On periodic Ambrosetti-Prodi-type problems

Feliz Minhós; Nuno Oliveira; Feliz Minhós; Nuno Oliveira

doi:10.3934/math.2023654

AIMS Mathematics

2023, Volume 8, Issue 6: 12986-12999. doi: 10.3934/math.2023654

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Research article Special Issues

On periodic Ambrosetti-Prodi-type problems

Feliz Minhós ^{1
,
,},
Nuno Oliveira ²

1.
Departament of Mathematics, School of Sciences and Technology, Research Centre in Mathematics and Applications (CIMA), Institute for Research and Advanced Training (IIFA), University of Évora. Rua Romão Ramalho, 59, 7000-671 Évora, Portugal
2.
Research Centre in Mathematics and Applications (CIMA), Institute for Research and Advanced Training (IIFA), University of Évora. Rua Romão Ramalho, 59, 7000-671 Évora, Portugal

Received: 23 December 2022 Revised: 09 March 2023 Accepted: 17 March 2023 Published: 31 March 2023
MSC : 34B15, 34B18, 34L30

This work presents a discussion of Ambrosetti-Prodi-type second-order periodic problems. In short, the existence, non-existence, and multiplicity of solutions will be discussed on the parameter $\lambda$ . The arguments rely on a Nagumo condition, to guarantee an apriori bound on the first derivative, lower and upper-solutions method, and the Leray-Schauder's topological degree theory. There are two types of new results based on the parameter's variation: An existence and non-existence theorem and a multiplicity theorem, proving the existence of a bifurcation point. An application to a damped and forced pendulum is studied, suggesting a method to estimate the critical values of the parameter.

Keywords:

Citation: Feliz Minhós, Nuno Oliveira. On periodic Ambrosetti-Prodi-type problems[J]. AIMS Mathematics, 2023, 8(6): 12986-12999. doi: 10.3934/math.2023654

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Abstract

1. Introduction

Nonlinear differential equations are important mathematical tools for describing natural phenomena. The integrability and exact solutions of nonlinear evolution equations have always been of great concern to scientists ^[1,2]. The exact solution of nonlinear models plays an important role in describing some complex nonlinear phenomena. Some researchers have worked on solving the exact analytic solutions of the nonlinear extended models. They proposed many methods and also obtained more new types of exact solutions ^[3,4,5,6,7].

The study of integrable systems is an important research topic in disciplines such as physics and mathematics. The trace identity proposed by Tu is a simple and powerful tool for generating integrable hierarchies of soliton equations and their corresponding Hamiltonian structures ^[8]. By using the trace identity, based on Lie algebras, some isospectral integrable hierarchies and the corresponding Hamiltonian structures were constructed ^{[9,10,11,12,13]}. Some methods had been developed in deriving (2+1)-dimensional integrable systems, such as the TAH scheme. The TAH scheme is an effective method for generating (2+1)-dimensional soliton hierarchies. Based on this method, some (2+1)-dimensional integrable hierarchies and the corresponding Hamiltonian structures were obtained ^{[14,15,16,17,18,19]}.

In order to obtain more integrable sysytems, Zhang et al. proposed an approach for generating nonisospectral integrable hierarchies under the assumption $\lambda_t = \sum_{i\geq 0}k_i(t)\lambda^{-i}$ ^[20,21]. By using this method, ^[22,23] constructed some multi-component integrable hierarchies associated with multi-component non-semisimple Lie algebras. Moreover, based on Lie superalgebras, ^[24,25] investigated some nonisospectral super integrable hierarchies and the corresponding super Hamiltonian structures.

Matrix spectral problems and zero curvature equations play an important role in exploring the mathematical properties of associated soliton equations ^[26,27]. The semi-direct sum decomposition of Lie algebras provide a helpful way to construct the integrable couplings of soliton systems ^[28,29,30].

This paper is arranged as follows. In Section 2, based on Lie algebra $\mathfrak{sp}(4)$ , we construct the generalized Lie algebra $G\mathfrak{sp}(4)$ , and obtain the nonisospectral integrable hierarchies and their Hamiltonian structures. In Section 3, based on the semi-direct sum decomposition of Lie algebras, the nonisospectral integrable coupling hierarchies and the corresponding Hamiltonian structures are obtained. In Sections 4 and 5, we obtain the nonisospectral integrable hierarchies and their coupling systems associated with the generalized Lie algebra $G\mathfrak{so}(5)$ . In Section 6, by using $\mathfrak{sp}(4)\cong \mathfrak{so}(3, 2)$ , we obtain that there are the same integrable hierarchies of these two generalized Lie algebras, and there are also the same integrable couplings.

2. The nonisospectral integrable hierarchy associated with the generalized Lie algebra $G\mathfrak{sp}(4)$

The compact real form $\mathfrak{sp}(4)$ of complex symplectic Lie algebra $\mathfrak{sp}(4, \mathbb{R})$ is defined as ^[31,32]

$\mathfrak{sp}(4) = \{x\in \mathfrak{gl}(4, \mathbb{R})|Hx+x^\mathrm{T}H = 0\},$

where $H = \left(\begin{array}{cc} 0 & I_2 \\ -I_2 & 0 \\ \end{array} \right)$ , $I_2$ is the $2\times 2$ identity matrix, and $x^\mathrm{T}$ represents the transposition of $x$ . We introduce the generalized Lie algebra $\mathfrak{sp}(4)$ , that admits a basis set as follows:

$\begin{equation} \begin{split} E_1 = e_{11}-e_{33}, \ \ E_2 = e_{22}-e_{44}, \ \ E_3 = e_{12}-e_{43}, \ \ E_4 = \varepsilon e_{21}-\varepsilon e_{34}, \\ E_5 = \varepsilon e_{14}+\varepsilon e_{23}, \ \ E_6 = e_{32}+e_{41}, \ \ E_7 = \varepsilon e_{13}, \ \ E_8 = e_{31}, \ \ E_9 = \varepsilon e_{24}, \ \ E_{10} = e_{42}, \end{split} \end{equation}$

(2.1)

where $\varepsilon\in \mathbb{R}$ , and $e_{ij}$ is a $4\times 4$ matrix with 1 in the $(i, j)$ -th position and zero elsewhere, which satisfy the commutative relations

$\begin{eqnarray*} & & [E_1, E_2] = 0, \ [E_1, E_3] = E_3, \ [E_1, E_4] = -E_4, \ [E_1, E_5] = E_5, \ [E_1, E_6] = -E_6, \\ & & [E_1, E_7] = 2E_7, \ [E_1, E_8] = -2E_8, \ [E_1, E_9] = [E_1, E_{10}] = 0, \ [E_2, E_3] = -E_3, \\ & & [E_2, E_4] = E_4, \ [E_2, E_5] = E_5, \ [E_2, E_6] = -E_6, \ [E_2, E_7] = [E_2, E_8] = 0, \\ & & [E_2, E_9] = 2E_9, \ [E_2, E_{10}] = -2E_{10}, \ [E_3, E_4] = \varepsilon(E_1-E_2), \ [E_3, E_5] = 2E_7, \\ & & [E_3, E_6] = -2E_{10}, \ [E_3, E_7] = 0, \ [E_3, E_8] = -E_6, \ [E_3, E_9] = E_5, \ [E_3, E_{10}] = 0, \\ & & [E_4, E_5] = 2E_9, \ [E_4, E_6] = -2\varepsilon E_8, \ [E_4, E_7] = E_5, \ [E_4, E_8] = [E_4, E_9] = 0, \\ & & [E_4, E_{10}] = -\varepsilon E_6, \ [E_5, E_6] = \varepsilon(E_1\!+\!E_2), \ [E_5, E_7] = 0, \ [E_5, E_8] = E_4, \ [E_5, E_9] = 0, \\ & & [E_5, E_{10}] = \varepsilon E_3, \ [E_6, E_7] = -\varepsilon E_3, \ [E_6, E_8] = [E_6, E_{10}] = 0, \ [E_6, E_9] = -E_4, \\ & & [E_7, E_8] = \varepsilon E_1, \ [E_7, E_9] = [E_7, E_{10}] = [E_8, E_9] = [E_8, E_{10}] = 0, \ [E_9, E_{10}] = \varepsilon E_2. \end{eqnarray*}$

We can construct different generalized Lie algebras $G\mathfrak{sp}(4)$ by adding the real number $\varepsilon$ to different elements of the Lie algebra $\mathfrak{sp}(4)$ . Here, we will only discuss one of these cases.

Consider the linear nonisospectral problem

$\begin{eqnarray*} \left\{ \begin{array}{ll} \varphi_x = U_1\varphi, \\ \varphi_t = V_1\varphi, \\ \lambda_t = \sum\nolimits_{i\geq 0}k_i(t)\lambda^{-i}, \\ \end{array} \right. \end{eqnarray*}$

where

$\begin{eqnarray} & & U_1 = \left( \begin{array}{cccc} \lambda & 0 & \varepsilon u_3 & \varepsilon u_1 \\ 0 & \lambda & \varepsilon u_1 & \varepsilon u_5 \\ u_4 & u_2 & -\lambda & 0 \\ u_2 & u_6 & 0 & -\lambda \\ \end{array} \right), \\ & & V_1 = \left( \begin{array}{cccc} a & c & \varepsilon g & \varepsilon e \\ \varepsilon d & b & \varepsilon e & \varepsilon p \\ h & f & -a & -\varepsilon d \\ f & q & -c & -b \\ \end{array} \right) = \sum\limits_{i\geq 0}\left( \begin{array}{cccc} a_i & c_i & \varepsilon g_i & \varepsilon e_i \\ \varepsilon d_i & b_i & \varepsilon e_i & \varepsilon p_i \\ h_i & f_i & -a_i & -\varepsilon d_i \\ f_i & q_i & -c_i & -b_i \\ \end{array} \right)\lambda^{-i}. \end{eqnarray}$

(2.2)

By solving stationary the zero curvature representation

$\begin{eqnarray} V_{1x} = \frac{\partial U_1}{\partial \lambda}\lambda_t+[U_1, V_1], \end{eqnarray}$

(2.3)

we can obtain

$\begin{eqnarray} \left\{ \begin{array}{llllllllll} a_{ix} = \varepsilon u_1f_i-\varepsilon u_2e_i+\varepsilon u_3h_i-\varepsilon u_4g_i+k_i(t), \\ b_{ix} = \varepsilon u_1f_i-\varepsilon u_2e_i+\varepsilon u_5q_i-\varepsilon u_6p_i+k_i(t), \\ c_{ix} = \varepsilon u_1q_i-\varepsilon u_2g_i+\varepsilon u_3f_i-\varepsilon u_6e_i, \\ d_{ix} = u_1h_i-u_2p_i-u_4e_i+u_5f_i, \\ e_{ix} = 2\lambda e_i-u_1a_i-u_1b_i-\varepsilon u_3d_i-u_5c_i, \\ f_{ix} = -2\lambda f_i+u_2a_i+u_2b_i+u_4c_i+\varepsilon u_6d_i, \\ g_{ix} = 2\lambda g_i-2u_1c_i-2u_3a_i, \\ h_{ix} = -2\lambda h_i+2\varepsilon u_2d_i+2u_4a_i, \\ p_{ix} = 2\lambda p_i-2\varepsilon u_1d_i-2u_5b_i, \\ q_{ix} = -2\lambda q_i+2u_2c_i+2u_6b_i. \end{array} \right. \end{eqnarray}$

(2.4)

By taking initial values

$\begin{eqnarray*} a_0 = \alpha(t), \ b_0 = \beta(t), \ c_0 = d_0 = e_0 = f_0 = g_0 = h_0 = p_0 = q_0 = k_0(t) = 0, \end{eqnarray*}$

one has

$\begin{eqnarray*} & & a_1 = b_1 = k_1(t)x, \ c_1 = \frac{\varepsilon}{2}\partial^{-1}(u_1u_6+u_2u_3)(\beta-\alpha), \ g_1 = u_3\alpha, \ p_1 = u_5\beta, \ q_1 = u_6\beta, \\ & & d_1 = \frac{1}{2}\partial^{-1}(u_1u_4+u_2u_5)(\alpha-\beta), \ e_1 = \frac{1}{2}u_1(\alpha+\beta), \ f_1 = \frac{1}{2}u_2(\alpha+\beta), \ h_1 = u_4\alpha, \\ & & e_2 = \frac{1}{4}u_{1x}(\alpha+\beta)+\frac{\varepsilon}{4}\left[u_{3}\partial^{-1}(u_1u_4+u_2u_5)-u_{5}\partial^{-1}(u_1u_6+u_2u_3)\right](\alpha\!-\!\beta)+u_1k_1(t)x, \\ & & f_2\! = \!-\frac{1}{4}u_{2x}(\alpha+\beta)\!+\!\frac{\varepsilon}{4}\left[u_{4}\partial^{-1}(u_1u_6+u_2u_3)-u_{6}\partial^{-1}(u_1u_4+u_2u_5)\right](\beta\!-\!\alpha)+u_2k_1(t)x, \\ & & g_2 = \frac{1}{2}u_{3x}\alpha+\frac{\varepsilon}{2}u_{1}\partial^{-1}(u_1u_6+u_2u_3)(\beta-\alpha)+u_3k_1(t)x, \\ & & h_2 = -\frac{1}{2}u_{4x}\alpha+\frac{\varepsilon}{2}u_{2}\partial^{-1}(u_1u_4+u_2u_5)(\beta-\alpha)+u_4k_1(t)x, \\ & & p_2 = \frac{1}{2}u_{5x}\beta-\frac{\varepsilon}{2}u_{1}\partial^{-1}(u_1u_4+u_2u_5)(\beta-\alpha)+u_5k_1(t)x, \\ & & q_2 = -\frac{1}{2}u_{6x}\beta+\frac{\varepsilon}{2}u_{2}\partial^{-1}(u_1u_6+u_2u_3)(\beta-\alpha)+u_6k_1(t)x, \\ & & \cdots\cdots \end{eqnarray*}$

where $\alpha(t)$ is an integral constant. Noting that

$\begin{eqnarray*} & & V^{(n)}_{1+}\! = \!\sum^{n}_{i = 0}(a_i, b_i, c_i, d_i, e_i, f_i, g_i, h_i, p_i, q_i)^{\mathrm{T}}, \ \ V^{(n)}_{1-}\! = \!\sum^{\infty}_{i = n+1}(a_i, b_i, c_i, d_i, e_i, f_i, g_i, h_i, p_i, q_i)^{\mathrm{T}}, \\ & & \lambda^{(n)}_{+, x} = \sum^{n}_{i = 0}k_i(t)\lambda^{n-i}, \ \ \lambda^{(n)}_{-, x} = \sum^{\infty}_{i = n+1}k_i(t)\lambda^{n-i}, \end{eqnarray*}$

it follows that one has

$\begin{eqnarray*} -V^{(n)}_{1+, x}+\frac{\partial U_1}{\partial \lambda} \lambda^{(n)}_{t, +}+[U_1, V^{(n)}_{1+}] = (0, 0, 0, 0, -2e_{n+1}, 2f_{n+1}, -2g_{n+1}, 2h_{n+1}, -2p_{n+1}, 2q_{n+1})^{\mathrm{T}}. \end{eqnarray*}$

