The theory of integrals and derivatives of non-integer order goes back to Leibniz, Liouville, Riemann, Grunwald and Letnikov. The fractional analysis attracted the interest of many researchers, because fractional analysis has numerous applications: kinetic theories [1, 2], statistical mechanics [3, 4], dynamics in complex media [5, 6], and many others [7-9]. The main advantage of fractional derivative in comparison with classical integer-order models is that it provides an effective instrument for the description of memory and hereditary properties of various materials and progress. Also, the advantages of the fractional derivatives become apparent in modeling mechanical and electrical properties of real materials, as well as in the description of rheological properties of rocks, and in many other fields.

Tu proposed a Lie algebras and trace identity to constructing the integrable system and Hamiltonian structure of integrable systems [10]. Afterwards, Ma called the method as Tu scheme [11]. From then on, many integrable system and theirs Hamiltonian structure were obtained [12, 13]. Then how to obtain new fractional integrable hierarchies of the fractional soliton equations is an important and interesting work in soliton theory.In [14], Wu firstly proposed the generalized Tu formula and research for the Hamiltonian structures of fractional AKNS hierarchy, Yu presented the generalized fractional KN equation hierarchy and its fractional Hamiltonian structure [15].Integrable couplings [16] were coupled systems of integrable equations, which has been introduced when we study of Virasoro symmetric algebras. It is an important and interesting topic to search for integrable couplings in soliton theory.

Let us consider an integrable evolution equation

$\quad D^{\beta}_{t}u=K(u). $

(1.1)

An integrable coupling of eq.(1.1)

$\quad D^{\beta}_{t}\bar{u}=\bar{K}_{1}(\bar{u})=\left(\begin{array}{c} K(u)\\ S_1(u,u_1) \\\end{array}\right),\bar{ u}=\left(\begin{array}{c} u\\ u_1\\\end{array}\right) $

(1.2)

is called nonlinear, if $ S_1(u,u_1) $ is nonlinear with respect to the sub-vector $ u_1 $ of dependent variables [17]. An integrable system of the form

$\quad D^{\beta}_{t}\bar{u}=\bar{K}_{2}(\bar{u})=\left(\begin{array}{c} K(u)\\ S_1(u,u_1) \\ S_2(u,u_1,u_2) \\\end{array}\right),\bar{ u}=\left(\begin{array}{c} u\\ u_1\\ u_2\\\end{array}\right) $

(1.3)

is called a bi-integrable coupling of eq.(1.1). Note that in(1.3), $ S_1 $ does not depend on $ u_2 $, and the whole systemis of triangular form. In this paper, we would like to explore some mathematical structures of Lie algebras and zero curvature equations, to construct bi-integrable couplings and their Hamiltonian structures by using the generalized fractional trace variational identity associated with the enlarged Lax pairs. In this paper, we shall introduce a kind of explicit Lie algebra for which fractional couplings of Kaup-Newell hierarchy can be generated. Then we construct the fractional Hamiltonian structures of the fractional integrable couplings of Kaup-Newell hierarchy by using generalized fractional trace variational identity.

Kolwankar and Gangal [18, 19] deflned the local fractional derivative as

$D^{\alpha}_{x^{+}}f(x)=\lim_{y\to x^{+}}\frac{1}{\Gamma(1-\alpha)}\frac{d}{dy}\int_{x}^{y}\frac{(f(\xi)-f(x))}{(y-\xi)^{\alpha}}d\xi\quad(0＜\alpha＜1) .$

(2.1)

Chen et al. [20] gave the necessary conditions for the following relationship

$D^{\alpha}_{x^{+}}f(x)=\lim_{y\to x^{+}}\frac{\Gamma(1+\alpha)(f(y)-f(x))}{(y-x)^{\alpha}}\quad(0＜\alpha＜1) .$

(2.2)

We adopt derivative (2.2) for simplicity. Some fractional derivative properties are proposed as follows.

