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A Kind of Generalized Integrable Couplings and Their Bi-Hamiltonian Structure

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Abstract

We introduce a Lie algebra \(\widetilde {g}\) which can be used to construct integrable couplings of some spectral problems. As two examples, the non-semisimple Lie algebra \(\widetilde {g}\) is applied to enlarge the spectral problems of an extended Ablowitz-Kaup-Newell-Segur (AKNS) spectral problem and a generalized D-Kaup-Newell (D-KN) spectral problem. It follows that we obtain two generalized integrable couplings by solving these expanded zero-curvature equations. Finally, we find that the integrable hierarchies that we obtain have bi-Hamiltonian structures of combinatorial form, thereby showing their Liouville integrability.

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Funding

This work was supported by the National Natural Science Foundation of China (grant No.11971475).

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Authors and Affiliations

  1. School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, People’s Republic of China

    Haifeng Wang & Yufeng Zhang

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  1. Haifeng Wang
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  2. Yufeng Zhang
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Correspondence to Yufeng Zhang.

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Appendices

Appendix: A

$$ \begin{array}{ll} &{}{\Psi}_{11}{=}\frac{1}{2}\partial{-}p{-}q\partial^{{-}1}r{-}2\eta q\partial^{{-}1}pu_{2}+\eta q\partial^{{-}1}u_{2}\partial{-}2\eta^{2}q\partial^{{-}1}(u_{1}r+u_{2}q)u_{2}+[{-}\eta\partial q+2\eta pq{-}\varepsilon u_{1}{-}2\eta^{2}q(u_{1}r+u_{2}q)]\partial^{{-}1}u_{2},\\ &{}{\Psi}_{12}{=-}q\partial^{-1}q-2\eta q\partial^{-1}pu_{1}-\eta q\partial^{-1}u_{1}\partial-2\eta^{2}q\partial^{-1}(u_{1}r+u_{2}q)u_{1}+[-\eta\partial q+2\eta pq-\varepsilon u_{1}-2\eta^{2}q(u_{1}r+u_{2}q)]\partial^{-1}u_{1},\\ &{}{\Psi}_{13}{=-}\varepsilon q\partial^{-1}u_{2}-2\eta q\partial^{-1}pr+\eta q\partial^{-1}r\partial-2\eta^{2}q\partial^{-1}(u_{1}r+u_{2}q)r+[-\eta\partial q+2\eta pq-\varepsilon u_{1}-2\eta^{2}q(u_{1}r+u_{2}q)]\partial^{-1}r,\\ &{}{\Psi}_{14}{=-}\varepsilon q\partial^{-1}u_{1}-2\eta q\partial^{-1}pq-\eta q\partial^{-1}q\partial-2\eta^{2}q\partial^{-1}(u_{1}r+u_{2}q)q+[-\eta\partial q+2\eta pq-\varepsilon u_{1}-2\eta^{2}q(u_{1}r+u_{2}q)]\partial^{-1}q, \end{array} $$
