A multiple Riccati equations rational expansion method and novel solutions of the Broer Kaup Kupershmidt system

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1 Chaos, Solitons and Fractals 30 (006) A multiple Riccati equations rational expansion method and novel solutions of the Broer Kaup Kupershmidt system Qi Wang a,c, *, Yong Chen b a Department of Applied Mathematics, Dalian University of Technology, Dalian 11604, China b Nonlinear Science Center, Ningbo University, Ningbo 31511, China c Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing , China Accepted 30 August 005 Abstract To construct exact solutions of nonlinear partial differential equation, a multiple Riccati equations rational expansion method (MRERE) is presented and a series of novel solutions of the Broer Kaup Kupershmidt system are found. The novel solutions obtained by MRERE method include solutions of hyperbolic (solitary) function and triangular periodic functions appearing at the same time. Ó 005 Elsevier Ltd. All rights reserved. 1. Introduction In the past decades, both mathematicians and physicists have devoted considerable effort to the study of solitons and related issue of the construction of solutions to nonlinear partial differential equations (PDEs) [1 10]. Recently, we present various rational expansion methods [10,11] to construct rational formal exact solutions of nonlinear PDEs. Advantages of these rational expansion methods are that the more general rational ansätz form or more subequation is used to reduce the target equation, the more general rational styles of exact solutions of nonlinear PDEs can be found. However in traditional subequation methods [6 11] the variables used in an ansätz always satisfy the same subequation or subequations. The present work is motivated by the desire to extend our work [10,11] to set up a new arithmetic, named multiple Riccati equations rational expansion method (MRERE), to construct new styles of solutions of nonlinear PDEs. We use two or more variables which satisfy different Riccati equations, in which different parameter is chosen independently. To our pleasantly surprised, in this way, we can construct many families of novel solution of some nonlinear PDEs, in which hyperbolic (solitary) function and triangular periodic functions can appear in a solution at same time. We use this new arithmetic to the Broer Kaup Kupershmidt system to test the validity of our new method. As a result, we find some novel solutions. To my knowledge, these new styles of solutions do not be found before. * Corresponding author. address: wangqi_dlut@yahoo.com.cn (Q. Wang) /$ - see front matter Ó 005 Elsevier Ltd. All rights reserved. doi: /j.chaos

2 198 Q. Wang, Y. Chen / Chaos, Solitons and Fractals 30 (006) Multiple Riccati equation rational expansion method In the following we would like to outline the main steps of our method: Step 1. Given a system of polynomial PDE with constant coefficients, with some physical fields u i (x,y,t) in three variables x, y, t, Dðu i u it u ix u iy u itt u ixt u iyt u ixx u iyy u ixy...þ¼0 ð:1þ use the wave transformation u i ðx y tþ ¼U i ðnþ n ¼ kðx ly ktþ ð:þ where k, l and k are constants to be determined later. Then the nonlinear partial differential system (.1) is reduced to a nonlinear ordinary differential system: HðU i U 0 i U 00 i...