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The coupling integrable couplings of the generalized coupled Burgers equation hierarchy and its Hamiltonian structure

  • Yuan WeiEmail author
  • Li Yin
  • Xin Long
Open Access
Research

Abstract

In this paper, we mainly give the Lie algebras E, F and H of three kinds and their commutator, respectively. Next, we establish three isospectral problems with the help of their corresponding loop algebras , , and . Based on on the Tu scheme, coupling integrable couplings of three kinds for the generalized coupled Burgers equation hierarchy are obtained. Finally, we obtain the Hamiltonian structure of one of the coupling integrable couplings of the generalized coupled Burgers equation hierarchy by using the quadratic-form identity.

Keywords

Isospectral problem Burgers equation hierarchy Quadratic-form identity Hamiltonian structure 

MSC

35B06 

1 Introduction

The development of soliton theory has undergone a rapid development in the 1960s. Integrable couplings are a new important and interesting topic in the field of soliton theory [1, 2, 3]. A lot of integrable nonlinear evolution equations, such as the Schroedinger equation and the KdV equation, were discussed. The notion of integrable couplings was first introduced by Virasoro [4, 5]. Integrable couplings are coupled systems of integrable equations which contain given integrable equations as their sub-systems [6]. Integrable couplings have much richer mathematical structures than scalar integrable equations [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. Recently, Inc, Yusuf, Aliyu and Baleanu discussed a Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation, the generalized shallow water wave equation, the time fractional dispersive long-wave equation and the time fractional generalized Burgers–Huxley equation. They also studied a time fractional third-order variant Boussinesq system and gave a symmetry analysis, explicit solutions, conservation laws and numerical approximations [32, 33, 34, 35, 36, 37]. So it is important to study integrable couplings in soliton theory. Zhang even proposed an efficient method for constructing nonlinear evolution equations and their resulting Hamiltonian structure. Ma called it the Tu scheme. Hence, Hu developed the trace identity, and got an efficient method to work out the soliton equations and the Hamiltonian structure. In [38] Zhang gave three kinds of coupling integrable couplings of the KdV hierarchy of evolution equations. The three Lie algebras E, F and H in [38] can be used to obtain the other coupling integrable couplings of the soliton equations. In this paper, we introduce three higher-dimensional Lie algebras and their corresponding loop algebras , , and , and consider three Lax pairs for the zero curvature equation
$$ V_{x}=[U,V]. $$
(1.1)
With the help of the loop algebras, we obtain three kinds of coupling integrable couplings of the generalized coupled Burgers equation hierarchy. And the one coupling integrable coupling of the generalized coupled Burgers equation hierarchy has a Hamiltonian structure obtained by employing the quadratic-form identity [39].

2 Three higher-dimensional Lie algebras

Let
$$E=\operatorname{span}\{e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7} \}. $$
For \(\forall a=\sum^{7}_{i=1}a_{i}e_{i}\), \(b=\sum^{7}_{i=1}b_{i}e_{i} \in E\), we have
[ a , b ] = ( 2 a 3 b 2 2 a 2 b 3 2 a 1 b 3 2 a 3 b 1 2 a 1 b 2 2 a 2 b 1 a 1 b 4 a 4 b 1 + a 2 b 5 a 5 b 2 + a 3 b 5 a 5 b 3 a 5 b 1 a 1 b 5 + a 2 b 4 a 4 b 2 + a 4 b 3 a 3 b 4 a 1 b 6 a 6 b 1 + a 2 b 7 a 7 b 2 + a 3 b 7 a 7 b 3 a 7 b 1 a 7 b 1 + a 2 b 6 a 6 b 2 + a 6 b 3 a 3 b 6 ) . Open image in new window
(2.1)
It is easy to see that E is a Lie algebra equipped with the commutator (2.1).
Denote
$$F=\operatorname{span}\{f_{1},f_{2},f_{3},f_{4},f_{5},f_{6},f_{7},f_{8},f_{9} \}. $$
For \(\forall a=\sum^{7}_{i=1}a_{i}f_{i}\), \(b=\sum^{7}_{i=1}b_{i}f_{i} \in L_{2}\), we easily obtain
[ a , b ] = ( 2 a 3 b 2 2 a 2 b 3 2 a 1 b 3 2 a 3 b 1 2 a 1 b 2 2 a 2 b 1 2 a 7 b 2 2 a 2 b 7 + 2 a 3 b 6 2 a 6 b 3 2 a 3 b 8 2 a 8 b 3 + 2 a 9 b 2 2 a 2 b 9 2 a 1 b 7 2 a 7 b 1 + 2 a 4 b 3 2 a 3 b 4 2 a 1 b 6 2 a 6 b 1 + 2 a 4 b 2 2 a 2 b 4 2 a 1 b 9 2 a 9 b 1 + 2 a 5 b 3 2 a 3 b 5 2 a 1 b 8 2 a 8 b 1 + 2 a 5 b 2 2 a 2 b 5 ) Open image in new window
(2.2)
and F is a Lie algebra equipped with the commutator (2.2).
The Lie algebras E and F are all expanding Lie algebra of the well-known Lie algebra
$$R^{3}=\bigl\{ r=(r_{1},r_{2},r_{3})^{T},r_{i} \in R\bigr\} , $$
which along with the commutator leads to
[ a , b ] = ( 2 a 3 b 2 2 a 2 b 3 2 a 1 b 3 2 a 3 b 1 2 a 1 b 2 2 a 2 b 1 ) a = ( a 1 , a 2 , a 3 ) T , b = ( b 1 , b 2 , b 3 ) T R 3 . Open image in new window
(2.3)
Assume that \(L_{s}\) is a s-dim Lie algebra,
K ( L s ) = { A = ( A 1 A 2 A 3 ) , A i L s , i = 1 , 2 , 3 } . Open image in new window
For
B = ( B 1 B 2 B 3 ) K ( L s ) Open image in new window
and
$$[A,B]=\bigl([A_{1},B_{1}],[A_{1},B_{2}]+ [A_{2},B_{1}],[A_{1},B_{3}]+ [A_{3},B_{1}]+[A_{3}+B_{3}] \bigr)^{T} \in K(L_{s}), $$
\(K(L_{s})\) is a Lie algebra. By taking \(s=3\) and \([A_{1},B_{1}]\) is the same as (2.3), then we can get another Lie algebra \(K(L_{s})\), which can be denoted H. The Lie algebra H is isomorphic to the Lie algebra F,
$$H=\operatorname{span}\{f_{1},f_{2},f_{3},f_{4},f_{5},f_{6},f_{7},f_{8},f_{9} \}, $$
along with the commutator
[ a , b ] = ( 2 a 3 b 2 2 a 2 b 3 2 a 1 b 3 2 a 3 b 1 2 a 1 b 2 2 a 2 b 1 2 a 3 b 5 2 a 5 b 3 + 2 a 6 b 2 2 a 2 b 6 2 a 1 b 6 2 a 6 b 1 + 2 a 4 b 3 2 a 3 b 4 2 a 1 b 5 2 a 5 b 1 + 2 a 4 b 2 2 a 2 b 4 2 a 3 b 8 2 a 8 b 3 + 2 a 9 b 2 2 a 2 b 9 + 2 a 9 b 8 2 a 8 b 9 2 a 1 b 9 2 a 9 b 1 + 2 a 7 b 3 2 a 3 b 7 + 2 a 7 b 9 2 a 9 b 7 2 a 1 b 8 2 a 8 b 1 + 2 a 7 b 2 2 a 2 b 7 + 2 a 7 b 8 2 a 8 b 7 ) , Open image in new window
(2.4)
where \(a=\sum_{i=1}^{9}a_{i}e_{i}\), \(b=\sum_{i=1}^{9}b_{i}e_{i}\).

