A Generalized Extended F -Expansion Method and Its Application in (2+1)-Dimensional Dispersive Long Wave Equation

Commun. Theor. Phys. (Beijing, China) 6 (006) pp. 580 586 c International Academic Publishers Vol. 6, No., October 15, 006 A Generalized Extended F -Expansion Method and Its Application in (+1)-Dimensional Dispersive Long Wave Equation HUANG Wen-Hua 1,, 1 Department of Physics, College of Science, Huzhou University, Huzhou 313000, China Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 0007, China (Received December 8, 005) Abstract A new generalized extended F -expansion method is presented for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. As an application of this method, we study the (+1)-dimensional dispersive long wave equation. With the aid of computerized symbolic computation, a number of doubly periodic wave solutions expressed by various Jacobi elliptic functions are obtained. In the limit cases, the solitary wave solutions are derived as well. PACS numbers: 0.30.Jr, 03.65.Ge, 05.5.Yv, 0.30.Nk Key words: (+1)-dimensional dispersive long wave equation, extended F -expansion, Jacobi elliptic function, periodic wave solution 1 Introduction Nonlinear evolution equations (NEEs) are widely used to describe the physical mechanism of the natural or social phenomena. Searching for and constructing exact solutions for NEEs is an important work in nonlinear science. In recent years, the doubly periodic wave solutions in terms of the Jacobi elliptic functions for NEEs attract considerable interests due to the elegant properties of the elliptic function. Various Jacobi elliptic function solutions for a wide class of NEEs in mathematical physics have been found by many means such as Jacobi elliptic function expansion method, [1 5] the extended Jacobi elliptic function expansion method, [6 9] the sinh-gordon equation expansion method, [10] the mixed dn-sn method, [11] etc. More recently, the F -expansion method is presented to construct periodic wave solutions of NEEs, [1 1] which can be thought of as a concentration of Jacobi elliptic function expansion since F here stands for any Jacobi elliptic functions. Furthermore, the F -expansion method also has been extended to obtain not only the single nondegenerative Jacobi elliptic function solutions, but also the combined nondegenerative Jacobi elliptic solutions and their corresponding degenerative solutions. [15 19] In this paper, we will propose a new generalized extended F -expansion method to solve NEEs. As an example to illustrate this method, we applied it to study the (+1)-dimensional dispersive long wave equation (DLWE), many doubly periodic wave solutions are obtained. In the limit cases, the solitary wave solutions are derived as well. This paper is organized as follows. In Sec., we introduce the new generalized extended F -expansion method. In Sec. 3, we apply the generalized method to DLWE and give out many exact