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Khalique Boundary Value Problems 2013, 2013:41http://www.boundaryvalueproblems.com/content/2013/1/41

RESEARCH Open Access

On the solutions and conservation laws of the( + )-dimensional higher-order Broer-KaupsystemChaudry Masood Khalique*

*Correspondence:[email protected] Institute for SymmetryAnalysis and MathematicalModelling, Department ofMathematical Sciences, North-WestUniversity, Mafikeng Campus,Private Bag X 2046, Mmabatho,2735, Republic of South Africa

AbstractIn this paper we obtain exact solutions of the (1 + 1)-dimensional higher-orderBroer-Kaup system which was obtained from the Kadomtsev-Petviashvili equation bythe symmetry constraints. The methods used to determine the exact solutions of theunderlying system are the Lie group analysis and the simplest equation method. Thesolutions obtained are the solitary wave solutions. Moreover, we derive theconservation laws of the (1 + 1)-dimensional higher-order Broer-Kaup system byemploying the multiplier approach and the new conservation theorem.

Keywords: the (1 + 1)-dimensional higher-order Broer-Kaup system; integrability; Liegroup analysis; simplest equation method; solitary waves; conservation laws

1 IntroductionIn this paper we study the ( + )-dimensional higher-order Broer-Kaup system

ut + (uxx + u – uux + uv

)x = , (.a)

vt + (vxx + uv + uvx + v

)x = , (.b)

which was first introduced by Lou and Hu [] by considering the symmetry constraints ofthe Kadomtsev-Petviashvili equation. The system (.a) and (.b) is in fact an extensionof the well-known ( + )-dimensional Broer-Kaup system [–]

ut – uxx + uux – vx = , (.a)

vt + vxx – (uv)x = , (.b)

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 (.a) and (.b) withthe help of a gauge transformation of the spectral problem and as an application obtainedsome new explicit soliton-like solutions. Recently, Huang et al. [] presented a new N-fold Darboux transformations of the ( + )-dimensional higher-order Broer-Kaup systemwith the help of a gauge transformation of the spectral problem and found new explicitmulti-soliton solutions of the system (.a) and (.b).

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Khalique Boundary Value Problems 2013, 2013:41 of 18http://www.boundaryvalueproblems.com/content/2013/1/41

In the latter half of the nineteenth century, Sophus Lie (-) developed one ofthemost powerful methods to determine solutions of differential equations. This method,known as the Lie group analysis method, systematically unifies and extends well-knownad hoc techniques to construct explicit solutions of differential equations. It has proved tobe a versatile tool for solving nonlinear problems described by the differential equationsarising in mathematics, physics and in other scientific fields of study. For the theory andapplication of the Lie group analysis methods, see, e.g., the Refs. [–].Conservation laws play a vital role in the solution process of differential equations. Find-

ing conservation laws of the system of differential equations is often the first step towardsfinding the solution []. Also, the conservation laws are useful in the numerical integra-tion of partial differential equations [], for example, to control numerical errors. Thedetermination of conservation laws of the Korteweg de Vries equation, in fact, initiatedthe discovery of a number of methods to solve evolutionary equations []. Moreover,conservation laws play an important role in the theories of non-classical transformations[, ], normal forms and asymptotic integrability []. Recently, in [] the conservedquantity was used to determine the unknown exponent in the similarity solution whichcannot 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 sys-tem (.a) and (.b). Furthermore, conservation lawswill be computed for (.a) and (.b)using the two approaches: the new conservation theorem due to Ibragimov [] and themultiplier method [, ].

2 Symmetry reductions and exact solutions of (1.1a) and (1.1b)The symmetry group of the (+)-dimensional higher-order Broer-Kaup system (.a) and(.b) will be generated by the vector field of the form

X = ξ (t,x,u, v)∂

∂t+ ξ (t,x,u, v)

∂

∂x+ η(t,x,u, v)

∂

∂u+ η(t,x,u, v)

∂

∂v.