According to (2.4), it is easy to show that we have the recursion relations

$\begin{eqnarray*} \left(\begin{array}{cccccc} 2\varepsilon f_{n+1} \\ 2\varepsilon e_{n+1} \\ \varepsilon h_{n+1} \\ \varepsilon g_{n+1} \\ \varepsilon q_{n+1} \\ \varepsilon p_{n+1} \\ \end{array}\right) = L_1\left(\begin{array}{cccccc} 2\varepsilon f_{n} \\ 2\varepsilon e_{n} \\ \varepsilon h_{n} \\ \varepsilon g_{n} \\ \varepsilon q_{n} \\ \varepsilon p_{n} \\ \end{array}\right)+\left(\begin{array}{cccccc} 2\varepsilon u_2 \\ 2\varepsilon u_1 \\ \varepsilon u_4 \\ \varepsilon u_3 \\ \varepsilon u_6 \\ \varepsilon u_5 \\ \end{array}\right)k_{n}(t)x, \end{eqnarray*}$

where the recurrence operator $L_1$ is defined as

$\begin{eqnarray*} L_1 = \left( \begin{array}{cccccc} l_{11} & l_{12} & l_{13} & l_{14} & l_{15} & l_{16} \\ l_{21} & l_{22} & l_{23} & l_{24} & l_{25} & l_{26} \\ l_{31} & l_{32} & l_{33} & l_{34} & l_{35} & l_{36} \\ l_{41} & l_{42} & l_{43} & l_{44} & l_{45} & l_{46} \\ l_{51} & l_{52} & l_{53} & l_{54} & l_{55} & l_{56} \\ l_{61} & l_{62} & l_{63} & l_{64} & l_{65} & l_{66} \\ \end{array} \right), \end{eqnarray*}$

and

$\begin{eqnarray*} & & l_{11}\! = \!-\frac{\partial}{2}\!+\!\frac{\varepsilon}{2}(2u_2\partial^{-1}u_1\!+\!u_4\partial^{-1}u_3\!+\!u_6\partial^{-1}u_5), l_{12}\! = \!-\frac{\varepsilon}{2}(2u_2\partial^{-1}u_2\!+\!u_4\partial^{-1}u_6\!+\!u_6\partial^{-1}u_4), \\ & & l_{21}\! = \!\varepsilon u_1\partial^{-1}u_1\!+\!\frac{\varepsilon}{2}(u_3\partial^{-1}u_5\!+\!u_5\partial^{-1}u_3), \ l_{22}\! = \!\frac{\partial}{2}\!-\!\varepsilon u_1\partial^{-1}u_2\!-\!\frac{\varepsilon}{2}(u_3\partial^{-1}u_4\!+\!u_5\partial^{-1}u_6), \\ & & l_{31} = \frac{1}{2}l_{15} = \frac{\varepsilon}{2}(u_2\partial^{-1}u_5+u_4\partial^{-1}u_1), \ \ l_{32} = \frac{1}{2}l_{14} = -\frac{\varepsilon}{2}(u_2\partial^{-1}u_4+u_4\partial^{-1}u_2), \\ & & l_{41} = \frac{1}{2}l_{23} = \frac{\varepsilon}{2}(u_1\partial^{-1}u_3+u_3\partial^{-1}u_1), \ \ l_{42} = \frac{1}{2}l_{26} = -\frac{\varepsilon}{2}(u_1\partial^{-1}u_6+u_3\partial^{-1}u_2), \\ & & l_{51} = \frac{1}{2}l_{13} = \frac{\varepsilon}{2}(u_2\partial^{-1}u_3+u_6\partial^{-1}u_1), \ \ l_{52} = \frac{1}{2}l_{16} = -\frac{\varepsilon}{2}(u_2\partial^{-1}u_6+u_6\partial^{-1}u_2), \\ & & l_{61} = \frac{1}{2}l_{25} = \frac{\varepsilon}{2}(u_1\partial^{-1}u_5+u_5\partial^{-1}u_1), \ \ l_{62} = \frac{1}{2}l_{24} = -\frac{\varepsilon}{2}(u_1\partial^{-1}u_4+u_5\partial^{-1}u_2), \\ & & l_{33} = -\frac{\partial}{2}+\varepsilon u_2\partial^{-1}u_1+\varepsilon u_4\partial^{-1}u_3, \ \ l_{34} = -\varepsilon u_4\partial^{-1}u_4, \ \ l_{35} = 0, \ \ l_{36} = -\varepsilon u_2\partial^{-1}u_2, \\ & & l_{43} = \varepsilon u_3\partial^{-1}u_3, \ \ l_{44} = \frac{\partial}{2}-\varepsilon u_1\partial^{-1}u_2-\varepsilon u_3\partial^{-1}u_4, \ \ l_{45} = \varepsilon u_1\partial^{-1}u_1, \ \ l_{46} = 0, \\ & & l_{53} = 0, \ \ l_{54} = -\varepsilon u_2\partial^{-1}u_2, \ \ l_{55} = -\frac{\partial}{2}+\varepsilon u_2\partial^{-1}u_1+\varepsilon u_6\partial^{-1}u_5, \ \ l_{56} = -\varepsilon u_6\partial^{-1}u_6, \\ & & l_{63} = \varepsilon u_1\partial^{-1}u_1, \ \ l_{64} = 0, \ \ l_{65} = \varepsilon u_5\partial^{-1}u_5, \ \ l_{66} = \frac{\partial}{2}-\varepsilon u_1\partial^{-1}u_2-\varepsilon u_5\partial^{-1}u_6. \end{eqnarray*}$

Taking $V_1^{(n)} = V_{1, +}^{(n)}$ , then the zero curvature equation

$\begin{eqnarray*} -V^{(n)}_{1x}+\frac{\partial U_1}{\partial u} u_t+\frac{\partial U_1}{\partial \lambda} \lambda^{(n)}_{t, +}+[U_1, V_1^{(n)}] = 0 \end{eqnarray*}$

leads to the following nonisospectral hierarchy

$\begin{eqnarray} u_{t_n} & = & \left(\begin{array}{cccccc} u_1 \\ u_2 \\ u_3 \\ u_4 \\ u_5 \\ u_6 \\ \end{array}\right)_{t_n}\! = \!\left(\begin{array}{cccccc} -2e_{n+1} \\ 2f_{n+1} \\ -2g_{n+1} \\ 2h_{n+1} \\ -2p_{n+1} \\ 2q_{n+1} \\ \end{array}\right)\! = \!J_1\left(\begin{array}{cccccc} 2\varepsilon f_{n+1} \\ 2\varepsilon e_{n+1} \\ \varepsilon h_{n+1} \\ \varepsilon g_{n+1} \\ \varepsilon q_{n+1} \\ \varepsilon p_{n+1} \\ \end{array}\right) \\ & = & J_1L_1\left(\begin{array}{cccccc} 2\varepsilon f_{n} \\ 2\varepsilon e_{n} \\ \varepsilon h_{n} \\ \varepsilon g_{n} \\ \varepsilon q_{n} \\ \varepsilon p_{n} \\ \end{array}\right)+J_1\left(\begin{array}{cccccc} 2\varepsilon u_2 \\ 2\varepsilon u_1 \\ \varepsilon u_4 \\ \varepsilon u_3 \\ \varepsilon u_6 \\ \varepsilon u_5 \\ \end{array}\right)k_{n}(t)x, \end{eqnarray}$

(2.5)

where the Hamiltonian operator $J_1$ is

$\begin{eqnarray*} J_1 = \frac{1}{\varepsilon}\left( \begin{array}{cccccc} 0 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -2 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -2 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ \end{array} \right). \end{eqnarray*}$

To furnish Hamiltonian structures, we use the trace identity, and have

$\begin{eqnarray*} & & \langle V_1, \frac{\partial U_1}{\partial \lambda}\rangle = 2a+2b, \ \langle V_1, \frac{\partial U_1}{\partial u_1}\rangle = 2\varepsilon f, \ \langle V_1, \frac{\partial U_1}{\partial u_2}\rangle = 2\varepsilon e, \\ & & \langle V_1, \frac{\partial U_1}{\partial u_3}\rangle = \varepsilon h, \ \langle V_1, \frac{\partial U_1}{\partial u_4}\rangle = \varepsilon g, \ \langle V_1, \frac{\partial U_1}{\partial u_5}\rangle = \varepsilon q, \ \langle V_1, \frac{\partial U_1}{\partial u_6}\rangle = \varepsilon p. \end{eqnarray*}$

Substituting the above formulas into the trace identity yields

$\begin{eqnarray*} \frac{\delta}{\delta u}\displaystyle {\int}(2a+2b)dx = \lambda^{-\gamma}\frac{\partial}{\partial \lambda}\lambda^\gamma \left(\begin{array}{cccccc} 2\varepsilon f \\ 2\varepsilon e \\ \varepsilon h \\ \varepsilon g \\ \varepsilon q \\ \varepsilon p \\ \end{array}\right). \end{eqnarray*}$

Balancing coefficients of each power of $\lambda$ in the above equality gives rise to

$\begin{eqnarray*} \frac{\delta}{\delta u}\displaystyle {\int}(2a_{n+1}+2b_{n+1})dx = (\gamma-n) \left(\begin{array}{cccccc} 2\varepsilon f_n \\ 2\varepsilon e_n \\ \varepsilon h_n \\ \varepsilon g_n \\ \varepsilon q_n \\ \varepsilon p_n \\ \end{array}\right). \end{eqnarray*}$

Taking $n = 1$ , gives $\gamma = 0$ . Thus, we see

$u_t = J_1\frac{\delta H_{n+1}^{(1)}}{\delta u} = J_1L_1\frac{\delta H_{n}^{(1)}}{\delta u}+J_1M_1k_n(t)x, \ \ \ H_{n+1}^{(1)} = -2\displaystyle {\int}(\frac{a_{n+2}+b_{n+2}}{n+1})dx, \ \ n\geq 0,$

where $M_1 = (2\varepsilon u_2, 2\varepsilon u_1, \varepsilon u_4, \varepsilon u_3, \varepsilon u_6, \varepsilon u_5)^{\mathrm{T}}$ .

3. The nonisospectral integrable coupling hierarchy associated with Lie algebra $\mathfrak{sp}(4)$

3.1. Preliminaries

We generalize the semisimple Lie algebra $g$ to the non-semisimple Lie algebra $\bar{g}$ . This has the block matrix form ^[29,30]

$\begin{eqnarray} M(A, B) = \left( \begin{array}{cc} A & \varepsilon B \\ B & A \\ \end{array} \right), \end{eqnarray}$

(3.1)

where $\varepsilon\in \mathbb{R}$ , which is different from Section 2, and $A, B$ are two arbitrary matrices with the same order. Non-semisimple Lie algebra $\bar{g}$ has two subalgebras $\widetilde{g} = \{M(A, O)\}$ and $\widetilde{g_c} = \{M(O, B)\}$ , that forms the semi-direct sum of Lie algebras $\bar{g} = \widetilde{g}\oplus_s \widetilde{g_c}$ . That is, $[\widetilde{g}, \widetilde{g_c}] = \{[A, B]\rvert A\in \widetilde{g}, B\in \widetilde{g_c}\}$ , $[\widetilde{g}, \widetilde{g}]\subseteq \widetilde{g}, \ [\widetilde{g_c}, \widetilde{g_c}]\subseteq \widetilde{g}, \ [\widetilde{g}, \widetilde{g_c}]\subseteq \widetilde{g_c}$ .

We introduce the enlarged spectral problems

$\begin{eqnarray*} \left\{ \begin{array}{ll} \varphi_x = \bar{U}\varphi = \left( \begin{array}{cc} U_1 & \varepsilon U_2 \\ U_2 & U_1 \\ \end{array} \right), \\ \varphi_t = \bar{V}\varphi = \left( \begin{array}{cc} V_1 & \varepsilon V_2 \\ V_2 & V_1 \\ \end{array} \right), \\ \lambda_t = \sum\limits_{i\geq 0}k_i(t)\lambda^{-i}. \\ \end{array} \right. \end{eqnarray*}$

From the corresponding enlarged stationary zero curvature equation

$\begin{eqnarray} \bar{V}_{x} = \frac{\partial \bar{U}}{\partial \lambda}\lambda_t+[\bar{U}, \bar{V}], \end{eqnarray}$

(3.2)

it is easy to have

$\begin{eqnarray*} \left\{ \begin{array}{ll} V_{1, x} = [U_1, V_1]+\varepsilon[U_2, V_2], \\ V_{2, x} = [U_2, V_1]+[U_1, V_2]. \\ \end{array} \right. \end{eqnarray*}$

3.2. The nonisospectral integrable couplings

In this section, we will construct the nonisospectral integrable coupling hierarchies associated with Lie algebra $\mathfrak{sp}(4)$ . We consider the nonisospectral problem

$\begin{eqnarray*} \left\{ \begin{array}{ll} \varphi_x = \bar{U}_1\varphi, \\ \varphi_t = \bar{V}_1\varphi, \\ \lambda_t = \sum\nolimits_{i\geq 0}k_i(t)\lambda^{-i}, \\ \end{array} \right. \end{eqnarray*}$

where

$\begin{eqnarray} & & \bar{U}_1 = \left( \begin{array}{cc} U'_1 & \varepsilon U'_2 \\ U'_2 & U'_1 \\ \end{array} \right), U'_1 = \left( \begin{array}{cccc} \lambda & 0 & u_3 & u_1 \\ 0 & \lambda & u_1 & u_5 \\ u_4 & u_2 & -\lambda & 0 \\ u_2 & u_6 & 0 & -\lambda \\ \end{array} \right), U'_2 = \left( \begin{array}{cccc} 0 & 0 & u'_3 & u'_1 \\ 0 & 0 & u'_1 & u'_5 \\ u'_4 & u'_2 & 0 & 0 \\ u'_2 & u'_6 & 0 & 0 \\ \end{array} \right), \\ & & \bar{V}_1 = \!\!\left( \begin{array}{cc} V'_1 & \varepsilon V'_2 \\ V'_2 & V'_1 \\ \end{array} \right)\!, V'_1 = \!\!\left( \begin{array}{cccc} a & c & g & e \\ d & b & e & p \\ h & f & -a & -d \\ f & q & -c & -b \\ \end{array} \right)\!, V'_2 = \!\left( \begin{array}{cccc} a' & c' & g' & e' \\ d' & b' & e' & p' \\ h' & f' & -a' & -d' \\ f' & q' & -c' & -b' \\ \end{array} \right)\!, \end{eqnarray}$

(3.3)

where $u_1, u_2, \cdots, u_6$ and $a, b, c, d, e, f, g, h, p, q$ are different from (2.2).

