A generalized method and general form solutions to the Whitham Broer Kaup equation

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1 Chaos, Solitons and Fratals (004) A generalized method and general form solutions to the Whitham Broer Kaup equation Yong Chen a,b,, *, Qi Wang b,, Biao Li b, a Department of Physis, Shanghai Jiao Tong University, Shanghai 00030, China b Department of Applied Mathematis, Dalian University of Tehnology, Dalian 604, China MM Key Lab, Chinese Aademy of Sienes, Beijing 00080, China Aepted 5 February 004 Communiated by Prof. M. Wadati Abstrat Based on a more general transformation presented in this paper, a generalized method for finding more types travelling wave solutions of nonlinear evolution equations (NLEEs) is presented and implemented in a omputer algebrai system. As an appliation of the method, Whitham Broer Kaup (WBK) equation is studied to illustrate the method. As a result, We annot only suessfully reover the previously known travelling wave solutions found by Fan s method [J. Phys. A 35 (00) 6853 Comput. Phys. Commun. 53 (003) 7], but also obtain some new formal solutions. The solutions obtained in this paper inlude polynomial, exponential, solitary wave, rational, triangular periodi, Jaobi and Weierstrass doubly periodi solutions. Ó 004 Elsevier Ltd. All rights reserved.. Introdution It is well known that the investigation of the travelling wave solutions of nonlinear evolution equations (NLEEs), whih desribe many important phenomena and dynamial proesses in physis, hemistry, biology, et., plays an important role in the study of soliton theory. In the past several deades, both mathematiians and physiists have made many attempts in this diretion. Many effetive methods have been presented, suh as inverse sattering transform method [,], Baklund transformation [3,4], Darboux transformation [5,6], Hirota bilinear method [7,8], the variable separation method [9], Homogeneous balane method [0] and similarity redution method [] et. The tanh method [] is onsidered to be one of most straightforward and effetive algorithm to obtain solitary wave solutions for a large NLEEs. Muh work [3 6] has been onentrated on the various extensions and appliations of the tanh method, suh as the extended tanh method by Fan [3], improved tanh method by Yan [5], Generalized extended tanh-funtion method by Chen et al. [6]. The basi purpose of above work is to simplify the routine alulation of the method or obtain more general solutions. In [7], Fan develop a new algebrai method with symboli omputation for obtaining the above-mentioned various travelling wave solutions in a unified way. Compared with most of the existing methods, the proposed method not only gives a unified formulation to onstrut various travelling wave solutions, but also provides a guideline to lassify the various types of the travelling wave solutions aording to the values of some parameters. The present work is motivated by the desire to extend the transformation in [7] to more general transformations and use symboli omputation to solve WBK equation [8 0] * Corresponding author. addresses: henyong@dlut.edu.n (Y. Chen), libiao@dlut.edu.n (B. Li) /$ - see front matter Ó 004 Elsevier Ltd. All rights reserved. doi:0.06/j.haos

2 676 Y. Chen et al. / Chaos, Solitons and Fratals (004) u t þ uu x þ v x þ bu xx ¼ 0 v t þ u x v þ uv x þ au xxx b v xx ¼ 0 ð:þ where a b 6¼ 0 are all onstants. Under Boussinesq approximation, Whitham [8], Broer [9] and Kaup [0] obtained nonlinear WBK equation. It is not diffiult to see that when parameters a and b take different onstants, system (.) inludes many important mathematial and physial equations, suh as when a ¼ 0, b 6¼ 0 system (.) beomes lassial long wave equation that desribe shallow water with dispersive [], and when a ¼, b ¼ 0 system beomes variant Boussinesq equation []. Many mathematiians and physiists have devoted onsider effort to the study on WBK equation and make new developments in the regards (see, e.g., [ 4] for detail).. Summary of the generalized method In the following we would like to outline the main steps of our general method: Step. For a given of nonlinear evolution equations (NLEEs) with some physial fields u i ðx y tþ in three variables x, y, t, F i ðu i u it u ix u iy u itt u ixt u iyt u ixx u iyy u ixy...þ¼0 ði ¼... nþ ð:þ by using the wave transformation u i ðx y tþ ¼u i ðnþ n ¼ kðx þ ly ktþ ð:þ where k, l and k are onstants to be determined later. Then the nonlinear partial differential equation (.) is redued to a nonlinear ordinary differential equation (ODE) G m ðu i u 0 i u00 i...