On the solutions and conservation laws of the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system

Boundary Value Problems20132013:41

DOI: 10.1186/1687-2770-2013-41

Received: 12 September 2012

Accepted: 8 February 2013

Published: 28 February 2013

Abstract

In this paper we obtain exact solutions of the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-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 ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system by employing the multiplier approach and the new conservation theorem.

Keywords

the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system integrability Lie group analysis simplest equation method solitary waves conservation laws

1 Introduction

In this paper we study the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system
u t + 4 ( u x x + u 3 3 u u x + 6 u v ) x = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ1_HTML.gif
(1.1a)
v t + 4 ( v x x + 3 u 2 v + 3 u v x + 3 v 2 ) x = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ2_HTML.gif
(1.1b)
which was first introduced by Lou and Hu [1] 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 ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional Broer-Kaup system [24]
u t u x x + 2 u u x 2 v x = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ3_HTML.gif
(1.2a)
v t + v x x 2 ( u v ) x = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ4_HTML.gif
(1.2b)

which is used to model the bi-directional 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 soliton-like solutions. Recently, Huang et al. [6] presented a new N-fold Darboux transformations of the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-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. [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 non-classical 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 ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-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 [19] and the multiplier method [20, 21].

2 Symmetry reductions and exact solutions of (1.1a) and (1.1b)

The symmetry group of the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system (1.1a) and (1.1b) will be generated by the vector field of the form
X = ξ 1 ( t , x , u , v ) t + ξ 2 ( t , x , u , v ) x + η 1 ( t , x , u , v ) u + η 2 ( t , x , u , v ) v . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equa_HTML.gif
Applying the third prolongation pr ( 3 ) X http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq2_HTML.gif [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:
X 1 = x , X 2 = t , X 3 = 3 t t x x + u u + 2 v v . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equb_HTML.gif

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 [11]. The adjoint transformations are given by
Ad ( exp ( ϵ X i ) ) X j = X j ϵ [ X i , X j ] + 1 2 ϵ 2 [ X i , [ X i , X j ] ] . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equc_HTML.gif
Here [ X i , X j ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq3_HTML.gif is the commutator given by
[ X i , X j ] = X i X j X j X i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equd_HTML.gif
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 )

 

X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq4_HTML.gif

X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq5_HTML.gif

X 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq6_HTML.gif

X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq7_HTML.gif

0

0

X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq8_HTML.gif

X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq9_HTML.gif

0

0

3 X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq10_HTML.gif

X 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq11_HTML.gif

X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq7_HTML.gif

3 X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq12_HTML.gif

0

Table 2

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

Ad

X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq4_HTML.gif

X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq5_HTML.gif

X 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq6_HTML.gif

X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq7_HTML.gif

X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq7_HTML.gif

X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq9_HTML.gif

X 3 + ϵ X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq13_HTML.gif

X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq9_HTML.gif

X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq7_HTML.gif

X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq9_HTML.gif

X 3 + 3 ϵ X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq14_HTML.gif

X 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq11_HTML.gif

e ϵ X 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq15_HTML.gif

e 3 ϵ X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq16_HTML.gif

X 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq11_HTML.gif

From Tables 1 and 2 one can obtain an optimal system of one-dimensional subalgebras given by { ν X 1 + X 2 , X 2 , X 3 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq17_HTML.gif.

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).

Case 1. ν X 1 + X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq18_HTML.gif

The symmetry ν X 1 + X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq19_HTML.gif gives rise to the group-invariant solution
u = F ( z ) , v = G ( z ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ5_HTML.gif
(2.1)
where z = x ν t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq20_HTML.gif is an invariant of the symmetry ν X 1 + X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq18_HTML.gif. Substitution of (2.1) into (1.1a) and (1.1b) results in the system of ODEs
4 F ( z ) 12 F ( z ) F ( z ) + 24 G ( z ) F ( z ) ν F ( z ) + 12 F ( z ) 2 F ( z ) 12 F ( z ) 2 + 24 F ( z ) G ( z ) = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ6_HTML.gif
(2.2a)
4 G ( z ) + 12 F ( z ) G ( z ) + 24 F ( z ) G ( z ) F ( z ) + 12 F ( z ) G ( z ) + 12 F ( z ) 2 G ( z ) ν G ( z ) + 24 G ( z ) G ( z ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ7_HTML.gif
(2.2b)