We solve the enlarged stationary zero curvature equation by means of

$\begin{eqnarray} \bar{V}_{1x} = \frac{\partial \bar{U}_1}{\partial \lambda}\lambda_t+[\bar{U}_1, \bar{V}_1], \end{eqnarray}$

(3.4)

which yields

$\begin{eqnarray} \left\{ \begin{array}{llllllllll} a_{ix} = u_1f_i-u_2e_i+u_3h_i-u_4g_i+\varepsilon u'_1f'_i-\varepsilon u'_2e'_i+\varepsilon u'_3h'_i-\varepsilon u'_4g'_i+k_i(t), \\ b_{ix} = u_1f_i-u_2e_i+u_5q_i-u_6p_i+\varepsilon u'_1f'_i-\varepsilon u'_2e'_i+\varepsilon u'_5q'_i-\varepsilon u'_6p'_i+k_i(t), \\ c_{ix} = u_1q_i-u_2g_i+u_3f_i-u_6e_i+\varepsilon u'_1q'_i-\varepsilon u'_2g'_i+\varepsilon u'_3f'_i-\varepsilon u'_6e'_i, \\ d_{ix} = u_1h_i-u_2p_i-u_4e_i+u_5f_i+\varepsilon u'_1h'_i-\varepsilon u'_2p'_i-\varepsilon u'_4e'_i+\varepsilon u'_5f'_i, \\ e_{ix} = 2\lambda e_i-u_1a_i-u_1b_i-u_3d_i-u_5c_i-\varepsilon u'_1a'_i-\varepsilon u'_1b'_i-\varepsilon u'_3d'_i-\varepsilon u'_5c'_i, \\ f_{ix} = -2\lambda f_i+u_2a_i+u_2b_i+u_4c_i+u_6d_i+\varepsilon u'_2a'_i+\varepsilon u'_2b'_i+\varepsilon u'_4c'_i+\varepsilon u'_6d'_i, \\ g_{ix} = 2\lambda g_i-2u_1c_i-2u_3a_i-2\varepsilon u'_1c'_i-2\varepsilon u'_3a'_i, \\ h_{ix} = -2\lambda h_i+2u_2d_i+2u_4a_i+2\varepsilon u'_2d'_i+2\varepsilon u'_4a'_i, \\ p_{ix} = 2\lambda p_i-2u_1d_i-2u_5b_i-2\varepsilon u'_1d'_i-2\varepsilon u'_5b'_i, \\ q_{ix} = -2\lambda q_i+2u_2c_i+2u_6b_i+2\varepsilon u'_2c'_i+2\varepsilon u'_6b'_i, \end{array} \right. \end{eqnarray}$

(3.5)

and

$\begin{eqnarray} \left\{ \begin{array}{llllllllll} a'_{ix} = u_1f'_i-u_2e'_i+u_3h'_i-u_4g'_i+u'_1f_i-u'_2e_i+u'_3h_i-u'_4g_i+k_i(t), \\ b'_{ix} = u_1f'_i-u_2e'_i+u_5q'_i-u_6p'_i+u'_1f_i-u'_2e_i+u'_5q_i-u'_6p_i+k_i(t), \\ c'_{ix} = u_1q'_i-u_2g'_i+u_3f'_i-u_6e'_i+u'_1q_i-u'_2g_i+u'_3f_i-u'_6e_i, \\ d'_{ix} = u_1h'_i-u_2p'_i-u_4e'_i+u_5f'_i+u'_1h_i-u'_2p_i-u'_4e_i+u'_5f_i, \\ e'_{ix} = 2\lambda e'_i-u_1a'_i-u_1b'_i-u_3d'_i-u_5c'_i-u'_1a_i-u'_1b_i-u'_3d_i-u'_5c_i, \\ f'_{ix} = -2\lambda f'_i+u_2a'_i+u_2b'_i+u_4c'_i+u_6d'_i+u'_2a_i+u'_2b_i+u'_4c_i+u'_6d_i, \\ g'_{ix} = 2\lambda g'_i-2u_1c'_i-2u_3a'_i-2u'_1c_i-2u'_3a_i, \\ h'_{ix} = -2\lambda h'_i+2u_2d'_i+2u_4a'_i+2u'_2d_i+2u'_4a_i, \\ p'_{ix} = 2\lambda p'_i-2u_1d'_i-2u_5b'_i-2u'_1d_i-2u'_5b_i, \\ q'_{ix} = -2\lambda q'_i+2u_2c'_i+2u_6b'_i+2u'_2c_i+2u'_6b_i. \end{array} \right. \end{eqnarray}$

(3.6)

By taking initial values

$\begin{eqnarray*} a_0 = \alpha(t), \ b_0 = \beta(t), \ c_0 = d_0 = e_0 = f_0 = g_0 = h_0 = p_0 = q_0 = k_0(t) = 0, \\ a'_0 = \alpha'(t), \ b'_0 = \beta'(t), \ c'_0 = d'_0 = e'_0 = f'_0 = g'_0 = h'_0 = p'_0 = q'_0 = 0, \end{eqnarray*}$

one has

$\begin{align*} a_1 = &b_1 = k_1(t)x, \ e_1 = \frac{1}{2}u_1(\alpha+\beta)+\frac{\varepsilon}{2}u'_1(\alpha'+\beta'), \ f_1 = \frac{1}{2}u_2(\alpha+\beta)+\frac{\varepsilon}{2}u'_2(\alpha'+\beta'), \\ g_1 = &u_3\alpha+\varepsilon u'_3\alpha', \ h_1 = u_4\alpha+\varepsilon u'_4\alpha', \ p_1 = u_5\beta+\varepsilon u'_5\beta', \ q_1 = u_6\beta+\varepsilon u'_6\beta', \\ c_1 = &\frac{1}{2}\partial^{-1}(u_1u_6+u_2u_3+\varepsilon u'_1u'_6+\varepsilon u'_2u'_3)(\beta-\alpha)+\frac{\varepsilon}{2}\partial^{-1}(u_1u'_6+u_2u'_3+ u'_1u_6+u'_2u_3)\\ &(\beta'-\alpha'), \\ d_1 = &\frac{1}{2}\partial^{-1}(u_1u_4+u_2u_5+\varepsilon u'_1u'_4+\varepsilon u'_2u'_5)(\alpha-\beta)+\frac{\varepsilon}{2}\partial^{-1}(u_1u'_4+u_2u'_5+u'_1u_4+u'_2u_5)\\ &(\alpha'-\beta'), \\ a'_1 = &b'_1 = k_1(t)x, \ e'_1 = \frac{1}{2}u'_1(\alpha+\beta)+\frac{1}{2}u_1(\alpha'+\beta'), \ f'_1 = \frac{1}{2}u'_2(\alpha+\beta)+\frac{1}{2}u_2(\alpha'+\beta'), \\ g'_1 = &u'_3\alpha+u_3\alpha', \ h'_1 = u'_4\alpha+u_4\alpha', \ p'_1 = u'_5\beta+u_5\beta', \ q'_1 = u'_6\beta+u_6\beta', \\ c'_1 = &\frac{1}{2}\partial^{-1}(u_1u'_6+u_2u'_3+ u'_1u_6+u'_2u_3)(\beta-\alpha)+\frac{1}{2}\partial^{-1}(u_1u_6+u_2u_3+\varepsilon u'_1u'_6+\varepsilon u'_2u'_3)\\ &(\beta'-\alpha'), \\ d'_1 = &\frac{1}{2}\partial^{-1}(u_1u'_4+u_2u'_5+u'_1u_4+u'_2u_5)(\alpha-\beta)+\frac{1}{2}\partial^{-1}(u_1u_4+u_2u_5+\varepsilon u'_1u'_4+\varepsilon u'_2u'_5)\\ &(\alpha'-\beta'), \\ & \cdots\cdots \end{align*}$

where $\alpha(t), \ \beta(t)$ are integral constants.

From the nonisospectral zero curvature equation

$\begin{eqnarray*} \frac{\partial \bar{U}_1}{\partial u}u_t+\frac{\partial \bar{U}_1}{\partial \lambda}\lambda_t-\bar{V}^{(n)}_{1x}+[\bar{U}_1, \bar{V}^{(n)}_1] = 0, \end{eqnarray*}$

we can obtain the following integrable couplings:

$\begin{eqnarray} \bar{u}_{t_n}\! = \!\left(\begin{array}{cc} \bar{u}_1 \\ \bar{u}_2 \\ \end{array}\right)_{t_n}\!, \ \bar{u}_{1t_n}\! = \!\left(\begin{array}{cccccc} u_1 \\ u_2 \\ u_3 \\ u_4 \\ u_5 \\ u_6 \\ \end{array}\right)_{t_n}\! = \!\left(\begin{array}{cccccc} -2e_{n+1} \\ 2f_{n+1} \\ -2g_{n+1} \\ 2h_{n+1} \\ -2p_{n+1} \\ 2q_{n+1} \\ \end{array}\right)\!, \ \bar{u}_{2t_n}\! = \!\left(\begin{array}{cccccc} u'_1 \\ u'_2 \\ u'_3 \\ u'_4 \\ u'_5 \\ u'_6 \\ \end{array}\right)_{t_n}\! = \!\left(\begin{array}{cccccc} -2e'_{n+1} \\ 2f'_{n+1} \\ -2g'_{n+1} \\ 2h'_{n+1} \\ -2p'_{n+1} \\ 2q'_{n+1} \\ \end{array}\right)\!. \end{eqnarray}$

(3.7)

To furnish Hamiltonian structures, we use the trace identity, and have

$\begin{eqnarray*} & & \langle V'_1, \frac{\partial U'_2}{\partial \lambda}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial \lambda}\rangle = 2(a'+b'), \ \langle V'_1, \frac{\partial U'_2}{\partial u_1}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u_1}\rangle = 2f', \\ & & \langle V'_1, \frac{\partial U'_2}{\partial u_2}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u_2}\rangle = 2e', \ \langle V'_1, \frac{\partial U'_2}{\partial u_3}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u_3}\rangle = h', \\ & & \langle V'_1, \frac{\partial U'_2}{\partial u_4}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u_4}\rangle = g', \ \langle V'_1, \frac{\partial U'_2}{\partial u_5}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u_5}\rangle = q', \\ & & \langle V'_1, \frac{\partial U'_2}{\partial u_6}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u_6}\rangle = p', \ \langle V'_1, \frac{\partial U'_2}{\partial u'_1}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u'_1}\rangle = 2f, \\ & & \langle V'_1, \frac{\partial U'_2}{\partial u'_2}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u'_2}\rangle = 2e, \ \langle V'_1, \frac{\partial U'_2}{\partial u'_3}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u'_3}\rangle = h, \\ & & \langle V'_1, \frac{\partial U'_2}{\partial u'_4}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u'_4}\rangle = g, \ \langle V'_1, \frac{\partial U'_2}{\partial u'_5}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u'_5}\rangle = q, \\ & & \langle V'_1, \frac{\partial U'_2}{\partial u'_6}\rangle+\langle V'_2, \frac{\partial U'_1}{\partial u'_6}\rangle = p. \end{eqnarray*}$

Substituting the above formulas into the trace identity, and balancing the coefficients of each power of $\lambda$ , we give the first form

$\begin{eqnarray} \frac{\delta}{\delta \bar{u}}\displaystyle {\int} 2(a'_{n+1}+b'_{n+1})dx = (\gamma-n) \left(\begin{array}{cccccc} M_1 \\ M_2 \\ \end{array}\right), \end{eqnarray}$

(3.8)

where

$\begin{eqnarray*} M_1 = \left(\begin{array}{cccccc} 2f'_n \\ 2e'_n \\ h'_n \\ g'_n \\ q'_n \\ p'_n \\ \end{array}\right), \ M_2 = \left(\begin{array}{cccccc} 2f_n \\ 2e_n \\ h_n \\ g_n \\ q_n \\ p_n \\ \end{array}\right). \end{eqnarray*}$

At the same time, we also have

$\begin{eqnarray*} & & \langle V'_1, \frac{\partial U'_1}{\partial \lambda}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial \lambda}\rangle = 2(a+b), \ \langle V'_1, \frac{\partial U'_1}{\partial u_1}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u_1}\rangle = 2f, \\ & & \langle V'_1, \frac{\partial U'_1}{\partial u_2}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u_2}\rangle = 2e, \ \langle V'_1, \frac{\partial U'_1}{\partial u_3}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u_3}\rangle = h, \\ & & \langle V'_1, \frac{\partial U'_1}{\partial u_4}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u_4}\rangle = g, \ \langle V'_1, \frac{\partial U'_1}{\partial u_5}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u_5}\rangle = q, \\ & & \langle V'_1, \frac{\partial U'_1}{\partial u_6}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u_6}\rangle = p, \ \langle V'_1, \frac{\partial U'_1}{\partial u'_1}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u'_1}\rangle = 2\varepsilon f', \\ & & \langle V'_1, \frac{\partial U'_1}{\partial u'_2}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u'_2}\rangle = 2\varepsilon e', \ \langle V'_1, \frac{\partial U'_1}{\partial u'_3}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u'_3}\rangle = \varepsilon h', \\ & & \langle V'_1, \frac{\partial U'_1}{\partial u'_4}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u'_4}\rangle = \varepsilon g', \ \langle V'_1, \frac{\partial U'_1}{\partial u'_5}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u'_5}\rangle = \varepsilon q', \\ & & \langle V'_1, \frac{\partial U'_1}{\partial u'_6}\rangle+\varepsilon \langle V'_2, \frac{\partial U'_2}{\partial u'_6}\rangle = \varepsilon p'. \end{eqnarray*}$

Substituting the above formulas into the trace identity, and balancing coefficients of each power of $\lambda$ , we give the second form

$\begin{eqnarray*} \frac{\delta}{\delta u}\displaystyle {\int} 2(a_{n+1}+b_{n+1})dx = (\gamma-n) \left(\begin{array}{cccccc} M_2 \\ \varepsilon M_1 \\ \end{array}\right), \end{eqnarray*}$

where $M_1, \ M_2$ are defined as (3.8).