Applying the third prolongation pr()X [] to (.a) and (.b) and solving the resultantoverdetermined system of linear partial differential equations one obtains the followingthree Lie point symmetries:

X =∂

∂x,

X =∂

∂t,

X = –t∂

∂t– x

∂

∂x+ u

∂

∂u+ v

∂

∂v.

2.1 One-dimensional optimal system of subalgebrasIn this subsection we present an optimal system of one-dimensional subalgebras for thesystem (.a) and (.b) to obtain an optimal system of group-invariant solutions. Themethod which we use here for obtaining a one-dimensional optimal system of subalge-bras is that given in []. The adjoint transformations are given by

Ad(exp(εXi)

)Xj = Xj – ε[Xi,Xj] +

ε

[Xi, [Xi,Xj]

]– · · · .



Table 1 Commutator table of the Lie algebra of the system (1.1a) and (1.1b)

X1 X2 X3X1 0 0 –X1X2 0 0 –3X2X3 X1 3X2 0

Table 2 Adjoint table of the Lie algebra of the system (1.1a) and (1.1b)

Ad X1 X2 X3X1 X1 X2 X3 + εX1X2 X1 X2 X3 + 3εX2X3 e–εX1 e–3εX2 X3

Here [Xi,Xj] is the commutator given by

[Xi,Xj] = XiXj –XjXi.

The commutator table of the Lie point symmetries of the system (.a) and (.b) and theadjoint representations of the symmetry group of (.a) and (.b) on its Lie algebra aregiven in Table and Table , respectively. Table and Table are used to construct anoptimal system of one-dimensional subalgebras for the system (.a) and (.b).From Tables and one can obtain an optimal system of one-dimensional subalgebras

given by {νX +X,X,X}.

2.2 Symmetry reductions of (1.1a) and (1.1b)In this subsection we use the optimal system of one-dimensional subalgebras calculatedabove to obtain symmetry reductions that transform (.a) and (.b) into a system of or-dinary differential equations (ODEs). Later, in the next subsection, we will look for exactsolutions of (.a) and (.b).Case . νX +X

The symmetry νX +X gives rise to the group-invariant solution

u = F(z), v =G(z), (.)

where z = x – νt is an invariant of the symmetry νX + X. Substitution of (.) into (.a)and (.b) results in the system of ODEs

F ′′′(z) – F(z)F ′′(z) + G(z)F ′(z) – νF ′(z)

+ F(z)F ′(z) – F ′(z) + F(z)G′(z) = , (.a)

G′′′(z) + F ′(z)G′(z) + F(z)G(z)F ′(z) + F(z)G′′(z)

+ F(z)G′(z) – νG′(z) + G(z)G′(z) = . (.b)

Case . X

The symmetry X gives rise to the group-invariant solution of the form

u = F(z), v =G(z), (.)



where z = x is an invariant of X and the functions F and G satisfy the following system ofODEs:

F ′′′(z) – F(z)F ′′(z) + G(z)F ′(z)

+ F(z)F ′(z) – F ′(z) + F(z)G′(z) = ,

G′′′(z) + F ′(z)G′(z) + F(z)G(z)F ′(z)

+ F(z)G′′(z) + F(z)G′(z) + G(z)G′(z) = .

Case . X

By solving the corresponding Lagrange system for the symmetry X, one obtains aninvariant z = xt–/ and the group-invariant solution of the form

u = t–/F(z), v = t–/G(z), (.)

where the functions F and G satisfy the following system of ODEs:

F ′(z)z – F ′(z)G(z) – F ′′′(z) – F ′(z)F(z)

– G′(z)F(z) + F(z) + F ′′(z)F(z) + F ′(z) = ,

–G′(z)F(z) + G(z) – F ′(z)G′(z) – G′(z)G(z)

– F ′(z)F(z)G(z) +G′(z)z – G′′′(z) – G′′(z)F(z) = .