Case 2. X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq9_HTML.gif

The symmetry X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq9_HTML.gif gives rise to the group-invariant solution of the form
u = F ( z ) , v = G ( z ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ8_HTML.gif
(2.3)
where z = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq21_HTML.gif is an invariant of X 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq9_HTML.gif and the functions F and G satisfy the following system of ODEs:
4 F ( z ) 12 F ( z ) F ( z ) + 24 G ( z ) F ( z ) + 12 F ( z ) 2 F ( z ) 12 F ( z ) 2 + 24 F ( z ) G ( z ) = 0 , 4 G ( z ) + 12 F ( z ) G ( z ) + 24 F ( z ) G ( z ) F ( z ) + 12 F ( z ) G ( z ) + 12 F ( z ) 2 G ( z ) + 24 G ( z ) G ( z ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Eque_HTML.gif

Case 3. X 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq11_HTML.gif

By solving the corresponding Lagrange system for the symmetry X 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq11_HTML.gif, one obtains an invariant z = x t 1 / 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq22_HTML.gif and the group-invariant solution of the form
u = t 1 / 3 F ( z ) , v = t 2 / 3 G ( z ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ9_HTML.gif
(2.4)
where the functions F and G satisfy the following system of ODEs:
F ( z ) z 72 F ( z ) G ( z ) 12 F ( z ) 36 F ( z ) F ( z ) 2 72 G ( z ) F ( z ) + F ( z ) + 36 F ( z ) F ( z ) + 36 F ( z ) 2 = 0 , 36 G ( z ) F ( z ) 2 + 2 G ( z ) 36 F ( z ) G ( z ) 72 G ( z ) G ( z ) 72 F ( z ) F ( z ) G ( z ) + G ( z ) z 12 G ( z ) 36 G ( z ) F ( z ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equf_HTML.gif

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 ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-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
F ( z ) = i = 0 M A i ( H ( z ) ) i , G ( z ) = i = 0 M B i ( H ( z ) ) i , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ10_HTML.gif
(2.5)

where H ( z ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq23_HTML.gif satisfies the Bernoulli and Riccati equations, M is a positive integer that can be determined by balancing procedure as in [24] and A 0 , , A M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq24_HTML.gif, B 0 , , B M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq25_HTML.gif 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
H ( z ) = a H ( z ) + b H 2 ( z ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ11_HTML.gif
(2.6)
where a and b are constants. Its solution is given by
H ( z ) = a { cosh [ a ( z + C ) ] + sinh [ a ( z + C ) ] 1 b cosh [ a ( z + C ) ] b sinh [ a ( z + C ) ] } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equg_HTML.gif

where C is a constant of integration.

For the Riccati equation
H ( z ) = a H 2 ( z ) + b H ( z ) + c , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ12_HTML.gif
(2.7)
where a, b and c are constants, we will use the solutions
H ( z ) = b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equh_HTML.gif
and
H ( z ) = b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equi_HTML.gif

where θ 2 = b 2 4 a c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq26_HTML.gif 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 M = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq27_HTML.gif, so the solutions of (2.2a) and (2.2b) are of the form
F ( z ) = A 0 + A 1 H + A 2 H 2 , G ( z ) = B 0 + B 1 H + B 2 H 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ13_HTML.gif
(2.8)

Substituting (2.8) into (2.2a) and (2.2b) and making use of (2.6) and then equating all coefficients of the functions H i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq28_HTML.gif to zero, we obtain an algebraic system of equations in terms of A 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq29_HTML.gif, A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq30_HTML.gif, A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq31_HTML.gif and B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq32_HTML.gif, B 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq33_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq34_HTML.gif. Solving the system of algebraic equations with the aid of Mathematica, we obtain the following cases.