(ⅰ) The generalized Leibniz product law. If $ f(x),g(x) $ are $ \alpha $ order differentiable functions, one can have

$D^{\alpha}_{x}(f(x)g(x))=g(x)D^{\alpha}_{x}f(x)+f(x)D^{\alpha}_{x}g(x).$

(2.3)

Proof  By the simplicity definition (2.2), we have

$\begin{split}D^{\alpha}_{x}(f(x)g(x))&= \lim_{y\to x^{+}}\frac{\Gamma(1+\alpha)(f(y)g(y)-f(x)g(x))}{(y-x)^{\alpha}}\\&= \lim_{y\to x^{+}}\frac{\Gamma(1+\alpha)(f(y)g(y)-f(x)g(y)+f(x)g(y)-f(x)g(x))}{(y-x)^{\alpha}}\\&= \lim_{y\to x^{+}}\frac{\Gamma(1+\alpha)(f(y)-f(x))g(y)}{(y-x)^{\alpha}}+\\& \lim_{y\to x^{+}}\frac{\Gamma(1+\alpha)(g(y)-g(x))f(x)}{(y-x)^{\alpha}}\\&=g(x)D^{\alpha}_{x}f(x)+f(x)D^{\alpha}_{x}g(x),\end{split}$

(2.4)

the proof is completed.

(ⅱ) The Leibniz Formula for fractional differentiable functions reads.

$_{0}I^{\alpha}_{x}D^{\alpha}_{x}f(x)=f(x)-f(0) \quad (0 ＜ a ＜1) ,$

(2.5)

where $ _{0}I^{\alpha}_{x} $ denotes the Riemann-Liouville integration, which is defined as

$_{0}I^{\alpha}_{x}f(x)=\frac{1}{\Gamma(\alpha)}\int_{0}^{x}(x-\xi)^{\alpha-1}f(\xi)d\xi\quad(0＜ a ＜1) .$

(2.6)

Therefore from the defined fractional integration, properties (ⅰ) and (ⅱ), the integration by parts can be used during the fractional calculus

$_{0}I^{\alpha}_{x}[g(x)D^{\alpha}_{x}f(x)]=f(x)g(x)|_{0}^{x}-_{0}I^{\alpha}_{x}[f(x)D^{\alpha}_{x}g(x)].$

(2.7)

(ⅲ) Fractional variational derivative

$\frac{\delta L}{\delta y}=\frac{\partial L}{\partial y}+\sum_{k=1}(-1) ^{k} (D^{\alpha}_{x})^{k}(\frac{\partial L}{\partial(D^{\alpha}_{x})^{k}y}),$

(2.8)

where $ k $ is a positive integer. Properties (ⅱ) and (ⅲ) can be proved similarly, we omit these proofs in this paper.

To generate bi-integrable couplings, we introduce a kind of block matrices

$M({A_1},{A_2},{A_3}) = \left( { \begin{array}{*{20}{c}} {{A_1}}&{{A_2}}&{{A_3}}\\ 0&{{A_1} + \xi {A_2}}&{{A_2}}\\ 0&0&{{A_1}} \end{array} } \right),$

(3.1)

where $ \xi $ be an arbitrary fixed constant, which could be zero. $ A_i=A_i(λ),i=1,2,3 $ are square matrices of the same order, depending on the free parameter $ λ $. Then we have the commutator relation

$[M(A_1,A_2,A_3) ,M(B_1,B_2,B_3)]=M(C_1,C_2,C_3) $

(3.2)

with

$\left\{ {\begin{array}{*{20}{l}} {{C_1} = [{A_1},{B_1}],}\\ {{C_2} = [{A_1},{B_2}] + [{A_2},{B_1}] + \xi [{A_2},{B_2}],}\\ {{C_3} = [{A_1},{B_3}] + [{A_3},{B_1}] + [{A_2},{B_2}].} \end{array}} \right.$

(3.3)

This closure property implies that all block matrices defined by(3.1) form a matrix Lie algebra. Such matrix Lie algebras create a basis for us to generate nonlinear Hamiltonian bi-integrable couplings. The block $ A_1 $ corresponds to the original integrable equation, and the other two blocks $ A_2 $ and $ A_3 $ are used to generate the supplementary vector fields $ S_1 $ and $ S_2 $.The commutator $ [A_2,B_2] $ yields nonlinear terms in the resulting bi-integrable couplings.