So, we can obtain the Hamiltonian structures of integrable couplings, which consists of the following two components. The first component has the form

$\begin{eqnarray*} \bar{u}_{t_n} = \left(\begin{array}{cc} \bar{u}_1 \\ \bar{u}_2 \\ \end{array}\right)_{t_n} = \bar{J}_1\left(\begin{array}{cc} K_1 \\ K_2 \\ \end{array}\right) = \bar{J}_1\frac{\delta \bar{H}_{1, m}}{\delta \bar{u}}, \ K_1 = \left(\begin{array}{cccccc} 2f'_{n+1} \\ 2e'_{n+1} \\ h'_{n+1} \\ g'_{n+1} \\ q'_{n+1} \\ p'_{n+1} \\ \end{array}\right), \ K_2 = \left(\begin{array}{cccccc} 2f_{n+1} \\ 2e_{n+1} \\ h_{n+1} \\ g_{n+1} \\ q_{n+1} \\ p_{n+1} \\ \end{array}\right), \end{eqnarray*}$

where $\bar{J}_1 = \left(\begin{array}{cc} O & J_1 \\ J_1 & O \end{array} \right)$ , and $J_1$ is defined as (2.5). The second component has the form

$\begin{eqnarray*} \bar{u}_{t_n} = \left(\begin{array}{cc} \bar{u}_1 \\ \bar{u}_2 \\ \end{array}\right)_{t_n} = \bar{J}_2\left(\begin{array}{cc} K_2 \\ \varepsilon K_1 \\ \end{array}\right) = \bar{J}_2\frac{\delta \bar{H}_{2, m}}{\delta \bar{u}}, \ \ \bar{J}_2 = \left( \begin{array}{cc} J_1 & O \\ O & \frac{1}{\varepsilon}J_1 \end{array} \right). \end{eqnarray*}$

From (3.5) and (3.6), we can obtain the recursion relations

$\begin{eqnarray*} \left(\begin{array}{cccccccccccc} -2e_{n+1} \\ 2f_{n+1} \\ -2g_{n+1} \\ 2h_{n+1} \\ -2p_{n+1} \\ 2q_{n+1} \\ -2e'_{n+1} \\ 2f'_{n+1} \\ -2g'_{n+1} \\ 2h'_{n+1} \\ -2p'_{n+1} \\ 2q'_{n+1} \\ \end{array}\right) = \bar{L}_1\left(\begin{array}{cccccccccccc} -2e_{n} \\ 2f_{n} \\ -2g_{n} \\ 2h_{n} \\ -2p_{n} \\ 2q_{n} \\ -2e'_{n} \\ 2f'_{n} \\ -2g'_{n} \\ 2h'_{n} \\ -2p'_{n} \\ 2q'_{n} \\ \end{array}\right)+\left(\begin{array}{cccccccccccc} -2u_1-2\varepsilon u'_1 \\ 2u_2+2\varepsilon u'_2 \\ -2u_3-2\varepsilon u'_3 \\ 2u_4+2\varepsilon u'_4 \\ -2u_5-2\varepsilon u'_5 \\ 2u_6+2\varepsilon u'_6 \\ -2u_1-2u'_1 \\ 2u_2+2u'_2 \\ -2u_3-2u'_3 \\ 2u_4+2u'_4 \\ -2u_5-2u'_5 \\ 2u_6+2u'_6 \\ \end{array}\right)k_n(t)x, \end{eqnarray*}$

where the recurrence operator $\bar{L}_1$ is defined as

$\begin{eqnarray*} \bar{L}_1\! = \!\left( \begin{array}{cc} L'_1 & L'_2 \\ \frac{1}{\varepsilon}L'_2 & L'_1 \\ \end{array} \right), \ L'_1\! = \!\!\left( \begin{array}{cccccc} l_{11} & l_{12} & l_{13} & l_{14} & l_{15} & l_{16} \\ l_{21} & l_{22} & l_{23} & l_{24} & l_{25} & l_{26} \\ l_{31} & l_{32} & l_{33} & l_{34} & l_{35} & l_{36} \\ l_{41} & l_{42} & l_{43} & l_{44} & l_{45} & l_{46} \\ l_{51} & l_{52} & l_{53} & l_{54} & l_{55} & l_{56} \\ l_{61} & l_{62} & l_{63} & l_{64} & l_{65} & l_{66} \\ \end{array} \right)\!, \ L'_2\! = \!\!\left( \begin{array}{cccccc} l'_{11} & l'_{12} & l'_{13} & l'_{14} & l'_{15} & l'_{16} \\ l'_{21} & l'_{22} & l'_{23} & l'_{24} & l'_{25} & l'_{26} \\ l'_{31} & l'_{32} & l'_{33} & l'_{34} & l'_{35} & l'_{36} \\ l'_{41} & l'_{42} & l'_{43} & l'_{44} & l'_{45} & l'_{46} \\ l'_{51} & l'_{52} & l'_{53} & l'_{54} & l'_{55} & l'_{56} \\ l'_{61} & l'_{62} & l'_{63} & l'_{64} & l'_{65} & l'_{66} \\ \end{array} \right)\!, \end{eqnarray*}$

and

$\begin{eqnarray*} & & l_{11}\! = \!\frac{1}{2}(\partial-2u_1\partial^{-1}u_2\!-\!u_3\partial^{-1}u_4\!-\!u_5\partial^{-1}u_6-2\varepsilon u'_1\partial^{-1}u'_2\!-\!\varepsilon u'_3\partial^{-1}u'_4\!-\!\varepsilon u'_5\partial^{-1}u'_6), \\ & & l_{12}\! = \!-\frac{1}{2}(2u_1\partial^{-1}u_1\!+\!u_3\partial^{-1}u_5\!+\!u_5\partial^{-1}u_3+2\varepsilon u'_1\partial^{-1}u'_1\!+\!\varepsilon u'_3\partial^{-1}u'_5\!+\!\varepsilon u'_5\partial^{-1}u'_3), \\ & & l_{13} = \frac{1}{2}l_{51}\! = \!-\frac{1}{2}(u_1\partial^{-1}u_4\!+\!u_5\partial^{-1}u_2+\varepsilon u'_1\partial^{-1}u'_4\!+\!\varepsilon u'_5\partial^{-1}u'_2), \\ & & l_{14} = \frac{1}{2}l_{32}\! = \!-\frac{1}{2}(u_1\partial^{-1}u_3\!+\!u_3\partial^{-1}u_1+\varepsilon u'_1\partial^{-1}u'_3\!+\!\varepsilon u'_3\partial^{-1}u'_1), \\ & & l_{15} = \frac{1}{2}l_{31}\! = \!-\frac{1}{2}(u_1\partial^{-1}u_6\!+\!u_3\partial^{-1}u_2+\varepsilon u'_1\partial^{-1}u'_6\!+\!\varepsilon u'_3\partial^{-1}u'_2), \\ & & l_{16} = \frac{1}{2}l_{52}\! = \!-\frac{1}{2}(u_1\partial^{-1}u_5\!+\!u_5\partial^{-1}u_1+\varepsilon u'_1\partial^{-1}u'_5\!+\!\varepsilon u'_5\partial^{-1}u'_1), \\ & & l'_{11}\! = \!-\frac{\varepsilon}{2}(2u_1\partial^{-1}u'_2\!+\!u_3\partial^{-1}u'_4\!+\!u_5\partial^{-1}u'_6+2u'_1\partial^{-1}u_2\!+\!u'_3\partial^{-1}u_4\!+\!u'_5\partial^{-1}u_6), \\ & & l'_{12}\! = \!-\frac{\varepsilon}{2}(2u_1\partial^{-1}u'_1\!+\!u_3\partial^{-1}u'_5\!+\!u_5\partial^{-1}u'_3+2u'_1\partial^{-1}u_1\!+\!u'_3\partial^{-1}u_5\!+\!u'_5\partial^{-1}u_3), \\ & & l'_{13} = \frac{1}{2}l'_{51}\! = \!-\frac{\varepsilon}{2}(u_1\partial^{-1}u'_4\!+\!u_5\partial^{-1}u'_2+ u'_1\partial^{-1}u_4\!+\!u'_5\partial^{-1}u_2), \\ & & l'_{14} = \frac{1}{2}l'_{32}\! = \!-\frac{\varepsilon}{2}(u_1\partial^{-1}u'_3\!+\!u_3\partial^{-1}u'_1+ u'_1\partial^{-1}u_3\!+\!u'_3\partial^{-1}u_1), \\ & & l'_{15} = \frac{1}{2}l'_{31}\! = \!-\frac{\varepsilon}{2}(u_1\partial^{-1}u'_6\!+\!u_3\partial^{-1}u'_2+ u'_1\partial^{-1}u_6\!+\!u'_3\partial^{-1}u_2), \\ & & l'_{16} = \frac{1}{2}l'_{52}\! = \!-\frac{\varepsilon}{2}(u_1\partial^{-1}u'_5\!+\!u_5\partial^{-1}u'_1+ u'_1\partial^{-1}u_5\!+\!u'_5\partial^{-1}u_1), \\ & & l_{21}\! = \!\frac{1}{2}(2u_2\partial^{-1}u_2\!+\!u_4\partial^{-1}u_6\!+\!u_6\partial^{-1}u_4+2\varepsilon u'_2\partial^{-1}u'_2\!+\!\varepsilon u'_4\partial^{-1}u'_6\!+\!\varepsilon u'_6\partial^{-1}u'_4), \\ & & l_{22}\! = \!\frac{1}{2}(-\partial+2u_2\partial^{-1}u_1\!+\!u_4\partial^{-1}u_3\!+\!u_6\partial^{-1}u_5+2\varepsilon u'_2\partial^{-1}u'_1\!+\!\varepsilon u'_4\partial^{-1}u'_3\!+\!\varepsilon u'_6\partial^{-1}u'_5), \\ & & l_{23} = \frac{1}{2}l_{41}\! = \!\frac{1}{2}(u_2\partial^{-1}u_4\!+\!u_4\partial^{-1}u_2+\varepsilon u'_2\partial^{-1}u'_4\!+\!\varepsilon u'_4\partial^{-1}u'_2), \\ & & l_{24} = \frac{1}{2}l_{62}\! = \!\frac{1}{2}(u_2\partial^{-1}u_3\!+\!u_6\partial^{-1}u_1+\varepsilon u'_2\partial^{-1}u'_3\!+\!\varepsilon u'_6\partial^{-1}u'_1), \\ & & l_{25} = \frac{1}{2}l_{61}\! = \!\frac{1}{2}(u_2\partial^{-1}u_6\!+\!u_6\partial^{-1}u_2+\varepsilon u'_2\partial^{-1}u'_6\!+\!\varepsilon u'_6\partial^{-1}u'_2), \\ & & l_{26} = \frac{1}{2}l_{42}\! = \!\frac{1}{2}(u_2\partial^{-1}u_5\!+\!u_4\partial^{-1}u_1+\varepsilon u'_2\partial^{-1}u'_5\!+\!\varepsilon u'_4\partial^{-1}u'_1), \\ & & l'_{21}\! = \!\frac{\varepsilon}{2}(2u_2\partial^{-1}u'_2\!+\!u_4\partial^{-1}u'_6\!+\!u_6\partial^{-1}u'_4+2u'_2\partial^{-1}u_2\!+\!u'_4\partial^{-1}u_6\!+\!u'_6\partial^{-1}u_4), \\ & & l'_{22}\! = \!\frac{\varepsilon}{2}(2u_2\partial^{-1}u'_1\!+\!u_4\partial^{-1}u'_3\!+\!u_6\partial^{-1}u'_5+2u'_2\partial^{-1}u_1\!+\!u'_4\partial^{-1}u_3\!+\!u'_6\partial^{-1}u_5), \\ & & l'_{23} = \frac{1}{2}l'_{41}\! = \!\frac{\varepsilon}{2}(u_2\partial^{-1}u'_4\!+\!u_4\partial^{-1}u'_2+ u'_2\partial^{-1}u_4\!+\!u'_4\partial^{-1}u_2), \\ & & l'_{24} = \frac{1}{2}l'_{62}\! = \!\frac{\varepsilon}{2}(u_2\partial^{-1}u'_3\!+\!u_6\partial^{-1}u'_1+ u'_2\partial^{-1}u_3\!+\!u'_6\partial^{-1}u_1), \\ & & l'_{25} = \frac{1}{2}l'_{61}\! = \!\frac{\varepsilon}{2}(u_2\partial^{-1}u'_6\!+\!u_6\partial^{-1}u'_2+ u'_2\partial^{-1}u_6\!+\!u'_6\partial^{-1}u_2), \\ & & l'_{26} = \frac{1}{2}l'_{42}\! = \!\frac{\varepsilon}{2}(u_2\partial^{-1}u'_5\!+\!u_4\partial^{-1}u'_1+ u'_2\partial^{-1}u_5\!+\!u'_4\partial^{-1}u_1), \ l_{35}\! = \!0, \\ & & l_{33}\! = \!\frac{\partial}{2}-u_1\partial^{-1}u_2\!-\!u_3\partial^{-1}u_4-\varepsilon u'_1\partial^{-1}u'_2\!-\!\varepsilon u'_3\partial^{-1}u'_4, \ l_{34}\! = \!-(u_3\partial^{-1}u_3\!+\!\varepsilon u'_3\partial^{-1}u'_3), \\ & & l_{36}\! = \!-(u_1\partial^{-1}u_1\!+\!\varepsilon u'_1\partial^{-1}u'_1), \ l'_{33}\! = \!-\varepsilon(u_1\partial^{-1}u'_2\!+\!u_3\partial^{-1}u'_4\!+\!u'_1\partial^{-1}u_2\!+\!u'_3\partial^{-1}u_4), \\ & & l'_{34}\! = \!-\varepsilon(u_3\partial^{-1}u'_3\!+\!u_3\partial^{-1}u'_3), \ l'_{35}\! = \!0, \ l'_{36}\! = \!-\varepsilon(u_1\partial^{-1}u'_1\!+\!u_1\partial^{-1}u'_1), \\ & & l_{43}\! = \!u_4\partial^{-1}u_4\!+\varepsilon u'_4\partial^{-1}u'_4, \ l_{44}\! = \!-\frac{\partial}{2}+u_4\partial^{-1}u_3\!+\!u_2\partial^{-1}u_1+\varepsilon u'_4\partial^{-1}u'_3\!+\!\varepsilon u'_2\partial^{-1}u'_1, \\ & & l_{45}\! = \!u_2\partial^{-1}u_2\!+\varepsilon u'_2\partial^{-1}u'_2, \ l_{46}\! = \!0, \ l'_{43}\! = \!\varepsilon(u_4\partial^{-1}u'_4+\!u'_4\partial^{-1}u_4), \\ & & l'_{44}\! = \!\varepsilon(u_4\partial^{-1}u'_3\!+\!u_2\partial^{-1}u'_1\!+\!u'_4\partial^{-1}u_3+\!u'_2\partial^{-1}u_1), \ l'_{45}\! = \!\varepsilon(u_2\partial^{-1}u'_2\!+\!u_2\partial^{-1}u'_2), \ l'_{46}\! = \!0, \\ & & l_{54}\! = \!-(u_1\partial^{-1}u_1\!+\varepsilon u'_1\partial^{-1}u'_1), \ l_{55}\! = \!\frac{\partial}{2}-u_1\partial^{-1}u_2\!-\!u_5\partial^{-1}u_6-\varepsilon u'_1\partial^{-1}u'_2\!-\!\varepsilon u'_5\partial^{-1}u'_6, \\ & & l_{56}\! = \!-(u_5\partial^{-1}u_5\!+\varepsilon u'_5\partial^{-1}u'_5), \ l_{53}\! = l'_{53} = 0, \ l'_{54}\! = \!-\varepsilon(u_1\partial^{-1}u'_1+\!u'_1\partial^{-1}u_1), \\ & & l'_{55}\! = \!-\varepsilon(u_1\partial^{-1}u'_2\!+\!u_5\partial^{-1}u'_6\!+\!u'_1\partial^{-1}u_2+\!u'_5\partial^{-1}u_6), \ l'_{56}\! = \!-\varepsilon(u_5\partial^{-1}u'_5\!+\!u_5\partial^{-1}u'_5), \\ & & l_{63}\! = \!u_2\partial^{-1}u_2\!+\varepsilon u'_2\partial^{-1}u'_2, \ l_{64} = l'_{64}\! = \!0, \ l_{65}\! = \!u_6\partial^{-1}u_6\!+\varepsilon u'_6\partial^{-1}u'_6, \\ & & l_{66}\! = \!-\frac{\partial}{2}+u_2\partial^{-1}u_1\!+\!u_6\partial^{-1}u_5+\varepsilon u'_2\partial^{-1}u'_1\!+\!\varepsilon u'_6\partial^{-1}u'_5, \ l'_{63}\! = \!\varepsilon(u_2\partial^{-1}u'_2\!+u'_2\partial^{-1}u_2), \\ & & l'_{65}\! = \!\varepsilon(u_6\partial^{-1}u'_6\!+u'_6\partial^{-1}u_6), \ l'_{66}\! = \!\varepsilon(u_2\partial^{-1}u'_1\!+\!u_6\partial^{-1}u'_5+u'_2\partial^{-1}u_1\!+\!u'_6\partial^{-1}u_5). \end{eqnarray*}$