Case 1
A 0 = 1 6 ( ± 3 a ± 3 a 2 + ν ) , A 1 = b A 0 ( 8 a 2 + ν 12 A 0 2 ) a ( 4 a 2 ν ) , A 2 = 0 , B 0 = 0 , B 1 = 2 a b 2 + 3 a b A 1 a A 1 2 6 b , B 2 = b B 1 a . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equj_HTML.gif
Thus, a solution of our ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system (1.1a) and (1.1b) is
u ( t , x ) = A 0 + A 1 a { cosh [ a ( z + C ) ] + sinh [ a ( z + C ) ] 1 b cosh [ a ( z + C ) ] b sinh [ a ( z + C ) ] } + A 2 a 2 { cosh [ a ( z + C ) ] + sinh [ a ( z + C ) ] 1 b cosh [ a ( z + C ) ] b sinh [ a ( z + C ) ] } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ14_HTML.gif
(2.9a)
v ( t , x ) = B 0 + B 1 a { cosh [ a ( z + C ) ] + sinh [ a ( z + C ) ] 1 b cosh [ a ( z + C ) ] b sinh [ a ( z + C ) ] } + B 2 a 2 { cosh [ a ( z + C ) ] + sinh [ a ( z + C ) ] 1 b cosh [ a ( z + C ) ] b sinh [ a ( z + C ) ] } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ15_HTML.gif
(2.9b)

where z = x ν t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq20_HTML.gif and C is a constant of integration.

Case 2
a = ± ν 2 , A 0 = ± ν 2 , A 1 = 6 a b A 0 ± b 2 ν 2 + 36 a 2 b 2 A 0 2 12 b 2 ν A 0 2 ν , A 2 = 0 , B 0 = 0 , B 1 = 2 a b 2 + 3 a b A 1 a A 1 2 6 b , B 2 = 1 6 ( 2 b 2 + 3 b A 1 A 1 2 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equk_HTML.gif
and so a solution of the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system (1.1a) and (1.1b) is
u ( t , x ) = A 0 + A 1 a { cosh [ a ( z + C ) ] + sinh [ a ( z + C ) ] 1 b cosh [ a ( z + C ) ] b sinh [ a ( z + C ) ] } + A 2 a 2 { cosh [ a ( z + C ) ] + sinh [ a ( z + C ) ] 1 b cosh [ a ( z + C ) ] b sinh [ a ( z + C ) ] } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ16_HTML.gif
(2.10a)
v ( t , x ) = B 0 + B 1 a { cosh [ a ( z + C ) ] + sinh [ a ( z + C ) ] 1 b cosh [ a ( z + C ) ] b sinh [ a ( z + C ) ] } + B 2 a 2 { cosh [ a ( z + C ) ] + sinh [ a ( z + C ) ] 1 b cosh [ a ( z + C ) ] b sinh [ a ( z + C ) ] } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ17_HTML.gif
(2.10b)

where z = x ν t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq20_HTML.gif 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 M = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq27_HTML.gif, so the solutions of the ODE system (2.2a) and (2.2b) are of the form
F ( z ) = A 0 + A 1 H + A 2 H 2 , G ( z ) = B 0 + B 1 H + B 2 H 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ18_HTML.gif
(2.11)

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 A 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq29_HTML.gif, A 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq30_HTML.gif, A 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq31_HTML.gif, B 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq32_HTML.gif, B 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq33_HTML.gif, B 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq34_HTML.gif by equating all coefficients of the functions H i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq28_HTML.gif to zero. Solving the algebraic equations, one obtains the following cases.

Case 1
A 0 = 1 6 ( ± 3 b ± 3 b 2 + 4 a c + ν ) , A 1 = a ( 4 b 2 + 4 a c + ν 12 A 0 2 ) 12 b A 0 , A 2 = 0 , B 0 = 1 2 ( a c + c A 1 ) , B 1 = 1 2 ( a b + b A 1 ) , B 2 = a B 1 b , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equl_HTML.gif
and hence the solutions of the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system (1.1a) and (1.1b) are
u ( t , x ) = A 0 + A 1 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } + A 2 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ19_HTML.gif
(2.12a)
v ( t , x ) = B 0 + B 1 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } + B 2 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ20_HTML.gif
(2.12b)
and
u ( t , x ) = A 0 + A 1 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } + A 2 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ21_HTML.gif
(2.13a)
v ( t , x ) = B 0 + B 1 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } + B 2 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ22_HTML.gif
(2.13b)

where z = x ν t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq20_HTML.gif and C is a constant of integration.

