Exact solutions and conservation laws of a -dimensional B-type Kadomtsev-Petviashvili equation
© Abudiab and Khalique; licensee Springer 2013
Received: 3 May 2013
Accepted: 26 June 2013
Published: 23 July 2013
In this paper we study a -dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation. This equation is an extension of the well-known Kadomtsev-Petviashvili equation, which describes weakly dispersive and small amplitude waves propagating in quasi-two-dimensional media. We first obtain exact solutions of the BKP equation using the multiple-exp function and simplest equation methods. Furthermore, the conservation laws for the BKP equation are constructed by using the multiplier method.
It is well known that many phenomena in science and engineering, especially in fluid mechanics, solid state physics, plasma physics, plasma waves and biology, are described by the nonlinear partial differential equations (NLPDEs). Therefore the investigation of exact solutions of NLPDEs plays an important role in the study of NLPDEs. For this reason, during the last few decades, researchers have established several methods to find exact solutions to NLPDEs. Some of these methods include the inverse scattering transform method , the Bäcklund transformation , the Darboux transformation , the Hirota bilinear method , the -expansion method , the homogeneous balance method , the variable separation approach , the tri-function method [8, 9], the sine-cosine method , the Jacobi elliptic function expansion method [11, 12], the exp-function expansion method  and the Lie symmetry method [14–16].
where α is a real-valued constant. This is a nonlinear wave equation in three spatial and one temporal coordinate .
was studied in [18–20] by different approaches. In  a new form of the -dimensional BKP equation given by (1.1) was investigated and it was shown, using the simplified form of the Hirota method, that one- and two-soliton solutions exist for (1.1). Also, specific constraints were developed that guarantee the existence of multiple soliton solutions for (1.1).
In this paper we employ the multiple exp-function method  and the simplest equation method [22, 23] to obtain some exact solutions of (1.1). In addition to this, conservation laws are constructed for (1.1) using the multiplier method .
2 Exact solutions of (1.1)
In this section we employ two methods of solution.
2.1 Exact solutions using the multiple exp-function method
where θ is any root of .
and θ is any root of .
2.2 The simplest equation method
In this subsection we use the simplest equation method and obtain exact solutions of (1.1). This method was introduced by Kudryashov  and modified by Vitanov . The simplest equations we use in this paper are the Bernoulli and Riccati equations. Their solutions can be written in elementary functions. For details, see, for example, .
where satisfies the Bernoulli and Riccati equations, M is a positive integer that can be determined by balancing procedure as in  and are parameters to be determined.
2.2.1 Solutions of (1.1) using the Bernoulli equation as the simplest equation
where and C is a constant of integration.
2.2.2 Solutions of (1.1) using the Riccati equation as the simplest equation
where and C is a constant of integration.
3 Conservation laws
In this section we construct conservation laws for -dimensional generalized B-type Kadomtsev-Petviashvili equation (1.1). The multiplier method will be used [15, 24, 26]. First we recall some results that will be used in the computation of conserved vectors.
and equation (3.3) defines a local conservation law of system (3.1).
After obtaining the multipliers, we can calculate the conserved vectors by using a homotopy formula .
3.2 Construction of conservation laws for (1.1)
Remark 1 Due to the presence of the arbitrary function f in the multiplier, one can obtain infinitely many conservation laws.
4 Concluding remarks
In this paper we studied -dimensional generalized B-type Kadomtsev-Petviashvili equation (1.1). Exact solutions of the BKP equation were found using two distinct methods, namely the multiple-exp function method and the simplest equation method. Also, the conservation laws for the BKP equation were derived by using the multiplier method.
MA and CMK would like to thank the organizing committee of the International Conference on the Theory, Methods and Application of Nonlinear Equations, held at Texas A&M University-Kingsville, USA, for their kind hospitality during the conference.
