1 Introduction

The study of solitons in regard to integrable systems has facilitated a deeper understanding of mathematics and physics. Many well-known nonlinear partial differential equations have been found to have soliton solutions, for example, the Korteweg–de Vries equation and the sine-Gordon equation. It is known that zero curvature equations associated with simple Lie algebras generate classical integrable systems [1], and semisimple Lie algebras generate non-coupled systems of classical integrable systems. It is our business to further develop the study of non-semisimple Lie algebras in relation to integrable couplings. Soliton hierarchies, and specifically, integrable couplings and bi-integrable couplings, provide valuable new insights into the classification of multi-component integrable systems [2,3,4,5,6].

It is known that zero curvature equations on semidirect sums of matrix loop algebras generate integrable couplings [7, 8], and the associated variational identity [9, 10] is used to furnish Hamiltonian structures and bi-Hamiltonian structures of the resulting integrable couplings and bi-integrable couplings [11,12,13,14,15,16,17]. An important step in generating Hamiltonian structures is to search for non-degenerate, symmetric, and ad-invariant bilinear forms on the underlying loop algebras [13, 18] as the trace identity proposed by Gui-Zhang Tu [18, 19] is ineffective for non-semisimple Lie algebras which possess a degenerate Killing form. Semidirect sums of loop algebras bring various interesting integrable couplings and bi-integrable couplings [20,21,22,23,24], including higher-dimensional local bi-Hamiltonian integrable couplings [25,26,27,28,29], greatly enriching multi-component integrable systems. Recently, it has been of interest to study new integrable couplings and bi-integrable couplings generated from spectral problems associated with so\((3,{\mathbb R})\) [14].

Integrable couplings enlarge an original integrable system and often times retain its properties [2, 4]. Bi-integrable couplings then take the integrable coupling system and enlarge that system. Again, the original properties frequently are maintained. An important feature is if a soliton hierarchy has infinitely many commuting symmetries and conserved densities, the integrable coupling and then bi-integrable coupling generally will too [14,15,16,17, 30, 31]. A bi-integrable coupling system is a natural way of extending a well-behaved integrable system. We show that the bi-integrable couplings of an original spectral problem associated with so(3, \(\mathbb {R}\)) will preserve bi-Hamiltonian structures, i.e., Liouville integrability, of the integrable couplings associated with the same spectral problem [32].

A zero curvature representation of a system of the form

$$\begin{aligned} u_t=K(u)=K(x,t,u,u_x,u_{xx}, \ldots ), \end{aligned}$$
(1)

where u is a column vector of dependent variables and means there exists a Lax pair [33] \(U=U(u, \lambda )\) and \(V=V(u, \lambda )\) in a matrix loop algebra such that the zero curvature equation,

$$\begin{aligned} U_t-V_x+[U,V]=0, \end{aligned}$$
(2)

will generate system (1) [19]. The integrable coupling of system (1) is an integrable system of the form ([25, 26] for definition):

$$\begin{aligned} \bar{u}_t=\bar{K}_1(\bar{u})=\left[ \begin{matrix} K(u)\\ S(u, u_1) \end{matrix} \right] , \qquad \bar{u} = \left[ \begin{matrix} u\\ u_1 \end{matrix} \right] , \end{aligned}$$
(3)

where \(u_1\) is a new column vector of dependent variables. An integrable system of the form

$$\begin{aligned} \bar{u}_t=\bar{K}_1(\bar{u})=\left[ \begin{matrix} K(u)\\ S_1(u, u_1) \\ S_2(u, u_1,u_2) \end{matrix} \right] , \qquad \bar{u} = \left[ \begin{matrix} u\\ u_1 \\ u_2 \end{matrix} \right] , \end{aligned}$$
(4)

is called a bi-integrable coupling of (1). Note that in (4), \(S_2\) depends on \(u_2\), but \(S_1\) does not. Now, we use zero curvature equations in order to generate bi-integrable couplings and associated Hamiltonian structures, through appropriate variational identities.

We will proceed with Sects. 2 through 6. In Sect. 2, we recall a soliton hierarchy presented in [32] for a matrix spectral problem in so\((3,{\mathbb R})\). In Sect. 3, we construct bi-integrable couplings from the results in Sect. 2 using an enlarged matrix loop algebra. We then use the corresponding variational identity to present the Hamiltonian structure of the bi-integrable coupling system in Sect. 4. In Sect. 5, infinitely many symmetries and conserved functionals are discussed. We finish the paper with a couple open questions.

2 A Soliton Hierarchy Associated with so(3, \({\mathbb R})\)

Let us recall the a soliton hierarchy [32] given by the spectral problem

$$\begin{aligned} \phi _x =U\phi ~,~~~~U=U(u,\lambda )=\left[ \begin{array}{ccc} 0 &{}\quad q &{}\quad \lambda \\ -q &{}\quad 0 &{}\quad -p \\ -\lambda &{}\quad p &{}\quad 0 \end{array} \right] \in \bar{so}(3), \end{aligned}$$
(5)

where

$$\begin{aligned} u=\left[ \begin{matrix} p\\ q \end{matrix}\right] , ~~ \phi =\left[ \begin{matrix} \phi _1\\ \phi _2 \end{matrix}\right] , \end{aligned}$$