Case 2
b = ± ν 6 , a = ν 18 c , A 0 = ± ν 6 , A 1 = a A 0 + a 8 b 2 + A 0 2 2 b , A 2 = 0 , B 0 = 4 a ν 3 ν A 1 + 36 b A 0 A 1 108 a , B 1 = A 1 ( 3 a b + a A 0 2 b A 1 ) 6 a , B 2 = 1 6 ( 2 a 2 + 3 a A 1 A 1 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equm_HTML.gif
In this case the solutions of the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system (1.1a) and (1.1b) are given by
u ( t , x ) = A 0 + A 1 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } + A 2 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ23_HTML.gif
(2.14a)
v ( t , x ) = B 0 + B 1 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } + B 2 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ24_HTML.gif
(2.14b)
and
u ( t , x ) = A 0 + A 1 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } + A 2 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ25_HTML.gif
(2.15a)
v ( t , x ) = B 0 + B 1 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } + B 2 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ26_HTML.gif
(2.15b)

where z = x ν t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq20_HTML.gif and C is a constant of integration.

Case 3
a = 4 b 2 ν 16 c , A 0 = ± b , A 1 = 2 a A 0 b , A 2 = 0 , B 0 = 88 a b 2 + 6 a ν + 12 b 2 A 1 3 ν A 1 + 32 b A 0 A 1 96 a , B 1 = A 1 ( 3 a b + a A 0 2 b A 1 ) 6 a , B 2 = 1 6 ( 2 a 2 + 3 a A 1 A 1 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equn_HTML.gif
The solutions in this case are
u ( t , x ) = A 0 + A 1 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } + A 2 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ27_HTML.gif
(2.16a)
v ( t , x ) = B 0 + B 1 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } + B 2 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ28_HTML.gif
(2.16b)
and
u ( t , x ) = A 0 + A 1 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } + A 2 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ29_HTML.gif
(2.17a)
v ( t , x ) = B 0 + B 1 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } + B 2 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ30_HTML.gif
(2.17b)

where z = x ν t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq20_HTML.gif and C is a constant of integration.

Case 4
a = 4 b 2 ν 4 c , A 0 = 0 , A 1 = ± a , A 2 = 0 , B 0 = 1 8 ( 4 b 2 + ν + 4 c A 1 ) , B 1 = 1 2 ( a b + b A 1 ) , B 2 = a B 1 b , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equo_HTML.gif
and so the solutions are
u ( t , x ) = A 0 + A 1 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } + A 2 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ31_HTML.gif
(2.18a)
v ( t , x ) = B 0 + B 1 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } + B 2 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ32_HTML.gif
(2.18b)
and
u ( t , x ) = A 0 + A 1 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } + A 2 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ33_HTML.gif
(2.19a)
v ( t , x ) = B 0 + B 1 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } + B 2 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ34_HTML.gif
(2.19b)

where z = x ν t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq20_HTML.gif and C is a constant of integration.

Case 5
a = 4 b 2 ν 16 c , A 0 = 1 4 ( ± 2 b ± ν ) , A 1 = a ( 4 b 2 ν + 16 A 0 2 ) 16 b A 0 , A 2 = 0 , B 0 = 8 a b 2 + 2 a ν + 12 b 2 A 1 3 ν A 1 16 c A 1 2 96 a , B 1 = b A 1 , B 2 = a B 1 b . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equp_HTML.gif
The solutions are
u ( t , x ) = A 0 + A 1 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } + A 2 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ35_HTML.gif
(2.20a)
v ( t , x ) = B 0 + B 1 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } + B 2 { b 2 a θ 2 a tanh [ 1 2 θ ( z + C ) ] } 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ36_HTML.gif
(2.20b)
and
u ( t , x ) = A 0 + A 1 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } + A 2 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ37_HTML.gif
(2.21a)
v ( t , x ) = B 0 + B 1 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } + B 2 { b 2 a θ 2 a tanh ( 1 2 θ z ) + sech ( θ z 2 ) C cosh ( θ z 2 ) 2 a θ sinh ( θ z 2 ) } 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ38_HTML.gif
(2.21b)

where z = x ν t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq20_HTML.gif and C is a constant of integration.

