1 µ 1 sn (k(x + ly + λt + sn(k(x + ly + λt, (18a lb 1 µ 3 1 sn (k(x + ly + λt + lb 1 µ 1 v 3 = A 0 µ 1 sn (k(x + ly + λt + sn(k(x + ly + λt + µ 4 1 lb 1 sn 4 (k(x + ly + λt lb 1 (µ 1 sn (k(x + ly + λt + sn(k(x + ly + λt, (18b where k = (4µ 1 + 4µ 1 m 4µ 14 4m 1 b 1, λ = a 0 + b 1 µ 1, A 0 = ( 4 + lb 1 µ 1 + lb 1 m + lb 1 /4, a 0, b 1, µ 1, l are arbitrary constants Family 4 u 4 = a 0 + b 1 µ + sn(k(x + ly + λt, (19a lb 1 µ ( 1 m + µ m µ 4 m µ µ m +1 v 4 = A 0 + (µ + sn(k(x + ly + λt lb 1 (µ + sn(k(x + ly + λt, (19b where k = (4µ 4 m 4µ 4µ m + 4 1 b 1, µ, b 1, a 0, l are arbitrary constants, λ = b 1m µ 3 b 1 µ b 1 µ m + a 0 µ 4 m a 0 µ a 0 µ m + a 0 (µ 4 m µ µ m + 1,
No Further Extended Jacobi Elliptic Function Rational Expansion Method and New Families of 03 Family 5 A 0 = (3lb 1 µ 4 + 8µ 6 4µ 4 4µ 8 lb 1 µ 6 m 4 4µ 4 + lb 1 + 8µ 4 4((µ 4 µ m µ + 1 ( 3lb 1 µ 4 8µ + 6lb 1 µ 8µ 6 lb 1 + 16µ 4 m 4((µ 4 µ m µ + 1 u 5 = a 0 + km sn (k(x + ly + λt kµ m µ + sn(k(x + ly + λt v 5 = A 0 4k lµ m sn (k(x + ly + λt + 4k lm µ 3 µ + sn(k(x + ly + λt, (0a k lm sn 4 (k(x + ly + λt µ 4 k lm (µ + sn(k(x + ly + λt, (0b where λ = a 0 + kµ m, A 0 = k lm µ + lk 1 + k lm, µ, a 0, l, k are arbitrary constants 3 cn ξ and nc ξ Rational Expansion Now we consider the ansätz (5b For Eq (14, the ansätz becomes a 1 cn (ξ + b 1 U(ξ = a 0 + µ 1 cn (ξ + µ + cn(ξ, V (ξ = A A 1 cn (ξ + B 1 0 + µ 1 cn (ξ + µ + cn(ξ + A cn 4 (ξ + B (µ 1 cn (ξ + µ + cn(ξ, (1 where a 0, a 1, b 1, A 0, A 1, A, B 1, B, µ 1, µ are constants to be determined later Following the same steps in Sec 31, we can obtain the following doubly periodic solutions: where Family 6 u 6 = a 0 + v 6 = A 0 + a 0, b 1, l are arbitrary constants, Family 7 (1 + i 3b 1 cn (k(x + ly + λt + b 1 (cn (k(x + ly + λt + cn(k(x + ly + λt + 1, b 1 (6m +im 3 5l 3+i 3+6m 3lb 1 (3m +im 3 3+i 3+6m cn (k(x + ly + λt + lb1 (8im 3 5 3i 3 ( 3+i 3+6m cn (k(x + ly + λt + cn(k(x + ly + λt + 1 (a cn 4 (k(x + ly + λt + 3lb1 (im 3 1 i 3 3+i 3+6m (cn (k(x + ly + λt + cn(k(x + ly + λt + 1, (b 1 + i 3 k = 1 + 4i 3 + 4m b 1, λ = 1b 1m + 4ib 1 m 3 7b 1 ib 1 3 + 1a0 m 6a 0 + ia 0 3 (6m 3 + i 3 A 0 = (54lb 1 + 54ilb 1 3 16m 6 + ( 189lb 1 + 34 45ilb 1 3 108i 3m 4 1(18m 6 + (9i 3 7m 4 + ( 9i 3 + 9m + i 3 + ( 108 + 13lb 1 + 108i 3 1ilb 1 3m 15lb 1 4i 3 + 13ilb 1 3 1(18m 6 + (9i 3 7m 4 + ( 9i 3 + 9m + i 3 u 7 = a 0 + µ 1 µ 14 m + µ 1 m + µ 14 m k cn(k(x + ly + λt µ 1 cn(k(x + ly + λt + 1 v 7 = A 0 + ( k lµ 1 4k