By applying Lie algebra E, F and H, we can obtain three kinds of coupling integrable couplings of the generalized coupled Burgers equation hierarchy.

3 Three kinds of coupling integrable couplings of the generalized coupled Burgers equation hierarchy

In this section, we shall show how to get three kinds of coupling integrable couplings of the generalized coupled Burgers equation hierarchy by making use of the Lie algebra E, F and H.

(I) The first kind of coupling integrable couplings of the generalized coupled Burgers equation hierarchy.

The loop algebra of Lie algebra E is given by \(\widetilde{E}=\operatorname{span}\{ e_{i}(n)=e_{i}\lambda^{n},i=1,2,3,4,5,6,7\}\), \([e_{i}(m),e_{j}(n)]=[e_{i},e_{j}]\lambda^{m+n}\), \(m,n\in Z\), \(i,j=1,2,3,4,5,6,7\), equipped with the commutator (2.1). Consider a Lax pair for zero curvature equation as follows:
$$ \textstyle\begin{cases} U=e_{1}(1)+qh_{2}(0)+h_{2}(1)+rh_{3}(0)+u_{1}h_{4}(0)+u_{2}h_{5}(0)+s_{1}h_{6}(0)+s_{2}h_{7}(0), \\ V=\sum_{m\geq 0}(V_{1m}h_{1}(-m)+V_{2m}h_{2}(-m)+V_{3m}h_{3}(-m)+V_{4m}h_{4}(-m) \\ \hphantom{V={}}{}+V_{5m}h_{5}(-m)+V_{6m}h_{6}(-m)+V_{7m}h_{7}(-m)). \end{cases} $$
(3.1)
The stationary zero equation
$$ V_{x}=[U,V] $$
(3.2)
changes into
$$ \textstyle\begin{cases} V_{1,mx}=-2V_{3,m+1}-2qV_{3m}+2rV_{2m}, \\ V_{2,mx}=2V_{3m}-2rV_{1m}, \\ V_{3,mx}=-2V_{1,m+1}-2qV_{1m}+2V_{2m}, \\ V_{4,mx}=V_{5,m+1}+qV_{5m}+V_{4m}-u_{1}V_{1m}-u_{2}V_{2m}+rV_{5m}-u_{2}V_{3m}, \\ V_{5,mx}=V_{4,m+1}+qV_{4m}+u_{2}V_{1m}-V_{5m}-u_{1}V_{2m}+u_{1}V_{3m}-rV_{4m}, \\ V_{6,mx}=V_{7,m+1}+qV_{7m}+V_{6m}-s_{1}V_{1m}-s_{2}V_{2m}+rV_{7m}-s_{2}V_{3m}, \\ V_{7,mx}=V_{6,m+1}+qV_{6m}+s_{2}V_{1m}-V_{7m}-s_{1}V_{2m}+s_{1}V_{3m}-rV_{6m}. \end{cases} $$
(3.3)
Denoting
$$\begin{aligned} V_{+}^{(n)} =&\sum_{m=0}^{n} \bigl(V_{1m}h_{1}(n-m)+V_{2m}h_{2}(n-m)+V_{3m}h_{3}(n-m)+V_{4m}h_{4}(n-m) \\ &{}+V_{5m}h_{5}(n-m)+V_{6m}h_{6}(n-m)+V_{7m}h_{7}(n-m) \bigr), \end{aligned}$$
the stationary zero curvature equation (3.2) can be decomposed into
$$-V_{+x}^{(n)}+\bigl[U,V_{+}^{(n)} \bigr]=V_{-x}^{(n)}-\bigl[U,V_{-}^{(n)} \bigr]. $$
Simple computation results in
$$\begin{aligned} -V_{+x}^{(n)}+\bigl[U,V_{+}^{(n)} \bigr] =&2V_{3,n+1}h_{1}(0)+2V_{1,n+1}h_{3}(0)-V_{5,n+1}h_{4}(0)-V_{4,n+1}h_{5}(0) \\ &{}-V_{7,n+1}h_{6}(0)-V_{6,n+1}h_{7}(0). \end{aligned}$$
Taking \(V^{(n)}=V_{+}^{(n)}-\frac{1}{r}V_{3,n+1}h_{2}(0)\), we have
$$\begin{aligned} -V_{x}^{(n)}+\bigl[U,V^{(n)}\bigr] =&\biggl( \frac{1}{r}V_{3,n+1}\biggr)_{x}h_{2}(0)- \frac {1}{r}V_{2,n+1x}h_{3}(0) \\ &{}+\biggl(-V_{5,n+1}+ \frac{u_{2}}{r}V_{3,n+1}\biggr)h_{4}(0) \\ &{}+ \biggl(-V_{4,n+1}+\frac{u_{1}}{r}V_{3,n+1}\biggr)h_{5}(0)+\biggl(-V_{7,n+1}+\frac {s_{2}}{r}V_{3,n+1} \biggr)h_{6}(0) \\ &{}+\biggl(-V_{6,n+1}+\frac {s_{1}}{r}V_{3,n+1} \biggr)h_{7}(0). \end{aligned}$$
Hence the zero curvature equation
$$ U_{t}-V_{x}^{(n)}+ \bigl[U,V^{(n)}\bigr]=0 $$
(3.4)
gives rise to the Lax integrable hierarchy
u t = ( q r u 1 u 2 s 1 s 2 ) t = ( ( 1 r V 3 , n + 1 ) 1 r ( V 2 , n + 1 ) V 5 , n + 1 u 2 r V 3 , n + 1 V 4 , n + 1 u 1 r V 3 , n + 1 V 7 , n + 1 s 2 r V 3 , n + 1 V 6 , n + 1 s 1 r V 3 , n + 1 ) . Open image in new window
(3.5)
When we take \(s_{1}=s_{2}=0\), (3.5) can be reduced to an integrable coupling of the generalized coupled Burgers equation hierarchy,
u t = ( q r u 1 u 2 ) t = ( ( 1 r V 3 , n + 1 ) 1 r ( V 2 , n + 1 ) V 5 , n + 1 u 2 r V 3 , n + 1 V 4 , n + 1 u 1 r V 3 , n + 1 ) . Open image in new window
(3.6)
And (3.4) can be reduced to another generalized coupled Burgers equation hierarchy by taking \(u_{1}=u_{2}=0\),
u t = ( q r s 1 s 2 ) t = ( ( 1 r V 3 , n + 1 ) 1 r ( V 2 , n + 1 ) V 7 , n + 1 s 2 r V 3 , n + 1 V 6 , n + 1 s 1 r V 3 , n + 1 ) . Open image in new window
(3.7)
So we call (3.5) the first kind of coupling integrable couplings of the generalized coupled Burgers equation hierarchy.