A profile of the solution (2.21a) and (2.21b) is given in Figure 1.
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Fig1_HTML.jpg
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 ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-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 [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 k th-order system of PDEs of n independent variables x = ( x 1 , x 2 , , x n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq35_HTML.gif and m dependent variables u = ( u 1 , u 2 , , u m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq36_HTML.gif
E α ( x , u , u ( 1 ) , , u ( k ) ) = 0 , α = 1 , , m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ39_HTML.gif
(3.1)
where u ( 1 ) , u ( 2 ) , , u ( k ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq37_HTML.gif denote the collections of all first, second, …, k th-order partial derivatives, that is, u i α = D i ( u α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq38_HTML.gif, u i j α = D j D i ( u α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq39_HTML.gif, …, respectively, with the total derivative operator with respect to x i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq40_HTML.gif given by
D i = x i + u i α u α + u i j α u j α + , i = 1 , , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ40_HTML.gif
(3.2)

where the summation convention is used whenever appropriate.

The Euler-Lagrange operator, for each α, is given by [810]
δ δ u α = u α + s 1 ( 1 ) s D i 1 D i s u i 1 i 2 i s α , α = 1 , , m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ41_HTML.gif
(3.3)
and the Lie-Bäcklund operator is
X = ξ i x i + η α u α , ξ i , η α A , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ42_HTML.gif
(3.4)
where A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq41_HTML.gif is the space of differential functions. The operator (3.4) is an abbreviated form of the infinite formal sum
X = ξ i x i + η α u α + s 1 ζ i 1 i 2 i s α u i 1 i 2 i s α , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ43_HTML.gif
(3.5)
where the additional coefficients are determined uniquely by the prolongation formulae
ζ i α = D i ( W α ) + ξ j u i j α , ζ i 1 i s α = D i 1 D i s ( W α ) + ξ j u j i 1 i s α , s > 1 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ44_HTML.gif
(3.6)
in which W α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq42_HTML.gif is the Lie characteristic function given by
W α = η α ξ i u j α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ45_HTML.gif
(3.7)
The Lie-Bäcklund operator (3.5) can be written in a characteristic form as
X = ξ i D i + W α u α + s 1 D i 1 D i s ( W α ) u i 1 i 2 i s α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ46_HTML.gif
(3.8)
The Noether operators associated with the Lie-Bäcklund symmetry operator X are given by
N i = ξ i + W α δ δ u i α + s 1 D i 1 D i s ( W α ) δ δ u i i 1 i 2 i s α , i = 1 , , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ47_HTML.gif
(3.9)
where the Euler-Lagrange operators with respect to derivatives of u α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq43_HTML.gif are obtained from (3.3) by replacing u α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq43_HTML.gif by the corresponding derivatives. For example,
δ δ u i α = u i α + s 1 ( 1 ) s D j 1 D j s u i j 1 j 2 j s α , i = 1 , , n , α = 1 , , m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ48_HTML.gif
(3.10)
and the Euler-Lagrange , Lie-Bäcklund and Noether operators are connected by the operator identity
X + D i ( ξ i ) = W α δ δ u α + D i N i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ49_HTML.gif
(3.11)
The n-tuple vector T = ( T 1 , T 2 , , T n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq44_HTML.gif, T j A http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq45_HTML.gif, j = 1 , , n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq46_HTML.gif, is a conserved vector of (3.1) if T i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq47_HTML.gif satisfies
D i T i | ( 3.1 ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ50_HTML.gif
(3.12)

The equation (3.12) defines a local conservation law of the system (3.1).

