In this paper we obtain exact solutions of the -dimensional higher-order Broer-Kaup system which was obtained from the Kadomtsev-Petviashvili equation by the symmetry constraints. The methods used to determine the exact solutions of the underlying system are the Lie group analysis and the simplest equation method. The solutions obtained are the solitary wave solutions. Moreover, we derive the conservation laws of the -dimensional higher-order Broer-Kaup system by employing the multiplier approach and the new conservation theorem.
Keywords:the -dimensional higher-order Broer-Kaup system; integrability; Lie group analysis; simplest equation method; solitary waves; conservation laws
which was first introduced by Lou and Hu  by considering the symmetry constraints of the Kadomtsev-Petviashvili equation. The system (1.1a) and (1.1b) is in fact an extension of the well-known -dimensional Broer-Kaup system [2-4]
which is used to model the bi-directional propagation of long waves in shallow water. In , Fan derived a unified Darboux transformation for the system (1.1a) and (1.1b) with the help of a gauge transformation of the spectral problem and as an application obtained some new explicit soliton-like solutions. Recently, Huang et al. presented a new N-fold Darboux transformations of the -dimensional higher-order Broer-Kaup system with the help of a gauge transformation of the spectral problem and found new explicit multi-soliton solutions of the system (1.1a) and (1.1b).
In the latter half of the nineteenth century, Sophus Lie (1842-1899) developed one of the most powerful methods to determine solutions of differential equations. This method, known as the Lie group analysis method, systematically unifies and extends well-known ad hoc techniques to construct explicit solutions of differential equations. It has proved to be a versatile tool for solving nonlinear problems described by the differential equations arising in mathematics, physics and in other scientific fields of study. For the theory and application of the Lie group analysis methods, see, e.g., the Refs. [7-12].
Conservation laws play a vital role in the solution process of differential equations. Finding conservation laws of the system of differential equations is often the first step towards finding the solution . Also, the conservation laws are useful in the numerical integration of partial differential equations , for example, to control numerical errors. The determination of conservation laws of the Korteweg de Vries equation, in fact, initiated the discovery of a number of methods to solve evolutionary equations . Moreover, conservation laws play an important role in the theories of non-classical transformations [15,16], normal forms and asymptotic integrability . Recently, in  the conserved quantity was used to determine the unknown exponent in the similarity solution which cannot be obtained from the homogeneous boundary conditions.
In this paper, we use the Lie group analysis approach along with the simplest equation method to obtain exact solutions of the -dimensional higher-order Broer-Kaup system (1.1a) and (1.1b). Furthermore, conservation laws will be computed for (1.1a) and (1.1b) using the two approaches: the new conservation theorem due to Ibragimov  and the multiplier method [20,21].
2 Symmetry reductions and exact solutions of (1.1a) and (1.1b)
Applying the third prolongation  to (1.1a) and (1.1b) and solving the resultant overdetermined system of linear partial differential equations one obtains the following three Lie point symmetries:
2.1 One-dimensional optimal system of subalgebras
In this subsection we present an optimal system of one-dimensional subalgebras for the system (1.1a) and (1.1b) to obtain an optimal system of group-invariant solutions. The method which we use here for obtaining a one-dimensional optimal system of subalgebras is that given in . The adjoint transformations are given by
The commutator table of the Lie point symmetries of the system (1.1a) and (1.1b) and the adjoint representations of the symmetry group of (1.1a) and (1.1b) on its Lie algebra are given in Table 1 and Table 2, respectively. Table 1 and Table 2 are used to construct an optimal system of one-dimensional subalgebras for the system (1.1a) and (1.1b).
Table 1. Commutator table of the Lie algebra of the system (1.1a) and (1.1b)
Table 2. Adjoint table of the Lie algebra of the system (1.1a) and (1.1b)
2.2 Symmetry reductions of (1.1a) and (1.1b)
In this subsection we use the optimal system of one-dimensional subalgebras calculated above to obtain symmetry reductions that transform (1.1a) and (1.1b) into a system of ordinary differential equations (ODEs). Later, in the next subsection, we will look for exact solutions of (1.1a) and (1.1b).
where the functions F and G satisfy the following system of ODEs:
2.3 Exact solutions using the simplest equation method
In this subsection we use the simplest equation method, which was introduced by Kudryashov [22,23] and modified by Vitanov  (see also ), to solve the ODE system (2.2a) and (2.2b), and as a result we will obtain the exact solutions of our -dimensional higher-order Broer-Kaup system (1.1a) and (1.1b). Bernoulli and Riccati equations will be used as the simplest equations.