- Ablowitz MJ, Clarkson PA: Soliton, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge; 1991.View ArticleGoogle Scholar
- Gu CH: Soliton Theory and Its Application. Zhejiang Science and Technology Press, Zhejiang; 1990.Google Scholar
- Matveev VB, Salle MA: Darboux Transformation and Soliton. Springer, Berlin; 1991.View ArticleGoogle Scholar
- Hirota R: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge; 2004.View ArticleGoogle Scholar
- Wang M, Xiangzheng LX, Jinliang ZJ: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
- 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
- Lou SY, Lu JZ: Special solutions from variable separation approach: Davey-Stewartson equation. J. Phys. A, Math. Gen. 1996, 29: 4209-4215. 10.1088/0305-4470/29/14/038MathSciNetView ArticleGoogle Scholar
- Yan ZY: The new tri-function method to multiple exact solutions of nonlinear wave equations. Phys. Scr. 2008., 78: Article ID 035001Google Scholar
- Yan ZY: Periodic, solitary and rational wave solutions of the 3D extended quantum Zakharov-Kuznetsov equation in dense quantum plasmas. Phys. Lett. A 2009, 373: 2432-2437. 10.1016/j.physleta.2009.04.018View ArticleGoogle Scholar
- Wazwaz M: The tanh and sine-cosine method for compact and noncompact solutions of nonlinear Klein-Gordon equation. Appl. Math. Comput. 2005, 167: 1179-1195. 10.1016/j.amc.2004.08.006MathSciNetView ArticleGoogle Scholar
- Lu DC: Jacobi elliptic functions solutions for two variant Boussinesq equations. Chaos Solitons Fractals 2005, 24: 1373-1385. 10.1016/j.chaos.2004.09.085MathSciNetView ArticleGoogle Scholar
- Yan ZY:Abundant families of Jacobi elliptic functions of the -dimensional integrable Davey-Stewartson-type equation via a new method. Chaos Solitons Fractals 2003, 18: 299-309. 10.1016/S0960-0779(02)00653-7MathSciNetView ArticleGoogle Scholar
- He JH, Wu XH: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 2006, 30: 700-708. 10.1016/j.chaos.2006.03.020MathSciNetView ArticleGoogle Scholar
- Bluman GW, Kumei S Applied Mathematical Sciences 81. In Symmetries and Differential Equations. Springer, New York; 1989.View ArticleGoogle Scholar
- Olver PJ Graduate Texts in Mathematics 107. In Applications of Lie Groups to Differential Equations. 2nd edition. Springer, Berlin; 1993.View ArticleGoogle Scholar
- Adem KR, Khalique CM:Exact solutions and conservation laws of a -dimensional nonlinear KP-BBM equation. Abstr. Appl. Anal. 2013., 2013: Article ID 791863 10.1155/2013/791863Google Scholar
- Wazwaz AM:Two forms of -dimensional B-type Kadomtsev-Petviashvili equation: multiple-soliton solutions. Phys. Scr. 2012., 86: Article ID 035007Google Scholar
- Wazwaz AM:Distinct kinds of multiple-soliton solutions for a -dimensional generalized B-type Kadomtsev-Petviashvili equation. Phys. Scr. 2011., 84: Article ID 055006Google Scholar
- Shen HF, Tu MH: On the constrained B-type Kadomtsev-Petviashvili equation: Hirota bilinear equations and Virasoro symmetry. J. Math. Phys. 2011., 52: Article ID 032704Google Scholar
- Ma WX, Abdeljabbar A, Asaad MG:Wronskian and Grammian solutions to a -dimensional generalized KP equation. Appl. Math. Comput. 2011, 217: 10016-10023. 10.1016/j.amc.2011.04.077MathSciNetView ArticleGoogle Scholar
- Ma WX, Huang T, Zhang Y: A multiple exp-function method for nonlinear differential equations and its applications. Phys. Scr. 2010., 82: Article ID 065003Google Scholar
- Kudryashov NA: Simplest equation method to look for exact solutions of nonlinear differential equations. Chaos Solitons Fractals 2005, 24: 1217-1231. 10.1016/j.chaos.2004.09.109MathSciNetView ArticleGoogle Scholar
- Vitanov NK: Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of PDEs with polynomial nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 2050-2060. 10.1016/j.cnsns.2009.08.011MathSciNetView ArticleGoogle Scholar
- Anco SC, Bluman GW: Direct construction method for conservation laws of partial differential equations. Part I: examples of conservation law classifications. Eur. J. Appl. Math. 2002, 13: 545-566.MathSciNetGoogle Scholar
- Adem AR, Khalique CM: Symmetry reductions, exact solutions and conservation laws of a new coupled KdV system. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 3465-3475. 10.1016/j.cnsns.2012.01.010MathSciNetView ArticleGoogle Scholar
- Anthonyrajah M, Mason DP: Conservation laws and invariant solutions in the Fanno model for turbulent compressible flow. Math. Comput. Appl. 2010, 15: 529-542.MathSciNetGoogle 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.