2.3 Exact solutions using the simplest equation methodIn this subsection we use the simplest equation method, which was introduced byKudryashov [, ] andmodified by Vitanov [] (see also []), to solve the ODE system(.a) and (.b), and as a resultwewill obtain the exact solutions of our (+)-dimensionalhigher-order Broer-Kaup system (.a) and (.b). Bernoulli and Riccati equations will beused as the simplest equations.Let us consider the solutions of the ODE system (.a) and (.b) in the form

F(z) =M∑i=

Ai(H(z)

)i, G(z) =M∑i=

Bi(H(z)

)i, (.)

whereH(z) satisfies the Bernoulli and Riccati equations,M is a positive integer that can bedetermined by balancing procedure as in [] and A, . . . ,AM , B, . . . ,BM are parametersto be determined. It is well known that the Bernoulli and Riccati equations are nonlinearODEs whose solutions can be written in terms of elementary functions.We consider the Bernoulli equation

H ′(z) = aH(z) + bH(z), (.)

where a and b are constants. Its solution is given by

H(z) = a{

cosh[a(z +C)] + sinh[a(z +C)] – b cosh[a(z +C)] – b sinh[a(z +C)]

},

where C is a constant of integration.



For the Riccati equation

H ′(z) = aH(z) + bH(z) + c, (.)

where a, b and c are constants, we will use the solutions

H(z) = –ba

–θ

atanh

[θ (z +C)

]

and

H(z) = –ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

),

where θ = b – ac > and C is a constant of integration.

.. Solutions of (.a) and (.b) using the Bernoulli equation as the simplest equationThe balancing procedure [] yieldsM = , so the solutions of (.a) and (.b) are of theform

F(z) = A +AH +AH, G(z) = B + BH + BH. (.)

Substituting (.) into (.a) and (.b) and making use of (.) and then equating all co-efficients of the functions Hi to zero, we obtain an algebraic system of equations in termsof A, A, A and B, B, B. Solving the system of algebraic equations with the aid ofMathematica, we obtain the following cases.Case

A =

(±a± √√–a + ν

),

A =bA(a + ν – A

)a(a – ν)

,

A = ,

B = ,

B =–ab + abA – aA

b

,

B =bB

a.

Thus, a solution of our ( + )-dimensional higher-order Broer-Kaup system (.a) and(.b) is

u(t,x) =A +Aa{


}

+Aa{


}

, (.a)



v(t,x) =B + Ba{


}

+ Ba{


}

, (.b)

where z = x – νt and C is a constant of integration.Case

a = ±√

ν

,

A = ±√

ν

,

A =abA ±

√bν + abA

– bνA

ν,

A = ,

B = ,

B =–ab + abA – aA

b

,

B =

(–b + bA –A

),

and so a solution of the ( + )-dimensional higher-order Broer-Kaup system (.a) and(.b) is

u(t,x) =A +Aa{


}

+Aa{


}

, (.a)

v(t,x) =B + Ba{


}

+ Ba{


}

, (.b)

where z = x – νt and C is a constant of integration.

.. Solutions of (.a) and (.b) using Riccati equation as the simplest equationThe balancing procedure yieldsM = , so the solutions of theODE system (.a) and (.b)are of the form

F(z) = A +AH +AH, G(z) = B + BH + BH. (.)

Substituting (.) into (.a) and (.b) and making use of (.), we obtain an algebraicsystem of equations in terms of A, A, A, B, B, B by equating all coefficients of thefunctions Hi to zero. Solving the algebraic equations, one obtains the following cases.



Case

A =

(±b± √√–b + ac + ν

),

A = –a(–b + ac + ν – A

)bA

,

A = ,

B =(–ac + cA),

B =(–ab + bA),

B =aB

b,

and hence the solutions of the ( + )-dimensional higher-order Broer-Kaup system (.a)and (.b) are

u(t,x) =A +A

{–ba

–θ

atanh

[θ (z +C)

]}

+A

{–ba

–θ

atanh

[θ (z +C)

]}

, (.a)

v(t,x) =B + B

{–ba

–θ

atanh

[θ (z +C)

]}

+ B

{–ba

–θ

atanh

[θ (z +C)

]}

(.b)

and

u(t,x) =A +A

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

+A

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

, (.a)

v(t,x) =B + B

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

+ B

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

, (.b)


b = ±√

ν

,

a = –ν

c,

A = ±√

ν

,



A =aA + a

√b +A

b,

A = ,

B =aν – νA + bAA

a,

B =A(ab + aA – bA)

a,

B =

(–a + aA –A

).