$$ \begin{array}{ll} &{}{\Psi}_{21}{=}r\partial^{-1}r+2\eta r\partial^{-1}pu_{2}-\eta r\partial^{-1}u_{2}\partial+2\eta^{2}r\partial^{-1}(u_{1}r+u_{2}q)u_{2}+[-\eta\partial r-2\eta pr+\varepsilon u_{2}+2\eta^{2}r(u_{1}r+u_{2}q)]\partial^{-1}u_{2},\\ &{}{\Psi}_{22}{=-}\frac{1}{2}\partial-p+r\partial^{-1}q+2\eta r\partial^{-1}pu_{1}+\eta r\partial^{-1}u_{1}\partial{+}2\eta^{2}r\partial^{-1}(u_{1}r{+}u_{2}q)u_{1}{+}[{-}\eta\partial r{-}2\eta pr{+}\varepsilon u_{2}{+}2\eta^{2}r(u_{1}r{+}u_{2}q)]\partial^{-1}u_{1},\\ &{}{\Psi}_{23}{=}\varepsilon r\partial^{-1}u_{2}+2\eta r\partial^{-1}pr-\eta r\partial^{-1}r\partial+2\eta^{2}r\partial^{-1}(u_{1}r+u_{2}q)r+[-\eta\partial r-2\eta pr+\varepsilon u_{2}+2\eta^{2}r(u_{1}r+u_{2}q)]\partial^{-1}r,\\ &{}{\Psi}_{24}{=}\varepsilon r\partial^{-1}u_{1}+2\eta r\partial^{-1}pq+\eta r\partial^{-1}q\partial+2\eta^{2}r\partial^{-1}(u_{1}r+u_{2}q)q+[-\eta\partial r-2\eta pr+\varepsilon u_{2}+2\eta^{2}r(u_{1}r+u_{2}q)]\partial^{-1}q, \end{array} $$
$$ \begin{array}{ll} &{}{\Psi}_{31}{=}{-}u_{1}\partial^{{-}1}r{-}2\eta u_{1}\partial^{{-}1}pu_{2}+\eta u_{1}\partial^{{-}1}u_{2}\partial{-}2\eta^{2}u_{1}\partial^{{-}1}(u_{1}r+u_{2}q)u_{2}+[{-}\eta\partial u_{1}+2\eta pu_{1}{-}q{-}2\eta^{2}u_{1}(u_{1}r+u_{2}q)]\partial^{{-}1}u_{2},\\ &{}{\Psi}_{32}{=}{-}u_{1}\partial^{{-}1}q{-}2\eta u_{1}\partial^{{-}1}pu_{1}{-}\eta u_{1}\partial^{{-}1}u_{1}\partial{-}2\eta^{2}u_{1}\partial^{{-}1}(u_{1}r+u_{2}q)u_{1}+[{-}\eta\partial u_{1}+2\eta pu_{1}{-}q{-}2\eta^{2}u_{1}(u_{1}r+u_{2}q)]\partial^{{-}1}u_{1},\\ &{}{\Psi}_{33}{=}\frac{1}{2}\partial{-}p{-}\varepsilon u_{1}\partial^{{-}1}u_{2}{-}2\eta u_{1}\partial^{{-}1}pr{+}\eta u_{1}\partial^{{-}1}r\partial{{-}}2\eta^{2}u_{1}\partial^{{-}1}(u_{1}r+u_{2}q)r+[{-}\eta\partial u_{1}{+}2\eta pu_{1}{-}q{-}2\eta^{2}u_{1}(u_{1}r+u_{2}q)]\partial^{{-}1}r,\\ &{}{\Psi}_{34}{=}{-}\varepsilon u_{1}\partial^{{-}1}u_{1}{-}2\eta u_{1}\partial^{{-}1}pq{-}\eta u_{1}\partial^{{-}1}q\partial{-}2\eta^{2}u_{1}\partial^{{-}1}(u_{1}r+u_{2}q)q+[{-}\eta\partial u_{1}+2\eta pu_{1}{-}q{-}2\eta^{2}u_{1}(u_{1}r+u_{2}q)]\partial^{{-}1}q, \end{array} $$
$$ \begin{array}{ll} &{}{\Psi}_{41}{=}u_{2}\partial^{-1}r{+}2\eta r\partial^{-1}pu_{2}-\eta u_{2}\partial^{-1}u_{2}\partial{+}2\eta^{2}u_{2}\partial^{-1}(u_{1}r{+}u_{2}q)q{+}[-\eta\partial u_{2}-2\eta pu_{2}{+}r{+}2\eta^{2}u_{2}(u_{1}r{+}u_{2}q)]\partial^{-1}u_{2},\\ &{}{\Psi}_{42}{=}u_{2}\partial^{-1}q{+}2\eta r\partial^{-1}pu_{1}{+}\eta u_{2}\partial^{-1}u_{1}\partial{+}2\eta^{2}u_{2}\partial^{-1}(u_{1}r{+}u_{2}q)u_{1}{+}[-\eta\partial u_{2}-2\eta pu_{2}{+}r{+}2\eta^{2}u_{2}(u_{1}r{+}u_{2}q)]\partial^{-1}u_{1},\\ &{}{\Psi}_{43}{=}\varepsilon u_{2}\partial^{-1}u_{2}{+}2\eta r\partial^{-1}pr-\eta u_{2}\partial^{-1}r\partial{+}2\eta^{2}u_{2}\partial^{-1}(u_{1}r{+}u_{2}q)r{+}[-\eta\partial u_{2}-2\eta pu_{2}{+}r{+}2\eta^{2}u_{2}(u_{1}r{+}u_{2}q)]\partial^{-1}r,\\ &{}{\Psi}_{44}{=-}\frac{1}{2}\partial{-}p{+}\varepsilon u_{2}\partial^{-1}u_{1}{{+}}2\eta r\partial^{-1}pq{{+}}\eta u_{2}\partial^{-1}q\partial{+}2\eta^{2}u_{2}\partial^{-1}(u_{1}r{+}u_{2}q)q{+}[-\eta\partial u_{2}-2\eta pu_{2}{+}r{+}2\eta^{2}u_{2}(u_{1}r{+}u_{2}q)]\partial^{-1}q. \end{array} $$