solutions. Summary and discussion will be presented finally. Description of Generalized Extended F -Expansion Method For a given NEE system with some physical fields u i (x, y, t) (i = 1,,..., ) in three variable x, y, t, we first make the travelling wave transformation G i (u i, u it, u ix, u iy, u itt, u ixt, u iyt, u ixx, u iyy, u ixy,...) = 0, (1) u i (x, y, t) = u i (ξ), ξ = k(x + ly + ωt), () where k, l, and ω are constants to be determined later, then equation (1) is reduced to a nonlinear ordinary differential equation (ODE): P i = (u i, u i, u i,...) = 0. (3) We express the solution of ODE (3) by the new more general ansatz, m i { u i (ξ) = a i0 + F (ξ) j 1[ F (ξ) a ij ( 1 F (ξ) + G(ξ) + r) j + b G(ξ) ]} ij ( 1 F (ξ) + G(ξ) + r) j, () j=1 The project supported in part by National Natural Science Foundation of China under Grant No. 107071 and the Science Research Foundation of Huzhou University under Grant No. KX105 E-mail addresses: whhuanghz@hutc.zj.cn, whhuang00cn@yahoo.com.cn

No. A Generalized Extended F -Expansion Method and Its Application in (+1)-Dimensional Dispersive 581 where m i is an integer to be determined by balancing the highest-order derivative terms with the nonlinear terms in Eq. (3), and a ij (j = 0, 1,,..., m i ), b ij (j = 1,,..., m i ), 1,, r are constants to be determined later. The functions F (ξ) and G(ξ) satisfy the following relations: F (ξ) = P 1 F (ξ) + Q 1 F (ξ) + R 1 (F (ξ) = P 1 F 3 (ξ) + Q 1 F (ξ)), G (ξ) = P G (ξ) + Q G (ξ) + R (G (ξ) = P G 3 (ξ) + Q G(ξ)), G (ξ) = µf (ξ) + ν (G (ξ) = µf (ξ)f (ξ)/g(ξ)), (5) R 1 = (Q 1 Q ) 9P R, µ = P 1, ν = Q 1 Q, ν 0, (6) 9P 1 P 3P where F (ξ) = df (ξ)/dξ, G (ξ) = dg(ξ)/dξ, and P 1, P, Q 1, Q, R 1, R, µ, ν are constants. F (ξ) and G(ξ) can be different Jacobi elliptic functions with known parameters P i (i = 1, ), Q i (i = 1, ), R i (i = 1, ), µ, and ν. Substituting Eq. () with Eq. (5) into Eq. (3) and collecting coefficients of polynomials of F (ξ), G(ξ), F (ξ) with the aid of Maple or Mathematica, then setting each coefficient to zero, we can deduce a set of over-determined nonlinear algebraic equations. With the help of Maple, solving the equations, then we can determine a ij, b ij, k, l, ω, 1,, and r, and at the same time substitute different kinds of Jacobi elliptic functions into Eqs. (5) and (6) to get the relations between the parameters and the modulus m (0 m 1) of Jacobi elliptic function. Finally, substituting a ij, b ij, k, l, ω, 1, and r into Eq. () with the corresponding solutions of F (ξ) and G(ξ), we can get the exact Jacobi elliptic function periodic wave solutions of the given NEE (1). As is known to all, when the modulus m 1, Jacobi elliptic functions can degenerate as hyperbolic functions and when m 0, the Jacobi elliptic functions degenerate as trigonometric functions. So by this method we can get many other new exact solutions to NEE (1). Remark 1 By the description above, we can see that