In this case the solutions of the (+ )-dimensional higher-order Broer-Kaup system (.a)and (.b) are given by

u(t,x) =A +A

{–ba

–θ

atanh

[θ (z +C)

]}

+A

{–ba

–θ

atanh

[θ (z +C)

]}

, (.a)

v(t,x) =B + B

{–ba

–θ

atanh

[θ (z +C)

]}

+ B

{–ba

–θ

atanh

[θ (z +C)

]}

(.b)

and

u(t,x) =A +A

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

+A

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

, (.a)

v(t,x) =B + B

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

+ B

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

, (.b)


a =b – ν

c,

A = ±b,

A =aA

b,

A = ,

B =–ab + aν + bA – νA + bAA

a,



B =A(ab + aA – bA)

a,

B =

(–a + aA –A

).

The solutions in this case are

u(t,x) =A +A

{–ba

–θ

atanh

[θ (z +C)

]}

+A

{–ba

–θ

atanh

[θ (z +C)

]}

, (.a)

v(t,x) =B + B

{–ba

–θ

atanh

[θ (z +C)

]}

+ B

{–ba

–θ

atanh

[θ (z +C)

]}

(.b)

and

u(t,x) =A +A

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

+A

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

, (.a)

v(t,x) =B + B

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

+ B

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

, (.b)


a =b – ν

c,

A = ,

A = ±a,

A = ,

B =(–b + ν + cA

),

B =(–ab + bA),

B =aB

b,



and so the solutions are

u(t,x) =A +A

{–ba

–θ

atanh

[θ (z +C)

]}

+A

{–ba

–θ

atanh

[θ (z +C)

]}

, (.a)

v(t,x) =B + B

{–ba

–θ

atanh

[θ (z +C)

]}

+ B

{–ba

–θ

atanh

[θ (z +C)

]}

(.b)

and

u(t,x) =A +A

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

+A

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

, (.a)

v(t,x) =B + B

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

+ B

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

, (.b)


a =b – ν

c,

A =(±b± √

ν),

A =a(b – ν + A

)bA

,

A = ,

B =–ab + aν + bA – νA – cA

a

,

B = bA,

B =aB

b.

The solutions are

u(t,x) =A +A

{–ba

–θ

atanh

[θ (z +C)

]}

+A

{–ba

–θ

atanh

[θ (z +C)

]}

, (.a)



Figure 1 Profile of solitary waves (2.21a) and (2.21b).

v(t,x) =B + B

{–ba

–θ

atanh

[θ (z +C)

]}

+ B

{–ba

–θ

atanh

[θ (z +C)

]}

(.b)

and

u(t,x) =A +A

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

+A

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

, (.a)

v(t,x) =B + B

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

+ B

{–ba

–θ

atanh

(θz

)+

sech( θz )

C cosh( θz ) –

aθsinh( θz

)

}

, (.b)

where z = x – νt and C is a constant of integration.A profile of the solution (.a) and (.b) is given in Figure .

3 Conservation laws of (1.1a) and (1.1b)In this section, we derive conservation laws for the (+)-dimensional higher-order Broer-Kaup system (.a) and (.b). Two different approaches will be used. Firstly, we use thenew conservation method due to Ibragimov [] and then employ the multiplier method[, ]. We now present some preliminaries that we will need later in this section.

3.1 PreliminariesIn this subsection we briefly present the notation and pertinent results which we utilizebelow. For details the reader is referred to [–, –, ].