We show the specific calculation of the first row of operator Ψ as follows:

$$ \begin{array}{ll} &2b_{m+1}-4\eta qe_{m+1}\\ =&b_{m,x}+2a_{m}q+2\varepsilon e_{m}u_{1}-2b_{m}p-4\eta q\partial^{-1}(u_{1}c_{m+1}-u_{2}b_{m+1}+qg_{m+1}-rf_{m+1})\\ =&b_{m,x}+2q\partial^{-1}(qc_{m}-rb_{m}+\varepsilon u_{1}g_{m}-\varepsilon u_{2}f_{m})+2\varepsilon u_{1}\partial^{-1}(u_{1}c_{m}-u_{2}b_{m}+qg_{m}-rf_{m})-2b_{m}p\\ &-4\eta q\partial^{-1}u_{1}(-\frac{1}{2}c_{m,x}+a_{m}r+\varepsilon e_{m}u_{2}-c_{m}p)+4\eta q\partial^{-1}u_{2}(\frac{1}{2}b_{m,x}+a_{m}q+\varepsilon e_{m}u_{1}-b_{m}p)\\ &-4\eta q\partial^{-1}q(-\frac{1}{2}g_{m,x}+a_{m}u_{2}+e_{m}r-g_{m}p)+4\eta q\partial^{-1}r(\frac{1}{2}f_{m,x}+a_{m}u_{1}+e_{m}q-f_{m}p)\\ =&b_{m,x}-2b_{m}p+2q\partial^{-1}qc_{m}-2q\partial^{-1}rb_{m}+2\varepsilon q\partial^{-1}u_{1}g_{m}-2\varepsilon q\partial^{-1}u_{2}f_{m} +2\varepsilon u_{1}\partial^{-1}u_{1}c_{m}-2\varepsilon u_{1}\partial^{-1}u_{2}b_{m}+2\varepsilon u_{1}\partial^{-1}qg_{m}\\ &-2\varepsilon u_{1}\partial^{-1}rf_{m}-4\eta q\partial^{-1}p(u_{2}b_{m}-u_{1}c_{m}+rf_{m}-qg_{m})+2\eta q\partial^{-1}(u_{2}b_{m,x}+u_{1}c_{m,x}+rf_{m,x}+qg_{m,x})\\ =&\frac{1}{2}\partial(2b_{m}-4\eta qe_{m})-p(2b_{m}-4\eta qe_{m})-q\partial^{-1}q(-2c_{m}+4\eta re_{m})-q\partial^{-1}r(2b_{m}-4\eta qe_{m})- \varepsilon q\partial^{-1}u_{1}(-2g_{m}+4\eta u_{2}e_{m})\\ &-\varepsilon q\partial^{-1}u_{2}(2f_{m}-4\eta u_{1}e_{m})\\ =&-\varepsilon u_{1}\partial^{-1}u_{1}(-2c_{m}+4\eta re_{m})-\varepsilon u_{1}\partial^{-1}u_{2}(2b_{m}-4\eta qe_{m})-\varepsilon u_{1}\partial^{-1}q(-2g_{m}+4\eta u_{2}e_{m})-\varepsilon u_{1}\partial^{-1}r(2f_{m}-4\eta u_{1}e_{m})\\ &-2\eta q\partial^{-1}pu_{2}(2b_{m}-4\eta