4. The nonisospectral integrabl hierarchy of the generalized Lie algebra $G\mathfrak{so}(5)$

In this section, based on Lie algebra $\mathfrak{so}(5)$ ^[31,33], we introduce the generalized Lie algebra $G\mathfrak{so}(5)$ , that admits a basis set as follows:

$\begin{equation} \begin{split} E_1 = e_{22}-e_{44}, \ \ E_2 = e_{33}-e_{55}, \ \ E_3 = \varepsilon e_{32}-\varepsilon e_{45}, \ \ E_4 = e_{23}-e_{54}, \ E_5 = e_{52}-e_{43}, \\ E_6 = \varepsilon e_{25}-\varepsilon e_{34}, \ E_7 = e_{13}+e_{51}, \ E_8 = \varepsilon e_{15}+\varepsilon e_{31}, \ E_9 = \varepsilon e_{12}+\varepsilon e_{41}, \ E_{10} = e_{14}+e_{21}, \end{split} \end{equation}$

(4.1)

where $e_{ij}$ is a $5\times 5$ matrix with 1 in the $(i, j)$ -th position and zero elsewhere, which satisfy the commutative relations

$\begin{eqnarray*} & & [E_1, E_2] = 0, \ [E_1, E_3] = -E_3, \ [E_1, E_4] = E_4, \ [E_1, E_5] = -E_5, \ [E_1, E_6] = E_6, \\ & & [E_1, E_7] = [E_1, E_8] = 0, \ [E_1, E_9] = -E_9, \ [E_1, E_{10}] = E_{10}, \ [E_2, E_3] = E_3, \\ & & [E_2, E_4] = -E_4, \ [E_2, E_5] = -E_5, \ [E_2, E_6] = E_6, \ [E_2, E_7] = -E_7, \ [E_2, E_8] = E_8, \\ & & [E_2, E_9]\! = \![E_2, E_{10}]\! = \!0, \ [E_3, E_4]\! = \!\varepsilon(E_2-E_1), \ [E_3, E_5]\! = \![E_3, E_6]\! = \!0, \ [E_3, E_7]\! = \!-E_9, \\ & & [E_3, E_8] = [E_3, E_9] = 0, \ [E_3, E_{10}] = E_8, \ [E_4, E_5] = [E_4, E_6] = [E_4, E_7] = 0, \\ & & [E_4, E_8]\! = \!\varepsilon E_{10}, \ [E_4, E_9]\! = \!-\varepsilon E_7, \ [E_4, E_{10}]\! = \!0, \ [E_5, E_6]\! = \!-\varepsilon(E_1+E_2), \ [E_5, E_7]\! = \!0, \\ & & [E_5, E_8]\! = \!-E_9, \ [E_5, E_9]\! = \!0, \ [E_5, E_{10}]\! = \! E_7, \ [E_6, E_7]\! = \!\varepsilon E_{10}, \ [E_6, E_8]\! = \![E_6, E_{10}]\! = \!0, \\ & & [E_6, E_9] = -\varepsilon E_8, \ [E_7, E_8] = -\varepsilon E_2, \ [E_7, E_9] = \varepsilon E_5, \ [E_7, E_{10}] = -E_4, \ [E_8, E_9] = \varepsilon E_3, \\ & & [E_8, E_{10}] = -E_6, \ [E_9, E_{10}] = -\varepsilon E_1. \end{eqnarray*}$

Consider the nonisospectral problem

$\begin{eqnarray*} \left\{ \begin{array}{ll} \varphi_x = U_2\varphi, \\ \varphi_t = V_2\varphi, \\ \lambda_t = \sum\nolimits_{i\geq 0}k_i(t)\lambda^{-i}, \\ \end{array} \right. \end{eqnarray*}$

where

$\begin{eqnarray} & & U_2 = \left( \begin{array}{ccccc} 0 & \varepsilon u_5 & u_3 & u_6 & \varepsilon u_4 \\ u_6 & \lambda & 0 & 0 & \varepsilon u_2 \\ \varepsilon u_4 & 0 & \lambda & -\varepsilon u_2 & 0 \\ \varepsilon u_5 & 0 & -u_1 & -\lambda & 0 \\ u_3 & u_1 & 0 & 0 & -\lambda \\ \end{array} \right), \\ & & V_2\! = \!\!\left( \begin{array}{ccccc} 0 & \varepsilon p & g & q & \varepsilon h \\ q & a & d & 0 & \varepsilon f \\ \varepsilon h & \varepsilon c & b & -\varepsilon f & 0 \\ \varepsilon p & 0 & -e & -a & -\varepsilon c \\ g & e & 0 & -d & -b \\ \end{array} \right)\!\! = \!\sum\limits_{i\geq 0}\!\left( \begin{array}{ccccc} 0 & \varepsilon p_i & g_i & q_i & \varepsilon h_i \\ q_i & a_i & d_i & 0 & \varepsilon f_i \\ \varepsilon h_i & \varepsilon c_i & b_i & -\varepsilon f_i & 0 \\ \varepsilon p_i & 0 & -e_i & -a_i & -\varepsilon c_i \\ g_i & e_i & 0 & -d_i & -b_i \\ \end{array} \right)\!\lambda^{-i}, \end{eqnarray}$

(4.2)

here $u_1, u_2, \cdots, u_6$ and $a, b, c, d, e, f, g, h, p, q$ are different from (2.2) and (3.3).

By solving the stationary zero curvature representation

$\begin{eqnarray} V_{2x} = \frac{\partial U_2}{\partial \lambda}\lambda_t+[U_2, V_2], \end{eqnarray}$

(4.3)

we can obtain

$\begin{eqnarray} \left\{ \begin{array}{llllllllll} a_{ix} = -\varepsilon u_1f_i+\varepsilon u_2e_i-\varepsilon u_5q_i+\varepsilon u_6p_i+k_i(t), \\ b_{ix} = -\varepsilon u_1f_i+\varepsilon u_2e_i-\varepsilon u_3h_i+\varepsilon u_4g_i+k_i(t), \\ c_{ix} = \varepsilon u_4p_i-\varepsilon u_5h_i, \\ d_{ix} = -u_3q_i+u_6g_i, \\ e_{ix} = -2\lambda e_i+u_1a_i+u_1b_i+\varepsilon u_3p_i-\varepsilon u_5g_i, \\ f_{ix} = 2\lambda f_i-u_2a_i-u_2b_i-u_4q_i+u_6h_i, \\ g_{ix} = -\lambda g_i+u_1q_i+u_3b_i+\varepsilon u_5d_i-u_6e_i, \\ h_{ix} = \lambda h_i-\varepsilon u_2p_i-u_4b_i+\varepsilon u_5f_i-u_6c_i, \\ p_{ix} = -\lambda p_i-u_1h_i+u_3c_i+u_4e_i+u_5a_i, \\ q_{ix} = \lambda q_i+\varepsilon u_2g_i-\varepsilon u_3f_i-\varepsilon u_4d_i-u_6a_i. \end{array} \right. \end{eqnarray}$

(4.4)

By taking initial values

$\begin{eqnarray*} a_0 = \alpha(t), \ b_0 = \beta(t), \ c_0 = d_0 = e_0 = f_0 = g_0 = h_0 = p_0 = q_0 = k_0(t) = 0, \end{eqnarray*}$

one has

$\begin{eqnarray*} & & a_1 = b_1 = k_1(t)x, \ c_1 = \varepsilon\partial^{-1}u_4u_5(\alpha-\beta), d_1 = \partial^{-1}u_3u_6(\beta-\alpha), \ e_1 = \frac{1}{2}u_1(\alpha+\beta), \\ & & f_1 = \frac{1}{2}u_2(\alpha+\beta), \ g_1 = u_3\beta, \ h_1 = u_4\beta, \ p_1 = u_5\alpha, \ q_1 = u_6\alpha, \\ & & e_2 = -\frac{1}{4}u_{1x}(\alpha+\beta)+\frac{\varepsilon}{2}u_3u_5(\alpha-\beta)+u_1k_1(t)x, \\ & & f_2 = \frac{1}{4}u_{2x}(\alpha+\beta)+\frac{1}{2}u_4u_6(\alpha-\beta)+u_2k_1(t)x, \\ & & g_2 = -u_{3x}\beta+\frac{1}{2}u_{1}u_6(\alpha-\beta)+\varepsilon u_5\partial^{-1}u_3u_6(\beta-\alpha)+u_3k_1(t)x, \\ & & h_2 = u_{4x}\beta+\frac{\varepsilon}{2}u_2u_5(\alpha-\beta)+\varepsilon u_6\partial^{-1}u_4u_5(\alpha-\beta)+u_4k_1(t)x, \\ & & p_2 = -u_{5x}\alpha+\frac{1}{2}u_{1}u_4(\alpha-\beta)+\varepsilon u_3\partial^{-1}u_4u_5(\alpha-\beta)+u_5k_1(t)x, \\ & & q_2 = u_{6x}\alpha+\frac{\varepsilon}{2}u_2u_3(\alpha-\beta)+\varepsilon u_4\partial^{-1}u_3u_6(\beta-\alpha)+u_6k_1(t)x, \\ & & \cdots\cdots \end{eqnarray*}$

where $\alpha(t)$ is an integral constant. Noting that

$\begin{eqnarray*} & & V^{(n)}_{2+}\! = \!\sum^{n}_{i = 0}(a_i, b_i, c_i, d_i, e_i, f_i, g_i, h_i, p_i, q_i)^{\mathrm{T}}, \ \ V^{(n)}_{2-}\! = \!\sum^{\infty}_{i = n+1}(a_i, b_i, c_i, d_i, e_i, f_i, g_i, h_i, p_i, q_i)^{\mathrm{T}}, \\ & & \lambda^{(n)}_{+, x} = \sum^{n}_{i = 0}k_i(t)\lambda^{n-i}, \ \ \lambda^{(n)}_{-, x} = \sum^{\infty}_{i = n+1}k_i(t)\lambda^{n-i}, \end{eqnarray*}$

it follows that one has

$\begin{eqnarray*} -V^{(n)}_{2+, x}+\frac{\partial U_2}{\partial \lambda} \lambda^{(n)}_{t, +}+[U_2, V^{(n)}_{2+}] = (0, 0, 0, 0, 2e_{n+1}, -2f_{n+1}, g_{n+1}, -h_{n+1}, p_{n+1}, -q_{n+1})^{\mathrm{T}}. \end{eqnarray*}$