.. Fundamental operators and their relationshipConsider a kth-order system of PDEs of n independent variables x = (x,x, . . . ,xn) and mdependent variables u = (u,u, . . . ,um)

Eα(x,u,u(), . . . ,u(k)) = , α = , . . . ,m, (.)



where u(),u(), . . . ,u(k) denote the collections of all first, second, . . . , kth-order partialderivatives, that is, uα

i = Di(uα), uαij = DjDi(uα), . . . , respectively, with the total derivative

operator with respect to xi given by

Di =∂

∂xi+ uα

i∂

∂uα+ uα

ij∂

∂uαj+ · · · , i = , . . . ,n, (.)

where the summation convention is used whenever appropriate.The Euler-Lagrange operator, for each α, is given by [–]

δ

δuα=

∂

∂uα+

∑s≥

(–)sDi · · ·Dis∂

∂uαii···is

, α = , . . . ,m, (.)

and the Lie-Bäcklund operator is

X = ξ i ∂

∂xi+ ηα ∂

∂uα, ξ i,ηα ∈A, (.)

whereA is the space of differential functions. The operator (.) is an abbreviated form ofthe infinite formal sum

X = ξ i ∂

∂xi+ ηα ∂

∂uα+

∑s≥

ζ αii···is

∂

∂uαii···is

, (.)

where the additional coefficients are determined uniquely by the prolongation formulae

ζ αi =Di

(W α

)+ ξ juα

ij ,

ζ αi···is =Di · · ·Dis

(W α

)+ ξ juα

ji···is , s > , (.)

in whichW α is the Lie characteristic function given by

W α = ηα – ξ iuαj . (.)

The Lie-Bäcklund operator (.) can be written in a characteristic form as

X = ξ iDi +W α ∂

∂uα+

∑s≥

Di · · ·Dis(W α

) ∂

∂uαii···is

. (.)

The Noether operators associated with the Lie-Bäcklund symmetry operator X are givenby

Ni = ξ i +W α δ

δuαi+

∑s≥

Di · · ·Dis(W α

) δ

δuαiii···is

, i = , . . . ,n, (.)

where the Euler-Lagrange operators with respect to derivatives of uα are obtained from(.) by replacing uα by the corresponding derivatives. For example,

δ

δuαi=

∂

∂uαi+

∑s≥

(–)sDj · · ·Djs∂

∂uαijj···js

, i = , . . . ,n,α = , . . . ,m, (.)



and the Euler-Lagrange , Lie-Bäcklund and Noether operators are connected by the oper-ator identity

X +Di(ξ i) =W α δ

δuα+DiNi. (.)

The n-tuple vector T = (T ,T, . . . ,Tn), Tj ∈ A, j = , . . . ,n, is a conserved vector of (.) ifTi satisfies

DiTi|(.) = . (.)

The equation (.) defines a local conservation law of the system (.).

.. Multiplier methodA multiplier �α(x,u,u(), . . .) has the property that

�αEα =DiTi (.)

hold identically. We consider multipliers of the third-order, that is,

�α = �α(t,x,u, v,ux, vx,uxx, vxx,uxxx, vxxx).

The right-hand side of (.) is a divergence expression. The determining equation for themultiplier �α is

δ(�αEα)δuα

= . (.)

The conserved vectors are calculated via a homotopy formula [, , ] once the multi-pliers are obtained.

.. Variational method for a system and its adjointA system of adjoint equations for the system of kth-order differential equations (.) isdefined by []

E*α(x,u, v, . . . ,u(k), v(k)) = , α = , . . . ,m, (.)

where

E*α(x,u, v, . . . ,u(k), v(k)) =

δ(vβEβ )δuα

, α = , . . . ,m, v = v(x) (.)

and v = (v, v, . . . , vm) are new dependent variables.The following results are given in Ibragimov [] and recalled here.Assume that the system of equations (.) admits the symmetry generator

X = ξ i ∂

∂xi+ ηα ∂

∂uα. (.)



Then the system of adjoint equations (.) admits the operator

Y = ξ i ∂

∂xi+ ηα ∂

∂uα+ ηα

*∂

∂vα, ηα

* = –[λα

βvβ + vαDi

(ξ i)], (.)

where the operator (.) is an extension of (.) to the variable vα and the λαβ are obtain-

able from

X(Eα) = λβαEβ . (.)