if the functions F and G in ansatz () are replaced by the concrete forms of different Jacobi elliptic functions, our method can reduce to the extended Jacobian elliptic function expansion algorithm presented in Ref. [6] when 1 = = 0, r = 1 and the extended Jacobi elliptic function rational expansion method in Ref. [9] when r = 1 respectively. On the other hand, as 1 = = 0, r = 1 this method reduces to the extended F -expansion method in Ref. [15] directly. So this method is more general, convenient and powerful. Furthermore, if we do not reduce Eq. (1) to an ODE, but directly assume that equation (1) has the formal solution m i { u i (ξ) = a i0 (x, y, t) + F j 1[ F a ij (x, y, t) ( 1 F + G + r) j + b G ]} ij(x, y, t) ( 1 F + G + r) j, (7) j=1 where F = F (ψ(x, y, t)), G = G(ψ(x, y, t)), and a ij (x, y, t), b ij (x, y, t) (j = 0, 1,,..., m i ), ψ(x, y, t) are functions to be determined later, then we may obtain more type of doubly periodic solutions of NEE (1). In this paper, we only consider to construct the exact solutions of NEE (1) in the travelling wave form. 3 Exact Travelling Wave Solutions for DLWE Now, let us consider the (+1)-dimensional dispersive long wave equation (DLWE), i.e., u yt + v xx + (uu x ) y = 0, v t + u x + (uv) x + u xxy = 0. (8) The system DLWE (8) was first derived by Boiti et al. [0] as a compatibility for a weak Lax pair. In recent years, many authors have studied this system by various methods and obtained abundant physically significant solutions. [6,9,1 ] According to the above method, to seek exact solution of Eqs. (8), we first make the travelling wave transformation then equation (8) reduces to the following ODEs: u(x, y, t) = U(ξ), v(x, y, t) = V (ξ), ξ = k(x + ly + ωt), (9) ωlu ξξ + V ξξ + lu ξ + luu ξξ = 0, ωv ξ + U ξ + (UV ) ξ + k lu ξξξ = 0. (10) Balancing the highest derivative terms with nonlinear terms in Eqs. (10), we suppose that equation (10) has formal solution in the form F U = a 0 + a 1 1 F + G + r + b G 1 1 F + G + r, F V = A 0 + A 1 1 F + G + r + B G 1 1 F + G + r + A ( 1 F + G + r) + B F G ( 1 F + G + r), (11) where a 0, a 1, b 1, A 0, A 1, A, B 1, B, 1,, and r are constants to be determined. With the aid of Maple, substituting Eq. (11) with Eq. (5) into Eq. (10) yields a set of algebraic equations for F i G j, (i = 0, 1,..., 8; j = 0, 1). Setting the F

58 HUANG Wen-Hua Vol. 6 coefficients of F i G j to zero yields a set of over-determined algebraic equations with respect to a 0, a 1, b 1, A 0, A 1, A, B 1, and B. Solving the over-determined algebraic equations results in the following results: a 0 = ω, a 1 = ±rk P 1, A 0 = 1 k lq 1, A = P 1 lr k, b 1 = A 1 = B 1 = B = 1 = = 0, (1) kr 1 Q 1 + k 3 a 0 = ω 1R 1 ±r 1 r Q 1 + 1 R, a 1 = ± k 1 + r 1 P 1 r r Q 1 + 1 R 1 + r P 1, b 1 = B 1 = B = = 0, 6 A 0 = 1k R1l r ( 1 r Q 1 + 1 R 1 + r P 1 ) r Q 1 ( 1 + r k P 1 l) + 1R 1 + r P 1 1 r Q 1 + 1 R 1 + r 3k 1lR 1 ( 1Q 1 + r P 1 ) P 1 1 r Q 1 + 1 