When we set \(V_{1,0}=0\), \(V_{2,0}=\beta \), \(V_{3,0}=V_{4,0}=V_{5,0}=V_{6,0}=V_{7,0}=0\), we can obtain \(V_{1,1}=\beta\), \(V_{2,1}=0\), \(V_{3,1}=\beta r\), \(V_{4,1}=\beta u_{1}\), \(V_{5,1}=\beta u_{2} \), \(V_{6,1}=\beta s_{1}\), \(V_{7,1}=\beta s_{2} , \ldots \) .

Hence, when we take \(n=2\) in Eq. (3.5), we obtain the first kind of the coupling integrable couplings of the generalized coupled Burgers equation, that is,
$$ \textstyle\begin{cases} q_{t}=-\beta(\frac{r_{xx}}{4r}+\frac{q_{x}}{r}+q^{2}+\frac {r^{2}}{2})_{x}, \\ r_{t}=\beta(\frac{r_{xx}}{2r}+\frac{q_{x}}{r}-q_{x}r-2qr_{x}), \\ u_{1t}=\beta u_{2xx}-2\beta u_{1x}q-\beta u_{1}q_{x}-\beta u_{2x}-\frac {\beta}{2}u_{1}r_{x}-\beta u_{1x}r-\beta\frac{u_{2}r_{xx}}{4r}-\beta \frac{u_{2}q_{x}}{2r}, \\ u_{2t}=\beta u_{1xx}-2\beta u_{2x}q-\beta u_{2}q_{x}+\beta u_{1x}+\frac {\beta}{2}u_{2}r_{x}+\beta u_{2x}r-\beta\frac{u_{1}r_{xx}}{4r}-\beta \frac{u_{1}q_{x}}{2r}, \\ s_{1t}=\beta s_{2xx}-2\beta s_{1x}q-\beta s_{1}q_{x}-\beta s_{2x}-\frac {\beta}{2}s_{1}r_{x}-\beta s_{1x}r-\beta\frac{s_{2}r_{xx}}{4r}-\beta \frac{s_{2}q_{x}}{2r}, \\ s_{2t}=\beta s_{1xx}-2\beta s_{2x}q-\beta s_{2}q_{x}+\beta s_{1x}+\frac {\beta}{2}s_{2}r_{x}+\beta s_{2x}r-\beta\frac{s_{1}r_{xx}}{4r}-\beta \frac{s_{1}q_{x}}{2r}. \end{cases} $$
(3.8)
When we take \(u_{1}=u_{2}=s_{1}=s_{2}=0\), \(\beta=-2\) in (3.9), we can get equations as follows:
$$ \textstyle\begin{cases} q_{t}=(\frac{r_{xx}}{2r}+\frac{q_{x}}{r}+2q^{2}+r^{2})_{x}, \\ r_{t}=-\frac{r_{xx}}{r} q_{xx}-\frac{2q_{x}}{r}+2q_{x}r+4qr_{x}. \end{cases} $$
(3.9)
And when we take \(r=-1\) in (3.8), we can get the famous generalized Burgers equation
$$ q_{t}=4qq_{x}-q_{xx}. $$
(3.10)

(II) The second kind of coupling integrable couplings of the generalized coupled Burgers equation hierarchy.