lm µ 1 3 + 4k lm µ 1 + 4k lµ 1 3 cn(k(x + ly + λt µ 1 cn(k(x + ly + λt + 1,, (3a + l(µ 1 + µ 1 4 m µ 1 m µ 1 4 + m k cn (k(x + ly + λt (µ 1 cn(k(x + ly + λt + 1, (3b where l, k, a 0, µ 1 are arbitrary constants, λ = kµ 1 + km µ 1 3 km µ 1 kµ 1 3 a 0 µ1 µ 14 m + µ 1 m + µ 14 m µ1 µ 14 m + µ 1 m + µ 14 m, A 0 = (lk µ 1 6 k l + 6k lµ 1 6µ 1 4 k lm 4 + lk µ 1 6 µ 1 + µ 1 4 3µ 1 4 k l (µ 14 µ 1 + 1m + µ 1 µ 1 4
04 ZHANG Yuan-Yuan, WANG Qi, and ZHANG Hong-Qing Vol 45 Family 8 + (9µ 1 4 k l + k l µ 1 4 6k lµ 1 4lk µ 1 6 1 + µ 1 m (µ 14 µ 1 + 1m + µ 1 µ 1 4 u 8 = a 0 v 8 = A 0 + b 1 µ 1 cn (k(x + ly + λt b 1 µ 1 cn (k(x + ly + λt + cn(k(x + ly + λt, (4a µ 1 3 lb 1 cn (k(x + ly + λt + b 1 µ 1 µ 1 cn (k(x + ly + λt + cn(k(x + ly + λt µ 1 4 lb 1 cn 4 (k(x + ly + λt lb 1 (µ 1 cn (k(x + ly + λt + cn(k(x + ly + λt, (4b where k = ( 4 + 4m 1 b 1, λ = b 1 µ 1 a 0, µ 1, a 0, b 1, l are arbitrary constants, Family 9 A 0 = lb 1 µ 1 + lb 1 m µ 1 lb 1 + lb 1 m 4m + 4 4(m 1 u 9 = a 0 + 1 µ 4 m µ + µ m m k µ + cn(k(x + ly + λt v 9 = A 0 + 4lk µ 3 m k lµ + 4k lµ m µ + cn(k(x + ly + λt where a 0, µ, l, k are arbitrary constants,, (5a + l(µ 4 m + µ µ m 1 + m k (µ + cn(k(x + ly + λt, (5b λ = kµ 3 m + kµ kµ m a 0 1 µ4 m µ + µ m m 1 µ4 m µ + µ m m, Family 10 A 0 = ( k l + 6k lµ 6lk µ 4 + lk µ 6 m 4 + 1 µ k l (µ 4 µ + 1m + µ 1 + ( µ 4 + µ 1 6k lµ + 3k l + 3lk µ 4 m (µ 4 µ + 1m + µ 1 u 10 = a 0 + a 1 cn (k(x + ly + λt a 1 µ µ + cn(k(x + ly + λt v 10 = A 0 la 1 µ cn (k(x + ly + λt lµ 3 a 1 µ + cn(k(x + ly + λt where λ = a 0 + a 1 µ, k = ia 1 /m, µ, a 0, a 1, l are arbitrary constants, 33 sc ξ and cs ξ Rational Expansion, (6a A 0 = la 1 + la 1 m 4m + la 1 m µ 4m Now we consider the ansätz (5c For Eq (14, the ansätz becomes U(ξ = a 0 + la 1 cn 4 (k(x + ly + λt la 1 µ 4 (µ + cn(k(x + ly + λt, (6b a 1 sc (ξ + b 1 µ 1 sc (ξ + µ + sc(ξ, V (ξ = A A 1 sc (ξ + B 1 0 + µ 1 sc (ξ + µ + sc(ξ + A sc 4 (ξ + B (µ 1 sc (ξ + µ + sc(ξ, (7 where a 0, a 1, b 1, A 0, A 1, A, B 1, B, µ 1, µ are constants to be determined later Following the same steps in Sec 31, we can obtain the following doubly periodic solutions: Family 11 u 11 = m b 1 + λ + iλ 3 λm 1 + i (1 + i 3b 1 sc (k(x + ly + λt + b 1 + 3 m (sc (k(x + ly + λt + sc(k(x + ly + λt + 1, v 11 = A 0 + lb 1 (1+i 3(+i 3 3m 1+i sc (k(x + ly + λt + lb1 (m + i 3+im 3 3 m 1+i 3 m (sc (k(x + ly + λt + sc(k(x + ly + λt + 1 (8a lb 1 ( i 3+3m +im 3 1+i 3 m sc 4 (k(x + ly + λt ilb 1 3 (sc (k(x + ly + λt + sc(k(x + ly + λt + 1, (8b