þ¼0: ð:3þ Step. We introdue a new and more general ansates in the forms: 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffi P 9 < X u i ðnþ ¼a i0 þ Xmi r r a ij / j þ b : ij / j þ f ij / j l / l l¼0 þ k l/ l = ij ð:4þ j¼ l¼0 where the new variable / ¼ /n ð Þ satisfying sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi / 0 ¼ d/ dn ¼ X r l / l ð:5þ l¼0 and a i0 a ij b ij f ij k ij ði ¼... j ¼... m i Þ are onstants to be determined later. Step 3. The underlying mehanism for a series of fundamental solutions suh as polynomial, exponential, solitary wave, rational, triangular periodi, Jaobi and Weierstrass doubly periodi solutions to our is that differ effets that at to hange wave forms in many nonlinear equations, i.e. dispersion, dissipation and nonlinearity, either separately or various ombination are able to balane out. We define the degree of u i ðnþ as D½u i ðnþš ¼ n i, whih gives rise to the degrees of other expressions as D½u ðqþ i Š¼n i þ q D½u p i ðu ðqþ j Þ s Š¼n i p þðqþn j Þs: ð:6þ Therefore we an get the value of m i in Eq. (.4). If n i is a nonnegative integer, then we first make the transformation u i ¼ x ni i. Step 4. Substitute ffiffiffiffiffiffiffi Eq. (.4) into Eq. (.3) along with Eq. (.5) and then set all oeffiients of / p P q r l¼0 l/ l ðq ¼ 0 p ¼ 0...Þ to be zero to get an over-determined system of nonlinear algebrai equations with respet to k, l, k, a i0, a ij, b ij, f ij and k ij (i ¼... j ¼... m i ). Step 5. Solving the over-determined system of nonlinear algebrai equations by use of Maple, we would end up with the expliit expressions for k, l, k, a i0, a ij, b ij, f ij and k ij (i ¼... j ¼... m i ). Step 6. By using the results obtained in the above step, we an derive a series of fundamental solutions suh as polynomial, exponential, solitary wave, rational, triangular periodi, Jaobi and Weierstrass doubly periodi solutions. Beause we interested in solitary wave, Jaobi and Weierstrass doubly periodi solutions. On the other hand, tan and ot type solutions appear in pairs with tanh and oth type solutions respetively, polynomial, / j

3 rational triangular periodi solutions are omitted in this paper. By onsidering the different values of 0,,, 3 and (in this paper we only onsider the ase l ¼ 4 in Eqs. (.4) and (.5)), Eq. (.5) has many kinds of solitary wave, Jaobi and Weierstrass doubly periodi solutions whih are listed as follows: (i) Solitary wave solutions (a) Bell-shaped solitary wave solutions rffiffiffiffiffiffiffiffiffi / ¼ pffiffiffiffi seh ð nþ 0 ¼ ¼ 3 ¼ 0 > 0 < 0 ð:7þ Y. Chen et al. / Chaos, Solitons and Fratals (004) / ¼ pffiffi seh 3 n 0 ¼ ¼ ¼ 0 > 0: ð:8þ (b) Kink-shaped solitary wave solutions rffiffiffiffiffiffiffiffiffiffiffi / ¼ k rffiffiffiffiffiffiffiffiffi tanh n 0 ¼ ¼ 3 ¼ 0 4 < 0 > 0: ð:9þ () Solitary wave solutions seh pffiffiffiffi n / ¼ p ffiffiffiffiffiffiffiffi tanh pffiffiffiffi n 3 0 ¼ ¼ 0 > 0: ð:0þ (ii) Jaobi and Weierstrass doubly periodi solutions sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi m / ¼ n n ðm Þ m 4 < 0 > 0 0 ¼ m ð m Þ ðm Þ ð:þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m / ¼ dn n ð m Þ m 4 < 0 > 0 0 ¼ ð m Þ ð m Þ ð:þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffi m / ¼ sn n ðm þ Þ m 4 > 0 < 0 0 ¼ þ ðm þ Þ ð:3þ where m is a modulus. pffiffiffiffi 3 / ¼ } n g g 3 ¼ 0 3 > 0 ð:4þ where g ¼4 3 and g 3 ¼4 0 3 are alled invariants of Weierstrass ellipti funtion. The Jaobi ellipti funtions are doubly periodi and possess properties of triangular funtions: sn n þ n n ¼ dn n ¼ m sn n ð:5þ ðsnnþ 0 ¼ nndnn ðnnþ 00 ¼snn ðdnnþ 00 ¼m sn nnn: When m!, the Jaobi funtions degenerate to the hyperboli funtions, i.e. snn! tanh n nn! sehn: ð:6þ When m! 0, the Jaobi funtions degenerate to the triangular funtions, i.e. snn! sin n nn! os n: ð:7þ The more detailed notations for the Weierstrass and Jaobi ellipti funtions an be found in [5 7]. Remark. Compared with the method proposed by Fan [8], our ansates is more general than the ansates in [8]. When b ij ¼ f ij ¼ k ij ¼ 0 in Eq. (.4), Eq. (.4) beomes the ansates proposed by Fan. Remark. The method an be extend to find soliton-like solutions and more types double periodi solutions of PDE Eq. (.). Only will the restrition on nðx y tþ as merely a linear funtion x, y, t and the restritions on the oeffiients a i0, a ij, b ij, f ij, k ij and i as onstants be remove.

