Exact three-wave solutions for the -dimensional Boiti-Leon-Manna-Pempinelli equation
© Ma et al.; licensee Springer. 2013
Received: 11 July 2013
Accepted: 26 September 2013
Published: 18 November 2013
In this paper, we consider a -dimensional Boiti-Leon-Manna-Pempinelli equation. We employ the Hirota bilinear method to obtain the bilinear form of the -dimensional Boiti-Leon-Manna-Pempinelli equation. Based on the bilinear form, we derive exact three-wave solutions by using an extended three-soliton method. In addition, we also get the trajectory of some solution with the help of MAPLE.
Integrable systems and nonlinear evolution equations [1–9] have attracted much attention of mathematicians and physicists. Especially, exact solutions of nonlinear evolution equations play a pivotal role in the study of mathematical physical phenomena. Not only can these exact solutions describe many important phenomena in physics and other fields, but they can also help physicists to understand the mechanisms of the complicated physical phenomena. A variety of powerful methods have been employed to study nonlinear phenomena, such as the inverse scattering transform , the tanh function method , the extended tanh-function method , the homogeneous balance method , the auxiliary function method , and the exp-function method , the Pfaffian technique , the dressing method , the Bäcklund transformation method , the Darboux transformation , the generalized symmetry method, the tri-function method  and the -expansion method , the modified CK direct method .
Very recently, Dai et al. proposed a new technique called the three-wave approach to seek periodic solitary wave solutions for integrable equations . The method is to use Frobenius’ idea  to reduce the PDE into integrable ODEs. Frobenius’ idea was successfully used to establish the transformed rational function method  and to solve the KPP equation . In fact, the Tanh function method and the expansion method are special cases of the reduction idea raised in , say, the general Frobenius idea. Furthermore, a three-wave solution in -dimension was obtained by using the multiple exp-function method [27, 28]. With the rapid development of computer technology and the help of symbolic computation, this approach is of utmost simplicity. Hence, it can be applied to many kinds of nonlinear evolution equations and higher-dimensional soliton equations. Zitian Li obtained periodic cross-kink wave solutions, doubly periodic solitary wave solutions and breather type of two-solitary wave solutions for the -dimensional Jimbo-Miwa equation by this method . Wang applied the method to a higher dimensional KdV-type equation .
where and subscripts represent partial differentiation with respect to the given variable. Boiti et al.  also discussed the Painlevé property, Lax pairs and some exact solutions of -dimensional BLMP. Through the Bäcklund transformation, Bai and Zhao got some new solutions of the BLMP equation. By means of the multilinear variable separation approach, a general variable separation solution of the BLMP equation was derived in . Liu proposed a simple Bäcklund transformation of a potential BLMP system by using the standard truncated Painlevé expansion and symbolic computation, and a solution of the potential BLMP system with three arbitrary functions was given in . The symmetry, similarity reductions and new solutions of the -dimensional BLMP equation were obtained in . These solutions include rational function solutions, double-twisty function solutions, Jacobi oval function solutions and triangular cycle solutions. In , based on the binary Bell polynomials, the bilinear form for the BLMP equation was obtained. The new exact solutions were derived with an arbitrary function in y, and soliton interaction properties were discussed by the graphical analysis. The author in  discussed the BLMP equation and generalized breaking soliton equations by using the exponential function and obtained some new exact solutions of the equations. By using the modified Clarkson-Kruskal (CK) direct method, Li et al.  constructed a Bäcklund transformation of the -dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation. Laurent Delisle and Masoud Mosaddeghi proposed the study of the BLMP equation from two points of view: the classical and the super symmetric. They constructed new solutions of this equation from Wronskian formalism and the Hirota method in .
which was introduced by Darvishi in . We apply the extended three-soliton method to the -dimensional Boiti-Leon-Manna-Pempinelli equation, obtaining more exact solutions including a complexiton solution, periodic cross-kink solutions about it.
where P is a polynomial in its arguments. The solution method will also work for systems of nonlinear equations and high-dimensional ones.
where F is a polynomial in its arguments.
where , , and , , , () are free constants to be determined later.
Step 4. Solving the set of algebraic equations defined by Step 3 with the help of MAPLE, we can derive parameters , , , , , , (). Therefore, we can obtain abundant exact multi-wave solutions of Eq. (3).
3 Exact three-wave solutions for the -dimensional Boiti-Leon-Manna-Pempinelli equation
Solving the above algebraic equations with the help of MAPLE gives the following solutions.
where , , , , are arbitrary constants.
where and , , , , , are free constants.
where and , , , , , are free constants.
where , , and , , , , , , , , are free constants.
where , and , , , , , , , are free constants.
where , , , , , , , , and are free constants.
Remark 1 Noting if we set in Case 1 to Case 5 of the solutions above are special solutions of the equation, we can see that for an arbitrary function, is also a solution. However, the other cases are different.
Remark 2 Noting and , the solutions presented in this paper can be obtained by using the multiple exp-function. Furthermore, we can get an N-soliton solution just by modifying the ansatz and using the exp expanding method .