R 1 + r, P 1 A 1 = k l 1 r ( 1R 1 + r Q 1 ), A = k l r ( 1R 1 + 1Q 1 r + r P 1 ). (13) Substituting Eqs. (1) and (13) into Eq. (11) with Eq. (9), we derive two families of concentration formulas of travelling wave solutions to DLWE (8): (a) u = ω ± k P 1 F (ξ), v = 1 k lq 1 k lp 1 F (ξ), (1) where ξ = k(x + ly + ωt), k, l, ω are arbitrary constants; kr 1 Q 1 + k 3 (b) u = ω 1R 1 ±r 1 r Q 1 + 1 R ± k 1 r Q 1 + 1 R 1 + r P 1 F (ξ), 1 + r P 1 r( 1 F (ξ) + r) v = 6 1k R 1l r ( 1 r Q 1 + 1 R 1 + r P 1 ) r Q 1 ( 1 + r k P 1 l) + 1R 1 + r P 1 1 r Q 1 + 1 R 1 + r P 1 3k 1lR 1 ( 1Q 1 + r P 1 ) 1 r Q 1 + 1 R 1 + r P 1 + k l 1 ( 1R 1 + r Q 1 ) r F (ξ) k l( 1R 1 + 1Q 1 r + r P 1 ) ( 1 F (ξ) + r) r ( 1 F (ξ) + r) F (ξ), (15) where ξ = k(x + ly + ωt), while k, l, ω, and 1 are arbitrary constants and r 0. According to concentration formulas (1) and (15), setting F (ξ) to be possible Jacobian elliptic functions with suitable parameters P i, Q i, and R i, doubly periodic wave solutions expressed by various Jacobian elliptic functions to DLWE (8) can be obtained. Considering there exist twelve types of Jacobian elliptic function solution F (ξ) with known values of (R 1, Q 1, P 1 ), we can simultaneously construct twelve Jacobian elliptic function periodic wave solutions for the general forms of travelling solutions (1) and (15) respectively. For the concentration formula (1), substituting the values of (R 1, Q 1, P 1 ) and the corresponding Jacobi F (ξ) chosen into it, we derive the following twelve single nondegenerative Jacobian elliptic function solutions. (i) P 1 = m, Q 1 = (1 + m ), R 1 = 1, then F (ξ) = sn (ξ) or cd (ξ), we have u 1 = ω ± mk sn [k(x + ly + ωt)], v 1 = 1 + k l + k m l m k l sn [k(x + ly + ωt)], (16) u = ω ± mk cd [k(x + ly + ωt)], v = 1 + k l + k m l m k l cd [k(x + ly + ωt)] ; (17) (ii) P 1 = m, Q 1 = m 1, R 1 = 1 m, then F (ξ) = cn (ξ), yields u 3 = ω ± imk cn [k(x + ly + ωt)], v 3 = 1 k m l + k l + m k l cn [k(x + ly + ωt)] ; (18) (iii) P 1 = 1, Q 1 = m, R 1 = m 1, then F (ξ) = dn (ξ), yields u = ω ± ik dn [k(x + ly + ωt)], v = 1 k l + k m l + k l dn [k(x + ly + ωt)] ; (19) (iv) P 1 = 1, Q 1 = (1 + m ), R 1 = m, then F (ξ) = ns (ξ) or dc (ξ), we have u 5 = ω ± k ns [k(x + ly + ωt)], v 5 = 1 + k l + k m l k l ns [k(x + ly + ωt)], (0) u 6 = ω ± k dc [k(x + ly + ωt)], v 6 = 1 + k l + k m l k l dc [k(x + ly + ωt)] ; (1) (v) P 1 = 1 m, Q 1 = m 1, R 1 = m, then F (ξ) = nc (ξ), yields u 7 = ω ± k 1 m nc [k(x + ly + ωt)], v 7 = 1 k m l + k l k l(1 m )nc [k(x + ly + ωt)]; () (vi) P 1 = m 1, Q 1 = m, R 1 = 1, then F (ξ) = nd (ξ), yields u 8 = ω ± ik 1 m nd [k(x + ly + ωt)], v 8 = 1 k l + k m l + k l(1 m )nd [k(x + ly + ωt)] ; (3) (vii) P 1 = 1 m, Q 1 = m, R 1 = 1, then F (ξ) = sc (ξ), yields u 9 = ω ± k 1 m sc [k(x + ly + ωt)], v 9 = 1 k l + k m l k l(1 m )sc [k(x + ly + ωt)] ; () (viii) P 1 = m (1 m ), Q 1 = m 1, R 1 = 1, then F (ξ) = sd (ξ), yields u 10 = ω ± imk 1 m sd [k(x + ly + ωt)], v 10 = 1 k m l + k l + k m l(1 m )sd [k(x + ly + ωt)] ; (5) (ix) P 1 = 1, Q 1 = m, R 1 = 1 m, then F (ξ) = cs (ξ), yields u 11 = ω ± k cs [k(x + ly + ωt)], v 11 = 1 k l + k m l k l cs [k(x + ly + ωt)] ; (6)