Let us consider linear spectral problem

$D^{\alpha}_{x}\varphi=U(u,λ)\varphi,\quad D^{\beta}_{t}\varphi=V(u,λ)\varphi $

(3.4)

from the fractional zero curvature equation

$D^{\beta}_{t}U-D^{\alpha}_{x}V+[U,V]=0,$

(3.5)

we get an integrable system

$D^{\beta}_{t_n}u=K_n(u). $

(3.6)

Now we introduce an enlarged spectral matrix

$\bar U = \bar U(\bar u,\lambda ) = \left( {\begin{array}{*{20}{c}} {U(u,\lambda )}&{{U_1}({u_1},\lambda )}&{{U_2}({u_2},\lambda )}\\ 0&{U(u,\lambda ) + \xi {U_1}({u_1},\lambda )}&{{U_1}({u_1},\lambda )}\\ 0&0&{U(u,\lambda )} \end{array} } \right),$

(3.7)

where $ \bar{u} $ consists of $ u,u_1,u_2 $. From an enlarged generalized zero curvature equation

$D^{\beta}_{t}\bar{U}-D^{\alpha}_{x}\bar{V}+[\bar{U},\bar{V}]=0$

(3.8)

with

$\bar V = \bar V(\bar u,\lambda ) = \left( {\begin{array}{*{20}{c}} {V(u,\lambda )}&{{V_1}(u,{u_1},\lambda )}&{{V_2}(u,{u_1},{u_2},\lambda )}\\ 0&{V(u,\lambda ) + \xi {V_1}(u,{u_1},\lambda )}&{{V_1}(u,{u_1},\lambda )}\\ 0&0&{V(u,\lambda )} \end{array}} \right)$

(3.9)

gives rise to

$\left\{ {\matrix{ {D_t^\beta U - D_x^\alpha V + [U,V] = 0,} \hfill \cr {D_t^\beta {U_t} - D_x^\alpha {V_1} + [U,{V_1}] + [{U_1},V] + \xi [{U_1},{V_1}] = 0,} \hfill \cr {D_t^\beta {U_t} - D_x^\alpha {V_2} + [U,{V_2}] + [{U_2},V] + [{U_1},{V_1}] = 0,} \hfill \cr } } \right.$

(3.10)

i.e.,

$D^{\beta}_{t_n}\bar{u}=\left(\begin{array}{c} D^{\beta}_{t_n}u\\ D^{\beta}_{t_n}u_1\\ D^{\beta}_{t_n}u_2\\\end{array}\right)=\left(\begin{array}{c} K(u)\\ S_{1}(u,u_1) \\ S_{2}(u,u_1,u_2) \\\end{array}\right). $

(3.11)

This is a bi-integrable coupling of the evolution eq.(3.6), noting the zero curvature representation (3.5) of (3.4). Normally, it is nonlinear with respect to $ u_1 $ and $ u_2 $, thereby providing a nonlinear bi-integrable coupling.

We take a solution $ \bar{W} $ to the enlarged stationary zero curvature equation

$D^{\alpha}_{x}\bar{W}=[\bar{U},\bar{W}$]

(3.12)

apply the associated generalized fractional trace variational identity [18]

$\frac{\delta }{{\delta \bar u}}\langle \bar W,{\bar U_\lambda }\rangle = {\lambda ^{ - \gamma }}\frac{\partial }{{\partial \lambda }}{\lambda ^\gamma }\langle \bar W,{\bar U_{\bar u}}\rangle (\gamma \;{\rm{is}}\;{\rm{a}}\;{\rm{constant}})$

(3.13)

to furnish Hamiltonian structures for the bi-integrable couplings described above.

In the next section, we will apply the above computational paradigm to the Kaup-Newell hierarchy, thus generating ahierarchy of nonlinear Hamiltonian bi-integrable couplings for the Kaup-Newell equations. We remark that our general ideaworks for both positive and negative soliton hierarchies.

The Kaup-Newell spectral problem [21]

$D_x^\alpha \varphi = U(u,\lambda )\varphi ,U = \left( { \matrix{ { - \lambda } & {\lambda q} \cr r & \lambda \cr } } \right),u = \left( {\matrix{ q \cr r \cr } } \right).$

(4.1)

Setting

$W = \left( {\matrix{ a & {\lambda b} \cr c & { - a} \cr } } \right) = \sum\limits_{m \ge 0} {{W_m}} {\lambda ^{ - m}} = \sum\limits_{m \ge 0} {\left( {\matrix{ {{a_m}} & {\lambda {b_m}} & {} \cr {{c_m}} & { - {a_m}} & {} \cr } } \right)} {\lambda ^{ - m}},$