We have the loop algebra of the Lie algebra F:
$$\begin{aligned}& \widetilde {F}=\operatorname{span}\bigl\{ f_{i}(n)=f_{i} \lambda^{n},i=1,2,3,4,5,6,7,8,9\bigr\} , \\& \bigl[f_{i}(m),f_{j}(n)\bigr]=[f_{i},f_{j}] \lambda^{m+n},\quad m,n\in Z,i,j=1,2,3,4,5,6,7,8,9, \end{aligned}$$
of which the resulting commutators are defined the same as (2.2).
By using the loop algebra , we introduce an isospectral Lax pair for zero curvature equation as follows:
$$ \textstyle\begin{cases} U=f_{1}(0)+qf_{2}(0)+f_{2}(1)+rf_{3}(0)+u_{1}f_{4}(0)+u_{2}f_{5}(0)+s_{1}f_{7}(0)+s_{2}f_{9}(0), \\ V=\sum_{m\geq 0}(V_{1m}f_{1}(-m)+V_{2m}f_{2}(-m)+V_{3m}f_{3}(-m)+V_{4m}f_{4}(-m) \\ \hphantom{V={}}{}+V_{5m}f_{5}(-m)+V_{6m}f_{6}(-m)+V_{7m}f_{7}(-m)+V_{8m}f_{8}(-m)+V_{9m}f_{9}(-m)). \end{cases} $$
(3.11)
The stationary zero curvature
$$V_{x}=[U,V] $$
is presented by the recurrence relations as follows:
$$ \textstyle\begin{cases} V_{1,mx}=-2V_{3,m+1}-2qV_{3m}+2rV_{2m}, \\ V_{2,mx}=2V_{3m}-2rV_{1m}, \\ V_{3,mx}=-2V_{1,m+1}-2qV_{1m}+2V_{2m}, \\ V_{4,mx}=-2V_{7,m+1}-2qV_{7m}+2s_{1}V_{2m}+2rV_{6m}, \\ V_{5,mx}=-2V_{9,m+1}-2qV_{9m}+2s_{2}V_{2m}+2rV_{8m}, \\ V_{6,mx}=2V_{7m}-2s_{1}V_{1m}+2u_{1}V_{3m}-2rV_{4m}, \\ V_{7,mx}=-2V_{4,m+1}-2qV_{4m}++2u_{1}V_{2m}+2V_{6m}, \\ V_{8,mx}=2V_{9m}-2s_{2}V_{1m}+2u_{2}V_{3m}-2rV_{5m}, \\ V_{9,mx}=-2V_{5,m+1}-2qV_{5m}+2u_{2}V_{2m}+2V_{8m}. \end{cases} $$
(3.12)
Noting that
$$\begin{aligned} V_{+}^{(n)} =&\sum_{m=0}^{n} \bigl(V_{1m}f_{1}(n-m)+V_{2m}f_{2}(n-m)+V_{3m}f_{3}(n-m)+V_{4m}f_{4}(n-m) \\ &{}+V_{5m}f_{5}(n-m)+V_{6m}f_{6}(n-m)+V_{7m}f_{7}(n-m)+V_{8m}f_{8}(n-m)+V_{9m}f_{9}(n-m) \bigr), \end{aligned}$$
we obtain by a direct calculation
$$\begin{aligned} -V_{+x}^{(n)}+\bigl[U,V_{+}^{(n)} \bigr] =&2V_{3,n+1}f_{1}(0)+2V_{1,n+1}f_{3}(0)+2V_{7,n+1}f_{4}(0)+2V_{9,n+1}f_{5}(0) \\ &{}+2V_{4,n+1}f_{7}(0)+2V_{5,n+1}f_{9}(0). \end{aligned}$$
Taking \(V^{(n)}=V_{+}^{(n)}-\frac{1}{r}V_{3,n+1}f_{2}(0)\), after a calculation, we get
$$\begin{aligned} -V_{x}^{(n)}+\bigl[U,V^{(n)}\bigr] =&\biggl( \frac{1}{r}V_{3,n+1}\biggr)_{x}f_{2}(0)- \frac {1}{r}V_{2,n+1x}f_{3}(0)+\biggl(2V_{7,n+1}- \frac {2s_{1}}{r}V_{3,n+1}\biggr)f_{4}(0) \\ &{}+\biggl(2V_{9,n+1}-\frac{2s_{2}}{r}V_{3,n+1} \biggr)f_{5}(0) +\biggl(2V_{4,n+1}-\frac{2u_{1}}{r}V_{3,n+1} \biggr)f_{7}(0) \\ &{}+\biggl(2V_{5,n+1}-\frac {2u_{2}}{r}V_{3,n+1} \biggr)f_{9}(0). \end{aligned}$$
The zero curvature equation
$$ U_{t}-V_{x}^{(n)}+ \bigl[U,V^{(n)}\bigr]=0 $$
(3.13)
admits
u t = ( q r u 1 u 2 s 1 s 2 ) t = ( ( 1 r V 3 , n + 1 ) x 1 r V 2 , n + 1 x 2 V 7 , n + 1 + 2 s 1 r V 3 , n + 1 2 V 9 , n + 1 + 2 s 2 r V 3 , n + 1 2 V 4 , n + 1 + 2 u 1 r V 3 , n + 1 2 V 5 , n + 1 + 2 u 2 r V 3 , n + 1 ) , Open image in new window
(3.14)
which can be regarded as a composition of two integrable coupling of the generalized coupled Burgers equation hierarchy as follows:
u t = ( q r u 1 u 2 ) t = ( ( 1 r V 3 , n + 1 ) x 1 r V 2 , n + 1 x 2 V 7 , n + 1 + 2 s 1 r V 3 , n + 1 2 V 9 , n + 1 + 2 s 2 r V 3 , n + 1 ) , Open image in new window
(3.15)
u t = ( q r s 1 s 2 ) t = ( ( 1 r V 3 , n + 1 ) x 1 r V 2 , n + 1 x 2 V 4 , n + 1 + 2 u 1 r V 3 , n + 1 2 V 5 , n + 1 + 2 u 2 r V 3 , n + 1 ) . Open image in new window
(3.16)
When we set \(V_{1,0}=0\), \(V_{2,0}=\beta \), \(V_{3,0}=V_{4,0}=V_{5,0}=V_{6,0}=V_{7,0}=V_{8,0}=V_{9,0}=0\), we can obtain \(V_{1,1}=\beta\), \(V_{2,1}=0\), \(V_{3,1}=\beta r\), \(V_{4,1}=\beta u_{1}\), \(V_{5,1}=\beta u_{2} \), \(V_{6,1}=0\), \(V_{7,1}=\beta s_{1}\), \(V_{8,1}=0\), \(V_{9,1}=\beta s_{2} , \ldots\) .
Hence, when we take \(n=2\) in (3.14), we get the second kind of the coupling integrable couplings of the generalized coupled Burgers equation as follows:
$$ \textstyle\begin{cases} q_{t}=-\beta(\frac{r_{xx}}{4r}+\frac{q_{x}}{r}+q^{2}+\frac {r^{2}}{2})_{x}, \\ r_{t}=\beta(\frac{r_{xx}}{2r}+\frac{q_{x}}{r}-q_{x}r-2qr_{x}), \\ u_{1t}=-\frac{\beta}{2}s_{1xx}-2\beta u_{1x}q-\beta u_{1}q_{x}+2\beta u_{1}r-2\beta s_{1}r^{2}+\frac{\beta}{2} \frac{s_{1}r_{xx}}{r}+\beta \frac{s_{1}q_{x}}{r}, \\ u_{2t}=-\frac{\beta}{2}s_{2xx}-2\beta u_{2x}q-\beta u_{2}q_{x}+2\beta u_{2}r-2\beta s_{2}r^{2}+\frac{\beta}{2} \frac{s_{2}r_{xx}}{r}+\beta \frac{s_{2}q_{x}}{r}, \\ s_{1t}=-\frac{\beta}{2}u_{1xx}-2\beta s_{1x}q-\beta s_{1}q_{x}+2\beta u_{1}-2\beta s_{1}r+\frac{\beta}{2} \frac{u_{1}r_{xx}}{r}+\beta \frac {u_{1}q_{x}}{r}, \\ s_{2t}=-\frac{\beta}{2}u_{2xx}-2\beta s_{2x}q-\beta s_{2}q_{x}+2\beta u_{2}-2\beta s_{2}r+\frac{\beta}{2} \frac{u_{2}r_{xx}}{r}+\beta \frac {u_{2}q_{x}}{r}. \end{cases} $$
(3.17)
When take \(u_{1}=u_{2}=s_{1}=s_{2}=0\), \(r=-1\), \(\beta=-2\) in (3.17), we can also get the same as (3.10).

(III) The third kind of coupling integrable couplings of the generalized coupled Burgers equation hierarchy.