qe_{m})-2\eta q\partial^{-1}pu_{1}(-2c_{m}+4\eta re_{m})-2\eta q\partial^{-1}pq(-2g_{m}+4\eta u_{2}e_{m})\\ &-2\eta q\partial^{-1}pr(2f_{m}-4\eta u_{1}e_{m})-\eta q\partial^{-1}u_{1}\partial(-2c_{m}+4\eta re_{m})+\eta q\partial^{-1}u_{2}\partial(2b_{m}-4\eta qe_{m})-\eta q\partial^{-1}q\partial(-2g_{m}+4\eta u_{2}e_{m})\\ &+\eta q\partial^{-1}r\partial(2f_{m}-4\eta u_{1}e_{m})+2\eta\partial qe_{m}-4\eta pqe_{m}+4\eta^{2}q\partial^{-1}(u_{1}\partial r+u_{2}\partial q+q\partial u_{2}+r\partial u_{1})e_{m}\\ {=}&\{\frac{1}{2}\partial{-}p{-}q\partial^{-1}r{-}2\eta q\partial^{-1}pu_{2}{+}\eta q\partial^{-1}u_{2}\partial-2\eta^{2}q\partial^{-1}(u_{1}r{+}u_{2}q)u_{2}+[-\eta\partial q+2\eta pq-\varepsilon u_{1}-2\eta^{2}q(u_{1}r+u_{2}q)]\partial^{-1}u_{2}\}(2b_{m}-4\eta qe_{m})\\ &+\{-q\partial^{-1}q-2\eta q\partial^{-1}pu_{1}-\eta q\partial^{-1}u_{1}\partial-2\eta^{2}q\partial^{-1}(u_{1}r+u_{2}q)u_{1}+[-\eta\partial q+2\eta pq-\varepsilon u_{1}-2\eta^{2}q(u_{1}r+u_{2}q)]\partial^{-1}u_{1}\}(-2c_{m}+4\eta re_{m})\\ &+\{-\varepsilon q\partial^{-1}u_{2}-2\eta q\partial^{-1}pr+\eta q\partial^{-1}r\partial-2\eta^{2}q\partial^{-1}(u_{1}r+u_{2}q)r+[-\eta\partial q+2\eta pq-\varepsilon u_{1}-2\eta^{2}q(u_{1}r+u_{2}q)]\partial^{-1}r\}(2f_{m}-4\eta u_{1}e_{m})\\ &+\{-\varepsilon q\partial^{-1}u_{1}-2\eta q\partial^{-1}pq-\eta q\partial^{-1}q\partial-2\eta^{2}q\partial^{-1}(u_{1}r+u_{2}q)q+[-\eta\partial q+2\eta pq-\varepsilon u_{1}-2\eta^{2}q(u_{1}r+u_{2}q)]\partial^{-1}q\}(-2g_{m}+4\eta u_{2}e_{m})\\ =&{\Psi}_{11}(2b_{m}-4\eta qe_{m})+{\Psi}_{12}(-2c_{m}+4\eta re_{m})+{\Psi}_{13}(2f_{m}-4\eta u_{1}e_{m})+{\Psi}_{14}(-2g_{m}+4\eta u_{2}e_{m}). \end{array} $$