No. A Generalized Extended F -Expansion Method and Its Application in (+1)-Dimensional Dispersive 583 (x) P 1 = 1, Q 1 = m 1, R 1 = m (1 m ), then F (ξ) = ds (ξ), yields u 1 = ω ± k ds [k(x + ly + ωt)], v 1 = 1 k m l + k l k l ds [k(x + ly + ωt)]. (7) In the above solutions, k, l, and ω are arbitrary constants and i = 1. Similarly, substituting the values of (R 1, Q 1, P 1 ) and the corresponding Jacobi F (ξ) chosen into the concentrate formulas (15), we can derive other twelve Jacobi elliptic function solutions as follows: kr 1 (1 + m ) k 3 1 u 13 = ω + ±r 1 + m r 1 r 1 r m ± k 1 + m r 1 r 1 r m sn [k(x + ly + ωt)], r{ 1 sn [k(x + ly + ωt)] + r} 6 v 13 = 1k l r ( 1 + m r 1 r 1 r m ) + r (1 + m )( 1 + r k m l) 1 r m 1 + m r 1 r 1 r m + 3k 1l( 1 + 1m r m ) 1 + m r 1 r 1 r m + k l 1 ( 1 r r m ) r sn [k(x + ly + ωt)] { 1 sn [k(x + ly + ωt)] + r} k l( 1 + m r 1r 1r m ) r { 1 sn [k(x + ly + ωt)] + r} sn [k(x + ly + ωt)] ; (8) kr 1 (1 + m ) k 3 1 u 1 = ω + ±r 1 + m r 1 r 1 r m ± k 1 + m r 1 r 1 r m cd [k(x + ly + ωt)], r{ 1 cd [k(x + ly + ωt)] + r} 6 v 1 = 1k l r ( 1 + m r 1 r 1 r m ) + r (1 + m )( 1 + r k m l) 1 r m 1 + m r 1 r 1 r m + 3k 1l( 1 + 1m r m ) 1 + m r 1 r 1 r m + k l 1 ( 1 r r m ) r cd [k(x + ly + ωt)] { 1 cd [k(x + ly + ωt)] + r} k l( 1 + m r 1r 1r m ) r { 1 cd [k(x + ly + ωt)] + r} cd [k(x + ly + ωt)] ; (9) kr 1 (m 1) + k 3 u 15 = ω 1(1 m ) ±r 1 r m 1 r + 1 1 m m r ± k 1 r m 1 r + 1 1 m m r r{ 1 cn [k(x + ly + ωt)] + r} cn [k(x + ly + ωt)], v 15 = 6 1k (1 m ) l + r (m 1)( 1 r k m l) + r 1 r 1m r 6 m r ( 1 r m 1 r + 1 1 m m r ) 3k 1l(1 m )(m 1 1 r m ) 1 r m 1 r + 1 1 m m r + k l 1 ( 1 1m + r m r ) r cn [k(x + ly + ωt)] { 1 cn [k(x + ly + ωt)] + r} k l( 1r m 1r + 1 1m m r ) r { 1 cn [k(x + ly + ωt)] + r} cn [k(x + ly + ωt)] ; (30) kr 1 (1 + m ) k 3 u 16 = ω + 1m ±r 1 m + r 1 r 1 r m ± k 1 m + r 1 r 1 r m ns [k(x + ly + ωt)], r{ 1 ns [k(x + ly + ωt)] + r} 6 v 16 = 1k m l r ( 1 m + r 1 r 1 r m ) + r (1 + m )( 1 + r k P 1 l) 1m r 1 m + r 1 r 1 r m + 3k 1lm ( 1 + 1m r ) 1 m + r 1 r 1 r m + k l 1 ( 1m r r m ) r ns [k(x + ly + ωt)] { 1 ns [k(x + ly + ωt)] + r} k l( 1m + r 1r 1r m ) r { 1 ns [k(x + ly + ωt)] + r} ns [k(x + ly + ωt)] ; (31) kr 1 (1 + m ) k 3 u 17 = ω + 1m ±r 1 m + r 1 r 1 r m ± k 1 m + r 1 r 1 r m dc [k(x + ly + ωt)], r{ 1 dc [k(x + ly + ωt)] + r} 6 v 17 = 1k m l r ( 1 m + r 1 r 1 r m ) + r (1 + m )( 1 + r k P 1 l) 1m r 1 m + r 1 r 1 r m + 3k 1lm ( 1 + 1m r ) 1 m + r 1 r 1 r m + k l 1 ( 1m r r m ) r dc [k(x + ly + ωt)] { 1 dc [k(x + ly + ωt)] + r}