According to (4.4), it is easy to show that we have the recursion relations

$\begin{eqnarray*} \left(\begin{array}{cccccc} 2\varepsilon f_{n+1} \\ 2\varepsilon e_{n+1} \\ 2\varepsilon h_{n+1} \\ 2\varepsilon g_{n+1} \\ 2\varepsilon q_{n+1} \\ 2\varepsilon p_{n+1} \\ \end{array}\right) = L_2\left(\begin{array}{cccccc} 2\varepsilon f_{n} \\ 2\varepsilon e_{n} \\ 2\varepsilon h_{n} \\ 2\varepsilon g_{n} \\ 2\varepsilon q_{n} \\ 2\varepsilon p_{n} \\ \end{array}\right)+\left(\begin{array}{cccccc} 2\varepsilon u_2 \\ 2\varepsilon u_1 \\ 2\varepsilon u_4 \\ 2\varepsilon u_3 \\ 2\varepsilon u_6 \\ 2\varepsilon u_5 \\ \end{array}\right)k_{n}(t)x, \end{eqnarray*}$

where the recurrence operator $L_2$ is defined as

$\begin{eqnarray*} L_2 = \left( \begin{array}{cccccc} l_{11} & l_{12} & l_{13} & l_{14} & l_{15} & l_{16} \\ l_{21} & l_{22} & l_{23} & l_{24} & l_{25} & l_{26} \\ l_{31} & l_{32} & l_{33} & l_{34} & l_{35} & l_{36} \\ l_{41} & l_{42} & l_{43} & l_{44} & l_{45} & l_{46} \\ l_{51} & l_{52} & l_{53} & l_{54} & l_{55} & l_{56} \\ l_{61} & l_{62} & l_{63} & l_{64} & l_{65} & l_{66} \\ \end{array} \right), \end{eqnarray*}$

and

$\begin{eqnarray*} & & l_{11} = \frac{\partial}{2}-\varepsilon u_2\partial^{-1}u_1, \ l_{12} = \varepsilon u_2\partial^{-1}u_2, \ l_{13} = -\frac{\varepsilon}{2}u_2\partial^{-1}u_3-\frac{u_6}{2}, \ l_{14} = \frac{\varepsilon}{2}u_2\partial^{-1}u_4, \\ & & l_{15} = -\frac{\varepsilon}{2}u_2\partial^{-1}u_5+\frac{u_4}{2}, \ l_{16} = \frac{\varepsilon}{2}u_2\partial^{-1}u_6, \ l_{21} = -\varepsilon u_1\partial^{-1}u_1, \ l_{22} = -\frac{\partial}{2}+\varepsilon u_1\partial^{-1}u_2, \\ & & l_{23}\! = \!-\frac{\varepsilon}{2}u_1\partial^{-1}u_3, \ l_{24}\! = \!\frac{\varepsilon}{2}u_1\partial^{-1}u_4-\frac{\varepsilon}{2}u_5, \ l_{25}\! = \!-\frac{\varepsilon}{2}u_1\partial^{-1}u_5, \ l_{26}\! = \!\frac{\varepsilon}{2}u_1\partial^{-1}u_6+\frac{\varepsilon}{2}u_3, \\ & & l_{31} = -\varepsilon u_4\partial^{-1}u_1-\varepsilon u_5, \ l_{32} = \varepsilon u_4\partial^{-1}u_2, \ l_{33} = \partial-\varepsilon u_4\partial^{-1}u_3-\varepsilon u_6\partial^{-1}u_5, \ l_{35} = 0, \\ & & l_{34} = \varepsilon u_4\partial^{-1}u_4, \ l_{36} = \varepsilon u_2+\varepsilon u_6\partial^{-1}u_4, \ l_{41} = -\varepsilon u_3\partial^{-1}u_1, \ l_{42} = \varepsilon u_3\partial^{-1}u_2-u_6, \\ & & l_{43} = -\varepsilon u_3\partial^{-1}u_3, \ l_{44} = -\partial+\varepsilon u_3\partial^{-1}u_4+\varepsilon u_5\partial^{-1}u_6, \ l_{45} = u_1-\varepsilon u_5\partial^{-1}u_3, \ l_{46} = 0, \\ & & l_{51} = \varepsilon u_3-\varepsilon u_6\partial^{-1}u_1, \ l_{52} = \varepsilon u_6\partial^{-1}u_2, \ l_{53} = 0, \ l_{54} = -\varepsilon u_2+\varepsilon u_4\partial^{-1}u_6, \\ & & l_{55}\! = \!\partial\!-\!\varepsilon u_4\partial^{-1}u_3\!-\!\varepsilon u_6\partial^{-1}u_5, \ l_{56}\! = \!\varepsilon u_6\partial^{-1}u_6, \ l_{61}\! = \!-\varepsilon u_5\partial^{-1}u_1, \ l_{62}\! = \!u_4\!+\!\varepsilon u_5\partial^{-1}u_2, \\ & & l_{63} = -u_1-\varepsilon u_3\partial^{-1}u_5, \ l_{64} = 0, \ l_{65} = -\varepsilon u_5\partial^{-1}u_5, \ l_{66} = -\partial+\varepsilon u_3\partial^{-1}u_4+\varepsilon u_5\partial^{-1}u_6. \end{eqnarray*}$

Taking $V_2^{(n)} = V_{2, +}^{(n)}$ , the zero curvature equation

$\begin{eqnarray*} -V^{(n)}_{2x}+\frac{\partial U_2}{\partial u} u_t+\frac{\partial U_2}{\partial \lambda} \lambda^{(n)}_{t, +}+[U_2, V_2^{(n)}] = 0 \end{eqnarray*}$

leads to the nonisospectral hierarchy

$\begin{eqnarray} u_{t_n} & = & \left(\begin{array}{cccccc} u_1 \\ u_2 \\ u_3 \\ u_4 \\ u_5 \\ u_6 \\ \end{array}\right)_{t_n}\! = \!\left(\begin{array}{cccccc} 2e_{n+1} \\ -2f_{n+1} \\ g_{n+1} \\ -h_{n+1} \\ p_{n+1} \\ -q_{n+1} \\ \end{array}\right)\! = \!J_2\left(\begin{array}{cccccc} 2\varepsilon f_{n+1} \\ 2\varepsilon e_{n+1} \\ 2\varepsilon h_{n+1} \\ 2\varepsilon g_{n+1} \\ 2\varepsilon q_{n+1} \\ 2\varepsilon p_{n+1} \\ \end{array}\right) \\ & = & J_2L_2\left(\begin{array}{cccccc} 2\varepsilon f_{n} \\ 2\varepsilon e_{n} \\ 2\varepsilon h_{n} \\ 2\varepsilon g_{n} \\ 2\varepsilon q_{n} \\ 2\varepsilon p_{n} \\ \end{array}\right)+J_2\left(\begin{array}{cccccc} 2\varepsilon u_2 \\ 2\varepsilon u_1 \\ 2\varepsilon u_4 \\ 2\varepsilon u_3 \\ 2\varepsilon u_6 \\ 2\varepsilon u_5 \\ \end{array}\right)k_{n}(t)x, \end{eqnarray}$

(4.5)

where the Hamiltonian operator $J_2$ is

$\begin{eqnarray*} J_2 = \frac{1}{\varepsilon}\left( \begin{array}{cccccc} 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & -\frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2} \\ 0 & 0 & 0 & 0 & -\frac{1}{2} & 0 \\ \end{array} \right). \end{eqnarray*}$

To furnish Hamiltonian structures, we use the trace identity, and have

$\begin{eqnarray*} & & \langle V_2, \frac{\partial U_2}{\partial \lambda}\rangle = 2a+2b, \ \langle V_2, \frac{\partial U_2}{\partial u_1}\rangle = 2\varepsilon f, \ \langle V_2, \frac{\partial U_2}{\partial u_2}\rangle = 2\varepsilon e, \\ & & \langle V_2, \frac{\partial U_2}{\partial u_3}\rangle = 2\varepsilon h, \ \langle V_2, \frac{\partial U_2}{\partial u_4}\rangle = 2\varepsilon g, \ \langle V_2, \frac{\partial U_2}{\partial u_5}\rangle = 2\varepsilon q, \ \langle V_2, \frac{\partial U_2}{\partial u_6}\rangle = 2\varepsilon p. \end{eqnarray*}$

Substituting the above formulas into the trace identity and balancing coefficients of each power of $\lambda$ gives rise to

$\begin{eqnarray*} \frac{\delta}{\delta u}\displaystyle {\int}(2a_{n+1}+2b_{n+1})dx = (\gamma-n) \left(\begin{array}{cccccc} 2\varepsilon f_n \\ 2\varepsilon e_n \\ 2\varepsilon h_n \\ 2\varepsilon g_n \\ 2\varepsilon q_n \\ 2\varepsilon p_n \\ \end{array}\right). \end{eqnarray*}$

Taking $n = 1$ , gives $\gamma = 0$ . Thus, we see

$u_t = J_2\frac{\delta H_{n+1}^{(2)}}{\delta u} = J_2L_2\frac{\delta H_{n}^{(2)}}{\delta u}+J_2M_2k_n(t)x, \ \ \ H_{n+1}^{(2)} = -2\displaystyle {\int}(\frac{a_{n+2}+b_{n+2}}{n+1})dx, \ \ n\geq 0,$

where $M_2 = (2\varepsilon u_2, 2\varepsilon u_1, 2\varepsilon u_4, 2\varepsilon u_3, 2\varepsilon u_6, 2\varepsilon u_5)^{\mathrm{T}}$ .

5. The nonisospectral integrable coupling hierarchy

In this section, we will construct the nonisospectral integrable coupling hierarchies associated with Lie algebra $\mathfrak{so}(5)$ . We consider the nonisospectral problem

$\begin{eqnarray*} \left\{ \begin{array}{ll} \varphi_x = \bar{U}_2\varphi, \ \bar{U}_2 = \left( \begin{array}{cc} U'_3 & \varepsilon U'_4 \\ U'_4 & U'_3 \\ \end{array} \right), \\ \varphi_t = \bar{V}_2\varphi, \ \bar{V}_2 = \left( \begin{array}{cc} V'_3 & \varepsilon V'_4 \\ V'_4 & V'_3 \\ \end{array} \right), \\ \lambda_t = \sum\nolimits_{i\geq 0}k_i(t)\lambda^{-i}, \\ \end{array} \right. \end{eqnarray*}$

where

$\begin{eqnarray} & & U'_3 = \left( \begin{array}{ccccc} 0 & u_5 & u_3 & u_6 & u_4 \\ u_6 & \lambda & 0 & 0 & u_2 \\ u_4 & 0 & \lambda & -u_2 & 0 \\ u_5 & 0 & -u_1 & -\lambda & 0 \\ u_3 & u_1 & 0 & 0 & -\lambda \\ \end{array} \right), U'_4 = \left( \begin{array}{ccccc} 0 & u'_5 & u'_3 & u'_6 & u'_4 \\ u'_6 & 0 & 0 & 0 & u'_2 \\ u'_4 & 0 & 0 & -u'_2 & 0 \\ u'_5 & 0 & -u'_1 & 0 & 0 \\ u'_3 & u'_1 & 0 & 0 & 0 \\ \end{array} \right), \\ & & V'_3 = \left( \begin{array}{ccccc} 0 & p & g & q & h \\ q & a & d & 0 & f \\ h & c & b & -f & 0 \\ p & 0 & -e & -a & -c \\ g & e & 0 & -d & -b \\ \end{array} \right), V'_4 = \left( \begin{array}{ccccc} 0 & p' & g' & q' & h' \\ q' & a' & d' & 0 & f' \\ h' & c' & b' & -f' & 0 \\ p' & 0 & -e' & -a' & -c' \\ g' & e' & 0 & -d' & -b' \\ \end{array} \right), \end{eqnarray}$

(5.1)

where $u_1, u_2, \cdots, u_6, u'_1, u'_2, \cdots, u'_6,$ and $a, b, c, d, e, f, g, h, p, q, a', b', c', d', e', f', g', h', p', q'$ are different from (2.2), (3.3), and (4.2).

We solve the stationary zero curvature equation by means of

$\begin{eqnarray} \bar{V}_{2x} = \frac{\partial \bar{U}_2}{\partial \lambda}\lambda_t+[\bar{U}_2, \bar{V}_2] \end{eqnarray}$

(5.2)

which yields

$\begin{eqnarray} \left\{ \begin{array}{llllllllll} a_{ix} = -u_1f_i+u_2e_i-u_5q_i+u_6p_i-\varepsilon u'_1f'_i+\varepsilon u'_2e'_i-\varepsilon u'_5q'_i+\varepsilon u'_6p'_i+k_i(t), \\ b_{ix} = -u_1f_i+u_2e_i-u_3h_i+u_4g_i-\varepsilon u'_1f'_i+\varepsilon u'_2e'_i-\varepsilon u'_3h'_i+\varepsilon u'_4g'_i+k_i(t), \\ c_{ix} = u_4p_i-u_5h_i+\varepsilon u'_4p'_i-\varepsilon u'_5h'_i, \\ d_{ix} = -u_3q_i+u_6g_i-\varepsilon u'_3q'_i+\varepsilon u'_6g'_i, \\ e_{ix} = -2\lambda e_i+u_1a_i+u_1b_i+u_3p_i-u_5g_i+\varepsilon u'_1a'_i+\varepsilon u'_1b'_i+\varepsilon u'_3p'_i-\varepsilon u'_5g'_i, \\ f_{ix} = 2\lambda f_i-u_2a_i-u_2b_i-u_4q_i+u_6h_i-\varepsilon u'_2a'_i-\varepsilon u'_2b'_i-\varepsilon u'_4q'_i+\varepsilon u'_6h'_i, \\ g_{ix} = -\lambda g_i+u_1q_i+u_3b_i+u_5d_i-u_6e_i+\varepsilon u'_1q'_i+\varepsilon u'_3b'_i+\varepsilon u'_5d'_i-\varepsilon u'_6e'_i, \\ h_{ix} = \lambda h_i-u_2p_i-u_4b_i+u_5f_i-u_6c_i-\varepsilon u'_2p'_i-\varepsilon u'_4b'_i+\varepsilon u'_5f'_i-\varepsilon u'_6c'_i, \\ p_{ix} = -\lambda p_i-u_1h_i+u_3c_i+u_4e_i+u_5a_i-\varepsilon u'_1h'_i+\varepsilon u'_3c'_i+\varepsilon u'_4e'_i+\varepsilon u'_5a'_i, \\ q_{ix} = \lambda q_i+u_2g_i-u_3f_i-u_4d_i-u_6a_i+\varepsilon u'_2g'_i-\varepsilon u'_3f'_i-\varepsilon u'_4d'_i-\varepsilon u'_6a'_i, \end{array} \right. \end{eqnarray}$