4 678 Y. Chen et al. / Chaos, Solitons and Fratals (004) Exat solutions of WBK equation Aording to the above method, to seek the solutions of WBK, u t þ uu x þ v x þ bu xx ¼ 0 v t þ u x v þ uv x þ au xxx bv xx ¼ 0 ð3:þ we make the following transformation: uðx tþ ¼rðnÞ vðx tþ ¼sðnÞ n ¼ x kt ð3:þ where k is onstant to be determined later, and thus Eq. (3.) beomes ( kr 0 þ rr 0 þ s 0 þ br 00 ¼ 0 ks 0 þ rs 0 þ r 0 s þ ar 000 bs 00 ¼ 0: ð3:3þ Aording to Step in Setion, if a 6¼ 0 and b 6¼ 0, by balaning r 00 ðnþ and rðnþr 0 ðnþ in Eq. (3.3) and balaning s 00 ðnþ and r 0 ðnþsðnþ in Eq. (3.3), we suppose that Eq. (3.3) has the following formal solutions: 8 ffiffi ffiffiffiffiffiffi P 4 r ¼ a 0 þ a / þ b þ f P 4 / l¼0 i/ i l¼0 >< þ k i/i / ffiffi ffiffiffiffiffiffi P 4 s ¼ A 0 þ A / þ B þ F P 4 / l¼0 i/ i l¼0 >: þ K i/i þ A / / þ B þ F / / ffiffiffiffiffiffi P 4 l¼0 i/ i þ K ffiffi P 4 l¼0 i/i / where /ðnþ satisfies (.5), where a 0, a, b, f, k, A 0, A, B, F, K, A, B, F and K are onstants to be determined later. With the aid of Maple, substituting (3.4) along with (.5) into (3.3), yields a set of algebrai equations for ffiffiffiffiffiffiffi / i P j ffiffiffiffiffiffiffi ðnþ 4 l¼0 l/ l ði ¼ 0... j ¼ 0 Þ. Setting the oeffiients of these terms / i P j ðnþ 4 l¼0 l/ l to zero yields a set of over-determined algebrai equations with respet to a 0, a, b, f, k, A 0, A, B, F, K, A, B, F, K and k. By use of the Maple, solving the over-determined algebrai equations, then we get the following results: Case A ¼ b þ a b þ a b k ¼ b þ a k ¼ a 0 ð3:4þ 0 ¼ ¼ 3 ¼ a ¼ b ¼ f ¼ A 0 ¼ A ¼ B ¼ B ¼ F ¼ F ¼ K ¼ K ¼ 0: ð3:5þ Case B ¼ b þ a b þ a b k ¼ a 0 A ¼ b þ a 3 b þ a b ¼ a b þ a a F ¼a b b þ a b b þ a þ a þ b A ¼ b þ a k ¼ b þ a 0 ¼ b ¼ f ¼ A 0 ¼ B ¼ F ¼ K ¼ K ¼ 0: ð3:6þ Case 3 K ¼b b þ a b k ¼ a 0 k ¼ b þ a 0 ¼ b b þ a B ¼ b þ a b þ a b A ¼ b þ a 3 b þ a b b b þ a b b þ a B ¼ b 4 ¼ a ¼ f ¼ A 0 ¼ A ¼ F ¼ F ¼ K ¼ 0: ð3:7þ þ a

5 Case 4 A ¼ b þ a b þ a b k ¼ b þ a B ¼ b þ a 0 b þ a b k ¼ a 0 Y. Chen et al. / Chaos, Solitons and Fratals (004) ¼ 3 ¼ a ¼ b ¼ f ¼ A 0 ¼ A ¼ B ¼ F ¼ F ¼ K ¼ K ¼ 0: ð3:8þ Case 5 k ¼ a 0b þ b þ a A 0 ¼ b b þ b4 þ b a þ a a b b b K ¼ bb B ¼b a 0 ¼ b 4 b þ a B ¼ b a ¼ f ¼ k ¼ A ¼ A ¼ F ¼ F ¼ K ¼ 0: ð3:9þ From (3.), (3.5) and Cases 5, we obtain the following solutions for Eqs. (3.): Family. From Eqs. (3.5), we obtain the following solutions for the WBK equations, as follows: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ¼ a 0 b þ a seh pffiffiffiffi ð nþ ð3:0þ v ¼ b þ a b þ a b seh pffiffiffiffi ð nþ ð3:þ where n ¼ x kt, k ¼ a 0, > 0 and a 0 are arbitrary onstants. Family. From Eqs. (3.6), when ¼ 0, then we obtain the following solutions for the WBK equations, as follows: seh pffiffiffiffi n u ¼ a 0 þ a p ffiffiffiffiffiffiffiffi tanh pffiffiffiffi n 3 