\(\lambda \text{ is } \text{ a } \text{ spectral } \text{ parameter }, ~p=p(x,t),~q=q(x,t),\) and \(\bar{so}(3)\) is the special matrix loop algebra, i.e.,

$$\begin{aligned} \bar{\mathfrak {g}}=\bar{so}(3)=\left\{ A \in so(3) | \text{ entries } \text{ of } A \text{ are } \text{ Laurent } \text{ series } \text{ in } \lambda \right\} . \end{aligned}$$
(6)

Under the assumption that W is of the form

$$\begin{aligned} W=\left[ \begin{matrix} 0 &{}\quad c &{}\quad a \\ -c&{}\quad 0 &{}\quad -b \\ -a&{}\quad b &{}\quad 0 \end{matrix} \right] = \sum _{i\ge 0}\left[ \begin{matrix} 0 &{}\quad c_i &{}\quad a_i \\ -c_i&{}\quad 0 &{}\quad -b_i \\ -a_i&{}\quad b_i &{}\quad 0 \end{matrix} \right] \lambda ^{-i} = \sum _{i\ge 0}W_i\lambda ^{-i}, \end{aligned}$$
(7)

then the stationary zero curvature equation,

$$\begin{aligned} W_{x}=[U,W], \end{aligned}$$
(8)

determines the system of equations

$$\begin{aligned} {\left\{ \begin{array}{ll} a_x = pc-qb,\\ b_x = -\lambda c+qa,\\ c_x =-pa + \lambda b. \end{array}\right. } \end{aligned}$$
(9)

After setting a, b, c to appropriate Laurent expansions, system (9) equivalently generates

$$\begin{aligned} {\left\{ \begin{array}{ll} b_{i+1}=pa_i+c_{i,x},\\ c_{i+1}=-b_{i,x}+qa_i,\qquad i\ge 0.\\ a_{i+1,x}=pc_{i+1}-qb_{i+1},\\ \end{array}\right. } \end{aligned}$$
(10)

Next, we set the initial conditions as \(\{a_0=-1,~b_0=0=c_0 \}\) and take all constants of integration to be zero. We can present for \(1 \le i \le 4\):

$$\begin{aligned}&a_1=0, \quad c_1=-q, \quad b_1= -p,\\&a_2=\frac{1}{2}(p^2+q^2), \quad c_2=p_x, \quad b_2=-q_x,\\&a_3=pq_x-p_xq, \quad c_3=q_{xx}+\frac{1}{2}p^2q+\frac{1}{2}q^3, \quad b_3= p_{xx}+\frac{1}{2}p^3+\frac{1}{2}pq^2,\\&a_4=-\frac{3}{4}p^2q^2-\frac{3}{8}p^4 +\frac{1}{2}p_{x}^2-pp_{xx}-\frac{3}{8}q^4 +\frac{1}{2}q_{x}^2-qq_{xx}, \\&b_4=q_{xxx}+\frac{1}{2}(3p^2+3q^2)q_x, \quad c_4=-p_{xxx}-\frac{1}{2}(3p^2+3q^2)p_x. \end{aligned}$$

All functions \(\{ a_i, b_i, c_i | i \ge 0 \}\) are differential polynomials of u with respect to x.

The zero curvature equations are

$$\begin{aligned} U_{t_m}-V_x^{[m]} + \left[ U,V^{[m]}\right] =0 \quad \text{ with } \quad V^{[m]}= (\lambda ^mW)_{+}, \end{aligned}$$
(11)

where \(m \ge 0\), and, therefore, provide a hierarchy of soliton equations, i.e.,

$$\begin{aligned} u_{t_m}=K_m=\left[ \begin{matrix} -c_{m+1} \\ b_{m+1} \end{matrix} \right] ={\varPhi }^m \left[ \begin{matrix} q \\ -p \end{matrix} \right] =J\frac{\delta \mathcal {H}_m}{\delta u}, \end{aligned}$$
(12)

where \(m \ge 0\). The Hamiltonian operator J, the hereditary recursion operator \({\varPhi }\), and the Hamiltonian functions are defined as follows:

$$\begin{aligned} J= \left[ \begin{matrix} 0 &{} \quad -1 \\ 1 &{}\quad 0 \end{matrix} \right] ,\quad {\varPhi }=\left[ \begin{matrix} q\partial ^{-1}p &{}\quad \partial + q\partial ^{-1}q \\ -\partial - p\partial ^{-1}p &{}\quad -p\partial ^{-1}q \end{matrix} \right] ,\quad \mathcal {H}_m=\int -\frac{a_{m+2}}{m+1} \,\hbox {d}x, \end{aligned}$$
(13)

in which \(m \ge 0\) and \(\partial = \frac{\partial }{\partial x}\). The first nonlinear example is

$$\begin{aligned} u_{t_2}=K_2=\begin{bmatrix} -q_{xx}-\frac{1}{2}p^2q-\frac{1}{2}q^3 \\ p_{xx}+\frac{1}{2}p^3+\frac{1}{2}pq^2 \end{bmatrix}=J \begin{bmatrix} p_{xx}+\frac{1}{2}p^3+\frac{1}{2}pq^2 \\ q_{xx}+\frac{1}{2}p^2q+\frac{1}{2}q^3 \end{bmatrix}=J \frac{\delta \mathcal {H}_2}{\delta u}. \end{aligned}$$
(14)

3 Bi-Integrable Couplings

We construct Hamiltonian bi-integrable couplings for the soliton hierarchy by using a matrix loop Lie algebra. Define a triangular block matrix

$$\begin{aligned} M(A_1, A_2, A_3)=\left[ \begin{matrix} A_1 &{}\quad A_2 &{}\quad A_3 \\ 0 &{}\quad A_1 &{}\quad \alpha A_2 \\ 0 &{}\quad 0 &{}\quad A_1 \end{matrix}\right] . \end{aligned}$$
(15)