3.1.2 Multiplier method

A multiplier Λ α ( x , u , u ( 1 ) , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq48_HTML.gif has the property that
Λ α E α = D i T i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ51_HTML.gif
(3.13)
hold identically. We consider multipliers of the third-order, that is,
Λ α = Λ α ( t , x , u , v , u x , v x , u x x , v x x , u x x x , v x x x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equq_HTML.gif
The right-hand side of (3.13) is a divergence expression. The determining equation for the multiplier Λ α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq49_HTML.gif is
δ ( Λ α E α ) δ u α = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ52_HTML.gif
(3.14)

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 k th-order differential equations (3.1) is defined by [27]
E α ( x , u , v , , u ( k ) , v ( k ) ) = 0 , α = 1 , , m , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ53_HTML.gif
(3.15)
where
E α ( x , u , v , , u ( k ) , v ( k ) ) = δ ( v β E β ) δ u α , α = 1 , , m , v = v ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ54_HTML.gif
(3.16)

and v = ( v 1 , v 2 , , v m ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq50_HTML.gif 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
X = ξ i x i + η α u α . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ55_HTML.gif
(3.17)
Then the system of adjoint equations (3.15) admits the operator
Y = ξ i x i + η α u α + η α v α , η α = [ λ β α v β + v α D i ( ξ i ) ] , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ56_HTML.gif
(3.18)
where the operator (3.18) is an extension of (3.17) to the variable v α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq51_HTML.gif and the λ β α http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq52_HTML.gif are obtainable from
X ( E α ) = λ α β E β . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ57_HTML.gif
(3.19)

Theorem 1 [19]

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 components T i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq47_HTML.gif of the conserved vector T = ( T 1 , , T n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq53_HTML.gif are determined by
T i = ξ i L + W α δ L δ u i α + s 1 D i 1 D i s ( W α ) δ L δ u i i 1 i 2 i s α , i = 1 , , n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ58_HTML.gif
(3.20)
with Lagrangian given by
L = v α E α ( x , u , , u ( k ) ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ59_HTML.gif
(3.21)

3.2 Construction of conservation laws for (1.1a) and (1.1b)

We now construct conservation laws for the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system (1.1a) and (1.1b) using the two approaches.