þ¼0. ð:3þ Step. We introduce a new ansätz in terms of finite rational formal expansion in the following forms: P U i ðnþ ¼a i0 Xmi r j1 r j ¼j aj r j1 r j / r j1 ðnþw r j ðnþ ðl 1 /ðnþl wðnþ1þ j ð:4þ where a j r j1 r j, l 1 and l (r jn =0,1,...,j n = 1,) are constants to be determined later and the new variables / = /(n) and w = w(n) satisfy the Riccati equation, i.e., d/ dn ¼ h 1 h / dw dn ¼ h 3 h 4 w ð:5þ where h 1, h, h 3 and h 4 are constants. Step 3. Determine the m i of the rational formal polynomial solutions (.4) by, respectively, balancing the highest nonlinear terms and the highest-order partial derivative terms in the given system equations (see Refs. [6 11] for details), and then give the formal solutions. Step 4. Substitute (.4) into (.3) along with (.5) and then set all coefficients of / p (n)w q (n), (p = 0,1,,... q = 0,1,,...) of the resulting systemõs numerator to be zero to get an over-determined system of nonlinear algebraic equations with respect to k, l 1, l and a j r j1 r j (r jn =0,1,...,j n = 1,). Step 5. By solving the over-determined system of nonlinear algebraic equations by use of symbolic computation system Maple, we end up with the explicit expressions for k, l 1, l and a j r j1 r j (r jn =0,1,...,j n = 1,). Step 6. According to system (.), (.4), the conclusions in Step 5 and the general solutions of system (.5) which can be seen in Appendix A, we can obtain rational formal exact solutions of system (.1). Remark 1. The MRERE method is more general than various existing methods [6 8] for finding exact solutions of nonlinear PDEs. The appeal and success of the method lies in the fact that writing the exact solutions of a nonlinear partial differential system as polynomials of / and w whose derivations are in closed form, the equation can be changed into a nonlinear system of algebraic equations. The system can be solved with the help of symbolic computation. Note that: The projective Riccati equation expansion method which can also change an equation to a nonlinear system of algebraic equations is very like our method, but it dos not find the real reason why projective Riccati equation are in closed-form is that hyperbolic functions and triangular functions in solutions are in closed-form. So it is just particular case of our method. Remark. We can easily see that when h 1 5h 3 or h 5h 4, / and w satisfy the different Riccati equation, so hyperbolic functions and triangular functions can appear in a solution at the same time. For example, according to Appendix A, when h 1 ¼ 1, h ¼ 1 and h 3 ¼ h 4 ¼ 1, we can get following particular solution: u i ¼ a i0 Xmi u i ¼ a i0 Xmi Pr j1 r j ¼j aj r j1 r j ðtanhðnþisechðnþþ r j1 ðsecðnþtanðnþþ r j ðl 1 ðtanhðnþisechðnþþ l ðsecðnþtanðnþþ 1Þ j ð:6:1þ Pr j1 r j ¼j aj r j1 r j ðcothðnþcschðnþþ r j1 ðsecðnþtanðnþþ r j ðl 1 ðcothðnþcschðnþþ l ðsecðnþtanðnþþ 1Þ j ð:6:þ