It is known that block matrices of this form are closed under multiplication, i.e., constitute a Lie algebra [34]. The associated loop matrix Lie algebra \(\tilde{\mathfrak {g}}(\lambda )\) is formed by all block matrices of the type

$$\begin{aligned} \tilde{\mathfrak {g}}(\lambda )=\{ M(A_1, A_2, A_3) | M \, \text{ defined } \text{ by } \, (15), \text{ entries } \text{ of } \text{ A } \text{ are } \text{ Laurent } \text{ series } \text{ in } \lambda \}. \end{aligned}$$
(16)

A spectral matrix is chosen from \(\tilde{\mathfrak {g}}(\lambda )\) as

$$\begin{aligned} \bar{U}=\bar{U}(\bar{u},\lambda )=M(U,U_1,U_2), \qquad \bar{u}=(p,q,r,s,v,w)^\mathrm{T}, \end{aligned}$$
(17)

where U is defined as in (5) and the supplementary spectral matrices \(U_1\) and \(U_2\) are

$$\begin{aligned} U_1=U_{{1}}(u_{1})= & {} \left[ \begin{matrix} 0&{}\quad s&{}\quad 0 \\ -s&{}\quad 0&{}\quad -r \\ 0&{}\quad r&{}\quad 0\end{matrix} \right] , \qquad u_{{1}} = \left[ \begin{matrix} r\\ s\end{matrix} \right] , \end{aligned}$$
(18)
$$\begin{aligned} U_2=U_{{2}}(u_{2})= & {} \left[ \begin{matrix} 0&{}\quad w&{}\quad 0\\ -w&{}\quad 0&{}\quad -v \\ 0&{}\quad v&{}\quad 0\end{matrix} \right] , \qquad u_{{2}} = \left[ \begin{matrix} v\\ w\end{matrix} \right] . \end{aligned}$$
(19)

In order to solve the enlarged stationary zero curvature equation,

$$\begin{aligned} \bar{W}_x=[\bar{U},\bar{W}], \end{aligned}$$
(20)

we take the solution to be of the following form:

$$\begin{aligned} \bar{W}=\bar{W}(\bar{u},\lambda )=M(W,W_1,W_2) \in \tilde{\mathfrak {g}}(\lambda ), \end{aligned}$$
(21)

where W is defined by (7) and solves \(W_x=[U,W]\), and \(W_1\) and \(W_2\) are assumed to be

$$\begin{aligned} W_1=W_{{1}}(u,u_{1},\lambda )= \left[ \begin{matrix} 0&{}\quad g&{}\quad e\\ -g&{}\quad 0&{}\quad -f \\ -e&{}\quad f&{}\quad 0\end{matrix} \right] \in \bar{so}(3), \end{aligned}$$
(22)

and

$$\begin{aligned} W_2=W_{{2}}(u, u_1, u_{2}, \lambda )= \left[ \begin{matrix} 0&{}\quad g'&{}\quad e' \\ -g'&{}\quad 0&{}\quad -f' \\ -e'&{}\quad f'&{}\quad 0 \end{matrix} \right] \in \bar{so}(3). \end{aligned}$$
(23)

Equation (20) is equivalent to satisfying the following matrix equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} W_x=[U,W],\\ W_{1,x}=[U,W_1]+[U_1,W],\\ W_{2,x}=[U,W_2]+[U_2,W]+\alpha [U_1,W_1]. \end{array}\right. } \end{aligned}$$
(24)

The second and third equations in (24) generate

$$\begin{aligned} {\left\{ \begin{array}{ll} e_x = pg-qf+rc-sb,\\ f_x = -\lambda g+qe+sa,\\ g_x =-pe + \lambda f -ra, \end{array}\right. } \end{aligned}$$
(25)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} e'_x = -fs\alpha +gr\alpha -qf'+pg'-wb+vc,\\ f'_x =qe'-\lambda g' + wa + se\alpha ,\\ g'_x =-pe' + \lambda f' -re\alpha -va, \end{array}\right. } \end{aligned}$$
(26)

respectively. Plugging into recursion relations (25) and (26) into the Laurent expansions,

$$\begin{aligned} {\left\{ \begin{array}{ll} e=\displaystyle \sum _{i\ge 0}e_i\lambda ^{-i}, &{}f=\displaystyle \sum _{i\ge 0}f_i\lambda ^{-i},\quad g=\displaystyle \sum _{i\ge 0}g_i\lambda ^{-i},\\ e'=\displaystyle \sum _{i\ge 0}e'_i\lambda ^{-i}, &{}f'=\displaystyle \sum _{i\ge 0}f'_i\lambda ^{-i},\quad g'=\displaystyle \sum _{i\ge 0}g'_i\lambda ^{-i}, \end{array}\right. } \end{aligned}$$
(27)

we have

$$\begin{aligned} {\left\{ \begin{array}{ll} f_{i+1}=g_{i,x}+pe_i+ra_i,\\ g_{i+1}=-f_{i,x}+qe_i+sa_i,\\ e_{i+1,x}=pg_{i+1}-qf_{i+1}+rc_{i+1}-sb_{i+1},\\ f'_{i+1}=g'_{i,x}+pe'_i+va_i+\alpha rc_i,\\ g'_{i+1}=-f'_{i,x}+qe'_i+wa_i+\alpha sc_i,\\ e'_{i+1,x}=pg'_{i+1}-qf'_{i+1}-\alpha sf_{i+1}+\alpha rg_{i+1}-wb_{i+1}+vc_{i+1}, \end{array}\right. } \end{aligned}$$
(28)