(4.2)

the stationary zero curvature equation $ D^{\alpha}_{x}W=[U,W] $ gives

$\left\{ {\matrix{ {D_x^\alpha {a_m} = - r{b_{m + 1}} + q{c_{m + 1}},} \hfill \cr {D_x^\alpha {b_m} = - 2{b_{m + 1}} - 2q{a_m},} \hfill \cr {D_x^\alpha {c_m} = 2r{a_m} + 2{c_{m + 1}},} \hfill \cr {{b_0} = {c_0} = 0,{a_0} = 1,{b_1} = - q,{c_1} = - r,{a_1} = - {1 \over 2}qr,} \hfill \cr {{b_2} = {1 \over 2}D_x^\alpha q + {1 \over 2}{q^2}r,{c_2} = - {1 \over 2}D_x^\alpha r + {1 \over 2}q{r^2},{a_2} = {1 \over 4}rD_x^\alpha q - {1 \over 4}qD_x^\alpha r + {3 \over 8}{q^2}{r^2},\cdots .} \hfill \cr } .} \right.$

(4.3)

Let

$V^{(n)}=(\lambda^{n}W)_++\triangle_n,$

(4.4)

where $ \triangle_n=-a_ne_1 $. The fractional zero curvature equation

$D^{\beta}_{t_{n}}U-D^{\alpha}_{x}V^{(n)}+[U,V^{(n)}]=0$

(4.5)

generate the fractional Kaup-Newell hierarchy of soliton equations

$\begin{split}D^{\beta}_{t_{n}}u=K_n(u)=( D^{\alpha}_{x}b_{n},D^{\alpha}_{x}c_{n})^{T}=J\frac{\delta H_n}{\delta u},\quad n\geq 1\end{split}$

(4.6)

with the Hamiltonian operator $ J $, the Hamiltonian functions and the hereditary recursion operator $ L $ :

$\eqalign{ & J = \left( {\matrix{ 0 & {D_x^\alpha } \cr {D_x^\alpha } & 0 \cr } } \right),\quad {H_n} = {{2{a_n} - q{c_n}} \over n}, \cr & L = \left( {\matrix{ { - {1 \over 2}D_x^\alpha - {1 \over 2}D_x^\alpha qD_x^{ - \alpha }r} & { - {1 \over 2}D_x^\alpha qD_x^{ - \alpha }q} \cr { - {1 \over 2}D_x^\alpha rD_x^{ - \alpha }r} & {{1 \over 2}D_x^\alpha - {1 \over 2}D_x^\alpha rD_x^{ - \alpha }q} \cr } } \right). \cr} $

(4.7)

When we take $ n=2 $, hierarchy (4.6) can be reduced to 2-order fractional Kaup-Newell equations

$\left\{ {\matrix{ {D_{{t_2}}^\beta q = D_x^\alpha ({1 \over 2}D_x^\alpha q + {1 \over 2}{q^2}r),} \hfill \cr {D_{{t_2}}^\beta r = D_x^\alpha ( - {1 \over 2}D_x^\alpha r + {1 \over 2}q{r^2}).} \hfill \cr } } \right.$

(4.8)

We begin with an enlarged spectral matrix

$\bar U(\bar u) = \left( { \matrix{ U & {{U_1}} & {{U_2}} \cr 0 & {U + \xi {U_1}} & {{U_1}} \cr 0 & 0 & U \cr } } \right),\quad \bar u = \left( {\matrix{ q \cr r \cr {{p_1}} \cr {{p_2}} \cr {{w_1}} \cr {{w_2}} \cr } } \right),$

(4.9)

where $ U $ is defined as in (4.1) and the supplementary spectral matrices $ U_1 $ and $ U_2 $ read

$\eqalign{ & {U_1} = {U_1}({u_1}) = \left( { \matrix{ 0 & {\lambda {p_1}} \cr {{p_2}} & 0 \cr } } \right),{u_1} = \left( {\matrix{ {{p_1}} \cr {{p_2}} \cr } } \right),\cr & {U_2} = {U_2}({u_2}) = \left( { \matrix{ 0 & {\lambda {w_1}} \cr {{w_2}} & 0 \cr } } \right),{u_2} = \left( {\matrix{ {{w_1}} \cr {{w_2}} \cr } } \right). \cr} $

(4.10)

To solve the enlarged stationary zero curvature eq.(3.12), we take a solution of the following form

$\bar W(\bar u) = \left( { \matrix{ W & {{W_1}} & {{W_2}} \cr 0 & {W + \xi {W_1}} & {{W_1}} \cr 0 & 0 & W \cr } } \right),$

(4.11)

where $ W $, defined by (4.2), and

${W_1} = \left( {\matrix{ e & {\lambda f} \cr g & { - e} \cr } } \right),{W_2} = \left( {\matrix{ {e'} & {\lambda f'} \cr {g'} & { - e'} \cr } } \right)$

(4.12)