(5.3)

and

$\begin{eqnarray} \left\{ \begin{array}{llllllllll} a'_{ix} = -u_1f'_i+u_2e'_i-u_5q'_i+u_6p'_i-u'_1f_i+u'_2e_i-u'_5q_i+u'_6p_i+k_i(t), \\ b'_{ix} = -u_1f'_i+u_2e'_i-u_3h'_i+u_4g'_i-u'_1f_i+u'_2e_i-u'_3h_i+u'_4g_i+k_i(t), \\ c'_{ix} = u_4p'_i-u_5h'_i+u'_4p_i-u'_5h_i, \\ d'_{ix} = -u_3q'_i+u_6g'_i-u'_3q_i+u'_6g_i, \\ e'_{ix} = -2\lambda e'_i+u_1a'_i+u_1b'_i+u_3p'_i-u_5g'_i+u'_1a_i+u'_1b_i+ u'_3p_i-u'_5g_i, \\ f'_{ix} = 2\lambda f'_i-u_2a'_i-u_2b'_i-u_4q'_i+u_6h'_i-u'_2a_i-u'_2b_i-u'_4q_i+u'_6h_i, \\ g'_{ix} = -\lambda g'_i+u_1q'_i+u_3b'_i+u_5d'_i-u_6e'_i+u'_1q_i+u'_3b_i+ u'_5d_i-u'_6e_i, \\ h'_{ix} = \lambda h'_i-u_2p'_i-u_4b'_i+u_5f'_i-u_6c'_i-u'_2p_i-u'_4b_i+ u'_5f_i-u'_6c_i, \\ p'_{ix} = -\lambda p'_i-u_1h'_i+u_3c'_i+u_4e'_i+u_5a'_i-u'_1h_i+u'_3c_i+ u'_4e_i+u'_5a_i, \\ q'_{ix} = \lambda q'_i+u_2g'_i-u_3f'_i-u_4d'_i-u_6a'_i+u'_2g_i-u'_3f_i- u'_4d_i-u'_6a_i. \end{array} \right. \end{eqnarray}$

(5.4)

By taking initial values

one has

$\begin{align*} a_1 = &b_1 = k_1(t)x, \ e_1 = \frac{1}{2}u_1(\alpha+\beta)+\frac{\varepsilon}{2}u'_1(\alpha'+\beta'), \ f_1 = \frac{1}{2}u_2(\alpha+\beta)+\frac{\varepsilon}{2}u'_2(\alpha'+\beta'), \\ g_1 = &u_3\beta+\varepsilon u'_3\beta', \ h_1 = u_4\beta+\varepsilon u'_4\beta', \ p_1 = u_5\alpha+\varepsilon u'_5\alpha', \ q_1 = u_6\alpha+\varepsilon u'_6\alpha', \\ c_1 = &\partial^{-1}(u_4u_5+\varepsilon u'_4u'_5)(\alpha-\beta)+\varepsilon\partial^{-1}(u_4u'_5+u'_4u_5)(\alpha'-\beta'), \\ d_1 = &\partial^{-1}(u_3u_6+\varepsilon u'_3u'_6)(\beta-\alpha)+\varepsilon\partial^{-1}(u_3u'_6+u'_3u_6)(\beta'-\alpha'), \\ a'_1 = &b'_1 = k_1(t)x, \ e'_1 = \frac{1}{2}u'_1(\alpha+\beta)+\frac{1}{2}u_1(\alpha'+\beta'), \ f'_1 = \frac{1}{2}u'_2(\alpha+\beta)+\frac{1}{2}u_2(\alpha'+\beta'), \\ g'_1 = &u'_3\beta+u_3\beta', \ h'_1 = u'_4\beta+u_4\beta', \ p'_1 = u'_5\alpha+u_5\alpha', \ q'_1 = u'_6\alpha+u_6\alpha', \\ c'_1 = &\partial^{-1}(u_4u'_5+ u'_4u_5)(\alpha-\beta)+\partial^{-1}(u_4u_5+\varepsilon u'_4u'_5)(\alpha'-\beta'), \\ d'_1 = &\partial^{-1}(u_3u'_6+ u'_3u_6)(\beta-\alpha)+\partial^{-1}(u_3u_6+\varepsilon u'_3u'_6)(\beta'-\alpha'), \\ & \cdots\cdots \end{align*}$

where $\alpha(t), \ \beta(t)$ are integral constants.

From the nonisospectral zero curvature equation

$\begin{eqnarray*} \frac{\partial \bar{U}_2}{\partial u}u_t+\frac{\partial \bar{U}_2}{\partial \lambda}\lambda_t-\bar{V}^{(n)}_{2x}+[\bar{U}_2, \bar{V}^{(n)}_2] = 0, \end{eqnarray*}$

we can obtain the following integrable couplings

$\begin{eqnarray*} \bar{u}_{t_n} = \left(\begin{array}{cc} \bar{u}_3 \\ \bar{u}_4 \\ \end{array}\right)_{t_n}, \ \bar{u}_{3t_n} = \left(\begin{array}{cccccc} u_1 \\ u_2 \\ u_3 \\ u_4 \\ u_5 \\ u_6 \\ \end{array}\right)_{t_n} = \left(\begin{array}{cccccc} -2e_{n+1} \\ 2f_{n+1} \\ -2g_{n+1} \\ 2h_{n+1} \\ -2p_{n+1} \\ 2q_{n+1} \\ \end{array}\right), \ \bar{u}_{4t_n} = \left(\begin{array}{cccccc} u'_1 \\ u'_2 \\ u'_3 \\ u'_4 \\ u'_5 \\ u'_6 \\ \end{array}\right)_{t_n} = \left(\begin{array}{cccccc} -2e'_{n+1} \\ 2f'_{n+1} \\ -2g'_{n+1} \\ 2h'_{n+1} \\ -2p'_{n+1} \\ 2q'_{n+1} \\ \end{array}\right). \end{eqnarray*}$

To furnish Hamiltonian structures, we use the trace identity, and have

$\begin{eqnarray*} & & \langle V'_3, \frac{\partial U'_4}{\partial \lambda}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial \lambda}\rangle = 2(a'+b'), \ \langle V'_3, \frac{\partial U'_4}{\partial u_1}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u_1}\rangle = 2f', \\ & & \langle V'_3, \frac{\partial U'_4}{\partial u_2}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u_2}\rangle = 2e', \ \langle V'_3, \frac{\partial U'_4}{\partial u_3}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u_3}\rangle = 2h', \\ & & \langle V'_3, \frac{\partial U'_4}{\partial u_4}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u_4}\rangle = 2g', \ \langle V'_3, \frac{\partial U'_4}{\partial u_5}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u_5}\rangle = 2q', \\ & & \langle V'_3, \frac{\partial U'_4}{\partial u_6}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u_6}\rangle = 2p', \ \langle V'_3, \frac{\partial U'_4}{\partial u'_1}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u'_1}\rangle = 2f, \\ & & \langle V'_3, \frac{\partial U'_4}{\partial u'_2}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u'_2}\rangle = 2e, \ \langle V'_3, \frac{\partial U'_4}{\partial u'_3}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u'_3}\rangle = 2h, \\ & & \langle V'_3, \frac{\partial U'_4}{\partial u'_4}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u'_4}\rangle = 2g, \ \langle V'_3, \frac{\partial U'_4}{\partial u'_5}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u'_5}\rangle = 2q, \\ & & \langle V'_3, \frac{\partial U'_4}{\partial u'_6}\rangle+\langle V'_4, \frac{\partial U'_3}{\partial u'_6}\rangle = 2p. \end{eqnarray*}$

Substituting the above formulas into the trace identity, and balancing coefficients of each power of $\lambda$ , we give the first form

$\begin{eqnarray} \frac{\delta}{\delta \bar{u}}\displaystyle {\int} 2(a'_{n+1}+b'_{n+1})dx = (\gamma-n) \left(\begin{array}{cccccc} M_3 \\ M_4 \\ \end{array}\right), \end{eqnarray}$

(5.5)

where

$\begin{eqnarray*} M_3 = \left(\begin{array}{cccccc} 2f'_n \\ 2e'_n \\ 2h'_n \\ 2g'_n \\ 2q'_n \\ 2p'_n \\ \end{array}\right), \ M_4 = \left(\begin{array}{cccccc} 2f_n \\ 2e_n \\ 2h_n \\ 2g_n \\ 2q_n \\ 2p_n \\ \end{array}\right). \end{eqnarray*}$

At the same time, we also have

$\begin{eqnarray*} & & \langle V'_3, \frac{\partial U'_3}{\partial \lambda}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial \lambda}\rangle = 2(a+b), \ \langle V'_3, \frac{\partial U'_3}{\partial u_1}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u_1}\rangle = 2f, \\ & & \langle V'_3, \frac{\partial U'_3}{\partial u_2}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u_2}\rangle = 2e, \ \langle V'_3, \frac{\partial U'_3}{\partial u_3}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u_3}\rangle = 2h, \\ & & \langle V'_3, \frac{\partial U'_3}{\partial u_4}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u_4}\rangle = 2g, \ \langle V'_3, \frac{\partial U'_3}{\partial u_5}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u_5}\rangle = 2q, \\ & & \langle V'_3, \frac{\partial U'_3}{\partial u_6}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u_6}\rangle = 2p, \ \langle V'_3, \frac{\partial U'_3}{\partial u'_1}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u'_1}\rangle = 2\varepsilon f', \\ & & \langle V'_3, \frac{\partial U'_3}{\partial u'_2}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u'_2}\rangle = 2\varepsilon e', \ \langle V'_3, \frac{\partial U'_3}{\partial u'_3}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u'_3}\rangle = 2\varepsilon h', \\ & & \langle V'_3, \frac{\partial U'_3}{\partial u'_4}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u'_4}\rangle = 2\varepsilon g', \ \langle V'_3, \frac{\partial U'_3}{\partial u'_5}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u'_5}\rangle = 2\varepsilon q', \\ & & \langle V'_3, \frac{\partial U'_3}{\partial u'_6}\rangle+\varepsilon \langle V'_4, \frac{\partial U'_4}{\partial u'_6}\rangle = 2\varepsilon p'. \end{eqnarray*}$

Substituting the above formulas into the trace identity, and balancing coefficients of each power of $\lambda$ , we give the second form

$\begin{eqnarray*} \frac{\delta}{\delta u}\displaystyle {\int} 2(a_{n+1}+b_{n+1})dx = (\gamma-n) \left(\begin{array}{cccccc} M_4 \\ \varepsilon M_3 \\ \end{array}\right), \end{eqnarray*}$

where $M_3, \ M_4$ are defined as (5.5).

So, we can obtain the Hamiltonian structures of integrable couplings, that consists of the following two components. The first component has the form

$\begin{eqnarray*} \bar{u}_{t_n} = \left(\begin{array}{cc} \bar{u}_3 \\ \bar{u}_4 \\ \end{array}\right)_{t_n} = \bar{J}_3\left(\begin{array}{cc} K_3 \\ K_4 \\ \end{array}\right) = \bar{J}_3\frac{\delta \bar{H}_{3, m}}{\delta \bar{u}}, \ K_3 = \left(\begin{array}{cccccc} 2f'_{n+1} \\ 2e'_{n+1} \\ 2h'_{n+1} \\ 2g'_{n+1} \\ 2q'_{n+1} \\ 2p'_{n+1} \\ \end{array}\right), \ K_4 = \left(\begin{array}{cccccc} 2f_{n+1} \\ 2e_{n+1} \\ 2h_{n+1} \\ 2g_{n+1} \\ 2q_{n+1} \\ 2p_{n+1} \\ \end{array}\right), \end{eqnarray*}$

where $\bar{J}_3 = \left(\begin{array}{cc} O & J_2 \\ J_2 & O \end{array} \right)$ , and $J_2$ is defined as (4.5). The second component has the form

$\begin{eqnarray*} \bar{u}_{t_n} = \left(\begin{array}{cc} \bar{u}_3 \\ \bar{u}_4 \\ \end{array}\right)_{t_n} = \bar{J}_4\left(\begin{array}{cc} K_4 \\ \varepsilon K_3 \\ \end{array}\right) = \bar{J}_4\frac{\delta \bar{H}_{4, m}}{\delta \bar{u}}, \ \ \bar{J}_4 = \left( \begin{array}{cc} J_2 & O \\ O & \frac{1}{\varepsilon}J_2 \end{array} \right). \end{eqnarray*}$