where \(i \ge 0\). We take the initial data as \(\{ e_0=-1,f_0=g_0=0; e'_0=-1, f'_0=g'_0=0 \}\) and suppose that the integration constants are zero. Then, recursion relation (28) uniquely generates \(\{e_i, f_i, g_i,e'_i, f'_i, g'_i| i \ge 1\}\). We obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} e_1=0,\\ f_1=-p-r,\\ g_1=-q-s;\\ e_2=\frac{1}{2}p^2+\frac{1}{2}q^2+rp+sq,\\ f_2=-q_x-s_x,\\ g_2=p_x+r_x;\\ e_3=q_xp-qp_x-sp_x+rq_x+s_xp-r_xq,\\ f_3=p_{xx}+\frac{1}{2}p^3+\frac{1}{2}pq^2+\frac{3}{2}rp^2+psq+\frac{1}{2}rq^2+r_{xx},\\ g_3=q_{xx}+\frac{1}{2}q^3+\frac{1}{2}qp^2+\frac{3}{2}sq^2+qrp+\frac{1}{2}sp^2+s_{xx};\\ e_4=(-p-r)p_{xx}+(-q-s)q_{xx}-pr_{xx}-qs{xx}\frac{1}{2}p_x^2+p_x r_x+\frac{1}{2}q_x^2+q_x s_x\\ \qquad \;\; -\frac{3}{8}(p^2+q^2)(p^2+4pr+q(q+4s)),\\ f_4=q_{xxx}+s_{xxx}+\frac{1}{2}(3p^2+6pr+3q^2+6qs)q_x+\frac{1}{2}(3p^2+3q^2)s_x, \\ g_4=-p_{xxx}-r_{xxx}+\frac{1}{2}(-3p^2-6pr-3q^2-6qs)p_x+\frac{1}{2}(-3p^2-3q^2)r_x; \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\left\{ \begin{array}{ll} e_1'=0,\\ f_1'=-p-\alpha r-v,\\ g_1'=-q-\alpha s -w;\\ e_2'=\frac{1}{2}p^2+\frac{1}{2}q^2+\alpha rp+\alpha sq+vp+wq+\frac{1}{2}\alpha s^2+\frac{1}{2}\alpha r^2,\\ f_2'=-q_x-\alpha s_x-w_x,\\ g_2'=p_x+\alpha r_x+v_x;\\ e_3'=q_xp-qp_x-\alpha sp_x+\alpha rq_x+\alpha s_xp-r_xq-wp_x+vq_x+w_xp-\alpha qr_x\\ \qquad \;\;-\,v_xq+\alpha s_xr-{\alpha sr_x},\\ f_3'=p_{xx}+\frac{1}{2}p^3+\frac{1}{2}pq^2+\alpha \frac{3}{2}rp^2+\alpha psq+\alpha \frac{1}{2}rq^2+r_{xx}+\alpha \frac{3}{2}pr^2+\alpha rsq\\ \qquad \;\;+\,pqw+\frac{3}{2}vp^2+\frac{1}{2}vq^2+ \frac{1}{2}\alpha ps^2+\alpha r_{xx}+v_{xx},\\ g_3'=q_{xx}+\frac{1}{2}q^3+\frac{1}{2}qp^2+\alpha \frac{3}{2}sq^2+\alpha qrp+\alpha \frac{1}{2}sp^2+s_{xx}+\alpha \frac{3}{2}qs^2+\alpha srp\\ \qquad \;\;+\,pvq+\frac{3}{2}wq^2+\frac{1}{2}wp^2+ \frac{1}{2}\alpha pr^2+\alpha s_{xx}+w_{xx};\\ e_4'=(-\alpha r -p -v)p_{xx}+(-\alpha s - q -w)q_{xx}-\alpha (p+r)r_{xx}-\alpha (q+s)s_{xx}\\ \qquad \;\; -\,v_{xx}p-w_{xx}q+\frac{1}{2}p_x^2+(\alpha r_x +v_x)p_x\\ \qquad \;\; +\,\frac{1}{2}q_x^2+(\alpha s_x +w_x)q_x+\frac{1}{2}\alpha r_x^2+\frac{1}{2}\alpha s_x^2-\frac{3}{8}p^4+\frac{3}{2}(-\alpha r - v)p^3 \\ \qquad \;\; +\,\frac{1}{8}(-6q^2+(-12 \alpha s -12w)q - 18\alpha r^2 -6 \alpha s^2)p^2\\ \qquad \;\; -\,\frac{3}{2}q((\alpha r +v)q+2\alpha rs)p -\frac{3}{2}q^2\left( \frac{1}{4}q^2 +(\alpha s+w)q+\frac{1}{2}\alpha (r^2+3s^2)\right) ,\\ f_4'=q_{xxx}+\alpha s_{xxx}+w_{xxx}+\frac{1}{2}(3p^2+(6\alpha r+6v)p+3q^2+(6\alpha s+6w)q\\ \qquad \;\; +\,3\alpha r^2+3\alpha s^2)q_x+\frac{1}{2}(3\alpha p^2+6\alpha pr+3\alpha q^2+6\alpha q s)s_x\\ \qquad \;\; +\,\frac{1}{2}(3p^2+3q^2)w_x, \\ g_4'=-p_{xxx}-\alpha r_{xxx}-v_{xxx}+\frac{1}{2}(-3p^2-(6\alpha r+6v)p-3q^2-(6\alpha s+6w)q \\ \qquad \;\; -\,3\alpha r^2-3\alpha s^2)p_x+\frac{1}{2}(-3\alpha p^2-6\alpha pr-3\alpha q^2-6\alpha q s)r_x \\ \qquad \;\; -\,\frac{1}{2}(3p^2+3q^2)v_x. \end{array}\right. } \end{aligned}$$

These functions are differential polynomials in the variables p, q, r, s, v, and w.

