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Exact solutions and conservation laws of a -dimensional B-type Kadomtsev-Petviashvili equation
Advances in Difference Equations volume 2013, Article number: 221 (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].
The purpose of this paper is to study one such NLPDE, namely the -dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation, that is given by 
where α is a real-valued constant. This is a nonlinear wave equation in three spatial and one temporal coordinate .
It is well known that the Kadomtsev-Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in quasi-two-dimensional media . The KP hierarchy of B-type possesses many integrable structures as the KP hierarchy. The -dimensional nonlinear generalized BKP equation
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
In this subsection we employ the multiple exp-function method and obtain exact explicit one-wave and two-wave solutions of (1.1). For details of the method, the reader is referred to the paper , in which this method was introduced. So, following the method and using the notation of , for a one-wave solution, we have
and the resulting one-wave solution is
where θ is any root of .
Likewise, for a two-wave solution, we have
and the resulting two-wave solution is
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, .
Making use of the wave variable
where , are constants, the -dimensional generalized B-type Kadomtsev-Petviashvili (1.1) transforms to a fourth-order nonlinear ordinary differential equation (ODE)
Let us consider the solutions of ODE (2.1) in the form
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
The balancing procedure yields so the solutions of (2.1) are of the form
Substituting (2.3) into ODE (2.1) and making use of the Bernoulli equation and then equating the coefficients of the functions to zero, we obtain an algebraic system of equations. Solving this system with the aid of Mathematica, we obtain
As a result, a solution of (1.1) is
where and C is a constant of integration.
2.2.2 Solutions of (1.1) using the Riccati equation as the simplest equation
The balancing procedure yields , so the solutions of (2.1) are of the form
Substituting (2.4) into ODE (2.1) and making use of the Riccati equation, we obtain an algebraic system of equations by equating all coefficients of the functions to zero. Solving the algebraic equations, one obtains
Hence solutions of (1.1) are
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.
Consider a k th-order system of PDEs given by
with n independent variables and m dependent variables . Here denote the collections of all first, second, … , k th-order partial derivatives. That is, , respectively, where the total derivative operator with respect to is given by
The n-tuple , , , where is the space of differential functions, is a conserved vector of (3.1) if satisfies
and equation (3.3) defines a local conservation law of system (3.1).
The Euler-Lagrange operator, for each α, is defined as
A multiplier has the property that
hold identically. The right-hand side of (3.5) is a divergence expression. The determining equation for the multiplier is given by
After obtaining the multipliers, we can calculate the conserved vectors by using a homotopy formula .
3.2 Construction of conservation laws for (1.1)
We now construct conservation laws for -dimensional nonlinear BKP equation (1.1). We obtain a multiplier of the form
where C is an arbitrary constant and f is any solution of . Corresponding to the above multiplier, we obtain the following conserved vectors:
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.
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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.
The authors declare that they have no competing interests.
MA and CMK worked together in the derivation of the mathematical results. All authors read and approved the final manuscript.
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Abudiab, M., Khalique, C.M. Exact solutions and conservation laws of a -dimensional B-type Kadomtsev-Petviashvili equation. Adv Differ Equ 2013, 221 (2013). https://doi.org/10.1186/1687-1847-2013-221
- -dimensional generalized B-type Kadomtsev-Petviashvili equation
- multiple-exp function method
- simplest equation method
- conservation laws