Theorem [] Every Lie point, Lie-Bäcklund and nonlocal symmetry (.) admittedby the system of equations (.) gives rise to a conservation law for the system consisting ofequation (.) and adjoint equation (.),where the components Ti of the conserved vectorT = (T , . . . ,Tn) are determined by

Ti = ξ iL +W α δLδuα

i+

∑s≥

Di · · ·Dis(W α

) δLδuα

iii···is, i = , . . . ,n, (.)

with Lagrangian given by

L = vαEα(x,u, . . . ,u(k)). (.)

3.2 Construction of conservation laws for (1.1a) and (1.1b)We now construct conservation laws for the ( + )-dimensional higher-order Broer-Kaupsystem (.a) and (.b) using the two approaches.

.. Application of the multiplier methodFor the ( + )-dimensional higher-order Broer-Kaup system (.a) and (.b), after somelengthy calculations, we obtain the third-order multipliers

� = �(t,x,u, v,ux, vx,uxx, vxx,uxxx, vxxx)

and

� = �(t,x,u, v,ux, vx,uxx, vxx,uxxx, vxxx)

that are given by

� =C(tuv + tvx)

+C(vxu + vxxu + uxxv + vxv + uv + uv + uxvx + vxxx

)+C

(vxu + uv + v + vxx

)+C(uv + vx) +Cv +C, (.)

� =C(tu + tv – tux – x

)+C

(–uxv – uxu + uxxu + uv + u + v + ux – uxxx + vxx

)+C

(–uxu + uv + u + uxx

)+C

(u + v – ux

)+Cu +C, (.)



where Ci, i = , , , , , , are arbitrary constants. Corresponding to the above multi-pliers, we obtain the following seven local conserved vectors of the ( + )-dimensionalhigher-order Broer-Kaup system (.a) and (.b):

�t = tvxu – tuxv + tuv + tv – xv,

�x =

{tvxu – tuxuv + tvxxu – xvxu + tvxuv

– tuxvxu + tuxxuv – tvtu – tuxv + tutv + tvxxv

+ tuv + tuv – xuv + uv + tv – xv + tuxxvx

– tuxvxx – tvx + vx – xvxx},

�t =

{vu + vxu + vu – vuxu + vxxu + vvxu

+ uxvxu + vuxxu + vxxxu + v + vux – vux

+ vvxx – vuxxx},

�x =

{vu + vxu + vu – vuxu + vxxu

+ vvxu – uxvxu + vuxxu – vtu + vu

+ vuxu + vxu – vuxu + vxuxxu + vvxxu

– uxvxxu + vutu – vtxu + vvxu + uxvxu – vuxvxu

+ vuxxu – vuxuxxu + vxvxxu + uxxvxxu + vxutu – vvtu

+ uxvtu – vutxu – vtxxu + v + vuxx + vxx – uxvxuxx

+ vvxx + vut + vxxut + vxvt – uxxvt

– vxutx – vvtx + uxvtx + vutxx},

�t =

{vxu – uxuv + vxxu + uxxv + uv + uv

},

�x =

{vxu – uxuv + vxxu + vxuv – uxvxu + uxxuv

– vtu – uxuv + uxxvxu + vxxuv – uxvxxu + utuv – uvtx

+ uxxv – vutx + uv + uv + uv + utvx + vtux + uxxvxx},

�t = u,

�x =

{–uxu + uv + u + uxx

},

�t = uv,

�x =

{vxu – uxuv + vxxu + uxxv + uv + uv – uxvx

},

�t =

{vxu – uxv + uv + v

},

�x =

{vxu – uxuv + vxxu + vxuv – uxvxu + uxxuv

– vtu – uxv + utv + vxxv + uv + uv + v + uxxvx – uxvxx – vx},



�t = v,

�x =

{vxu + uv + v + vxx

}.

Remark Higher-order conservation laws of (.a) and (.b) can be computed by in-creasing the order of multipliers.