Similar to [35], for each integer \(m \ge 0\), we further introduce an enlarged Lax matrix

$$\begin{aligned} \bar{V}^{[m]}=(\lambda ^m \bar{W})_{+}=M\left( V^{[m]}, V_1^{[m]}, V_2^{[m]}\right) \in \tilde{\mathfrak {g}}(\lambda ), \end{aligned}$$
(29)

where \(V^{[m]}\) is defined by (11) and \(V_i^{[m]}=(\lambda ^m W_i)_+, i=1,2.\) The enlarged zero curvature equation,

$$\begin{aligned} \bar{U}_{t_m}-\bar{V}_x^{[m]} + \left[ \bar{U},\bar{V}^{[m]}\right] =0, \end{aligned}$$
(30)

gives the following matrix equations:

$$\begin{aligned} {\left\{ \begin{array}{ll} U_{1,t_m}-V_{1,x}^{[m]} + \left[ U,V_1^{[m]}\right] + \left[ U_1,V^{[m]}\right] =0,\\ U_{2,t_m}-V_{2,x}^{[m]} + \left[ U,V_2^{[m]}\right] + \left[ U_2,V^{[m]}\right] +\alpha \left[ U_1,V_1^{[m]}\right] =0, \end{array}\right. } \end{aligned}$$
(31)

along with the system in (11). The above equations then present the additional systems

$$\begin{aligned} \bar{v}_{t_m}=S_m=S_m(\bar{v})=\left[ \begin{matrix} S_{1,m}(u,u_1)\\ S_{2,m}(u,u_1,u_2) \end{matrix}\right] , \quad m \ge 0, \end{aligned}$$
(32)

where \(\bar{v}=(r,s,v,w)^\mathrm{T}\) and

$$\begin{aligned} S_{1,m}(u,u_1)=\left[ \begin{matrix} -g_{m+1}\\ f_{m+1} \end{matrix}\right] , \end{aligned}$$

and

$$\begin{aligned} S_{2,m}(u,u_1,u_2)=\left[ \begin{matrix} -g'_{m+1}\\ f'_{m+1} \end{matrix}\right] . \end{aligned}$$

Then the enlarged zero curvature equation generates a hierarchy of bi-integrable couplings,

$$\begin{aligned} \bar{u}_{t_m}=\left[ \begin{matrix} p\\ q\\ r\\ s\\ v\\ w \end{matrix}\right] _{t_m} = \left[ \begin{matrix} -c_{m+1}\\ b_{m+1}\\ -g_{m+1}\\ f_{m+1}\\ -g'_{m+1}\\ f'_{m+1} \end{matrix}\right] = \bar{K}_m(\bar{u}), \quad m \ge 0, \end{aligned}$$
(33)

for soliton hierarchy (12).

In particular, when \(m=2\), we have \(u_{t_2}=\bar{K_2}\), i.e.,

$$\begin{aligned}&\begin{bmatrix} p \\ q \\ r \\ s \\ v\\ \\ w\\ \\ \end{bmatrix}_{t_2} \nonumber \\&\quad = \begin{bmatrix} -q_{xx}-\frac{1}{2}p^2q-\frac{1}{2}q^3 \\ p_{xx}+\frac{1}{2}p^3+\frac{1}{2}pq^2 \\ -q_{xx}-\frac{1}{2}q^3-\frac{1}{2}qp^2-\frac{3}{2}sq^2-qrp-\frac{1}{2}sp^2-s_{xx}\\ p_{xx}+\frac{1}{2}p^3+\frac{1}{2}pq^2+\frac{3}{2}rp^2+psq+\frac{1}{2}rq^2+r_{xx}\\ -q_{xx}-\frac{1}{2}q^3-\frac{1}{2}qp^2-\alpha \frac{3}{2}sq^2-\alpha qrp-\alpha \frac{1}{2}sp^2-s_{xx}-\alpha \frac{3}{2}qs^2-\alpha srp\\ -pvq+\frac{3}{2}wq^2-\frac{1}{2}wp^2-\frac{1}{2}\alpha ps^2-\alpha s_{xx}-w_{xx}\\ p_{xx}+\frac{1}{2}p^3+\frac{1}{2}pq^2+\alpha \frac{3}{2}rp^2+\alpha psq+\alpha \frac{1}{2}rq^2+r_{xx}+\alpha \frac{3}{2}pr^2+\alpha rsq\\ +pqw+\frac{3}{2}vp^2+\frac{1}{2}vq^2+ \frac{1}{2}\alpha ps^2+\alpha r_{xx}+v_{xx} \end{bmatrix} .\nonumber \\ \end{aligned}$$
(34)