3.2.1 Application of the multiplier method

For the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system (1.1a) and (1.1b), after some lengthy calculations, we obtain the third-order multipliers
Λ 1 = Λ 1 ( t , x , u , v , u x , v x , u x x , v x x , u x x x , v x x x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equr_HTML.gif
and
Λ 2 = Λ 2 ( t , x , u , v , u x , v x , u x x , v x x , u x x x , v x x x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equs_HTML.gif
that are given by
Λ 1 = C 1 ( 24 t u v + 12 t v x ) + C 2 ( 6 v x u 2 + 4 v x x u + 2 u x x v + 6 v x v + 4 u 3 v + 12 u v 2 + 2 u x v x + v x x x ) + C 3 ( 3 v x u + 3 u 2 v + 3 v 2 + v x x ) + C 6 ( 2 u v + v x ) + C 5 v + C 4 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ60_HTML.gif
(3.22)
Λ 2 = C 1 ( 12 t u 2 + 24 t v 12 t u x x ) + C 2 ( 6 u x v 6 u x u 2 + 4 u x x u + 12 u 2 v + u 4 + 6 v 2 + 3 u x 2 u x x x + 2 v x x ) + C 3 ( 3 u x u + 6 u v + u 3 + u x x ) + C 6 ( u 2 + 2 v u x ) + C 5 u + C 7 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ61_HTML.gif
(3.23)
where C i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq54_HTML.gif, i = 1 , 2 , 3 , 4 , 5 , 6 , 7 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq55_HTML.gif are arbitrary constants. Corresponding to the above multipliers, we obtain the following seven local conserved vectors of the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system (1.1a) and (1.1b):
Φ 1 t = 6 t v x u 6 t u x v + 12 t u 2 v + 12 t v 2 x v , Φ 1 x = 2 { 72 t v x u 3 144 t u x u 2 v + 24 t v x x u 2 6 x v x u + 144 t v x u v 120 t u x v x u + 48 t u x x u v 3 t v t u 24 t u x 2 v + 3 t u t v + 48 t v x x v + 72 t u 4 v + 288 t u 2 v 2 6 x u 2 v + 3 u v + 96 t v 3 6 x v 2 + 24 t u x x v x 24 t u x v x x 24 t v x 2 + 2 v x 2 x v x x } , Φ 2 t = 1 6 { 6 v u 4 + 9 v x u 3 + 36 v 2 u 2 9 v u x u 2 + 8 v x x u 2 + 12 v v x u + 4 u x v x u + 12 v u x x u + 3 v x x x u + 12 v 3 + 6 v u x 2 12 v 2 u x + 6 v v x x 3 v u x x x } , Φ 2 x = 1 6 { 72 v u 6 + 72 v x u 5 + 792 v 2 u 4 288 v u x u 4 + 24 v x x u 4 + 864 v v x u 3 240 u x v x u 3 + 96 v u x x u 3 9 v t u 3 + 1296 v 3 u 2 + 216 v u x 2 u 2 + 216 v x 2 u 2 864 v 2 u x u 2 + 72 v x u x x u 2 + 288 v v x x u 2 72 u x v x x u 2 + 9 v u t u 2 8 v t x u 2 + 432 v 2 v x u + 72 u x 2 v x u 144 v u x v x u + 288 v 2 u x x u 144 v u x u x x u + 144 v x v x x u + 24 u x x v x x u + 8 v x u t u 12 v v t u + 12 u x v t u 12 v u t x u 3 v t x x u + 216 v 4 + 24 v u x x 2 + 24 v x x 2 24 u x v x u x x + 144 v 2 v x x + 12 v 2 u t + 3 v x x u t + 6 v x v t 3 u x x v t 3 v x u t x 6 v v t x + 3 u x v t x + 3 v u t x x } , Φ 3 t = 1 2 { 2 v x u 2 2 u x u v + v x x u + u x x v + 2 u 3 v + 6 u v 2 } , Φ 3 x = 1 2 { 24 v x u 4 72 u x u 3 v + 8 v x x u 3 + 144 v x u 2 v 72 u x v x u 2 + 24 u x x u 2 v 2 v t u 2 72 u x u v 2 + 24 u x x v x u + 48 v x x u v 24 u x v x x u + 2 u t u v u v t x + 24 u x x v 2 v u t x + 24 u 5 v + 168 u 3 v 2 + 144 u v 3 + u t v x + v t u x + 8 u x x v x x } , Φ 4 t = u , Φ 4 x = 4 { 3 u x u + 6 u v + u 3 + u x x } , Φ 5 t = u v , Φ 5 x = 4 { 3 v x u 2 3 u x u v + v x x u + u x x v + 3 u 3 v + 6 u v 2 u x v x } , Φ 6 t = 1 2 { v x u u x v + 2 u 2 v + 2 v 2 } , Φ 6 x = 1 2 { 24 v x u 3 48 u x u 2 v + 8 v x x u 2 + 48 v x u v 40 u x v x u + 16 u x x u v v t u 8 u x 2 v + u t v + 16 v x x v + 24 u 4 v + 96 u 2 v 2 + 32 v 3 + 8 u x x v x 8 u x v x x 8 v x 2 } , Φ 7 t = v , Φ 7 x = 4 { 3 v x u + 3 u 2 v + 3 v 2 + v x x } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equt_HTML.gif

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
24 ψ x u v 12 ϕ x u 2 12 ϕ x x u 24 ϕ x v ϕ t 12 v x ψ x 4 ϕ x x x = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ62_HTML.gif
(3.24a)
12 ψ x u 2 + 12 ψ x x u 24 ϕ x u 24 ψ x v ψ t + 12 u x ψ x 4 ψ x x x = 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ63_HTML.gif
(3.24b)
where ϕ = ϕ ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq56_HTML.gif and ψ = ψ ( t , x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq57_HTML.gif 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):
L = ϕ ( t , x ) { u t + 4 ( u x x + u 3 3 u u x + 6 u v ) x } + ψ ( t , x ) { v t + 4 ( v x x + 3 u 2 v + 3 u v x + 3 v 2 ) x } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equ64_HTML.gif
(3.25)
Because of the three Lie point symmetries of the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system (1.1a) and (1.1b), we have the following three cases to consider:
  1. (i)
    We first consider the Lie point symmetry generator X 1 = x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq58_HTML.gif of the ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-dimensional higher-order Broer-Kaup system (1.1a) and (1.1b). Corresponding to this symmetry, the Lie characteristic function is W = ( u x + v x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq59_HTML.gif. Thus, by using (3.20), the components of the conserved vector are given by
    T 1 t = u x ϕ v x ψ , T 1 x = 12 v x ψ x u 12 u x ϕ x u + u t ϕ + v t ψ ( t , x ) + 4 u x x ϕ x 4 u x ϕ x x + 4 v x x ψ x 4 v x ψ x x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equu_HTML.gif
     
