Abstract
In this paper we obtain exact solutions of the dimensional higherorder BroerKaup system which was obtained from the KadomtsevPetviashvili 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 higherorder BroerKaup system by employing the multiplier approach and the new conservation theorem.
Keywords:
the dimensional higherorder BroerKaup system; integrability; Lie group analysis; simplest equation method; solitary waves; conservation laws1 Introduction
In this paper we study the dimensional higherorder BroerKaup system
which was first introduced by Lou and Hu [1] by considering the symmetry constraints of the KadomtsevPetviashvili equation. The system (1.1a) and (1.1b) is in fact an extension of the wellknown dimensional BroerKaup system [24]
which is used to model the bidirectional propagation of long waves in shallow water. In [5], 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 solitonlike solutions. Recently, Huang et al.[6] presented a new Nfold Darboux transformations of the dimensional higherorder BroerKaup system with the help of a gauge transformation of the spectral problem and found new explicit multisoliton solutions of the system (1.1a) and (1.1b).
In the latter half of the nineteenth century, Sophus Lie (18421899) 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 wellknown 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. [712].
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 [7]. Also, the conservation laws are useful in the numerical integration of partial differential equations [13], 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 [14]. Moreover, conservation laws play an important role in the theories of nonclassical transformations [15,16], normal forms and asymptotic integrability [17]. Recently, in [18] 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 higherorder BroerKaup 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 [19] and the multiplier method [20,21].
2 Symmetry reductions and exact solutions of (1.1a) and (1.1b)
The symmetry group of the dimensional higherorder BroerKaup system (1.1a) and (1.1b) will be generated by the vector field of the form
Applying the third prolongation [11] 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 Onedimensional optimal system of subalgebras
In this subsection we present an optimal system of onedimensional subalgebras for the system (1.1a) and (1.1b) to obtain an optimal system of groupinvariant solutions. The method which we use here for obtaining a onedimensional optimal system of subalgebras is that given in [11]. The adjoint transformations are given by
Here is the commutator 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 onedimensional 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)
From Tables 1 and 2 one can obtain an optimal system of onedimensional subalgebras given by .
2.2 Symmetry reductions of (1.1a) and (1.1b)
In this subsection we use the optimal system of onedimensional 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).
The symmetry gives rise to the groupinvariant solution
where is an invariant of the symmetry . Substitution of (2.1) into (1.1a) and (1.1b) results in the system of ODEs
The symmetry gives rise to the groupinvariant solution of the form
where is an invariant of and the functions F and G satisfy the following system of ODEs:
By solving the corresponding Lagrange system for the symmetry , one obtains an invariant and the groupinvariant solution of the form
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 [24] (see also [25]), to solve the ODE system (2.2a) and (2.2b), and as a result we will obtain the exact solutions of our dimensional higherorder BroerKaup 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 [24] 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
and
where and C is a constant of integration.
2.3.1 Solutions of (1.1a) and (1.1b) using the Bernoulli equation as the simplest equation
The balancing procedure [24] 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.
Case 1
Thus, a solution of our dimensional higherorder BroerKaup system (1.1a) and (1.1b) is
where and C is a constant of integration.
Case 2
and so a solution of the dimensional higherorder BroerKaup system (1.1a) and (1.1b) is
where and C is a constant of integration.
2.3.2 Solutions of (1.1a) and (1.1b) using Riccati equation as the simplest equation
The balancing procedure yields , so the solutions of the ODE system (2.2a) and (2.2b) are of the form
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.
Case 1
and hence the solutions of the dimensional higherorder BroerKaup system (1.1a) and (1.1b) are
and
where and C is a constant of integration.
Case 2
In this case the solutions of the dimensional higherorder BroerKaup system (1.1a) and (1.1b) are given by
and
where and C is a constant of integration.
Case 3
The solutions in this case are
and
where and C is a constant of integration.
Case 4
and so the solutions are
and
where and C is a constant of integration.
Case 5
The solutions are
and
where and C is a constant of integration.
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 higherorder BroerKaup system (1.1a) and (1.1b). Two different approaches will be used. Firstly, we use the new conservation method due to Ibragimov [19] and then employ the multiplier method [20,21]. We now present some preliminaries that we will need later in this section.
3.1 Preliminaries
In this subsection we briefly present the notation and pertinent results which we utilize below. For details the reader is referred to [810,1921,27].
3.1.1 Fundamental operators and their relationship
Consider a kthorder system of PDEs of n independent variables and m dependent variables
where denote the collections of all first, second, …, kthorder partial derivatives, that is, , , …, respectively, with the total derivative operator with respect to given by
where the summation convention is used whenever appropriate.
The EulerLagrange operator, for each α, is given by [810]
and the LieBäcklund operator is
where is the space of differential functions. The operator (3.4) is an abbreviated form of the infinite formal sum
where the additional coefficients are determined uniquely by the prolongation formulae
in which is the Lie characteristic function given by
The LieBäcklund operator (3.5) can be written in a characteristic form as
The Noether operators associated with the LieBäcklund symmetry operator X are given by
where the EulerLagrange operators with respect to derivatives of are obtained from (3.3) by replacing by the corresponding derivatives. For example,
and the EulerLagrange , LieBäcklund and Noether operators are connected by the operator identity
The ntuple vector , , , is a conserved vector of (3.1) if satisfies
The equation (3.12) defines a local conservation law of the system (3.1).
3.1.2 Multiplier method
A multiplier has the property that
hold identically. We consider multipliers of the thirdorder, that is,
The righthand side of (3.13) is a divergence expression. The determining equation for the multiplier is
The conserved vectors are calculated via a homotopy formula [20,21,26] once the multipliers are obtained.
3.1.3 Variational method for a system and its adjoint
A system of adjoint equations for the system of kthorder differential equations (3.1) is defined by [27]
where
and are new dependent variables.
The following results are given in Ibragimov [19] 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
where the operator (3.18) is an extension of (3.17) to the variable and the are obtainable from
Theorem 1[19]
Every Lie point, LieBäcklund 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)
We now construct conservation laws for the dimensional higherorder BroerKaup system (1.1a) and (1.1b) using the two approaches.
3.2.1 Application of the multiplier method
For the dimensional higherorder BroerKaup system (1.1a) and (1.1b), after some lengthy calculations, we obtain the thirdorder multipliers
and
that are given by
where , are arbitrary constants. Corresponding to the above multipliers, we obtain the following seven local conserved vectors of the dimensional higherorder BroerKaup system (1.1a) and (1.1b):
Remark 1 Higherorder 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
where and are the new dependent variables. By recalling (3.21), we get the following Lagrangian for the system of equations (1.1a) and (1.1b) and (3.24a) and (3.24b):
Because of the three Lie point symmetries of the dimensional higherorder BroerKaup system (1.1a) and (1.1b), we have the following three cases to consider:
(i) We first consider the Lie point symmetry generator of the dimensional higherorder BroerKaup 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
(ii) The Lie point symmetry generator has the Lie characteristic function . Hence using (3.20), one can obtain the conserved vector whose components are
(iii) Finally, we consider the symmetry generator . For this case, the Lie characteristic function , and 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 higherorder BroerKaup 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 onedimensional 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.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
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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