Setting

$\begin{split}&e=\sum_{m\geq0}e_m\lambda^{-m},f=\sum_{m\geq0}f_m\lambda^{-m},g=\sum_{m\geq0}g_m\lambda^{-m},\\&e'=\sum_{m\geq0}e'_m\lambda^{-m},f'=\sum_{m\geq0}f'_m\lambda^{-m},g'=\sum_{m\geq0}g'_m\lambda^{-m},\end{split}$

(4.13)

which solves $ D^{\alpha}_{x}W= [U,W] $, we have

$\left\{ {\matrix{ {D_x^\alpha {e_m} = q{g_{m + 1}} - r{f_{m + 1}} + {p_1}{c_{m + 1}} - {p_2}{b_{m + 1}} + \beta {p_1}{g_{m + 1}} - \beta {p_2}{f_{m + 1}},} \hfill \cr {D_x^\alpha {f_m} = - 2{f_{m + 1}} - 2q{e_m} - 2{p_1}{a_m} - 2\beta {p_1}{e_m},} \hfill \cr {D_x^\alpha {g_m} = 2{g_{m + 1}} + 2r{e_m} + 2{u_2}{a_m} + 2{p_2}{e_m},} \hfill \cr {D_x^\alpha {{e'}_m} = q{{g'}_{m + 1}} - r{{f'}_{m + 1}} + {w_1}{c_{m + 1}} - {w_2}{b_{m + 1}} + {p_1}{g_{m + 1}} - {p_2}{f_{m + 1}},} \hfill \cr {D_x^\alpha {{f'}_m} = - 2{{f'}_{m + 1}} - 2q{{e'}_m} - 2{w_1}{a_m} - 2{p_1}{e_m},} \hfill \cr {D_x^\alpha {{g'}_m} = 2{{g'}_{m + 1}} + 2r{{e'}_m} + 2{w_2}{a_m} + 2{p_2}{e_m}.} \hfill \cr } } \right.$

(4.14)

We choose the initial data to be

$f_0=g_0=f'_0=g'_0=0,e_0=e'_0=1$

(4.15)

from the recursion relation (4.14), we can get

$\begin{split}&f_1=-q-(1+\beta)p_1,g_1=-r-(1+\beta)p_2,\\&e_1=-\frac{1}{2}(qr+qp_2+\beta qp_2+p_1r+\beta p_1r-\beta p_1p_2-\beta^{2}p_1p_2) ,\\&f'_1=-q-w_1-p_1,g'_1=-r-w_2-p_2,\\&e'_1=-\frac{1}{2}(qr+qw_2+qp_2+w_1r+ p_1r+p_1p_2+\beta p_1p_2) ,\cdots.\end{split}$

(4.16)

For each integer $ n\geq0 $, take

${\bar V^{(n)}} = \left( { \matrix{ {{V^{(n)}}} & {V_1^{(n)}} & {V_2^{(n)}} \cr 0 & {{V^{(n)}} + \xi V_1^{(n)}} & {V_1^{(n)}} \cr 0 & 0 & {{V^{(n)}}} \cr } } \right),$

(4.17)

where $ V^{(n)} $ define in (4.4), and $ V_1^{(n)} $, $ V_2^{(n)},$

$V_1^{(n)}=(\lambda^{n}W_1) _++\triangle_1,V_2^{(n)}=(\lambda^{n}W_2) _++\triangle_2 $

(4.18)

with $ \triangle_1=-e_ne_1,\triangle_2=-e'_ne_1 $, and then from the enlarged generalized zero curvature equation (3.8), we have

$D^{\beta}_{t_{n}}\bar{v}=S_{n}(\bar{v})=\left(\begin{array}{cc} S_{1n}(u,u_1) \\ S_{2n}(u,u_1,u_2) \\\end{array}\right),$

(4.19)

where

$\begin{split} &S_{1n}(u,u_1) =\left(\begin{array}{c} D^{\alpha}_{x}f_{n}\\ D^{\alpha}_{x}g_{n}\end{array}\right),S_{2n}(u,u_1,u_2) =\left(\begin{array}{c} D^{\alpha}_{x}f'_{n}\\ D^{\alpha}_{x}g'_{n}\end{array}\right). \end{split}$

(4.20)

Together with system (4.6), we present bi-integrable couplings of Kaup-Newell hierarchy