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi seh pffiffiffiffi n k þ 3 ffiffiffiffiffiffiffiffi seh ð pffiffiffiffi! u t p tanh pffiffiffiffi nþ þ p n 3 ffiffiffiffiffiffiffiffi tanhð pffiffiffiffi ð3:þ nþ3 A seh pffiffiffiffi n v ¼ ffiffiffiffiffiffiffiffi p tanh pffiffiffiffi seh ð pffiffiffiffi! nþ þ A p n 3 ffiffiffiffiffiffiffiffi tanhð pffiffiffiffi nþ3 F seh pffiffiffiffi! vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n p ffiffiffiffiffiffiffiffi 3 seh pffiffiffiffi tanh pffiffiffiffi n þ n 3 ffiffiffiffiffiffiffiffi seh ð pffiffiffiffi! u t p tanh pffiffiffiffi nþ þ p n 3 ffiffiffiffiffiffiffiffi tanhð pffiffiffiffi nþ3 ð3:3þ where n ¼ x kt, k ¼ a 0, A ¼ b þ a 3 b þ a b, F ¼a b þ a þ b, A ¼ a k ¼ b þ a, ¼ a, b þa > 0, a 0, a and 3 are arbitrary onstants. b þab b þa ffi b þa, Family 3. From Eqs. (3.7), when ¼ 0, then we an obtain the following solutions for the WBK equations, as follows: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi b 0 þ } 3 n g g 3 þ 3 } 3 pffiffiffi 3 n g g 3 u 3 ¼ a 0 þ pffiffiffi þ k } 3 n g pffiffiffi ð3:4þ g 3 } 3 n g g 3

6 680 Y. Chen et al. / Chaos, Solitons and Fratals (004) pffiffiffiffi 3 v 3 ¼ A } n g B B g 3 þ pffiffiffi þ } 3 ffiffiffi g 3 } p 3 g 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 0 þ } 3 ffiffiffi g 3 þ 3 } 3 p 3 g 3 þ k ffiffiffi } p ð3:5þ 3 g 3 where n ¼ x kt, k ¼ a 0, g ¼4 3, g 3 ¼4 0 3, K ¼b b þ a b, k ¼ ffi b þ a, B ¼ b þ a b þ a b, A ¼ b þ a 3 b þ a b, B ¼ b b þab b þa b þa, 0 ¼ b b þa 3 > 0, b, and a 0 are arbitrary onstants. Family 4. From Eqs. (3.8), we an obtain the following solutions for the WBK equations, as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi 0 m n ffiffi m n þ m4 n 4 m n ðm u 4 ¼ a 0 þ k Þ ðm Þ ffiffi ffiffiffi m n ð3:6þ ðm Þ m n v 4 ¼ A ffiffi ffiffi m ðm Þ ðm Þ n n B m n n ð3:7þ m m where n ¼ x kt, k ¼ a 0, k ¼ b þ a, A ¼ b þ a b þ a b, B ¼ b þ a 0 b þ a b, 0 ¼ m ðm Þ, ðm Þ > 0, and a 0 are arbitrary onstants. Family 5. From Eqs. (3.8), we an obtain the following solutions for the WBK equations, as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 m dn m n þ m4 4 dn m n ðm u 5 ¼ a 0 þ k Þ ðm Þ ffiffi m dn ð3:8þ ðm Þ m n v 5 ¼ A m ð m Þ ð m Þ dn n B m nd n ð3:9þ m m where n ¼ x kt, k ¼ a 0, k ¼ b þ a, A ¼ b þ a b þ a b, B ¼ b þ a 0 b þ a b, 0 ¼ ðm Þ, ðm Þ > 0, and a 0 are arbitrary onstants. Family 6. From Eqs. (3.8), we an obtain the following solutions for the WBK equations, as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 m sn m n þ þ m4 sn 4 m n þ ðm u 6 ¼ a 0 þ k þþ ðm þþ qffiffiffiffiffiffiffiffi ð3:0þ m sn n ðm þþ m þ m ðm þ Þ v 6 ¼A ðm þ Þ sn n B m ns n ð3:þ þ m m þ where n ¼ x kt, k ¼ a 0, k ¼ b þ a, A ¼ b þ a b þ a b, B ¼ b þ a 0 b þ a b, 0 ¼ m, ðm þþ < 0, and a 0 are arbitrary onstants. Family 7. From Eqs. (3.9), we an obtain the following solutions for the WBK equations, as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi ðm u 7 ¼ a 0 þ b Þ n n ð3:þ m m