4 Hamiltonian Structures

We have a systematic approach for generating Hamiltonian structures for the bi-integrable coupling in (33) using the variational identity over the enlarged matrix loop algebra \(\tilde{\mathfrak {g}}(\lambda )\) [13, 18]. The variational identity is as follows:

$$\begin{aligned} \frac{\delta }{\delta \bar{u}} \int \langle \bar{W}, \bar{U}_{\lambda } \rangle \hbox {d}x= \lambda ^{-\gamma } \frac{\partial }{\partial \lambda } \lambda ^{\gamma } \langle \bar{W}, \bar{U}_{\bar{u}} \rangle , \quad \gamma = \text{ constant }. \end{aligned}$$
(35)

As seen in [35], there is a convenient method to constructing a symmetric and ad-invariant bilinear form on \(\tilde{\mathfrak {g}}(\lambda )\) by rewriting the semidirect sum \(\tilde{\mathfrak {g}}(\lambda )\) into a vector form. First, we define a mapping

$$\begin{aligned} \sigma : \tilde{\mathfrak {g}}(\lambda ) \mapsto \mathbb {R}^9, A \mapsto (a_1, \ldots , a_9)^\mathrm{T} , \end{aligned}$$
(36)

where

$$\begin{aligned} A=M(A_1,A_2,A_3) \in \tilde{\mathfrak {g}}(\lambda ), \quad A_i = \left[ \begin{matrix} 0 &{}\quad a_{3i} &{}\quad a_{3i-2}\\ -a_{3i} &{}\quad 0 &{}\quad -a_{3i-1} \\ -a_{3i-2} &{}\quad a_{3i-1} &{}\quad 0 \end{matrix}\right] , \quad 1 \le i \le 3 . \end{aligned}$$
(37)

The map \(\sigma \) induces a Lie algebra structure on \(\mathbb {R}^9\) isomorphic to the enlarged matrix loop algebra \( \tilde{\mathfrak {g}}(\lambda )\). Thus, the corresponding Lie bracket \([ \cdot , \cdot ]\) on \(\mathbb {R}^9\) is generated by letting

$$\begin{aligned}{}[a,b]^\mathrm{T}=a^\mathrm{T} R(b), \end{aligned}$$
(38)

where \(a=(a_1, \ldots , a_9)^\mathrm{T} , b = (b_1, \ldots , b_9)^\mathrm{T} \in \mathbb {R}^9\) and

$$\begin{aligned} R(b)= M(R_1, R_2, R_3), \end{aligned}$$
(39)

with

$$\begin{aligned} R_i = \left[ \begin{matrix} 0 &{}\quad -b_{3i} &{}\quad b_{3i-1}\\ b_{3i} &{}\quad 0 &{}\quad -b_{3i-2}\\ -b_{3i-1} &{}\quad b_{3i-2} &{}\quad 0 \end{matrix}\right] , \quad 1 \le i \le 3 . \end{aligned}$$
(40)

There is an Lie isomorphism, \(\sigma \), between the Lie algebra \((\mathbb {R}^9, [\cdot ,\cdot ])\) with the enlarged matrix loop algebra \( \tilde{\mathfrak {g}}(\lambda )\).

We may find a bilinear form on \(\mathbb {R}^9\) by

$$\begin{aligned} \langle a, b \rangle = a^\mathrm{T} F b, \end{aligned}$$
(41)

where F is a constant matrix and the symmetric property of \(\langle \cdot , \cdot \rangle \) requires that

$$\begin{aligned} F^\mathrm{T}=F. \end{aligned}$$
(42)

The symmetric condition along with the ad-invariance property

$$\begin{aligned} \langle a, [b,c] \rangle = \langle [a,b],c \rangle , \end{aligned}$$

provides the condition

$$\begin{aligned} F(R(b))^\mathrm{T}=-R(b)F, \quad b \in \mathbb {R}^9. \end{aligned}$$
(43)

Upon solving the derived system of equations from (43) for an arbitrary vector \(b \in \mathbb {R}^9\), we find

$$\begin{aligned} F=\left[ \begin{matrix} \eta _1 &{}\quad \eta _2 &{}\quad \eta _3\\ \eta _2 &{}\quad \alpha \eta _3 &{}\quad 0\\ \eta _3 &{}\quad 0 &{}\quad 0 \end{matrix} \right] \otimes F_0, \end{aligned}$$
(44)

where

$$\begin{aligned} F_0=\left[ \begin{matrix} 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 1 \end{matrix} \right] , \end{aligned}$$
(45)

and \(\eta _i, 1 \le i \le 3\), are arbitrary constants. Thus, the bilinear form on the semidirect sum \(\tilde{\mathfrak {g}}(\lambda )\) of the two Lie subalgebras \(\tilde{g}\) and \(\tilde{g}_c\) is defined as

$$\begin{aligned} \langle A,B \rangle _{\tilde{\mathfrak {g}}(\lambda )}= & {} \langle \sigma (A), \sigma (B) \rangle _{\mathbb {R}^9}\nonumber \\= & {} \,(a_1, \ldots , a_9)F(b_1, \ldots , b_9)^\mathrm{T}\nonumber \\= & {} \,(a_1b_1+a_2b_2+a_3b_3)\eta _1 + (a_1b_4+a_2b_5 + a_3b_6 + a_4b_1 + a_5b_2\nonumber \\&+\, a_6b_3)\eta _2 + (\alpha a_4b_4 + \alpha a_5 b_5 + \alpha a_6b_6 + a_1b_7 + a_2b_8 + a_3b_9\nonumber \\&+\, a_7b_1 + a_8b_2 + a_9b_3)\eta _3, \end{aligned}$$
(46)

where A and B are two matrices in \(\tilde{\mathfrak {g}}(\lambda )\) presented by

$$\begin{aligned} {\left\{ \begin{array}{ll} A = \sigma ^{-1}((a_1, \ldots , a_9)^\mathrm{T}) \in \tilde{\mathfrak {g}}(\lambda ),\\ B = \sigma ^{-1}((b_1, \ldots , b_9)^\mathrm{T}) \in \tilde{\mathfrak {g}}(\lambda ). \end{array}\right. } \end{aligned}$$
(47)