By using the loop algebra of Lie algebra H as follows:
$$\widetilde{H}=\operatorname{span}\bigl\{ f_{i}(n)=f_{i} \lambda^{n},i=1,2,3,4,5,6,7,8,9\bigr\} , $$
along with the commutator
$$\bigl[f_{i}(m),f_{j}(n)\bigr]=[f_{i},f_{j}] \lambda^{m+n},\quad m,n\in Z,i,j=1,2,3,4,5,6,7,8,9, $$
we get the third type of coupling integrable couplings of the generalized coupled Burgers equation hierarchy.
A Lax pair for zero curvature equation is given as follows:
$$ \textstyle\begin{cases} U=f_{1}(0)+qf_{2}(0)+f_{2}(1)+rf_{3}(0)+u_{1}f_{4}(0)+u_{2}f_{6}(0)+s_{1}f_{7}(0)+s_{2}f_{9}(0), \\ V=\sum_{m\geq 0}(V_{1m}f_{1}(-m)+V_{2m}f_{2}(-m)+V_{3m}f_{3}(-m)+V_{4m}f_{4}(-m) \\ \hphantom{V={}}{}+V_{5m}f_{5}(-m)+V_{6m}f_{6}(-m)+V_{7m}f_{7}(-m)+V_{8m}f_{8}(-m)+V_{9m}f_{9}(-m)). \end{cases} $$
(3.18)
A solution to the stationary zero curvature equation
$$V_{x}=[U,V] $$
is presented now:
$$ \textstyle\begin{cases} V_{1,mx}=-2V_{3,m+1}-2qV_{3m}+2rV_{2m}, \\ V_{2,mx}=2V_{3m}-2rV_{1m}, \\ V_{3,mx}=-2V_{1,m+1}-2qV_{1m}+2V_{2m}, \\ V_{4,mx}=-2V_{6,m+1}-2qV_{6m}+2rV_{5m}+2u_{2}V_{2m}, \\ V_{5,mx}=2V_{6m}-2u_{2}V_{1m}+2u_{1}V_{3m}-2rV_{4m}, \\ V_{6,mx}=-2V_{4,m+1}-2qV_{4m}+2V_{5m}+2u_{1}V_{2m}, \\ V_{7,mx}=-2V_{9,m+1}-2qV_{9m}+2rV_{8m}+2s_{2}V_{2m}+2s_{2}V_{8m}, \\ V_{8,mx}=2V_{9m}-2s_{2}V_{1m}+2s_{1}V_{3m}-2rV_{7m}+2s_{1}V_{9m}-2s_{2}V_{7m}, \\ V_{9,mx}=-2V_{7,m+1}-2qV_{7m}+2V_{8m}+2s_{1}V_{2m}+2s_{1}V_{8m}. \end{cases} $$
(3.19)
Setting
$$\begin{aligned} V_{+}^{(n)} =&\sum_{m=0}^{n} \bigl(V_{1m}f_{1}(n-m)+V_{2m}f_{2}(n-m)+V_{3m}f_{3}(n-m)+V_{4m}f_{4}(n-m) \\ &{}+V_{5m}f_{5}(n-m)+V_{6m}f_{6}(n-m)+V_{7m}f_{7}(n-m)+V_{8m}f_{8}(n-m)+V_{9m}f_{9}(n-m) \bigr), \end{aligned}$$
we have
$$\begin{aligned}& -V_{+x}^{(n)}+\bigl[U,V_{+}^{(n)}\bigr] \\& \quad =2V_{3,n+1}f_{1}(0)+2V_{1,n+1}f_{3}(0)+2V_{6,n+1}f_{4}(0) \\& \qquad {}+2V_{4,n+1}f_{6}(0)+2V_{9,n+1}f_{7}(0)+2V_{7,n+1}f_{9}(0). \end{aligned}$$
Taking \(V^{(n)}=V_{+}^{(n)}-\frac{1}{r}V_{3,n+1}f_{2}(0)\), we obtain
$$\begin{aligned} -V_{x}^{(n)}+\bigl[U,V^{(n)}\bigr] =&\biggl( \frac{1}{r}V_{3,n+1}\biggr)_{x}f_{2}(0)- \frac {1}{r}V_{2,n+1x}f_{3}(0)+\biggl(2V_{6,n+1}-2 \frac {u_{2}}{r}V_{3,n+1}\biggr)f_{4}(0) \\ &{}+\biggl(2V_{4,n+1}-2 \frac{u_{1}}{r}V_{3,n+1}\biggr)f_{6}(0)+\biggl(2V_{9,n+1}-2\frac{s_{2}}{r}V_{3,n+1} \biggr)f_{7}(0) \\ &{}+\biggl(2V_{7,n+1}-2\frac {s_{1}}{r}V_{3,n+1} \biggr)f_{9}(0). \end{aligned}$$
Thus, the zero curvature equation
$$ U_{t}-V_{x}^{(n)}+ \bigl[U,V^{(n)}\bigr]=0 $$
(3.20)
admits the Lax integrable hierarchy
u t = ( q r u 1 u 2 s 1 s 2 ) t = ( ( 1 r V 3 , n + 1 ) x 1 r V 2 , n + 1 x 2 V 6 , n + 1 + 2 u 2 r V 3 , n + 1 2 V 4 , n + 1 + 2 u 1 r V 3 , n + 1 2 V 9 , n + 1 + 2 s 2 r V 3 , n + 1 2 V 7 , n + 1 + 2 s 1 r V 3 , n + 1 ) . Open image in new window
(3.21)
When we take \(s_{1}=s_{2}=0\) and \(u_{1}=u_{2}=0\) in (3.21), respectively, we get two integrable coupling of the generalized coupled Burgers equation hierarchy
u t = ( q r u 1 u 2 ) t = ( ( 1 r V 3 , n + 1 ) x 1 r V 2 , n + 1 x 2 V 6 , n + 1 + 2 u 2 r V 3 , n + 1 2 V 4 , n + 1 + 2 u 1 r V 3 , n + 1 ) , Open image in new window
(3.22)
u t = ( q r s 1 s 2 ) t = ( ( 1 r V 3 , n + 1 ) x 1 r V 2 , n + 1 x 2 V 9 , n + 1 + 2 s 2 r V 3 , n + 1 2 V 7 , n + 1 + 2 s 1 r V 3 , n + 1 ) . Open image in new window
(3.23)
So we call (3.21) the third type of coupling integrable couplings of the generalized coupled Burgers equation hierarchy.

Set \(V_{1,0}=0\), \(V_{2,0}=\beta \), \(V_{3,0}=V_{4,0}=V_{5,0}=V_{6,0}=V_{7,0}=V_{8,0}=V_{9,0}=0\), we can obtain \(V_{1,1}=\beta\), \(V_{2,1}=0\), \(V_{3,1}=\beta r\), \(V_{4,1}=\beta u_{1}\), \(V_{5,1}=0\), \(V_{6,1}=\beta u_{2} \), \(V_{7,1}=\beta s_{1}\), \(V_{8,1}=0\), \(V_{9,1}=\beta s_{2} , \ldots\) .

And when we take \(n=2\) in (3.21), we get the third hind of the coupling integrable couplings of the generalized coupled Burgers equation, that is,
$$ \textstyle\begin{cases} q_{t}=-\beta(\frac{r_{xx}}{4r}+\frac{q_{x}}{r}+q^{2}+\frac {r^{2}}{2})_{x}, \\ r_{t}=\beta(\frac{r_{xx}}{2r}+\frac{q_{x}}{r}-q_{x}r-2qr_{x}), \\ u_{1t}=-\frac{\beta}{2}u_{2xx}-2\beta u_{1x}q-\beta u_{1}q_{x}+2\beta u_{1}r-2\beta u_{2}r^{2}+\frac{\beta}{2} \frac{u_{2}r_{xx}}{r}+\beta \frac{u_{2}q_{x}}{r}, \\ u_{2t}=-\frac{\beta}{2}u_{1xx}-2\beta u_{2x}q-\beta u_{2}q_{x}+2\beta u_{1}-2\beta u_{2}r+\frac{\beta}{2} \frac{u_{1}r_{xx}}{r}+\beta \frac {u_{1}q_{x}}{r}, \\ s_{1t}=-\frac{\beta}{2}s_{2xx}-2\beta s_{1x}q-\beta s_{1}q_{x}+2\beta s_{1}r-2\beta s_{2}r^{2}-3\beta s_{2}^{2}r+\beta s_{1}^{2}r+2\beta s_{1}s_{2} \\ \hphantom{s_{1t}={}}{}-\beta s_{2}^{3}+\beta s_{1}^{2}s_{2}+\frac{\beta}{2} \frac {s_{2}r_{xx}}{r}+\beta\frac{s_{2}q_{x}}{r}, \\ s_{2t}=-\frac{\beta}{2}s_{1xx}-2\beta s_{2x}q-\beta s_{2}q_{x}+2\beta s_{1}-2\beta s_{2}r+3\beta s_{1}^{2}r-\beta s_{2}^{2}-2\beta s_{1}s_{2}r \\ \hphantom{s_{2t}={}}{}+\beta s_{1}^{3}-\beta s_{1}s_{2}^{2}+\frac{\beta}{2} \frac{s_{1}r_{xx}}{r}+\beta\frac{s_{1}q_{x}}{r}. \end{cases} $$
(3.24)

Similarly when take \(u_{1}=u_{2}=s_{1}=s_{2}=0\), \(n=2\), \(\alpha=-2\) in (3.24), we can obtain (3.10).