3 Q. Wang, Y. Chen / Chaos, Solitons and Fractals 30 (006) u i ¼ a i0 Xmi u i ¼ a i0 Xmi Pr j1 r j ¼j aj r j1 r j ðtanhðnþisechðnþþ r j1 ðcscðnþcotðnþþ r j ðl 1 ðtanhðnþisechðnþþ l ðcscðnþcotðnþþ 1Þ j ð:6:3þ Pr j1 r j ¼j aj r j1 r j ðcothðnþcschðnþþ r j1 ðcscðnþcotðnþþ r j. ðl 1 ðcothðnþcschðnþþ l ðcscðnþcotðnþþ 1Þ j ð:6:4þ These solutions have not been obtained by any other Riccati equation expansion methods or projective Riccati equation expansion methods. Remark 3. Recently, there are a lot of papers of subequation method focus their attentions on generalize auxiliary equations or get more solutions of existing auxiliary equations to get more generalized solutions of target equations. But, to our knowledge, these methods can all be summarized as following rational expansion form: uðnþ ¼a 0 Xm P r 1 r n¼j aj r 1 r n F r 1 1 F rn n P r 1 r n¼j bj r 1 r n F l 1 1 F ln n b 0 ð:7þ where a 0 a j r 1 r n b j r 1 r n and n are differentiable function to be determined and df i ¼ K dn iðf 1... F n Þ, where K i are polynomial of F i. It is clearly to see that (.7) is also satisfying solving the recurrent relation or derivative relation for the terms of polynomial for computation closed. And according to this formula, we think we can get more generalized solutions of target equations by using simple auxiliary equation or simple solutions of auxiliary equation. This work will be continued in future. 3. Exact solutions of the Broer Kaup Kupershmidt system Let us consider the Broer Kaup Kupershmidt (BKK) system, H ty H xxy ðhh x Þ y G xx ¼ 0 G t G xx ðhgþ x ¼ 0. ð3:1þ The BKK system may be derived from the parameter dependent symmetry constraint of the Kadomtsev Petviashvili (KP) equation [1]. For more details about the results about this system, the reader is advised to see the achievements in Refs. [1 16]. In order to get some families of rational formal wave solutions to the BKK system, by considering the wave transformations Hðx y tþ ¼ HðnÞ, Gðx y tþ ¼ GðnÞ and n = k(x + ly + kt), we change (3.1) to the form klh 00 lkh 000 lðhh 0 Þ 0 G 00 ¼ 0 kg 0 kg 00 ðghþ 0 ¼ 0. ð3:þ For the BKK system, by balancing the highest nonlinear terms and the highest-order partial derivative terms in (3.), we suppose (3.) have the following formal travelling wave solution: HðnÞ ¼a 0 a 1/ b 1 w l 1 / l w 1 GðnÞ ¼A 0 A 1/ B 1 w l 1 / l w 1 A / B /w C 1 w ð3:3þ ðl 1 / l w 1Þ where l 1, l, a 0, a i, b i, A 0, A i, B i and C 1 (i = 1,) are constants to be determined later and the new variables / and w satisfy (.5). With the aid of Maple, substituting (3.3) along with (.5) into (3.) and setting the coefficients of these terms / i w j to be zero yields a set of over-determined algebraic equations with respect to a 0, a 1, b 1, A 0, A 1, B 1 A, B, C 1, l 1, l and k. By use of the Maple soft package Charsets by Dongming Wang, which is based on the Wu-elimination method, solving the over-determined algebraic equations, we get the following results:

4 00 Q. Wang, Y. Chen / Chaos, Solitons and Fractals 30 (006) A 1 ¼ l 1ðkh 1 l 1 a 0 k kl h 3 Þb 1 l B 1 ¼ 1 l ðkh 1l 1 a 0 k kl h 3 Þb 1 l A ¼ lb 1l 1 ðh kl l 1 l kh 1 l 1 l kh 3 b 1 l 1 Þ B l ¼ ðl kh 1 l 1 b 1 l kh 3Þlb 1 l 1 a 1 ¼ b 1l 1 l l C 1 ¼ 1 lb 1ðh 4 k b 1 l kh 1 l 1 l kh 3Þ ð3:4þ where a 0, A 0, b 1, l 1, l, k, l and k are arbitrary constants. According to (3.3), (3.4) and the general solutions of (.5) listed in Appendix A, we will obtain the following wave solutions for BKK system. Note that: Since the solutions obtained here are so many, we just list some new solutions for the