No Further Extended Jacobi Elliptic Function Rational Expansion Method and New Families of 05 where l, b 1, λ are arbitrary constants, k = 1 1 + i 3 1 + i 3 m b 1, A 0 = ( 8 + 5(1 + i 3b 1 lm 6 + 6(1 + i 3b 1 l 8 8m 6 4(3 + 3i 3m 4 + 4( 3 + 3i 3m + 8 + (3lb 1 + 1 + 1i 3 15(1 + i 3b 1 lm 4 8m 6 4(3 + 3i 3m 4 + 4( 3 + 3i 3m + 8 Family 1 + (1 1i 3 3lb 1 (1 + i 3b 1 lm 8m 6 4(3 + 3i 3m 4 + 4( 3 + 3i 3m + 8 u 1 = a 0 + v 1 = A 0 + a 1 sc(k(x + ly + λt µ 1 sc(k(x + ly + λt + 1, (9a la 1 µ 1(µ 1 m + µ 14 +µ 1 µ 1 m +1 m sc(k(x + ly + λt la 1 sc (k(x + ly + λt (µ 1 sc(k(x + ly + λt + 1 (µ 1 sc(k(x + ly + λt + 1, (9b where l, λ, a 1, µ 1 are arbitrary constants, k = (4µ 14 + 8µ 1 4µ 1 m + 4 4m 1 a 1, Family 13 a 0 = λµ 1 4 + 4λµ 1 λµ 1 m + λ λm + a 1 µ 1 3 + a 1 µ 1 a 1 µ 1 m (µ 14 + µ 1 µ 1 m + 1 m A 0 = (8µ 1 + 4µ 1 4 + 4 + la 1 m 4 4(( µ 1 1m + µ 14 + µ 1 + 1 ( 3la 1 µ 1 4 8µ 1 6 4µ 1 6la 1 µ 1 4µ 1 4 8 3la 1 m 4(( µ 1 1m + µ 14 + µ 1 + 1 4µ 1 8 + 16µ 1 + la 1 µ 1 6 + 16µ 1 6 + 4 + 6la 1 µ 1 + 4µ 1 4 + 6la 1 µ 1 4 + la 1 4(( µ 1 1m + µ 14 + µ 1 + 1 u 13 = b 1 µ 1 λ + b 1µ 1 sc(k(x + ly + λt µ 1 sc(k(x + ly + λt + 1, (30a lb 1 µ 3 1 sc (k(x + ly + λt + lb 1 µ 1 v 13 = A 0 µ 1 sc (k(x + ly + λt + sc(k(x + ly + λt + lb 1 µ 4 1 sc 4 (k(x + ly + λt lb 1 (µ 1 sc(k(x + ly + λt + sc(k(x + ly + λt, (30b where k = b 1 /, A 0 = ( 4 lb 1 + lb 1 µ 1 + lb 1 m /4, l, λ, b 1, µ 1 are arbitrary constants Family 14 u 14 = a 0 + µ m + 1 + µ + µ 4 µ 4 m k µ + sc(k(x + ly + λt v 14 = A 0 ( µ + µ m + m k lµ µ + sc(k(x + ly + λt where l, k, λ, µ are arbitrary constants,,, (31a + l( µ 4 + µ 4 m µ + µ m 1k (µ + sc(k(x + ly + λt, (31b a 0 = km µ kµ λ µ m + 1 + µ + µ 4 µ 4 m kµ 3 + kµ 3 m µ m + 1 + µ + µ 4 µ 4 m, Family 15 A 0 = (3k lµ 4 + k lµ 6 m 4 + µ 4 + 1 + µ + k lµ 6 + 6k lµ 4 + 6k lµ + k l (µ 4 + µ m µ 4 µ 1 + ( µ k l 6k lµ 9k lµ 4 4k lµ 6 µ 4 m (µ 4 + µ m µ 4 µ 1 u 15 = λ + a 1 µ + a 1 sc (k(x + ly + λt a 1 µ µ + sc(k(x + ly + λt v 15 = A 0 la 1 µ sc (k(x + ly + λt + a 1 µ 3 l µ + sc(k(x + ly + λt, (3a la 1 sc 4 (k(x + ly + λt a 1 µ 4 l (µ + sc(k(x + ly + λt, (3b