4 Hamiltonian structure of coupling integrable couplings of the generalized coupled Burgers equation hierarchy

In this section we shall deduce the Hamiltonian forms of the coupling integrable couplings (3.21) by using the quadratic-form [39]. The commutator (2.4) can be written as
$$ [a,b]^{T}=a^{T}R(b), $$
(4.1)
where
R ( b ) = ( 0 2 b 3 2 b 2 0 2 b 6 2 b 5 0 2 b 9 2 b 8 2 b 3 0 2 b 1 2 b 6 0 2 b 4 2 b 9 0 2 b 7 2 b 2 2 b 1 0 2 b 5 2 b 4 0 2 b 8 2 b 7 0 0 0 0 0 2 b 3 2 b 2 0 0 0 0 0 0 2 b 3 0 2 b 1 0 0 0 0 0 0 2 b 2 2 b 1 0 0 0 0 0 0 0 0 0 0 0 2 b 3 + 2 b 9 2 b 2 + 2 b 8 0 0 0 0 0 0 2 b 3 2 b 9 0 2 b 1 2 b 7 0 0 0 0 0 0 2 b 2 + 2 b 8 2 b 1 2 b 7 0 ) . Open image in new window
(4.2)
The constant symmetric matrix F satisfying the matrix equation
$$ R(b)F=-\bigl(R(b)F\bigr)^{T},\qquad F=F^{T}, $$
(4.3)
shows that
F = ( 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 ) . Open image in new window
(4.4)
In the linear space \(R^{9}\), define a functional,
$$\begin{aligned} \{a,b\} =&a^{T}Fb \\ =&a_{1}b_{1}+a_{2}b_{2}-a_{3}b_{3}+a_{7}b_{7} \\ &{}+a_{8}b_{8}-a_{9}b_{9}+(a_{1}b_{4}+a_{4}b_{1})+(a_{1}b_{7}+a_{7}b_{1})+(a_{2}b_{5}+a_{5}b_{2}) \\ &{}+(a_{2}b_{8}+a_{8}b_{2})-(a_{3}b_{6}+a_{6}b_{3})-(a_{3}b_{9}+a_{9}b_{3}). \end{aligned}$$
(4.5)
The Lax pair U and V in (3.18) can be written as
$$ \textstyle\begin{cases} U=(1,q+\lambda,r,u_{1},0,u_{2},s_{1},0,s_{2})^{T}, \\ V=(V_{1},V_{2},V_{3},V_{4},V_{5},V_{6},V_{7},V_{8},V_{9}), \end{cases} $$
(4.6)
where \(V_{i}=\sum_{m\geq0}V_{im}\lambda^{-m}\), \(i=1,2,\ldots,9\). With the help of (4.2), we have
$$\begin{aligned}& \biggl\{ V,\frac{\partial U}{\partial\lambda}\biggr\} =V_{2}+V_{5}+V_{8}, \\& \biggl\{ V,\frac{\partial U}{\partial q}\biggr\} =V_{2}+V_{5}+V_{8}, \\& \biggl\{ V,\frac{\partial U}{\partial r}\biggr\} =-V_{3}-V_{6}-V_{9}, \\& \biggl\{ V,\frac{\partial U}{\partial u_{1}}\biggr\} =V_{1},\qquad \biggl\{ V, \frac{\partial U}{\partial u_{2}}\biggr\} =-V_{3}, \\& \biggl\{ V,\frac{\partial U}{\partial s_{1}}\biggr\} =V_{1}+V_{7},\qquad \biggl\{ V, \frac {\partial U}{\partial s_{2}}\biggr\} =-V_{3}-V_{9}, \end{aligned}$$
substituting the above into the quadratic-form identity yields
δ δ u ( V 2 + V 5 + V 8 ) = λ γ λ λ γ ( V 2 + V 5 + V 8 V 3 V 6 V 9 V 1 V 3 V 1 + V 7 V 3 V 9 ) , Open image in new window
(4.7)
where
δ δ u = ( δ δ q δ δ r δ δ u 1 δ δ u 2 δ δ s 1 δ δ s 2 ) . Open image in new window
Comparing the coefficients of \(\lambda^{-n-2}\) on both sides of (4.7), we get
δ δ u ( V 2 , n + 2 + V 5 , n + 2 + V 8 , n + 2 ) = ( γ n 1 ) ( V 2 , n + 1 + V 5 , n + 1 + V 8 , n + 1 V 3 , n + 1 V 6 , n + 1 V 9 , n + 1 V 1 , n + 1 V 3 , n + 1 V 1 , n + 1 + V 7 , n + 1 V 3 , n + 1 V 9 , n + 1 ) . Open image in new window
(4.8)
Taking \(n=0\) in (4.8), we obtain \(\gamma=0\).
So we get
δ H n + 2 δ u = ( V 2 , n + 1 + V 5 , n + 1 + V 8 , n + 1 V 3 , n + 1 V 6 , n + 1 V 9 , n + 1 V 1 , n + 1 V 3 , n + 1 V 1 , n + 1 + V 7 , n + 1 V 3 , n + 1 V 9 , n + 1 ) , Open image in new window
(4.9)
where
$$H_{n+2}=-\frac{V_{2,n+2}+V_{5,n+2}+V_{8,n+2}}{n+1}. $$