$D^{\beta}_{t_{n}}\bar{u}=\left(\begin{array}{cc}D^{\beta}_{t_{n}} q\\ D^{\beta}_{t_{n}}r\\ D^{\beta}_{t_{n}}p_1\\ D^{\beta}_{t_{n}}p_2\\ D^{\beta}_{t_{n}} w_1\\ D^{\beta}_{t_{n}} w_2\\\end{array}\right)=\bar{K}_n(\bar{u})=\left(\begin{array}{cc} K_n(u)\\ S_{1n}(u,u_1) \\ S_{2n}(u,u_1,u_2) \\\end{array}\right)=\left(\begin{array}{cc} D^{\alpha}_{x}b_{n}\\ D^{\alpha}_{x}c_{n}\\ D^{\alpha}_{x}f_{n}\\ D^{\alpha}_{x}g_{n}\\ D^{\alpha}_{x}f'_{n}\\ D^{\alpha}_{x}g'_{n} \end{array}\right),n\geq 1.$

(4.21)

When $ n=2 $, we can obtain the first nonlinear bi-integrable coupling of (4.8)

$\left\{ \begin{array}{*{35}{l}} D_{{{t}_{2}}}^{\beta }q= & D_{x}^{\alpha }(\frac{1}{2}D_{x}^{\alpha }q+\frac{1}{2}{{q}^{2}}r),\\ D_{{{t}_{2}}}^{\beta }r= & D_{x}^{\alpha }(-\frac{1}{2}D_{x}^{\alpha }r+\frac{1}{2}q{{r}^{2}}),\\ D_{{{t}_{2}}}^{\beta }{{p}_{1}}= & \frac{1}{2}D_{x}^{\alpha }[D_{x}^{\alpha }q+qr{{p}_{1}}+(1+\beta )D_{x}^{\alpha }{{p}_{1}}+(q+\beta {{P}_{1}}) \\ {} & (qr+q{{p}_{2}}+\beta q{{p}_{2}}+{{p}_{1}}r+\beta {{p}_{1}}r+\beta {{p}_{1}}{{p}_{2}}+{{\beta }^{2}}{{p}_{1}}{{p}_{2}})],\\ D_{{{t}_{2}}}^{\beta }{{p}_{2}}= & -\frac{1}{2}D_{x}^{\alpha }[D_{x}^{\alpha }r-qr{{p}_{2}}+(1+\beta )D_{x}^{\alpha }{{p}_{2}}+(r+\beta {{p}_{2}}) \\ {} & (qr+q{{p}_{2}}+\beta q{{p}_{2}}+{{p}_{1}}r+\beta {{p}_{1}}r+\beta {{p}_{1}}{{p}_{2}}+{{\beta }^{2}}{{p}_{1}}{{p}_{2}})],\\ D_{{{t}_{2}}}^{\beta }{{w}_{1}}= & D_{x}^{\alpha }[\frac{1}{2}D_{x}^{\alpha }(q+{{w}_{1}}+{{p}_{1}})+\\ {} &\frac{1}{2}{{q}^{2}}(r+{{w}_{2}}+{{p}_{2}})+\frac{1}{2}p_{1}^{2}(r+\beta r+\beta {{p}_{2}}+{{\beta }^{2}}{{p}_{2}}) \\ {} & +qr({{p}_{1}}+{{w}_{1}})+(1+\beta )q{{p}_{1}}{{p}_{2}}],\\ D_{{{t}_{2}}}^{\beta }{{w}_{2}}= & D_{x}^{\alpha }[-\frac{1}{2}(r+{{w}_{2}}+{{p}_{2}})+\\ {} &\frac{1}{2}{{r}^{2}}(q+{{w}_{1}}+{{p}_{1}})+\frac{1}{2}p_{2}^{2}(q+\beta q+\beta {{p}_{1}}+{{\beta }^{2}}{{p}_{1}}) \\ {} & +qr({{w}_{2}}+{{p}_{2}})+(1+\beta ){{p}_{1}}{{p}_{2}}r]. \\ \end{array} \right.$

(4.22)

In order to generate Hamiltonian structures of the obtained fractional nonlinear bi-integrable couplings, we have to compute non-degenerate, symmetric and ad-invariant bilinear forms on the adopted Lie algebra

$\bar g = \left( {\matrix{ {\left( {\matrix{ {{A_1}} & {{A_2}} & {{A_3}} \cr 0 & {{A_1} + \xi {A_2}} & {{A_2}} \cr 0 & 0 & {{A_1}} \cr } } \right)|{A_1},{A_2},{A_3} \in \widetilde {sl}(2)} \cr } } \right).$