Bilinear form (46) is symmetric and ad-invariant due to the isomorphism \(\sigma \). A bilinear form, defined by (46), is non-degenerate iff the determinant of F is not zero, i.e.,

$$\begin{aligned} \hbox {det}(F)=-\eta _3^9 \alpha ^3 \ne 0. \end{aligned}$$
(48)

Therefore, we choose \(\eta _3 \ne 0\) to obtain a non-degenerate, symmetric, and ad-invariant bilinear form over the enlarged matrix loop algebra \(\tilde{\mathfrak {g}}(\lambda )\).

Now, we compute

$$\begin{aligned} \langle \bar{W},\bar{U}_{\lambda } \rangle _{\tilde{\mathfrak {g}}(\lambda )}= a \eta _1+e\eta _2+e'\eta _3 \end{aligned}$$
(49)

and

$$\begin{aligned} \langle \bar{W},\bar{U}_{\bar{u}} \rangle _{\tilde{\mathfrak {g}}(\lambda )}= \left[ \begin{matrix} b \eta _1+f\eta _2+f'\eta _3\\ c \eta _1+g\eta _2+g'\eta _3\\ b \eta _2+\alpha f \eta _3\\ c\eta _2+\alpha g \eta _3\\ b \eta _3 \\ c \eta _3 \end{matrix} \right] . \end{aligned}$$
(50)

In addition, the formula \(\gamma =-\frac{\lambda }{2}\frac{d}{d\lambda } \text {ln}|\text {tr} (W^2)|\) [19] yields that the constant \(\gamma =0\), and thus, the corresponding variational identity is

$$\begin{aligned} \frac{\delta }{\delta \bar{u}} \int \frac{-a_{m+1} \eta _1 - e_{m+1}\eta _2 - e'_{m+1} \eta _3}{m}\hbox {d}x= \left[ \begin{matrix} b_m \eta _1+f_m\eta _2+f_m'\eta _3\\ c_m \eta _1+g_m\eta _2+g_m'\eta _3\\ b_m \eta _2+\alpha f_m \eta _3\\ c_m\eta _2+\alpha g_m \eta _3\\ b_m \eta _3 \\ c_m \eta _3 \end{matrix} \right] , \quad m \ge 1. \end{aligned}$$
(51)

We consequently obtain a Hamiltonian structure for hierarchy (33) of bi-integrable couplings,

$$\begin{aligned} \bar{u}_{t_m}=\bar{J} \frac{\delta \bar{\mathcal {H}}_m}{\delta \bar{u}}, \quad m \ge 0, \end{aligned}$$
(52)

with the Hamiltonian functionals,

$$\begin{aligned} \bar{\mathcal {H}}_m= \int \frac{-a_{m+2} \eta _1 - e_{m+2}\eta _2 - e'_{m+2} \eta _3}{m+1}\hbox {d}x, \end{aligned}$$
(53)

and the Hamiltonian operator,

$$\begin{aligned} \bar{J}= \left[ \begin{matrix} 0 &{} \quad \eta _1 &{} \quad 0 &{} \quad \eta _2 &{} \quad 0 &{} \quad \eta _3\\ -\eta _1 &{} \quad 0 &{} \quad -\eta _2 &{} \quad 0 &{} \quad -\eta _3 &{} \quad 0\\ 0 &{} \quad \eta _2 &{} \quad 0 &{} \quad \alpha \eta _3 &{} \quad 0 &{} \quad 0\\ -\eta _2 &{} \quad 0 &{} \quad -\alpha \eta _3 &{} \quad 0 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad \eta _3 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \\ -\eta _3 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 &{} \quad 0 \end{matrix} \right] ^{-1}, \end{aligned}$$
(54)

and note that \(\text {det}(\bar{J}) \ne 0\). In particular, when \(m=2\), the Hamiltonian functional is

$$\begin{aligned} \bar{\mathcal {H}}_2=\int \frac{1}{3}\left( -a_{4}\eta _1-e_4 \eta _2 -e_4' \eta _3\right) \hbox {d}x, \end{aligned}$$
(55)