So (3.21) can be written as a Hamiltonian form
u t = ( q r u 1 u 2 s 1 s 2 ) t = ( ( 1 r V 3 , n + 1 ) x 1 r V 2 , n + 1 x 2 V 6 , n + 1 + 2 u 2 r V 3 , n + 1 2 V 4 , n + 1 + 2 u 1 r V 3 , n + 1 2 V 9 , n + 1 + 2 s 2 r V 3 , n + 1 2 V 7 , n + 1 + 2 s 1 r V 3 , n + 1 ) = ( 0 0 0 1 r 0 0 0 0 2 2 r 0 0 0 2 0 2 u 2 r 0 2 1 r 2 r 2 u 2 r 0 2 + 2 s 2 r 2 s 1 r 0 0 0 2 2 s 2 r 0 2 0 0 2 2 s 1 r 2 0 ) × ( V 2 , n + 1 + V 5 , n + 1 + V 8 , n + 1 V 3 , n + 1 V 6 , n + 1 V 9 , n + 1 V 1 , n + 1 V 3 , n + 1 V 1 , n + 1 + V 7 , n + 1 V 3 , n + 1 V 9 , n + 1 ) = J δ H n + 2 δ u , Open image in new window
(4.10)
where \(\partial=\frac{\partial}{\partial x}\) and J is a Hamiltonian operator. We can obtain a recursive operator from (3.19),
L = ( l 11 l 12 l 13 l 14 l 15 l 16 l 21 l 22 l 23 l 24 l 25 l 26 l 31 l 32 l 33 l 34 l 35 l 36 l 41 l 42 l 43 l 44 l 45 l 46 l 51 l 52 l 53 l 54 l 55 l 56 l 61 l 62 l 63 l 64 l 65 l 66 ) , Open image in new window
(4.11)
where
$$\begin{aligned}& l_{11}=\frac{1}{2r}\partial-\partial^{-1}q\partial, \qquad l_{12}=\frac {1}{r}\partial-\partial^{-1}r \partial,\qquad l_{13}=\frac {u_{2}}{r}\partial - \partial^{-1}u_{1}\partial, \\& l_{14}= \frac{u_{1}}{r}\partial -\partial ^{-1}u_{2}\partial ,\qquad l_{15}=\frac{s_{2}}{r}\partial-\partial^{-1}s_{1} \partial ,\qquad l_{16}=\frac {s_{1}}{r}\partial- \partial^{-1}s_{2}\partial , \\& l_{21}=-r-\frac{1}{4}\partial\frac{1}{r}\partial,\qquad l_{22}=-q-\frac {1}{2}\partial\frac{1}{r}, \qquad l_{23}=2u_{2}\partial^{-1}r-\frac {1}{2} \partial\frac{u_{2}}{r},\qquad l_{24}=-\frac{1}{2}\partial \frac {u_{1}}{r} , \\& l_{25}=2s_{2}\partial^{-1}r+2s_{2} \partial^{-1}s_{2}-\frac {1}{2}\partial \frac{s_{2}}{r},\qquad l_{26}=-\frac{1}{2}\partial \frac {s_{1}}{r}+2s_{2}\partial^{-1}+2s_{2} \partial^{-1}s_{1} , \\& l_{31}=l_{32}=l_{35}=l_{36}=0,\qquad l_{33}=-q-2\partial^{-1}r,\qquad l_{34}= \frac {\partial}{2}-2\partial^{-1} , \\& l_{41}=l_{42}=l_{45}=l_{46}=0,\qquad l_{43}=\frac{\partial}{2}+2r\partial ^{-1}r,\qquad l_{34}=2r\partial^{-1}-q , \\& l_{51}=l_{52}=l_{53}=l_{54}=0,\qquad l_{55}=-q-2\partial^{-1}r-2\partial ^{-1}s_{2}-2s_{1} \partial^{-1}r-2s_{1}\partial^{-1}s_{2} , \\& l_{56}=\frac{1}{2}\partial-2\partial^{-1}-2\partial ^{-1}s_{1}-2s_{1}\partial^{-1}-2s_{1} \partial^{-1}s_{1} , \\& l_{61}=l_{62}=l_{63}=l_{64}=0,\qquad l_{65}=\frac{1}{2}\partial+2r\partial ^{-1}r+2r \partial^{-1}s_{2}+2s_{2}\partial^{-1}r+2s_{2} \partial ^{-1}s_{2} , \\& l_{66}=-q+2r\partial^{-1}+2r\partial^{-1}s_{1}+2s_{2} \partial ^{-1}+2s_{2}\partial^{-1}s_{1} . \end{aligned}$$
L satisfies
( V 2 , n + 1 + V 5 , n + 1 + V 8 , n + 1 V 3 , n + 1 V 6 , n + 1 V 9 , n + 1 V 1 , n + 1 V 3 , n + 1 V 1 , n + 1 + V 7 , n + 1 V 3 , n + 1 V 9 , n + 1 ) = L ( V 2 n + V 5 n + V 8 n V 3 n V 6 n V 9 n V 1 n V 3 n V 1 n + V 7 n V 3 n V 9 n ) . Open image in new window
The Hamiltonian form of the coupling integrable couplings (3.21) can be written as
u t = J δ H n + 2 δ u = J L n ( 0 β r β u 2 β s 2 β β r β + β s 1 β r β s 2 ) . Open image in new window
(4.12)