7 ffiffi ðm Þ v 7 ¼ A 0 B n n m m Y. Chen et al. / Chaos, Solitons and Fratals (004) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi 0 m n ffiffi m n þ m4 n 4 m n ðm K Þ ðm Þ ffiffiffi ð3:3þ n m ðm Þ n where n ¼ x kt, k ¼ a 0, A 0 ¼ðbþaÞ, K ¼ bb, B ¼ b, 0 ¼ b, b þa 4 ¼ m ðm Þ, 0 ðm Þ > 0, b and a 0 are arbitrary onstants. Family 8. From Eqs. (3.9), we an obtain the following solutions for the WBK equations, as follows: ffiffiffiffiffiffiffi ð m u 8 ¼ a 0 þ b Þ nd n ð3:4þ m m sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð m Þ v 8 ¼ A 0 B nd 0 m dn m n þ m4 4 dn m n ðm n K Þ ðm Þ m m ffiffi m ðm Þ dn ð3:5þ m n where n ¼ x kt, k ¼ a 0, A 0 ¼ðbþaÞ, K ¼ bb, B ¼ b, 0 ¼ b, b þa 4 ¼ ðm Þ, 0 ðm Þ > 0, b and a 0 are arbitrary onstants. Family 9. From Eqs. (3.9), we an obtain the following solutions for the WBK equations, as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm u 9 ¼ a 0 þ b þ Þ ns n ð3:6þ m m þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 0 m sn m n þ þ m4 sn 4 m n þ ðm v 8 ¼ A 0 B ðm þ Þ ns n K þþ ðm þþ m qffiffiffiffiffiffiffiffi ð3:7þ þ n m ðm þþ sn m, 0 ðm þþ < 0, b and a 0 are arbi- where n ¼ x kt, k ¼ a 0, A 0 ¼ðbþaÞ, K ¼ bb, B ¼ b, 0 ¼ 4 trary onstants. m þ m b b þa, ¼ Remark. Up to now we have obtained some new forms solutions whih annot be obtained by Fan s method, suh as the solutions of Families 9. When b ¼ f ¼ k ¼ B ¼ B ¼ F ¼ F ¼ K ¼ K ¼ 0, the form of the solutions of Families 9 beome the form whih an be obtained by Fan s method. This further shows our method is more general than Fan s method. Remark. The some kinds of solutions derived by the generalized transformation are singular soliton solution and Jaobi ellipti doubly periodi wave solution. There is muh urrent interest in the formation of so-alled hot spots or blow up of solutions. It appears that these singular solutions will model this physial phenomena. 4. Summary and onlusions In summary, based on a more general ansatz than one presented by Fan [8], we have presented the generalized method used to find more formal solutions of nonlinear evolution equations. To illustrate the method, the WBK equation are studied by the method and more types solutions are obtained, whih inlude polynomial, exponential, solitary wave, rational, triangular periodi, Jaobi and Weierstrass doubly periodi solutions of the WBK equation. The method an easily be extended to other NLEEs and is suffiient to seek more new formal solution of NLEEs. Aknowledgements We would express our sinere thanks to Prof. Wadati for his enthusiasti help and valuable suggestions. The work is supported by the Outstanding Youth Foundation (No. 9955).

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