Similarly, we can also get the Ψij(2 ≤ i ≤ 4, 1 ≤ j ≤ 4).

Appendix: B

$$ \begin{array}{ll} &{\Phi}_{11}={\Phi}_{12}={\Phi}_{13}={\Phi}_{18}={\Phi}_{19}={\Phi}_{110}=0,\ {\Phi}_{14}=\frac{r}{2},\ {\Phi}_{15}=\frac{q}{2},\ {\Phi}_{16}=\frac{\varepsilon u_{7}}{2},\ {\Phi}_{17}=\frac{\varepsilon u_{6}}{2},\\ &{\Phi}_{21}={\Phi}_{28}=0,\ {\Phi}_{22}=-q\partial^{-1}v-\varepsilon u_{6}\partial^{-1}u_{4},\ {\Phi}_{23}=-q\partial^{-1}s-\varepsilon u_{6}\partial^{-1}u_{3},\ {\Phi}_{24}=1-q\partial^{-1}r-\varepsilon u_{6}\partial^{-1}u_{7}, \\ &{\Phi}_{25}=-q\partial^{-1}q-\varepsilon u_{6}\partial^{-1}u_{6}, \ {\Phi}_{26}=-q\partial^{-1}(\varepsilon u_{3})-\varepsilon u_{6}\partial^{-1}r,\ {\Phi}_{27}=-q\partial^{-1}(\varepsilon u_{6})-\varepsilon u_{6}\partial^{-1}q, \\ &{\Phi}_{29}=-q\partial^{-1}(\varepsilon u_{4})-\varepsilon u_{6}\partial^{-1}v,\ {\Phi}_{210}=-q\partial^{-1}(\varepsilon u_{3})-\varepsilon u_{6}\partial^{-1}s,\\ &{\Phi}_{31}={\Phi}_{38}=0,\ {\Phi}_{32}=r\partial^{-1}v+\varepsilon u_{7}\partial^{-1}u_{7},\ {\Phi}_{33}=r\partial^{-1}s+\varepsilon u_{7}\partial^{-1}u_{3},\ {\Phi}_{34}=r\partial^{-1}r+\varepsilon u_{7}\partial^{-1}u_{7}, \\ &{\Phi}_{35}=1+r\partial^{-1}q+\varepsilon u_{7}\partial^{-1}u_{6}, \ {\Phi}_{36}=r\partial^{-1}(\varepsilon u_{7})+\varepsilon u_{7}\partial^{-1}r,\ {\Phi}_{37}=r\partial^{-1}(\varepsilon u_{6})+\varepsilon u_{7}\partial^{-1}q, \\ &{\Phi}_{39}=r\partial^{-1}(\varepsilon u_{4})+\varepsilon u_{7}\partial^{-1}v,\ {\Phi}_{310}=r\partial^{-1}(\varepsilon u_{3})+\varepsilon u_{7}\partial^{-1}s,\\ &{\Phi}_{41}={\Phi}_{48}=0,\ {\Phi}_{42}=\frac{\partial}{2}+p-s\partial^{-1}v-\varepsilon u_{3}\partial^{-1}u_{4},\ {\Phi}_{43}=-s\partial^{-1}s-\varepsilon u_{3}\partial^{-1}u_{3},\ {\Phi}_{44}=-s\partial^{-1}r-\varepsilon u_{3}\partial^{-1}u_{7}, \\ &{\Phi}_{45}=-s\partial^{-1}q-\varepsilon u_{3}\partial^{-1}u_{6}, \ {\Phi}_{46}=-s\partial^{-1}(\varepsilon u_{7})-\varepsilon u_{3}\partial^{-1}r,\ {\Phi}_{47}=-s\partial^{-1}(\varepsilon u_{6})-\varepsilon u_{3}\partial^{-1}q, \\ &{\Phi}_{49}=-\varepsilon u_{5}-s\partial^{-1}(\varepsilon u_{4})-\varepsilon u_{3}\partial^{-1}v,\ {\Phi}_{410}=-s\partial^{-1}(\varepsilon u_{3})-\varepsilon u_{3}\partial^{-1}s,\\ &{\Phi}_{51}={\Phi}_{58}=0,\ {\Phi}_{52}=v\partial^{-1}v+\varepsilon u_{4}\partial^{-1}u_{4},\ {\Phi}_{53}=-\frac{\partial}{2}+p+v\partial^{-1}s+\varepsilon u_{4}\partial^{-1}u_{3},\ {\Phi}_{54}=v\partial^{-1}r+\varepsilon u_{4}\partial^{-1}u_{7}, \\ &{\Phi}_{55}=v\partial^{-1}q+\varepsilon u_{4}\partial^{-1}u_{6}, \ {\Phi}_{56}=v\partial^{-1}(\varepsilon u_{7})+\varepsilon u_{4}\partial^{-1}r,\ {\Phi}_{57}=v\partial^{-1}(\varepsilon u_{6})+\varepsilon u_{4}\partial^{-1}q, \\ &{\Phi}_{59}=v\partial^{-1}(\varepsilon u_{4})+\varepsilon u_{4}\partial^{-1}v,\ {\Phi}_{510}=-\varepsilon u_{5}+v\partial^{-1}(\varepsilon u_{3})+\varepsilon u_{4}\partial^{-1}s,\\ &{\Phi}_{61}={\Phi}_{68}=0,\ {\Phi}_{62}=-u_{5}-s\partial^{-1}u_{4}-u_{3}\partial^{-1}u_{4},\ {\Phi}_{63}=-u_{3}\partial^{-1}s-s\partial^{-1}u_{3},\ {\Phi}_{64}=-u_{3}\partial^{-1}r- s\partial^{-1}u_{7}, \\ &{\Phi}_{65}=-u_{3}\partial^{-1}q-s\partial^{-1}u_{6}, \ {\Phi}_{66}=-u_{3}\partial^{-1}(\varepsilon u_{7})-s\partial^{-1}r,\ {\Phi}_{67}=-u_{3}\partial^{-1}(\varepsilon u_{6})-s\partial^{-1}q, \\ &{\Phi}_{69}=\frac{\partial}{2}+p-u_{3}\partial^{-1}(\varepsilon u_{4})-s\partial^{-1}v,\ {\Phi}_{610}=-u_{5}\partial^{-1}(\varepsilon u_{3})-s\partial^{-1}s,\\ &{\Phi}_{71}={\Phi}_{78}=0,\ {\Phi}_{72}=u_{4}\partial^{-1}v+v\partial^{-1}u_{4},\ {\Phi}_{73}=-u_{5}+u_{4}\partial^{-1}s+v\partial^{-1}u_{3},\ {\Phi}_{74}=u_{4}\partial^{-1}r+v\partial^{-1}u_{7}, \\ &{\Phi}_{75}=u_{4}\partial^{-1}q+v\partial^{-1}u_{6}, \ {\Phi}_{76}=u_{4}\partial^{-1}(\varepsilon u_{7})+v\partial^{-1}r,\ {\Phi}_{77}=u_{4}\partial^{-1}(\varepsilon u_{6})+v\partial^{-1}q, \\ &{\Phi}_{79}=u_{4}\partial^{-1}(\varepsilon u_{4})+v\partial^{-1}v,\ {\Phi}_{710}=-\frac{\partial}{2}+p+u_{4}\partial^{-1}(\varepsilon u_{3})+v\partial^{-1}s,\\ &{\Phi}_{81}={\Phi}_{82}={\Phi}_{83}={\Phi}_{88}={\Phi}_{89}={\Phi}_{810}=0,\ {\Phi}_{84}=\frac{u_{7}}{2},\ {\Phi}_{85}=\frac{u_{6}}{2},\ {\Phi}_{86}=-\frac{r}{2},\ {\Phi}_{87}=-\frac{\varepsilon q}{2},\\ &{\Phi}_{91}={\Phi}_{98}=0,\ {\Phi}_{92}=-u_{6}\partial^{-1}v-q\partial^{-1}u_{4},\ {\Phi}_{93}=-u_{6}\partial^{-1}s-q\partial^{-1}u_{3},\ {\Phi}_{94}=-u_{6}\partial^{-1}r-q\partial^{-1}u_{7}, \\ &{\Phi}_{95}=-u_{6}\partial^{-1}q-q\partial^{-1}u_{6}, \ {\Phi}_{96}=1-u_{6}\partial^{-1}(\varepsilon u_{7})-q\partial^{-1}r,\ {\Phi}_{97}=-u_{6}\partial^{-1}(\varepsilon u_{6})-q\partial^{-1}q, \\ &{\Phi}_{99}=-u_{6}\partial^{-1}(\varepsilon u_{4})-q\partial^{-1}v,\ {\Phi}_{910}=-u_{6}\partial^{-1}(\varepsilon u_{3})-q\partial^{-1}s,\\ &{\Phi}_{101}={\Phi}_{108}=0,\ {\Phi}_{102}=u_{7}\partial^{-1}v+r\partial^{-1}u_{4},\ {\Phi}_{103}=u_{7}\partial^{-1}s+r\partial^{-1}u_{3},\ {\Phi}_{104}=u_{7}\partial^{-1}r+r\partial^{-1}u_{7}, \\ &{\Phi}_{105}=u_{7}\partial^{-1}q+r\partial^{-1}u_{6}, \ {\Phi}_{106}=u_{7}\partial^{-1}(\varepsilon u_{7})+r\partial^{-1}r,\ {\Phi}_{107}=1+u_{7}\partial^{-1}(\varepsilon u_{6})+r\partial^{-1}q, \\ &{\Phi}_{109}=u_{7}\partial^{-1}(\varepsilon u_{4})+r\partial^{-1}v,\ {\Phi}_{1010}=u_{7}\partial^{-1}(\varepsilon u_{3})+r\partial^{-1}s. \end{array} $$

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Wang, H., Zhang, Y. A Kind of Generalized Integrable Couplings and Their Bi-Hamiltonian Structure. Int J Theor Phys 60, 1797–1812 (2021). https://doi.org/10.1007/s10773-021-04799-9

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