(5.1)

As usual, we transform the Lie algebra $ \bar{g} $ into a vector form through the mapping

$\sigma:\bar{g}\longrightarrow R^{9},\quad A\longmapsto(a_1,\cdots,a_9) ^{T}\in R^{9},$

(5.2)

where

$\eqalign{ & A = \left( { \matrix{ {{A_1}} & {{A_2}} & {{A_3}} \cr 0 & {{A_1} + \xi {A_2}} & {{A_2}} \cr 0 & 0 & {{A_1}} \cr } } \right) \in \bar g,\cr & {A_i} = \left( { \matrix{ {{a_{3i - 2}}} & {{a_{3i - 1}}} \cr {{a_{3i}}} & { - {a_{3i - 2}}} \cr } } \right),i = 1,2,3. \cr} $

(5.3)

A required bilinear form [22] on the underlying Lie algebra $ \bar{g} $ is given by

$\begin{split}\langle A,B\rangle_{\bar{g}}=&\eta_1(a_1b_1+\frac{1}{2}a_2b_3+\frac{1}{2}a_3b_2) +\eta_2[a_1b_4+\frac{1}{2}a_2b_6+\frac{1}{2}a_3b_5+a_4(b_1+\xi b_4) \\&+\frac{1}{2}a_5(b_3+\xi b_6) +\frac{1}{2}a_6(b_2+\xi b_5)]+\eta_3[2a_1b_7+a_2b_9+a_3b_8+2a_4b_4\\&+a_5b_6+a_6b_5+2a_7b_1+a_8b_3+a_9b_2],\end{split}$

(5.4)

where $ A $ and $ B $ are two block matrices of the form defined by (5.3), $ \eta_1,\eta_2,\eta_3 $ are constants.

Let us direct compute that

$\begin{split}&\langle\bar{W}\frac{\partial\bar{U}}{\partial\lambda}\rangle=\frac{1}{2}\eta_1(-2a+cq)+\\ &\frac{1}{2}\eta_2(cp_1-2e+gq+\beta gp_1) +\eta_3(cw_1+gp_1-2e'+qg'),\\&\langle\bar{W}\frac{\partial\bar{U}}{\partial q}\rangle=\\ &\frac{1}{2}\eta_1c\lambda+\frac{1}{2}\eta_2g\lambda+\eta_3g'\lambda,\langle\bar{W}\frac{\partial\bar{U}}{\partial r}\rangle=\frac{1}{2}\eta_1b\lambda+\frac{1}{2}\eta_2f\lambda+\eta_3f'\lambda,\langle\bar{W}\frac{\partial\bar{U}}{\partial w_1}\rangle=\eta_3c\lambda,\\&\langle\bar{W}\frac{\partial\bar{U}}{\partial p_1}\rangle=\\ &\frac{1}{2}\eta_2c\lambda+\frac{1}{2}\beta\eta_2g\lambda+\eta_3g\lambda,\langle\bar{W}\frac{\partial\bar{U}}{\partial p_2}\rangle=\\ &\frac{1}{2}\eta_2b\lambda+\frac{1}{2}\beta\eta_2f\lambda+\eta_3f\lambda,\langle\bar{W}\frac{\partial\bar{U}}{\partial w_2}\rangle=\eta_3b\lambda.\end{split}$

(5.5)

The corresponding fractional variational identity (3.13) leads to

$\begin{eqnarray}&&\frac{\delta}{\delta\bar{u}}\frac{\frac{1}{2}\eta_1(2a_n-qc_n)+\frac{1}{2}\eta_2(2e_n-p_1c_n-qg_n-\beta p_1g_n)+\eta_3(2e'_n-w_1c_n-p_1g_n-qg_n')}{n}\nonumber\\&=&(\frac{1}{2}\eta_1c_n+\frac{1}{2}\eta_2g_n+\eta_3g'_n,\frac{1}{2}\eta_1b_n+\frac{1}{2}\eta_2f_n+\eta_3f'_n,\frac{1}{2}\eta_2c_n+\frac{1}{2}\beta\eta_2g_n+\eta_3g_n,\nonumber\\&&\frac{1}{2}\eta_2b_n+\frac{1}{2}\beta\eta_2f_n+\eta_3f_n,\eta_3c_n,\eta_3b_n )^{T},n\geq1.\end{eqnarray}$

(5.6)

It is easy to see that $ \gamma=0 $. Threrfor, the Kaup-Newell bi-integrable couplings in (4.21) possess the following Hamiltonian structures