06 ZHANG Yuan-Yuan, WANG Qi, and ZHANG Hong-Qing Vol 45 where k = (4m 4 1 a 1, l, λ, a 1, µ are arbitrary constants, A 0 = (la 1 µ la 1 4m la 1 µ + la 1 + 4 4(m 1 Remark The solutions here, to our knowledge, are all new families of rational form Jacobi elliptic function solutions of the DLWE 4 Conclusion In short, we further extended Chen s method by introducing some new ansätze, and with the aid of Maple, implemented it in a computer algebraic system The validity of the method is tested by applying it to the (+1-dimensional dispersive long wave equation More importantly, we also obtain other new solutions When the modulus m 1, some of these obtained solutions degenerate as solitary wave solutions The method can be used to solve many other nonlinear equations or coupled ones References [1] CS Gardner, JM Greene, MD Kruskal, and RM Miura, Phys Rev Lett 19 (1967 1095 [] MJ Ablowitz and PA Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York (1991 [3] J Weiss, M Tabor, and G Garnevale, J Math Phys 4 (1983 5 [4] R Hirota, Phys Rev Lett 7 (1971 119 [5] W Malfliet, Am J Phys 60 (199 650 [6] W Malfliet and W Hereman, Phys Scr 54 (1996 563 [7] EJ Parkes and BR Duffy, Comput Phys Commun 98 (1996 88 [8] EJ Parkes and BR Duffy, Phys Lett A 9 (1997 17 [9] E Fan, Phys Lett A 77 (000 1 [10] E Fan, Phys Lett A 94 (00 6 [11] SA Elwakil, SK El-labany, MA Zahran, and R Sabry, Phys Lett A 99 (00 179 [1] ZY Yan, Phys Lett A 9 (001 100 [13] ZY Yan and HQ Zhang, Phys Lett A 85 (001 355 [14] ZS Lü and HQ Zhang, Phys Lett A 307 (003 69 [15] ZS Lü and HQ Zhang, Chaos, Solitons and Fractals 17 (003 669 [16] B Li, Y Chen, and HQ Zhang, Z Natur A 57 (00 874 [17] B Li, Y Chen, and HQ Zhang, Chaos, Solitons and Fractals 15 (003 647 [18] YT Gao and B Tian, Comput Phys Commun 133 (001 158 [19] B Tian and YT Gao, Z Natur A 57 (00 39 [0] Y Chen, B Li, and HQ Zhang, Commun Theor Phys (Being, China 40 (003 137 [1] Y Chen and B Li, Chaos, Solitons and Fractals 19 (004 977 [] B Li and Y Chen, Chaos, Solitons and Fractals 1 (004 41 [3] Q Wang, Y Chen, B Li, and HQ Zhang, Appl Math Comput 160 (005 77 [4] Q Wang, Y Chen, B Li, and HQ Zhang, Commun Theor Phys (Being, China 41 (004 81 [5] Q Wang, Y Chen, and HQ Zhang, Chaos, Solitons and Fractals 3 (005 477 [6] Q Wang, Y Chen, and HQ Zhang, Phys Lett A 340 (005 411 [7] Q Wang, Y Chen, and HQ Zhang, Commun Theor Phys (Being, China 43 (005 769 [8] Q Wang, Y Chen, and HQ Zhang, Commun Theor Phys (Being, China 43 (005 975 [9] Y Chen and Q Wang, Chaos, Solitons and Fractals 4 (005 745 [30] K Chandrasekharan, Elliptic Function, Springer, Berlin (1978 [31] Du Val Patrick, Elliptic Function and Elliptic Curves, Cambridge University Press, Cambridge (1973 [3] ZX Wang and XJ Xia, Special Functions, World Scientific, Singapore (1989