.. Application of the new conservation theoremThe adjoint equations of (.a) and (.b), by invoking (.), are given by

–ψxuv – φxu – φxxu – φxv – φt – vxψx – φxxx = , (.a)

–ψxu + ψxxu – φxu – ψxv –ψt + uxψx – ψxxx = , (.b)

where φ = φ(t,x) and ψ = ψ(t,x) are the new dependent variables. By recalling (.), weget the following Lagrangian for the system of equations (.a) and (.b) and (.a) and(.b):

L =φ(t,x){ut +

(uxx + u – uux + uv

)x

}+ψ(t,x)

{vt +

(vxx + uv + uvx + v

)x

}. (.)

Because of the three Lie point symmetries of the ( + )-dimensional higher-order Broer-Kaup system (.a) and (.b), we have the following three cases to consider:(i) We first consider the Lie point symmetry generator X = ∂x of the ( + )-dimensional

higher-order Broer-Kaup system (.a) and (.b). Corresponding to this symmetry, theLie characteristic function isW = –(ux + vx). Thus, by using (.), the components of theconserved vector are given by

Tt = –uxφ – vxψ ,

Tx = vxψxu – uxφxu + utφ + vtψ(t,x) + uxxφx – uxφxx + vxxψx – vxψxx.

(ii) The Lie point symmetry generator X = ∂t has the Lie characteristic function W =–(ut +vt). Hence using (.), one can obtain the conserved vector whose components are

Tt =

(vxuψ + uxuvψ + vxxuψ + uxvxψ + vxuφ + uxvφ

+ uxuφ – uxxuφ – uxφ + uxxxφ + vxvψ + vxxxψ),

Tx = –

(vtuψ + utuvψ – vtψxu + uψvtx + utvxψ + vtuφ

+ utvφ + utuφ + utφxu – uφutx – utuxφ + φutxx + vtvψ

+ψvtxx + utφxx – φxutx + vtψxx –ψxvtx).

(iii) Finally, we consider the symmetry generator X = –t∂t – x∂x + u∂u + v∂v. For thiscase, the Lie characteristic functionW = u+v+tut +tvt +xux+xvx, and by using (.),



the components of the conserved vector are given by

Tt = – tφuxu – tψvxu + φu – tvψuxu – tφvxu + tφuxxu

– tψvxxu + tφux + vψ + xφux – tvφux + xψvx – tvψvx

– tψuxvx – tφuxxx – tψvxxx,

Tx =φu

+ vψu + φxu + tφutu + tψvtu + vφu – φuxu

+ ψvxu + xuxφxu – vψxu – xvxψxu + φxxu + tvψutu + tφxutu

+ tφvtu – tψxvtu – tφutxu + tψvtxu

+ vψ – uxφx – vxψx + φuxx

– xφxuxx + ψvxx – xψxvxx + xuxφxx + vψxx + xvxψxx – xφut + tvφut

– tφuxut + tψvxut + tφxxut – xψvt + tvψvt

+ tψxxvt – tφxutx – tψxvtx

+ tφutxx + tψvtxx.

Remark The components of the conserved vectors contain the arbitrary solutionsφ andψ of adjoint equations (.a) and (.b), and hence one can obtain an infinite numberof conservation laws.

4 Concluding remarksIn this paper we have studied the ( + )-dimensional higher-order Broer-Kaup system(.a) and (.b). Similarity reductions and exact solutions, with the aid of the simplestequation method, were obtained based on optimal systems of one-dimensional subalge-bras for the underlying system.We have verified the correctness of the solutions obtainedhere by substituting them back into the system (.a) and (.b). Furthermore, conserva-tion laws for the system (.a) and (.b) were derived by using the multiplier method andthe new conservation theorem.

Competing interestsThe author declares that he has no competing interests.

AcknowledgementsThis 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.

Received: 12 September 2012 Accepted: 8 February 2013 Published: 28 February 2013

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doi:10.1186/1687-2770-2013-41Cite this article as: Khalique: On the solutions and conservation laws of the (1 + 1)-dimensional higher-orderBroer-Kaup system. Boundary Value Problems 2013 2013:41.


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