5 Conclusion

In this paper we get three kinds of coupling integrable couplings of the generalized coupled Burgers equation hierarchy, which are new results. And we obtain the Hamiltonian structure of one coupling integrable couplings of the generalized coupled Burgers equation hierarchy by using the quadratic-form identity.

6 Results and discussion

Firstly, we introduce three Lie algebras E, F and H. With the help of their corresponding loop algebras , , and , we establish three isospectral problems, respectively. Then, by taking advantage of the Tu scheme, we get three kinds of coupling integrable couplings of the generalized coupled Burgers equation hierarchy as follows:
$$\begin{aligned}& \textstyle\begin{cases} q_{t}=-\beta(\frac{r_{xx}}{4r}+\frac{q_{x}}{r}+q^{2}+\frac {r^{2}}{2})_{x}, \\ r_{t}=\beta(\frac{r_{xx}}{2r}+\frac{q_{x}}{r}-q_{x}r-2qr_{x}), \\ u_{1t}=-\frac{\beta}{2}u_{2xx}-2\beta u_{1x}q-\beta u_{1}q_{x}+2\beta u_{1}r-2\beta u_{2}r^{2}+\frac{\beta}{2} \frac{u_{2}r_{xx}}{r}+\beta \frac{u_{2}q_{x}}{r}, \\ u_{2t}=-\frac{\beta}{2}u_{1xx}-2\beta u_{2x}q-\beta u_{2}q_{x}+2\beta u_{1}-2\beta u_{2}r+\frac{\beta}{2} \frac{u_{1}r_{xx}}{r}+\beta \frac {u_{1}q_{x}}{r}, \\ s_{1t}=-\frac{\beta}{2}s_{2xx}-2\beta s_{1x}q-\beta s_{1}q_{x}+2\beta s_{1}r-2\beta s_{2}r^{2}-3\beta s_{2}^{2}r+\beta s_{1}^{2}r+2\beta s_{1}s_{2} \\ \hphantom{s_{1t}={}}{}-\beta s_{2}^{3}+\beta s_{1}^{2}s_{2}+\frac{\beta}{2} \frac {s_{2}r_{xx}}{r}+\beta\frac{s_{2}q_{x}}{r}, \\ s_{2t}=-\frac{\beta}{2}s_{1xx}-2\beta s_{2x}q-\beta s_{2}q_{x}+2\beta s_{1}-2\beta s_{2}r+3\beta s_{1}^{2}r-\beta s_{2}^{2}-2\beta s_{1}s_{2}r+\beta s_{1}^{3} \\ \hphantom{s_{2t}={}}{}-\beta s_{1}s_{2}^{2}+\frac{\beta}{2} \frac{s_{1}r_{xx}}{r}+\beta\frac{s_{1}q_{x}}{r}, \end{cases}\displaystyle \end{aligned}$$
(6.1)
$$\begin{aligned}& \textstyle\begin{cases} q_{t}=-\beta(\frac{r_{xx}}{4r}+\frac{q_{x}}{r}+q^{2}+\frac {r^{2}}{2})_{x}, \\ r_{t}=\beta(\frac{r_{xx}}{2r}+\frac{q_{x}}{r}-q_{x}r-2qr_{x}), \\ u_{1t}=\beta u_{2xx}-2\beta u_{1x}q-\beta u_{1}q_{x}-\beta u_{2x}-\frac {\beta}{2}u_{1}r_{x}-\beta u_{1x}r-\beta\frac{u_{2}r_{xx}}{4r}-\beta \frac{u_{2}q_{x}}{2r}, \\ u_{2t}=\beta u_{1xx}-2\beta u_{2x}q-\beta u_{2}q_{x}+\beta u_{1x}+\frac {\beta}{2}u_{2}r_{x}+\beta u_{2x}r-\beta\frac{u_{1}r_{xx}}{4r}-\beta \frac{u_{1}q_{x}}{2r}, \\ s_{1t}=\beta s_{2xx}-2\beta s_{1x}q-\beta s_{1}q_{x}-\beta s_{2x}-\frac {\beta}{2}s_{1}r_{x}-\beta s_{1x}r-\beta\frac{s_{2}r_{xx}}{4r}-\beta \frac{s_{2}q_{x}}{2r}, \\ s_{2t}=\beta s_{1xx}-2\beta s_{2x}q-\beta s_{2}q_{x}+\beta s_{1x}+\frac {\beta}{2}s_{2}r_{x}+\beta s_{2x}r-\beta\frac{s_{1}r_{xx}}{4r}-\beta \frac{s_{1}q_{x}}{2r}, \end{cases}\displaystyle \end{aligned}$$
(6.2)
$$\begin{aligned}& \textstyle\begin{cases} q_{t}=-\beta(\frac{r_{xx}}{4r}+\frac{q_{x}}{r}+q^{2}+\frac {r^{2}}{2})_{x}, \\ r_{t}=\beta(\frac{r_{xx}}{2r}+\frac{q_{x}}{r}-q_{x}r-2qr_{x}), \\ u_{1t}=-\frac{\beta}{2}s_{1xx}-2\beta u_{1x}q-\beta u_{1}q_{x}+2\beta u_{1}r-2\beta s_{1}r^{2}+\frac{\beta}{2} \frac{s_{1}r_{xx}}{r}+\beta \frac{s_{1}q_{x}}{r}, \\ u_{2t}=-\frac{\beta}{2}s_{2xx}-2\beta u_{2x}q-\beta u_{2}q_{x}+2\beta u_{2}r-2\beta s_{2}r^{2}+\frac{\beta}{2} \frac{s_{2}r_{xx}}{r}+\beta \frac{s_{2}q_{x}}{r}, \\ s_{1t}=-\frac{\beta}{2}u_{1xx}-2\beta s_{1x}q-\beta s_{1}q_{x}+2\beta u_{1}-2\beta s_{1}r+\frac{\beta}{2} \frac{u_{1}r_{xx}}{r}+\beta \frac {u_{1}q_{x}}{r}, \\ s_{2t}=-\frac{\beta}{2}u_{2xx}-2\beta s_{2x}q-\beta s_{2}q_{x}+2\beta u_{2}-2\beta s_{2}r+\frac{\beta}{2} \frac{u_{2}r_{xx}}{r}+\beta \frac {u_{2}q_{x}}{r}, \end{cases}\displaystyle \end{aligned}$$
(6.3)
$$\begin{aligned}& \textstyle\begin{cases} q_{t}=-\beta(\frac{r_{xx}}{4r}+\frac{q_{x}}{r}+q^{2}+\frac {r^{2}}{2})_{x}, \\ r_{t}=\beta(\frac{r_{xx}}{2r}+\frac{q_{x}}{r}-q_{x}r-2qr_{x}), \\ u_{1t}=-\frac{\beta}{2}u_{2xx}-2\beta u_{1x}q-\beta u_{1}q_{x}+2\beta u_{1}r-2\beta u_{2}r^{2}+\frac{\beta}{2} \frac{u_{2}r_{xx}}{r}+\beta \frac{u_{2}q_{x}}{r}, \\ u_{2t}=-\frac{\beta}{2}u_{1xx}-2\beta u_{2x}q-\beta u_{2}q_{x}+2\beta u_{1}-2\beta u_{2}r+\frac{\beta}{2} \frac{u_{1}r_{xx}}{r}+\beta \frac {u_{1}q_{x}}{r}, \\ s_{1t}=-\frac{\beta}{2}s_{2xx}-2\beta s_{1x}q-\beta s_{1}q_{x}+2\beta s_{1}r-2\beta s_{2}r^{2}-3\beta s_{2}^{2}r+\beta s_{1}^{2}r+2\beta s_{1}s_{2} \\ \hphantom{s_{1t}={}}{}-\beta s_{2}^{3}+\beta s_{1}^{2}s_{2}+\frac{\beta}{2} \frac {s_{2}r_{xx}}{r}+\beta\frac{s_{2}q_{x}}{r}, \\ s_{2t}=-\frac{\beta}{2}s_{1xx}-2\beta s_{2x}q-\beta s_{2}q_{x}+2\beta s_{1}-2\beta s_{2}r+3\beta s_{1}^{2}r-\beta s_{2}^{2}-2\beta s_{1}s_{2}r+\beta s_{1}^{3} \\ \hphantom{s_{2t}={}}{}-\beta s_{1}s_{2}^{2}+\frac{\beta}{2} \frac{s_{1}r_{xx}}{r}+\beta\frac{s_{1}q_{x}}{r}. \end{cases}\displaystyle \end{aligned}$$
(6.4)
Finally, we obtain the Hamiltonian structure of one of coupling integrable couplings of the generalized coupled Burgers equation hierarchy by using the quadratic-form identity as follows:
u t = J δ H n + 2 δ u = J L n ( 0 β r β u 2 β s 2 β β r β + β s 1 β r β s 2 ) . Open image in new window
(6.5)
In this paper we can only get the Hamiltonian structure of one of the coupling integrable couplings of the generalized coupled Burgers equation hierarchy by using the quadratic-form identity. How to get all the Hamiltonian structure of one coupling integrable couplings of the generalized coupled Burgers equation hierarchy is worthy of further studying.

7 Methods and experiment

Not applicable.

Notes

Acknowledgements

The authors are grateful to the anonymous referees and the editor for their careful corrections and valuable comments on the original version of this paper.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

Funding

The authors were supported by National Natural Science Foundation of China (Grant Nos. 11701320, 11705122), the Science and Technology Foundations of Shandong Province (Grant Nos. J16li52 and J14li54) and Science Foundations of Binzhou University (Grant Nos. BZXYL1104 and BZXYL1704).

Competing interests

The authors declare that they have no competing interests.

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© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.College of ScienceBinzhou UniversityBinzhou CityP.R. China
  2. 2.Department of Applied MathematicsChangsha University of Science and TechnologyChangshaP.R. China
  3. 3.Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in EngineeringChangshaP.R. China

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