In this paper, we obtained three-wave solutions to the -dimensional Boiti-Leon-Manna-Pempinelli equation with the extended three-soliton method. All the presented solutions show remarkable richness of the solution space of the -dimensional Boiti-Leon-Manna-Pempinelli equation and also that the -dimensional integrable system may have very rich dynamical behavior. The considered solutions are of complexiton type . There is also a generalized theory of the Bell polynomials method which describes the generalized bilinear differential equations [42, 43]. To our knowledge, our solutions are novel. They cannot be obtained just through the simple generalization of the -dimensional BLMP equation. In fact, the extended three-soliton method is entirely algorithmic and involves a large amount of tedious calculations. However, the method is direct, concise and effective. Therefore, we can apply the method to the variety of dynamics of a higher-dimensional nonlinear system and many other types of a nonlinear evolution equation in further work.
The authors are in debt to thank the anonymous referees for helpful suggestions. The work is supported by the National Natural Science Foundation of China (project No. 11371086), the Fund of Science and Technology Commission of Shanghai Municipality (project No. ZX201307000014) and the Fundamental Research Funds for the Central Universities.
- Korteweg D, de Vries G: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 1895, 39: 422-443.View ArticleGoogle Scholar
- Toda K, Yu S:The investigation into new equations in -dimensions. J. Nonlinear Math. Phys. 2001, 8: 272-277. 10.2991/jnmp.2001.8.s.47MathSciNetView ArticleGoogle Scholar
- Yu S, Toda K:Lax pairs, Painlevè properties and exact solutions of the alogero Korteweg-de Vries equation and a new -dimensional equation. J. Nonlinear Math. Phys. 2000, 7: 1-13. 10.2991/jnmp.2000.7.1.1MathSciNetView ArticleGoogle Scholar
- Li C-X, Ma WX, Liu X-J, Zeng Y-B: Wronskian solutions of the Boussinesq equation-solitons, negatons, positons and complexitons. Inverse Probl. 2007, 23: 279-296. 10.1088/0266-5611/23/1/015MathSciNetView ArticleGoogle Scholar
- Ma WX, Maruno K: Complexiton solutions of the Toda lattice equation. Physica A 2004, 343: 219-237.MathSciNetView ArticleGoogle Scholar
- Ma WX, You Y: Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions. Trans. Am. Math. Soc. 2005, 357: 1753-1778. 10.1090/S0002-9947-04-03726-2MathSciNetView ArticleGoogle Scholar
- Wazwaz A-M: New higher-dimensional fifth-order nonlinear equations with multiple soliton solutions. Phys. Scr. 2011., 84: Article ID 025007Google Scholar
- Wazwaz A-M:A new -dimensional Korteweg-de Vries equation and its extension to a new -dimensional Kadomtsev-Petviashvili equation. Phys. Scr. 2011., 84: Article ID 035010Google Scholar
- Asaad MG, Ma WX:Extended Gram-type determinant, wave and rational solutions to two -dimensional nonlinear evolution equations. Appl. Math. Comput. 2012, 219: 213-225. 10.1016/j.amc.2012.06.007MathSciNetView ArticleGoogle Scholar
- Ablowitz MJ, Clarkson PA: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York; 1991.View ArticleGoogle Scholar
- Parkes EJ, Duffy BR: An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. Phys. Commun. 1996, 98: 288-300. 10.1016/0010-4655(96)00104-XView ArticleGoogle Scholar
- Fan E: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 2000, 277: 212-218. 10.1016/S0375-9601(00)00725-8MathSciNetView ArticleGoogle Scholar
- Wang M, Zhou Y, Li Z: Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Phys. Lett. A 1996, 216: 67-75. 10.1016/0375-9601(96)00283-6View ArticleGoogle Scholar
- Zhang S, Xia T: A generalized new auxiliary equation method and its applications to nonlinear partial differential equations. Phys. Lett. A 2007, 363: 356-360. 10.1016/j.physleta.2006.11.035MathSciNetView ArticleGoogle Scholar
- He J-H: An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. Int. J. Mod. Phys. B 2008, 22: 3487-3578. 10.1142/S0217979208048668View ArticleGoogle Scholar
- Hirota R: Soliton solutions to the BKP equations. I. The Pfaffian technique. J. Phys. Soc. Jpn. 1989, 58: 2285-2296. 10.1143/JPSJ.58.2285MathSciNetView ArticleGoogle Scholar
- Zakharov V, Shabat A: Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II. Funct. Anal. Appl. 1979, 13: 166-174.MathSciNetView ArticleGoogle Scholar
- Satsuma J, Kaup DJ: A Bäcklund transformation for a higher order Korteweg-de Vries equation. J. Phys. Soc. Jpn. 1977, 43: 692-697. 10.1143/JPSJ.43.692MathSciNetView ArticleGoogle Scholar