58 HUANG Wen-Hua Vol. 6 k l( 1m + r 1r 1r m ) r { 1 dc [k(x + ly + ωt)] + r} dc [k(x + ly + ωt)] ; (3) kr 1 ( m ) + k 3 u 18 = ω 1(m 1) ±r k ± 1 r 1 r m + 1 m 1 r nd [k(x + ly + ωt)], 1 r 1 r m + 1 m 1 r r{ 1 nd [k(x + ly + ωt)] + r} 6 v 18 = 1k l(m 1) r ( 1 r 1 r m + 1 m 1 r ) r ( m )( 1 r k l) + 1m 1 r 1 r 1 r m + 1 m 1 r 3k 1l(m 1)( 1 1m r ) 1 r 1 r m + 1 m 1 + k l 1 ( 1m 1 + r r m ) r r nd [k(x + ly + ωt)] { 1 nd [k(x + ly + ωt)] + r} k l( 1r 1r m + 1m 1 r ) r { 1 nd [k(x + ly + ωt)] + r} nd [k(x + ly + ωt)] ; (33) kr 1 (m 1) k 3 u 19 = ω 1m ±r 1 r m 1 r 1 m + r m r k 1] r m 1 r 1 m + r m r ± nc [k(x + ly + ωt)], r{ 1 nc [k(x + ly + ωt)] + r} v 19 = 6 1k m l + r (m 1)( 1 + r k (1 m )l) r 1m + r 6 r 6 m r ( 1 r m 1 r 1 m + r m r ) + 3k 1lm ( 1m 1 + r r m ) 1 r m 1 r 1 m + r m r + k l 1 ( 1m + r m r ) r nc [k(x + ly + ωt)] { 1 nc [k(x + ly + ωt)] + r} k l( 1r m 1r 1m + r m r ) r { 1 nc [k(x + ly + ωt)] + r} nc [k(x + ly + ωt)] ; (3) u 0 = ω 1r 1r m 1 + m r r ±r 1 r ( m ) 1 + r (m 1) ± k 1 r 1 r m 1 + m r r nd [k(x + ly + ωt)], r{ 1 nd [k(x + ly + ωt)] + r} v 0 = 6 1k l r ( 1 r 1 r m 1 + m r r ) r ( m )( 1 + r k m l r k l) 1 + r m r 1 r 1 r m 1 + m r r + 3k 1l( 1 1m + r P 1 ) 1 r 1 r m 1 + m r r + k l 1 ( 1R 1 + r r m ) r nd [k(x + ly + ωt)] { 1 nd [k(x + ly + ωt)] + r} k l( 1r 1r m 1 + m r r ) r { 1 nd [k(x + ly + ωt)] + r} nd [k(x + ly + ωt)] ; (35) kr 1 ( m ) + k 3 1 u 1 = ω ±r 1 r 1 r m + 1 + r m r ± k 1 r 1 r m + 1 + r m r sc [k(x + ly + ωt)], r{ 1 sc [k(x + ly + ωt)] + r} v 1 = 6 1k l r ( 1 r 1 r m + 1 + r m r ) r ( m )( 1 + r k l r k m l) + 1 + r r m 1 r 1 r m + 1 + r m r u = ω 3k 1l( 1 1m + r 1 r m ) 1 r 1 r m + 1 + r m r + k l 1 ( 1R 1 + r r m ) r sc [k(x + ly + ωt)] { 1 sc [k(x + ly + ωt)] + r} k l( 1r 1r m + 1 + r m r ) r { 1 sc [k(x + ly + ωt)] + r} sc [k(x + ly + ωt)] ; (36) kr 1 (m 1) + k 3 1 ±r 1 r m 1 r + 1 m r + m r ± k 1 r m 1 r + 1 m r + m r r{ 1 sd [k(x + ly + ωt)] + r} sd [k(x + ly + ωt)], v = 6 1k l + r (m 1)( 1 r k m l r k m m l) + r 1 r 6 m + r 6 m r ( 1 r m 1 r + 1 m r + m r ) 3k 1l( 1m 1 r m + r m ) 1 r m 1 r + 1 m r + m r + k l 1 ( 1 + r m r ) r sd [k(x + ly + ωt)] { 1 sd [k(x + ly + ωt)] + r}

No. A Generalized Extended F -Expansion Method and Its Application in (+1)-Dimensional Dispersive 585 k l( 1r m 1r + 1 m r + m r ) r { 1 sd [k(x + ly + ωt)] + r} sd [k(x + ly + ωt)] ; (37) kr 1 ( m ) + k 3 u 3 = ω 1(1 m ) ±r 1 r 1 r m + 1 1 m + r ± k 1 r 1 r m + 1 1 m + r r{ 1 cs [k(x + ly + ωt)] + r} cs [k(x + ly + ωt)], 6 v 3 = 1k (1 m ) l r ( 1 r 1 r m + 1 1 m + r ) r ( m )( 1 + r k l) + 1 1m + r 1 r 1 r m + 1 1 m + r 3k 1l(1 m )( 1 1m + r ) 1 r 1 r m + 1 1 m + r + k l 1 ( 1 1m + r r m ) r cs [k(x + ly + ωt)] { 1 cs [k(x + ly + ωt)] + r} k l( 1r 1r m + 1 1m + r ) r { 1 cs [k(x + ly + ωt)] + r} cs [k(x + ly + ωt)] ; (38) kr 1 (m 1) + k 3 u = ω 1m (m 1) ±r 1 r m 1 r 1 m + 1 m + r v = ± k 1 r m 1 r 1 m + 1 m + r r{ 1 ds [k(x + ly + ωt)] + r} ds [k(x + ly + ωt)], 6 1k m (m 1) l r ( 1 r m 1 r 1 m + 1 m + r ) r (m 1)( 1 + r k l) + 1m (m 1) + r 1 r m 1 r 1 m + 1 m + r 3k 1lm (m 1)( 1m 1 + r ) 1 r m 1 r 1 m + 1 m + r + k l 1 ( 1m 1m + r m r ) r ds [k(x + ly + ωt)] { 1 ds [k(x + ly + ωt)] + r} k l( 1r m 1r 1m + 1m + r ) r { 1 ds [k(x + ly + ωt)] + r} ds [k(x + ly + ωt)]. (39) In solutions (8) (39), k, l, ω, 1 are arbitrary constants, i = 1 and r 0. Remark All the solutions we found above satisfy DLWE (8), checked by symbolic computation system Maple. Part of the Jacobi elliptic periodic solutions obtained from the concentration formula (1) can easily reproduce the corresponding same type Jacobi elliptic solutions found in Refs. [6] and [9] under proper parameter algebraic transformation and the known relations between different Jacobi ellitptic functions. The other solutions and those derived from the concentration formula (15), to our knowledge, are new families of rational formal periodic solution for DLWE (8). As we know, Jacobian elliptic functions can be degenerated as hyperbolic functions when the modulus m 1 and trigonometric functions when m 0, which is shown in detail in the following tabular: m sn ξ cn ξ dn ξ sc ξ sd ξ cd ξ ns ξ nc ξ nd ξ cs ξ ds ξ dc ξ m 1 tanh ξ sech ξ sech ξ sinh ξ sinh ξ 1 coth ξ cosh ξ cosh ξ csch ξ csch ξ 1 m 0 sin ξ cos ξ 1 tan ξ sin ξ cos ξ csc ξ sec ξ 1 cotξ csc ξ sec ξ So the doubly periodic wave solutions derived above can easily be generated into hyperbolic function solutions and trigonometric function solutions respectively. For simplification, here we omit listing all the concrete forms of the degenerated solutions. Especially, when the modulus m 1, some Jacobian elliptic function solutions can be degenerate into solitary wave solutions. Taking the nonsingular periodic wave solutions (16), (18), and (8), (30) as examples, we give the corresponding solitary wave solutions as follows: u 1 = ω ± k tanh [k(x + ly + ωt)], v 1 = 1 + k l + k l k l tanh [k(x + ly + ωt)] ; (0) u 3 = ω ± ik sech [k(x + ly + ωt)], v 3 = 1 k l + k l + k l sech [k(x + ly + ωt)] ; (1) u kr 1 k 3 1 13 = ω + ±r k 1 + ± 1 + r 1 r 1 r tanh [k(x + ly + ωt)], r 1 r 1r r{ 1 tanh [k(x + ly + ωt)] + r} v 13 6 = 1k l r ( 1 + r 1 r 1 r ) + r ( 1 + r k l) 1 r 1 + r 1 r 1 r

586 HUANG Wen-Hua Vol. 6 + 3k 1l( 1 + 1 r ) 1 + r 1 r + k l 1 ( 1 r r ) 1 r r tanh [k(x + ly + ωt)] { 1 tanh [k(x + ly + ωt)] + r} k l( 1 + r 1r 1r ) r { 1 tanh [k(x + ly + ωt)] + r} tanh [k(x + ly + ωt)] ; () u kr 1 15 = ω ±r k 1 r 1 r + 1 ± 1 r 1 r + 1 1 r sech [k(x + ly + ωt)], 1 r r{ 1 sech [k(x + ly + ωt)] + r} v 15 = r ( 1 r k l) + r 1 r 1 r 6 r ( 1 r 1 r + 1 1 r ) + k l 1 ( 1 1 + r r ) r sech [k(x + ly + ωt)] { 1 sech [k(x + ly + ωt)] + r} k l( 1r 1r + 1 1 r ) r { 1 sech [k(x + ly + ωt)] + r} sech [k(x + ly + ωt)]. (3) Summary In summary, based on the idea of extended Jacobi elliptic expansion method and extended F -function expansion method, we develop a new generalized extended F -expansion method for finding periodic wave solutions of NEEs in mathematical physics, which may be more general, convenient, and more powerful. We applied this method to solve the (+1)-dimensional dispersive long wave equation, a