  2. (ii)
    The Lie point symmetry generator X 2 = t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq60_HTML.gif has the Lie characteristic function W = ( u t + v t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq61_HTML.gif. Hence using (3.20), one can obtain the conserved vector whose components are
    T 2 t = 4 ( 3 v x u 2 ψ + 6 u x u v ψ + 3 v x x u ψ + 3 u x v x ψ + 6 v x u ϕ + 6 u x v ϕ + 3 u x u 2 ϕ 3 u x x u ϕ 3 u x 2 ϕ + u x x x ϕ + 6 v x v ψ + v x x x ψ ) , T 2 x = 4 ( 3 v t u 2 ψ + 6 u t u v ψ 3 v t ψ x u + 3 u ψ v t x + 3 u t v x ψ + 6 v t u ϕ + 6 u t v ϕ + 3 u t u 2 ϕ + 3 u t ϕ x u 3 u ϕ u t x 3 u t u x ϕ + ϕ u t x x + 6 v t v ψ + ψ v t x x + u t ϕ x x ϕ x u t x + v t ψ x x ψ x v t x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equv_HTML.gif
     
  3. (iii)
    Finally, we consider the symmetry generator X 3 = 3 t t x x + u u + 2 v v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq62_HTML.gif. For this case, the Lie characteristic function W = u + 2 v + 3 t u t + 3 t v t + x u x + x v x http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq63_HTML.gif, and by using (3.20), the components of the conserved vector are given by
    T 3 t = 36 t ϕ u x u 2 36 t ψ v x u 2 + ϕ u 72 t v ψ u x u 72 t ϕ v x u + 36 t ϕ u x x u 36 t ψ v x x u + 36 t ϕ u x 2 + 2 v ψ + x ϕ u x 72 t v ϕ u x + x ψ v x 72 t v ψ v x 36 t ψ u x v x 12 t ϕ u x x x 12 t ψ v x x x , T 3 x = 12 ϕ u 3 + 48 v ψ u 2 + 12 ϕ x u 2 + 36 t ϕ u t u 2 + 36 t ψ v t u 2 + 72 v ϕ u 36 ϕ u x u + 48 ψ v x u + 12 x u x ϕ x u 24 v ψ x u 12 x v x ψ x u + 4 ϕ x x u + 72 t v ψ u t u + 36 t ϕ x u t u + 72 t ϕ v t u 36 t ψ x v t u 36 t ϕ u t x u + 36 t ψ v t x u + 48 v 2 ψ 8 u x ϕ x 12 v x ψ x + 12 ϕ u x x 4 x ϕ x u x x + 16 ψ v x x 4 x ψ x v x x + 4 x u x ϕ x x + 8 v ψ x x + 4 x v x ψ x x x ϕ u t + 72 t v ϕ u t 36 t ϕ u x u t + 36 t ψ v x u t + 12 t ϕ x x u t x ψ v t + 72 t v ψ v t + 12 t ψ x x v t 12 t ϕ x u t x 12 t ψ x v t x + 12 t ϕ u t x x + 12 t ψ v t x x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_Equw_HTML.gif
     

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 ( 1 + 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-41/MediaObjects/13661_2012_Article_302_IEq1_HTML.gif-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.

Declarations

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.

Authors’ Affiliations

(1)
International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University

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© Khalique; licensee Springer. 2013

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