$D^{\beta}_{t_{n}}\bar{u}=\bar{K}_n(\bar{u}) =\bar{J}\frac{\delta \bar{H}_n}{\delta \bar{u}},\quad n\geq 0,$

(5.7)

where the Hamiltonian operator is

$\bar J = \left( {\matrix{ 0 & 0 & {{1 \over {{\eta _3}}}} \cr 0 & {{{2D_x^\alpha } \over {\beta {\eta _2} + 2{\eta _3}}}} & {{{ - {\eta _2}} \over {{\eta _3}(\beta {\eta _2} + 2{\eta _3})}}} \cr {{1 \over {{\eta _3}}}} & {{{ - {\eta _2}} \over {{\eta _3}(\beta {\eta _2} + 2{\eta _3})}}} & {{{\eta _2^2 - \beta {\eta _1}{\eta _2} - 2{\eta _1}{\eta _3}} \over {2{\eta _3}(\beta {\eta _2} + 2{\eta _3})}}} \cr } } \right) \otimes \left( {\matrix{ 0 & {D_x^\alpha } \cr {D_x^\alpha } & 0 \cr } } \right)$

(5.8)

with $ \otimes $ denotes the Kronecker product of matrices, and the fractional Hamiltonian functionals read

$\bar{H}_{n}=\frac{\frac{1}{2}\eta_1(2a_n-qc_n)+\frac{1}{2}\eta_2(2e_n-p_1c_n-qg_n-\beta p_1g_n)+\eta_3(2e'_n-w_1c_n-p_1g_n-qg_n')}{n}.$

(5.9)

A direct computation shows a recursion relation

$\bar{K}_{n+1}=\bar{L}\bar{K}_{n},$

(5.10)

where the recursion operator $ \bar{L} $ is given by

$\bar L = \left( { \matrix{ L & 0 & 0 \cr {{L_1}} & {L + \xi {L_1}} & 0 \cr {{L_2}} & {{L_1}} & L \cr } } \right)$

(5.11)

with $ L $ being given by (4.7) and

$\eqalign{ & {L_1} = - {1 \over 2}\left( { \matrix{ {D_x^\alpha qD_x^{ - \alpha }{p_2} + D_x^\alpha {p_1}D_x^{ - \alpha }(r + \beta {p_2})} & {D_x^\alpha qD_x^{ - \alpha }{p_1} + D_x^\alpha {p_1}D_x^{ - \alpha }(q + \beta {p_1})} \cr {D_x^\alpha rD_x^{ - \alpha }{p_2} + D_x^\alpha {p_2}D_x^{ - \alpha }(r + \beta {p_2})} & {D_x^\alpha rD_x^{ - \alpha }{p_1} + D_x^\alpha {p_2}D_x^{ - \alpha }(q + \beta {p_1})} \cr } } \right),\cr & {L_2} = - {1 \over 2}\left( { \matrix{ {D_x^\alpha qD_x^{ - \alpha }{w_2} + D_x^\alpha {w_1}D_x^{ - \alpha }r + D_x^\alpha {p_1}D_x^{ - \alpha }{p_2}} & {D_x^\alpha qD_x^{ - \alpha }{w_1} + D_x^\alpha {w_1}D_x^{ - \alpha }q + D_x^\alpha {p_1}D_x^{ - \alpha }{p_1}} \cr {D_x^\alpha rD_x^{ - \alpha }{w_2} + D_x^\alpha {w_2}D_x^{ - \alpha }r + D_x^\alpha {p_2}D_x^{ - \alpha }{p_2}} & {D_x^\alpha rD_x^{ - \alpha }{w_1} + D_x^\alpha {w_2}D_x^{ - \alpha }q + D_x^\alpha {p_2}D_x^{ - \alpha }{p_1}} \cr } } \right). \cr} $

A way to construct fractional nonlinear bi-integrable couplings of fractional soliton hierarchy is presented. The fractional variational identity has been generalized to the fractional zero-curvature equation. As an application, the fractional Kaup-Newell hierarchy gave a hierarchy of the fractional nonlinear bi-integrable couplings and fractional Hamiltonian structures. As its reduction, we gain the fractional nonlinear integrable couplings of the Kaup-Newell equations. The solution of reduced equations is a very important and difficult work, we will take great efforts in our next work. The obtained results supplement the existing theories on the perturbation equations and classical integrable couplings. The method can be generalized to other fractional integrable couplings.