where

$$\begin{aligned} \begin{aligned} -a_{4}\eta _1-e_4 \eta _2 -e_4' \eta _3=&\,\left( (\eta _1+\eta _2+\eta _3)p+(\alpha \eta _3+\eta _2)r+\eta _3 v\right) p_{xx}\\&+\,\left( (\eta _1+\eta _2+\eta _3)q+(\alpha \eta _3+\eta _2)s\eta _3 w\right) q_{xx}\\&+\,\left( (\alpha \eta _3+\eta _2)p+\eta _3 \alpha r\right) r_{xx}\left( (\alpha \eta _3+\eta _2)q+\eta _3 \alpha s\right) s_{xx}\\&+\,\eta _3 p v_{xx} \eta _3 q w_{xx}-\frac{1}{2}(\eta _1+\eta _2+\eta _3)p_x^2 \\&+\,\left( (-\alpha \eta _3 -\eta _2)r_x-\eta _3 v_x\right) p_x-(\eta _1+\eta _2+\eta _3)q_x^2 \\&+\,\left( (-\alpha \eta _3 -\eta _2)s_x-\eta _3 w_x\right) q_x-\frac{1}{2}\eta _3 \alpha r_x^2-\frac{1}{2}\eta _3 \alpha s_x^2\\&+\,\frac{3}{8}(\eta _1+\eta _2+\eta _3)p^4\frac{3}{2}\left( (\alpha \eta _3+\eta _2)r+\eta _3 v\right) p^3\\&+\,\frac{1}{8}\Bigg (6\left( \eta _1+\eta _2+\eta _3\right) q^2+\left( (12\alpha \eta _3+12\eta _2)s +12\eta _3 w\right) q \\&+\,18 \eta _3\left( r^2+\frac{1}{3}s^2\right) \alpha \Bigg )p^2+\frac{3}{2}(((\alpha \eta _3+\eta _2)r+\eta _3 v)q \\&+\,2\eta _3 \alpha rs)pq+\frac{3}{8}q^2((\eta _1+\eta _2+\eta _3)q^2+((4 \alpha \eta _3 \\&+\, 4\eta _2)s +4 \eta _3 w)q+2\alpha \eta _3 (r^2+3s^2)). \end{aligned} \end{aligned}$$
(56)

5 Symmetries and Conserved Functionals

We may solve the recursion relation of symmetries

$$\begin{aligned} \bar{K}_m=\bar{{\varPhi }} \bar{K}_{m-1}, \quad m \ge 0, \end{aligned}$$
(57)

for a recursion operator, \(\bar{{\varPhi }}\), to obtain

$$\begin{aligned} \bar{{\varPhi }}=\begin{bmatrix} {\varPhi }&0&0 \\ \; {\varPhi }_1&{\varPhi }&0 \\ \; {\varPhi }_2&\alpha {\varPhi }_1&{\varPhi }\end{bmatrix}, \end{aligned}$$
(58)

where \({\varPhi }\) is given by (13) and

$$\begin{aligned} {\varPhi }_1= \left[ \begin{matrix} q\partial ^{-1}r+s\partial ^{-1}p &{} q\partial ^{-1}s+s\partial ^{-1}q\\ -p\partial ^{-1}r-r\partial ^{-1}p &{} -p\partial ^{-1}s-r\partial ^{-1}q \end{matrix} \right] , \end{aligned}$$
(59)

and

$$\begin{aligned} {\varPhi }_2= \left[ \begin{matrix} q\partial ^{-1}v+w\partial ^{-1}p+\alpha s\partial ^{-1}r &{} q\partial ^{-1}w+w\partial ^{-1}q+\alpha s\partial ^{-1}s\\ -p\partial ^{-1}v-v\partial ^{-1}p-\alpha r\partial ^{-1}r &{} -p\partial ^{-1}w-v\partial ^{-1}q-\alpha r\partial ^{-1}s \end{matrix} \right] . \end{aligned}$$
(60)

It can be shown by a symbolic computation that \(\bar{{\varPhi }}\) is a hereditary operator [36, 37]. Therefore,

$$\begin{aligned} \bar{{\varPhi }}'(\bar{u})[\bar{{\varPhi }}\bar{T}_1]\bar{T}_2 - \bar{{\varPhi }} \bar{{\varPhi }}'(\bar{u})[\bar{T}_1]\bar{T}_2 \end{aligned}$$

is symmetric with respect to \(\bar{T}_1\) and \(\bar{T}_2\), and the two operators \(\bar{J}\) and \(\bar{M}=\bar{{\varPhi }}\bar{J}\) make a Hamiltonian pair [38], i.e., \(\bar{J}, \bar{M}\), and \(\bar{J}+ \bar{M}\) are all Hamiltonian operators. Thus, the hierarchy (33) of bi-integrable couplings possesses a bi-Hamiltonian structure [38, 39] and is Liouville integrable. It follows that there are infinitely many symmetries and conserved functionals:

$$\begin{aligned}{}[\bar{K}_m, \bar{K}_n]=0, \quad m,n \ge 0, \end{aligned}$$
(61)

and

$$\begin{aligned} \{\bar{\mathcal {H}}_m,\bar{\mathcal {H}}_n\}_{\bar{J}}=\{\bar{\mathcal {H}}_m,\bar{\mathcal {H}}_n\}_{\bar{M}}=0, \quad m,n \ge 0. \end{aligned}$$
(62)

6 Concluding Remarks

We have obtained a new class of bi-integrable couplings (33) for the soliton hierarchy (12) using on non-semisimple Lie algebra (16). We showed the resulting hierarchy of bi-integrable couplings possesses a bi-Hamiltonian structure and is Liouville integrable. It remains an open question how to generate a Hamiltonian structure for matrix loop algebra (15) when \(\alpha =0\) as the bilinear form presented in Sect. 4 is degenerate.

Some enlarged matrix loop algebras do not possess any non-degenerate, symmetric, and ad-invariant bilinear forms required in the variational identity. In the following example of a bi-integrable coupling,

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=K(u)\\ v_t=K'(u)[v]\\ w_t=K'(u)[w]. \end{array}\right. } \end{aligned}$$
(63)

where \(K'(u)\) denotes the Gateaux derivative, is there any Hamiltonian structure for this specific bi-integrable coupling?