BKK system to illustrate the efficiency of our method. Family 1. When h 1 ¼ h 3 ¼ 1 and h ¼ h 4 ¼ 1, then we obtain following solutions: H 1 ¼ a 0 b 1l 1 ðtanhðnþisechðnþþ b 1 l ðcothðnþcschðnþþ l ðl 1 ðtanhðnþisechðnþþ l ðcothðnþcschðnþþ 1Þ ð3:5:1þ G 1 ¼ A 0 ðkl 1 a 0 k kl Þb 1 lðl 1 ðtanhðnþisechðnþþ l ðcothðnþcschðnþþþ l ðl 1 ðtanhðnþisechðnþþ l ðcothðnþcschðnþþ 1Þ lb 1l 1 ð kl l 1 l k l 1 l k b 1l 1 ÞðtanhðnÞisechðnÞÞ 4l ðl 1ðtanhðnÞisechðnÞÞ l ðcothðnþcschðnþþ 1Þ ðl kl 1 b 1 l kþlb 1l 1 ðtanhðnþisechðnþþðcothðnþcschðnþþ l ðl 1 ðtanhðnþisechðnþþ l ðcothðnþcschðnþþ 1Þ lb 1 ðk b 1 l kl 1 l kþðcothðnþcschðnþþ ðl 1 ðtanhðnþisechðnþþ l ðcothðnþcschðnþþ 1Þ ð3:5:þ where n = k(x + ly + kt), a 0, A 0, b 1, l 1, l, k, l and k are arbitrary constants. Family. When h 1 ¼ h ¼ 1 and h 3 ¼ h 4 ¼ 1, then we obtain following solutions: H ¼ a 0 b 1l 1 ðsecðnþtanðnþþ b 1 l ðcscðnþcotðnþþ l ðl 1 ðsecðnþtanðnþþ l ðcscðnþcotðnþþ 1Þ ð3:6:1þ G ¼ A 0 ðkl 1 a 0 k kl Þb 1 lðl 1 ðsecðnþtanðnþþ l ðcscðnþcotðnþþþ l ðl 1 ðsecðnþtanðnþþ l ðcscðnþcotðnþþ 1Þ lb 1l 1 ðkl l 1 l k l 1 l k b 1l 1 ÞðsecðnÞtanðnÞÞ 4l ðl 1ðsecðnÞtanðnÞÞ l ðcscðnþcotðnþþ 1Þ ðl kl 1 b 1 l kþlb 1l 1 ðsecðnþtanðnþþðcscðnþcotðnþþ l ðl 1 ðsecðnþtanðnþþ l ðcscðnþcotðnþþ 1Þ lb 1ðk b 1 l kl 1 l kþðcscðnþcotðnþþ 4ðl 1 ðsecðnþtanðnþþ l ðcscðnþcotðnþþ 1Þ ð3:6:þ where n = k(x + ly + kt), a 0, A 0, b 1, l 1, l, k, l and k are arbitrary constants. Family 3. When h 1 ¼ 1, h ¼ 1 and h 3 ¼ h 4 ¼ 1, then we obtain following solutions: H 3 ¼ a 0 b 1l 1 ðtanhðnþisechðnþþ b 1 l ðsecðnþtanðnþþ l ðl 1 ðtanhðnþisechðnþþ l ðsecðnþtanðnþþ 1Þ ð3:7:1þ G 3 ¼ A 0 ð kl 1 a 0 k kl Þb 1 lðl 1 ðtanhðnþisechðnþþ l ðsecðnþtanðnþþþ l ðl 1 ðtanhðnþisechðnþþ l ðsecðnþtanðnþþ 1Þ lb 1l 1 ðkl l 1 l k l 1 l k b 1l 1 ÞðtanhðnÞisechðnÞÞ 4l ðl 1ðtanhðnÞisechðnÞÞ l ðsecðnþtanðnþþ 1Þ ð l kl 1 b 1 l kþlb 1l 1 ðtanhðnþisechðnþþðsecðnþtanðnþþ l ðl 1 ðtanhðnþisechðnþþ l ðsecðnþtanðnþþ 1Þ lb 1 ðk b 1 l kl 1 l kþðsecðnþtanðnþþ 4ðl 1 ðtanhðnþisechðnþþ l ðsecðnþtanðnþþ 1Þ where n = k(x + ly + kt), a 0, A 0, b 1, l 1, l, k, l and k are arbitrary constants. ð3:7:þ

5 Q. Wang, Y. Chen / Chaos, Solitons and Fractals 30 (006) Family 4. When h 1 ¼ 1, h ¼ 1 and h 3 ¼ h 4 ¼ 1, then we obtain following solutions: H 4 ¼ a 0 b 1l 1 ðcothðnþcschðnþþ b 1 l ðsecðnþtanðnþþ l ðl 1 ðcothðnþcschðnþþ l ðsecðnþtanðnþþ 1Þ G 4 ¼ A 0 ð kl 1 a 0 k kl Þb 1 lðl 1 ðcothðnþcschðnþþ l ðsecðnþtanðnþþþ l ðl 1 ðcothðnþcschðnþþ l ðsecðnþtanðnþþ 1Þ lb 1l 1 ðkl l 1 l k l 1 l k b 1l 1 ÞðcothðnÞcschðnÞÞ 4l ðl 1ðcothðnÞcschðnÞÞ l ðsecðnþtanðnþþ 1Þ ð l kl 1 b 1 l kþlb 1l 1 ðcothðnþcschðnþþðsecðnþtanðnþþ l ðl 1 ðcothðnþcschðnþþ l ðsecðnþtanðnþþ 1Þ lb 1ðk b 1 l kl 1 l kþðsecðnþtanðnþþ ðl 1 ðcothðnþcschðnþþ l ðsecðnþtanðnþþ 1Þ where n = k(x + ly + kt), a 0, A 0, b 1, l 1, l, k, l and k are arbitrary constants. Family 5. When h 1 ¼ 1, h ¼ 1 and h 3 ¼ h 4 ¼ 1, then we obtain following solutions: H 5 ¼ a 0 b 1l 1 ðtanhðnþisechðnþþ b 1 l ðcscðnþcotðnþþ l ðl 1 ðtanhðnþisechðnþþ l ðcscðnþcotðnþþ 1Þ G 5 ¼ A 0 ð kl 1 a 0 k kl Þb 1 lðl 1 ðtanhðnþisechðnþþ l ðcscðnþcotðnþþþ l ðl 1 ðtanhðnþisechðnþþ l ðcscðnþcotðnþþ 1Þ lb 1l 1 ðl k l 1 l k l 1 l k b 1l 1 ÞðtanhðnÞisechðnÞÞ 4l ðl 1ðtanhðnÞisechðnÞÞ l ðcscðnþcotðnþþ 1Þ ð l kl 1 b 1 l kþlb 1l 1 ðtanhðnþisechðnþþðcscðnþcotðnþþ l ðl 1 ðtanhðnþisechðnþþ l ðcscðnþcotðnþþ 1Þ lb 1ðk b 1 l kl 1 l kþðcscðnþcotðnþþ ðl 1 ðtanhðnþisechðnþþ l ðcscðnþcotðnþþ 1Þ where n = k(x + ly + kt), a 0, A 0, b 1, l 1, l, k, l and k are arbitrary constants. Family 6. When h 1 ¼ 1, h ¼ 1 and h 3 ¼ h 4 ¼ 1, then we obtain following solutions: H 6 ¼ a 0 b 1l 1 ðcothðnþcschðnþþ b 1 l ðcscðnþcotðnþþ l ðl 1 ðcothðnþcschðnþþ l ðcscðnþcotðnþþ 1Þ G 6 ¼ A 0 ð kl 1 a 0 k kl Þb 1 lðl 1 ðcothðnþcschðnþþ l ðcscðnþcotðnþþþ l ðl 1 ðcothðnþcschðnþþ l ðcscðnþcotðnþþ 1Þ lb 1l 1 ðl k l 1 l k l 1 l k b 1l 1 ÞðcothðnÞcschðnÞÞ 4l ðl 1ðcothðnÞcschðnÞÞ l ðcscðnþcotðnþþ 1Þ ð l kl 1 b 1 l kþlb 1l 1 ðcothðnþcschðnþþðcscðnþcotðnþþ l ðl 1 ðcothðnþcschðnþþ l ðcscðnþcotðnþþ 1Þ lb 1ðk b 1 l kl 1 l kþðcscðnþcotðnþþ ðl 1 ðcothðnþcschðnþþ l ðcscðnþcotðnþþ 1Þ where n = k(x + ly + kt), a 0, A 0, b 1, l 1, l, k, l and k are arbitrary constants. ð3:8:1þ ð3:8:þ ð3:9:1þ ð3:9:þ ð3:10:1þ ð3:10:þ 4. Conclusion In this method, a new algebraic MRERE method is presented to find new exact solution of nonlinear PDEs. The Broer Kaup Kupershmidt system is chosen to illustrate the method such that some novel solutions are found, which include solutions of hyperbolic (solitary) function and triangular periodic functions appearing at the same time. Of course, the algorithm can also be applied to many nonlinear PDEs in mathematical physics. Further work is to extend the MRERE method to construct solutions of soliton-like solution and triangular periodic functions solution appearing in a solution at the same time.

6 0 Q. Wang, Y. Chen / Chaos, Solitons and Fractals 30 (006) Acknowledgments The work was supported by the China Postdoctoral Science Foundation, Nature Foundation of Zhejiang Province of China (Y604056) and Ningbo Doctoral Foundation of China (005A610030). Appendix A are The general solutions of the Riccati Eq. (.5) df dn ¼ R 1 R F (1) when R 1 ¼ 1 and R ¼ 1, F ðnþ ¼tanhðnÞisechðnÞ F ðnþ ¼cothðnÞcschðnÞ () when R 1 ¼ R ¼ 1, F ðnþ ¼secðnÞtanðnÞ F ðnþ ¼cscðnÞcotðnÞ (3) when R 1 = 1 and R = 1, F ðnþ ¼tanhðnÞ F ðnþ ¼cothðnÞ. (4) when R 1 = R =1, F ðnþ ¼tanðnÞ (5) when R 1 = R = 1, F ðnþ ¼cotðnÞ (6) when R 1 = 0 and R 50, 1 F ðnþ ¼ R n c 0 where c 0 is an arbitrary constant. References [1] Ablowitz MJ, Clarkson PA. Soliton, nonlinear evolution equations and inverse scatting. New York: Cambridge University Press [] Wadati M. J Phys Soc Jpn :673 Wadati M. J Phys Soc Jpn :681 Wadati M, Sanuki H, Konno K. Prog Theor Phys :419 Konno K, Wadati M. Prog Theor Phys :165. [3] Chen Y, Li B, Zhang HQ. Chaos, Solitons & Fractals 00317:693 Chen Y, Yan ZY, Zhang HQ. Theor Math Phys 0013(1):970. [4] Matveev VA, Salle MA. Darboux transformations and solitons. Berlin, Heidelberg: Springer-Verlag [5] Lou SY, Lu JZ. J Phys A 19969:409. [6] Fan E. Phys Lett A 00077:1 Fan E. Phys Lett A 00185:373. [7] Conte R, Musette M. J Phys A: Math Gen 1995:5609. [8] Yan ZY. Phys Lett A 0019:100 Yan ZY. Chaos, Solitons & Fractals 00316:759. [9] Chen Y, Li B. Chaos, Solitons & Fractals 00419(4):977 Chen Y et al. Chin Phys 0031(9):940 Wang Q et al. Commun Theor Phys 00441(6):81. [10] Wang Q et al. Chaos, Solitons & Fractals 0053:477 Chen Y, Wang Q, Li B. Naturforsch A 00459a:59.

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