number of doubly periodic wave solutions expressed by various Jacobi elliptic functions are obtained with the aid of symbolic computation system Maple. When the modulus m 1 and m 0, their corresponding degenerative solutions expressed by hyperbolic functions and trigonometric functions are derived respectively. Especially, when the modulus m 1, solitary wave solutions can be obtained as well. Here the concentration formulas of the solutions to DLWE are expressed by only F (ξ) function, actually this method can be applied to construct exact periodic solution for more nonlinear models such as the coupled Drinfel d Sokolov Wilson equation. [5] (+1)-dimensional long-wave-short-wave resonance interaction equation, [6] etc., and the concentration formulas of the solutions can be expressed by both F and G generally. More work to explore these solutions to DLWE and apply the generalized extended F -expansion method to other nonlinear mathematical physics models is worthy of study further. References [1] S.K. Liu, Z.T. Fu, S.D. Liu, and Q. Zhao, Phys. Lett. A 89 (001) 69. [] S.D. Liu, Z.T. Fu, S.K. Liu, and Q. Zhao, Acta Phys. Sin. 51 (00) 718 (in Chinese). [3] Z.Y. Yan, Commun. Theor. Phys. (Beijing, China) 38 (00) 13. [] E.J. Parkes, B.R. Duffy, and P.C. Abbott, Phys. Lett. A 95 (00) 80. [5] Z.T. Fu, S.K. Liu, and Q. Zhao, Phys. Lett. A 90 (001) 7. [6] Z.Y. Yan, Comput. Phys. Commun. 153 (003) 15. [7] J.M. Zhu, Z.Y. Ma, J.P. Fang, C.L. Zheng, and J.F. Zhang, Chin. Phys. 13 (00) 798. [8] Y.X. Yu, Q. Wang, and H.Q. Zhang, Chaos, Solitons and Fractals 6 (005) 115. [9] Q. Wang, Y. Chen, and H.Q. Zhang, Phys. Lett. A 30 (005) 11. [10] Z.Y. Yan, Z. Naturforsch. A 59 (00) 3. [11] M.M. Hassan and A.H. Khaater, Z. Naturforsch. A 60 (005) 37. [1] Y.B. Zhou, M.L. Wang, and Y.M. Wang, Phys. Lett. A 308 (003) 31. [13] M.L. Wang, Y.M. Wang, and J.L. Zhang, Chin. Phys. 1 (003) 131. [1] Y.B. Zhou, M.L. Wang, and T.D. Miao, Phys. Lett. A 33 (00) 77. [15] M.L. Wang and X.Z. Li, Chaos, Solitons and Fractals (005) 157. [16] Y. Emmanuel, Phys. Lett. A 30 (005) 19. [17] M.L. Wang and X.Z. Li, Phys. Lett. A 33 (005) 8. [18] B.A. Li and M.L. Wang, Chin. Phys. 1 (005) 1698. [19] Y.M. Wang, X.Z. Li, and M.L. Wang, Commun. Theor. Phys. (Beijing, China) (005) 396; J. Chen, H.S. He, and K.Q. Yang, Commun. Theor. Phys. (Beijing, China) (005) 307. [0] P. Boiti, J.J.P. Leon, M. Manna, and F. Pempinelli, Inverse Problems 3 (1987) 105. [1] G. Paquin and P. Winternitz, Physica D 6 (1990) 1. [] S.Y. Lou, Math. Methods Appl. Sci. 18 (1995) 789. [3] X.Y. Tang, C.L. Chen, and S.Y. Lou, J. Phys. A: Math. Gen. 35 (00) L93. [] D.S. Li and H.Q. Zhang, Commun. Theor. Phys. (Beijing, China) 0 (003) 13; H.Y. Zhi, Z.S. Lu, and H.Q. Zhang, Commun. Theor. Phys. (Beijing, China) (00) 811. [5] X.Y. Jiao, J.H. Huan, and H.Q. Zhang, Commun. Theor. Phys. (Beijing, China) (005) 07. [6] W.H. Huang, J.F. Zhang, and Z.M. Sheng, Chin. Phys. 11 (00) 1101.