- Gu C, Hu H, Zhou Z: Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry. Kluwer Academic, London; 2005.View ArticleGoogle Scholar
- Yan Z: The new tri-function method to multiple exact solutions of nonlinear wave equations. Phys. Scr. 2008., 78: Article ID 035001Google Scholar
- Wang M, Li X, Zhang J:The -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 2008, 372: 417-423. 10.1016/j.physleta.2007.07.051MathSciNetView ArticleGoogle Scholar
- Lou S, Ma H-C:Non-Lie symmetry groups of -dimensional nonlinear systems obtained from a simple direct method. J. Phys. A, Math. Gen. 2005, 38: L129-L137. 10.1088/0305-4470/38/7/L04MathSciNetView ArticleGoogle Scholar
- Dai Z, Lin S, Fu H, Zeng X: Exact three-wave solutions for the KP equation. Appl. Comput. Math. 2010, 216: 1599-1604. 10.1016/j.amc.2010.03.013MathSciNetView ArticleGoogle Scholar
- Ma WX, Wu H, He J-S: Partial differential equations possessing Frobenius integrable decompositions. Phys. Lett. A 2007, 364: 29-32. 10.1016/j.physleta.2006.11.048MathSciNetView ArticleGoogle Scholar
- Ma WX, Lee J-H:A transformed rational function method and exact solutions to the dimensional Jimbo-Miwa equation. Chaos Solitons Fractals 2009, 42: 1356-1363. 10.1016/j.chaos.2009.03.043MathSciNetView ArticleGoogle Scholar
- Ma WX, Fuchssteiner B: Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation. Int. J. Non-Linear Mech. 1996, 31: 329-338. 10.1016/0020-7462(95)00064-XMathSciNetView ArticleGoogle Scholar
- Ma WX, Huang T, Zhang Y: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 2010., 82: Article ID 065003Google Scholar
- Ma WX, Zhu Z:Solving the -dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Comput. Math. 2012, 218: 11871-11879. 10.1016/j.amc.2012.05.049MathSciNetView ArticleGoogle Scholar
- Li Z, Dai Z, Liu J:Exact three-wave solutions for the -dimensional Jimbo-Miwa equation. Comput. Math. Appl. 2011, 61: 2062-2066. 10.1016/j.camwa.2010.08.070MathSciNetView ArticleGoogle Scholar
- Wang C, Dai Z, Liang L: Exact three-wave solution for higher dimensional KdV-type equation. Appl. Comput. Math. 2010, 216: 501-505. 10.1016/j.amc.2010.01.057MathSciNetView ArticleGoogle Scholar
- Boiti M, Leon JJ-P, Pempinelli F: On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions. Inverse Probl. 1986, 2: 271-279. 10.1088/0266-5611/2/3/005MathSciNetView ArticleGoogle Scholar
- Bai C-J, Zhao H:New solitary wave and Jacobi periodic wave excitations in -dimensional Boiti-Leon-Manna-Pempinelli system. Int. J. Mod. Phys. B 2008, 22: 2407-2420. 10.1142/S021797920803954XMathSciNetView ArticleGoogle Scholar
- Liu GT: Bäcklund transformation and new coherent structures of the potential BLMP system. J. Inn. Mong. Norm. Univ. 2008, 37: 145-148.Google Scholar
- Liu N, Liu X:Symmetries, new exact solutions and conservation laws of -dimensional Boiti-Leon-Manna-Pempinelli equation. Chin. J. Quantum Electron. 2008, 25: 546-552.Google Scholar
- Luo L: New exact solutions and Bäcklund transformation for Boiti-Leon-Manna-Pempinelli equation. Phys. Lett. A 2011, 375: 1059-1063. 10.1016/j.physleta.2011.01.009MathSciNetView ArticleGoogle Scholar
- Zhang LL: Exact solutions of breaking soliton equations and BLMP equation. J. Liaocheng Univ. Nat. Sci 2008, 21: 35-38.Google Scholar
- Li Y, Li D:New exact solutions for the -dimensional Boiti-Leon-Manna-Pempinelli equation. Appl. Math. Sci. 2012, 6: 579-587.MathSciNetGoogle Scholar
- Delisle L, Mosaddeghi M: Classical and SUSY solutions of the Boiti-Leon-Manna-Pempinelli equation. J. Phys. A, Math. Theor. 2013., 46: Article ID 115203Google Scholar
- Darvishi M, Najafi M, Kavitha L, Venkatesh M:Stair and step soliton solutions of the integrable and -Dimensional Boiti-Leon-Manna-Pempinelli Equations. Commun. Theor. Phys. 2012, 58: 785-794. 10.1088/0253-6102/58/6/01View ArticleGoogle Scholar
- Hirota R: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge; 2004.View ArticleGoogle Scholar
- Ma WX: Complexiton solutions to the Korteweg-de Vries equation. Phys. Lett. A 2002, 301: 35-44. 10.1016/S0375-9601(02)00971-4MathSciNetView ArticleGoogle Scholar
- Ma WX: Bilinear equations, Bell polynomials and linear superposition principle. J. Phys. Conf. Ser. 2013., 411: Article ID 012021Google Scholar
- Ma WX: Generalized bilinear differential equations. Stud. Nonlinear Sci. 2011, 2: 140-144.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.