Let us consider the solutions of the ODE system (2.2a) and (2.2b) 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. It is well known that the Bernoulli and Riccati equations are nonlinear ODEs whose solutions can be written in terms of elementary functions.
We consider the Bernoulli equation
where a and b are constants. Its solution is given by
where C is a constant of integration.
For the Riccati equation
where a, b and c are constants, we will use the solutions
2.3.1 Solutions of (1.1a) and (1.1b) using the Bernoulli equation as the simplest equation
The balancing procedure  yields , so the solutions of (2.2a) and (2.2b) are of the form
Substituting (2.8) into (2.2a) and (2.2b) and making use of (2.6) and then equating all coefficients of the functions to zero, we obtain an algebraic system of equations in terms of , , and , , . Solving the system of algebraic equations with the aid of Mathematica, we obtain the following cases.
2.3.2 Solutions of (1.1a) and (1.1b) using Riccati equation as the simplest equation
Substituting (2.11) into (2.2a) and (2.2b) and making use of (2.7), we obtain an algebraic system of equations in terms of , , , , , by equating all coefficients of the functions to zero. Solving the algebraic equations, one obtains the following cases.
The solutions in this case are
and so the solutions are
The solutions are
A profile of the solution (2.21a) and (2.21b) is given in Figure 1.
Figure 1. Profile of solitary waves (2.21a) and (2.21b).
3 Conservation laws of (1.1a) and (1.1b)
In this section, we derive conservation laws for the -dimensional higher-order Broer-Kaup system (1.1a) and (1.1b). Two different approaches will be used. Firstly, we use the new conservation method due to Ibragimov  and then employ the multiplier method [20,21]. We now present some preliminaries that we will need later in this section.
3.1.1 Fundamental operators and their relationship
where the summation convention is used whenever appropriate.
and the Lie-Bäcklund operator is
where the additional coefficients are determined uniquely by the prolongation formulae
The Lie-Bäcklund operator (3.5) can be written in a characteristic form as
The Noether operators associated with the Lie-Bäcklund symmetry operator X are given by
and the Euler-Lagrange , Lie-Bäcklund and Noether operators are connected by the operator identity
The equation (3.12) defines a local conservation law of the system (3.1).
3.1.2 Multiplier method
hold identically. We consider multipliers of the third-order, that is,
3.1.3 Variational method for a system and its adjoint
A system of adjoint equations for the system of kth-order differential equations (3.1) is defined by 
The following results are given in Ibragimov  and recalled here.
Assume that the system of equations (3.1) admits the symmetry generator
Then the system of adjoint equations (3.15) admits the operator
Every Lie point, Lie-Bäcklund and nonlocal symmetry (3.17) admitted by the system of equations (3.1) gives rise to a conservation law for the system consisting of equation (3.1) and adjoint equation (3.15), where the componentsof the conserved vectorare determined by
with Lagrangian given by
3.2 Construction of conservation laws for (1.1a) and (1.1b)
3.2.1 Application of the multiplier method
that are given by
Remark 1 Higher-order conservation laws of (1.1a) and (1.1b) can be computed by increasing the order of multipliers.
3.2.2 Application of the new conservation theorem
The adjoint equations of (1.1a) and (1.1b), by invoking (3.16), are given by
(i) We first consider the Lie point symmetry generator of the -dimensional higher-order Broer-Kaup system (1.1a) and (1.1b). Corresponding to this symmetry, the Lie characteristic function is . Thus, by using (3.20), the components of the conserved vector are given by
Remark 2 The components of the conserved vectors contain the arbitrary solutions ϕ and ψ of adjoint equations (3.24a) and (3.24b), and hence one can obtain an infinite number of conservation laws.
4 Concluding remarks
In this paper we have studied the -dimensional higher-order Broer-Kaup system (1.1a) and (1.1b). Similarity reductions and exact solutions, with the aid of the simplest equation method, were obtained based on optimal systems of one-dimensional subalgebras for the underlying system. We have verified the correctness of the solutions obtained here by substituting them back into the system (1.1a) and (1.1b). Furthermore, conservation laws for the system (1.1a) and (1.1b) were derived by using the multiplier method and the new conservation theorem.
The author declares that he has no competing interests.
This paper is dedicated to Prof. Ravi P. Agarwal on the occasion of his 65th birthday.
CMK would like to thank the Organizing Committee of ‘International Conference on Applied Analysis and Algebra (ICAAA2012)’ for their kind hospitality during the conference.
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