From (5.3) and (5.4), we obtain the following recursion relations

$\begin{eqnarray*} \left(\begin{array}{cccccccccccc} -2e_{n+1} \\ 2f_{n+1} \\ -2g_{n+1} \\ 2h_{n+1} \\ -2p_{n+1} \\ 2q_{n+1} \\ -2e'_{n+1} \\ 2f'_{n+1} \\ -2g'_{n+1} \\ 2h'_{n+1} \\ -2p'_{n+1} \\ 2q'_{n+1} \\ \end{array}\right) = \bar{L}_2\left(\begin{array}{cccccccccccc} -2e_{n} \\ 2f_{n} \\ -2g_{n} \\ 2h_{n} \\ -2p_{n} \\ 2q_{n} \\ -2e'_{n} \\ 2f'_{n} \\ -2g'_{n} \\ 2h'_{n} \\ -2p'_{n} \\ 2q'_{n} \\ \end{array}\right)+\left(\begin{array}{cccccccccccc} 2u_2+2u'_2 \\ 2u_1+2u'_1 \\ 2u_4+2u'_4 \\ 2u_3+2u'_3 \\ 2u_6+2u'_6 \\ 2u_5+2u'_5 \\ 2u_2+2\varepsilon u'_2 \\ 2u_1+2\varepsilon u'_1 \\ 2u_4+2\varepsilon u'_4 \\ 2u_3+2\varepsilon u'_3 \\ 2u_6+2\varepsilon u'_6 \\ 2u_5+2\varepsilon u'_5 \\ \end{array}\right)k_n(t)x, \end{eqnarray*}$

where the recurrence operator $\bar{L}_2$ is defined as

$\begin{eqnarray*} \bar{L}_2\! = \!\left(\! \begin{array}{cc} L'_3 & L'_4 \\ \frac{1}{\varepsilon}L'_4 & L'_3 \\ \end{array} \right), \ L'_3\! = \!\left( \begin{array}{cccccc} l_{11} & l_{12} & l_{13} & l_{14} & l_{15} & l_{16} \\ l_{21} & l_{22} & l_{23} & l_{24} & l_{25} & l_{26} \\ l_{31} & l_{32} & l_{33} & l_{34} & l_{35} & l_{36} \\ l_{41} & l_{42} & l_{43} & l_{44} & l_{45} & l_{46} \\ l_{51} & l_{52} & l_{53} & l_{54} & l_{55} & l_{56} \\ l_{61} & l_{62} & l_{63} & l_{64} & l_{65} & l_{66} \\ \end{array} \right), \ L'_4\! = \!\left( \begin{array}{cccccc} l'_{11} & l'_{12} & l'_{13} & l'_{14} & l'_{15} & l'_{16} \\ l'_{21} & l'_{22} & l'_{23} & l'_{24} & l'_{25} & l'_{26} \\ l'_{31} & l'_{32} & l'_{33} & l'_{34} & l'_{35} & l'_{36} \\ l'_{41} & l'_{42} & l'_{43} & l'_{44} & l'_{45} & l'_{46} \\ l'_{51} & l'_{52} & l'_{53} & l'_{54} & l'_{55} & l'_{56} \\ l'_{61} & l'_{62} & l'_{63} & l'_{64} & l'_{65} & l'_{66} \\ \end{array} \right), \end{eqnarray*}$

and

$\begin{eqnarray*} & & l_{11}\! = \!\frac{\partial}{2}\!-\! u_2\partial^{-1}u_1\!-\!\varepsilon u'_2\partial^{-1}u'_1, \ l_{12}\! = \! u_2\partial^{-1}u_2\!+\!\varepsilon u'_2\partial^{-1}u'_2, \ l_{13}\! = \!-\frac{1}{2}(u_2\partial^{-1}u_3\!+\!\varepsilon u'_2\partial^{-1}u'_3\!+\! u_6), \\ & & l_{14}\! = \!\frac{1}{2}(u_2\partial^{-1}u_4\!+\!\varepsilon u'_2\partial^{-1}u'_4), l_{15}\! = \!-\frac{1}{2}(u_2\partial^{-1}u_5\!+\!\varepsilon u'_2\partial^{-1}u'_5-u_4), l_{16}\! = \!\frac{1}{2}(u_2\partial^{-1}u_6\!+\!\varepsilon u'_2\partial^{-1}u'_6), \\ & & l'_{11} = -u_2\partial^{-1}u'_1-u'_2\partial^{-1}u_1, \ l'_{12} = u_2\partial^{-1}u'_2+u'_2\partial^{-1}u_2, \ l'_{13} = -\frac{1}{2}(u_2\partial^{-1}u'_3+u'_2\partial^{-1}u_3+u'_6), \\ & & l'_{14}\! = \!\frac{1}{2}(u_2\partial^{-1}u'_4\!+\! u'_2\partial^{-1}u_4), \ l'_{15}\! = \!-\frac{1}{2}(u_2\partial^{-1}u'_5\!+\! u'_2\partial^{-1}u_5\!-\! u'_4), \ l'_{16}\! = \!\frac{1}{2}(u_2\partial^{-1}u'_6\!+\! u'_2\partial^{-1}u_6), \\ & & l_{21}\! = \!-u_1\partial^{-1}u_1\!-\!\varepsilon u'_1\partial^{-1}u'_1, \ l_{22}\! = \!-\frac{\partial}{2}\!+\! u_1\partial^{-1}u_2\!+\!\varepsilon u'_1\partial^{-1}u'_2, \ l_{23}\! = \!-\frac{1}{2}(u_1\partial^{-1}u_3\!+\!\varepsilon u'_1\partial^{-1}u'_3), \\ & & l_{24}\!\! = \!\!\frac{1}{2}(u_1\partial^{-1}u_4\!\!+\!\!\varepsilon u'_1\partial^{-1}u'_4\!\!-\!\! u_5), l_{25}\!\! = \!\!-\frac{1}{2}(u_1\partial^{-1}u_5\!\!+\!\!\varepsilon u'_1\partial^{-1}u'_5), l_{26}\!\! = \!\!\frac{1}{2}(u_1\partial^{-1}u_6\!\!+\!\!\varepsilon u'_1\partial^{-1}u'_6\!\!+\!\! u_3), \\ & & l'_{21} = -u_1\partial^{-1}u'_1-u'_1\partial^{-1}u_1, \ l'_{22} = u_1\partial^{-1}u'_2+u'_1\partial^{-1}u_2, \ l'_{23} = -\frac{1}{2}(u_1\partial^{-1}u'_3+u'_1\partial^{-1}u_3), \\ & & l'_{24}\!\! = \!\!\frac{1}{2}(u_1\partial^{-1}u'_4\!\!+\!\! u'_1\partial^{-1}u_4\!\!-\!\! u'_5), l'_{25}\!\! = \!\!-\frac{1}{2}(u_1\partial^{-1}u'_5\!\!+\!\! u'_1\partial^{-1}u_5), l'_{26}\!\! = \!\!\frac{1}{2}(u_1\partial^{-1}u'_6\!\!+\!\! u'_1\partial^{-1}u_6\!\!+\!\! u'_3), \\ & & l_{31} = -u_4\partial^{-1}u_1-\varepsilon u'_4\partial^{-1}u'_1-u_5, \ l_{32} = u_4\partial^{-1}u_2+\varepsilon u'_4\partial^{-1}u'_2, \ l_{35} = l'_{35} = 0, \end{eqnarray*}$

$\begin{eqnarray*} & & l_{33} = \partial-u_4\partial^{-1}u_3-u_6\partial^{-1}u_5-\varepsilon u'_4\partial^{-1}u'_3-\varepsilon u'_6\partial^{-1}u'_5, \ l_{34} = u_4\partial^{-1}u_4+\varepsilon u'_4\partial^{-1}u'_4, \\ & & l_{36}\! = \! u_6\partial^{-1}u_4\!+\!\varepsilon u'_6\partial^{-1}u'_4\!+\! u_2, l'_{31}\! = \!-u_4\partial^{-1}u'_1\!-\! u'_4\partial^{-1}u_1\!-\! u'_5, l'_{32}\! = \! u_4\partial^{-1}u'_2\!+\! u'_4\partial^{-1}u_2, \\ & & l'_{33} = -u_4\partial^{-1}u'_3-u_6\partial^{-1}u'_5-u'_4\partial^{-1}u_3- u'_6\partial^{-1}u_5, \ l'_{34} = u_4\partial^{-1}u'_4+u'_4\partial^{-1}u_4, \\ & & l'_{36}\! = \! u_6\partial^{-1}u'_4\!+\! u'_6\partial^{-1}u_4\!+\! u'_2, l_{41}\! = \!-u_3\partial^{-1}u_1\!-\!\varepsilon u'_3\partial^{-1}u'_1, l_{42}\! = \! u_3\partial^{-1}u_2\!+\!\varepsilon u'_3\partial^{-1}u'_2\!-\! u_6, \\ & & l_{43} = -u_3\partial^{-1}u_3-\varepsilon u'_3\partial^{-1}u'_3, \ l_{44} = -\partial+u_3\partial^{-1}u_4+u_5\partial^{-1}u_6+\varepsilon u'_3\partial^{-1}u'_4+\varepsilon u'_5\partial^{-1}u'_6, \\ & & l_{45} = -u_5\partial^{-1}u_3+\varepsilon u'_5\partial^{-1}u'_3+u_1, \ l'_{41} = -u_3\partial^{-1}u'_1-u'_3\partial^{-1}u_1, \ l_{46} = l'_{46} = 0, \\ & & l'_{42} = u_3\partial^{-1}u'_2+u'_3\partial^{-1}u_2-u'_6, \ l'_{43} = -u_3\partial^{-1}u'_3-u'_3\partial^{-1}u_3, \ l_{53} = l'_{53} = 0, \\ & & l'_{44} = u_3\partial^{-1}u'_4+u_5\partial^{-1}u'_6+u'_3\partial^{-1}u_4+ u'_5\partial^{-1}u_6, \ l'_{45} = -u_5\partial^{-1}u'_3+u'_5\partial^{-1}u_3+u'_1, \\ & & l_{51}\! = \!-u_6\partial^{-1}u_1\!-\!\varepsilon u'_6\partial^{-1}u'_1\!+\!u_3, l_{52}\! = \! u_6\partial^{-1}u_2\!+\!\varepsilon u_6\partial^{-1}u'_2, l_{54}\! = \! u_4\partial^{-1}u_6\!+\!\varepsilon u'_4\partial^{-1}u'_6\!-\! u_2, \\ & & l_{55} = \partial-u_4\partial^{-1}u_3-u_6\partial^{-1}u_5-\varepsilon u'_4\partial^{-1}u'_3-\varepsilon u'_6\partial^{-1}u'_5, \ l_{56} = u_6\partial^{-1}u_6+\varepsilon u'_6\partial^{-1}u'_6, \\ & & l'_{51}\! = \!-u_6\partial^{-1}u'_1\!-\! u'_6\partial^{-1}u_1\!+\! u'_3, \ l'_{52}\! = \! u_6\partial^{-1}u'_2\!+\! u'_6\partial^{-1}u_2, \ l'_{54}\! = \! u_4\partial^{-1}u'_6\!+\! u'_4\partial^{-1}u_6\!-\! u'_2, \\ & & l'_{55} = -u_4\partial^{-1}u'_3-u_6\partial^{-1}u'_5-u'_4\partial^{-1}u_3- u'_6\partial^{-1}u_5, \ l'_{56} = u_6\partial^{-1}u'_6+u'_6\partial^{-1}u_6, \\ & & l_{61}\! = \!-u_5\partial^{-1}u_1\!-\!\varepsilon u'_5\partial^{-1}u'_1, l_{62}\! = \! u_5\partial^{-1}u_2\!+\!\varepsilon u_5\partial^{-1}u'_2\!+\! u_4, l_{63}\! = \!\!-u_3\partial^{-1}u_5\!\!-\!\!\varepsilon u'_3\partial^{-1}u'_5\!\!-\!\! u_1, \\ & & l_{64}\! = \!0, \ l_{65}\! = \!-u_5\partial^{-1}u_5\!-\!\varepsilon u'_5\partial^{-1}u'_5, \ l_{66}\! = \!-\partial\!+\! u_3\partial^{-1}u_4\!+\! u_5\partial^{-1}u_6\!+\!\varepsilon u'_3\partial^{-1}u'_4\!+\!\varepsilon u'_5\partial^{-1}u'_6, \\ & & l'_{61}\! = \!-u_5\partial^{-1}u'_1\!-\! u'_5\partial^{-1}u_1, \ l'_{62}\! = \! u_5\partial^{-1}u'_2\!+\! u'_5\partial^{-1}u_2\!+\! u'_4, \ l'_{63}\! = \!-u_3\partial^{-1}u'_5\!-\! u'_3\partial^{-1}u_5\!-\! u'_1, \\ & & l'_{64} = 0, \ l'_{65} = -u_5\partial^{-1}u'_5-u'_5\partial^{-1}u_5, \ l'_{66} = u_3\partial^{-1}u'_4+u_5\partial^{-1}u'_6+u'_3\partial^{-1}u_4+ u'_5\partial^{-1}u_6. \end{eqnarray*}$

6. The nonisospectral integrable hierarchy associated with Lie algebra $\mathfrak{so}(3, 2)$

6.1. The generalized Lie algebra $G\mathfrak{so}(3, 2)$

Lie algebra $\mathfrak{so}(3, 2)$ is defined as ^[31]

$\mathfrak{so}(3, 2) = \{x\in \mathfrak{gl}(5, \mathbb{R}\mid x = -I_{32}x^{\mathrm{T}}I_{32}, tr(x) = 0\},$

where $I_{32} = \left(\begin{array}{cc} -I_3 & 0 \\ 0 & I_2 \end{array} \right)$ .

So, elements of Lie algebra $\mathfrak{so}(3, 2)$ have the form

$\left( \begin{array}{cc} X_1 & X_2 \\ X^{\mathrm{T}}_2 & X_3 \end{array} \right),$

where $X^{\mathrm{T}}_1 = -X_1, \ X^{\mathrm{T}}_3 = -X_3$ , and $X_1, X_2, X_3$ are $3\times 3$ , $3\times 2$ , and $2\times 2$ real matrices, respectively. It is easy to get the elements of Lie algebra $\mathfrak{so}(3, 2)$ with the form

$\left( \begin{array}{ccccc} 0 & \lambda_1 & \lambda_2 & \lambda_5 & \lambda_6 \\ -\lambda_1 & 0 & \lambda_3 & \lambda_7 & \lambda_8 \\ -\lambda_2 & -\lambda_3 & 0 & \lambda_9 & \lambda_{10} \\ \lambda_5 & \lambda_7 & \lambda_9 & 0 & \lambda_4 \\ \lambda_6 & \lambda_8 & \lambda_{10} & -\lambda_4 & 0 \end{array} \right).$

We can obtain the bases of Lie algebra $\mathfrak{so}(3, 2)$ as

$\begin{equation} \begin{split} E_1 = e_{12}-e_{21}, \ E_2 = e_{13}-e_{31}, \ E_3 = e_{23}-e_{32}, \ E_4 = e_{45}-e_{54}, \ E_5 = e_{14}+e_{41}, \\ E_6 = e_{15}+e_{51}, \ E_7 = e_{24}+e_{42}, \ E_8 = e_{25}+e_{52}, \ E_9 = e_{34}+e_{43}, \ E_{10} = e_{35}+e_{53}, \end{split} \end{equation}$

(6.1)

where $e_{ij}$ is a $5\times 5$ matrix with 1 in the $(i, j)$ -th position and zero elsewhere. Next, we consider the generalized Lie algebra $G\mathfrak{so}(3, 2)$ , that admits a basis set as follows:

$\begin{eqnarray} & & E'_1\! = \! E_7\!+\! E_{10}, \ E'_2 = \!-E_7+E_{10}, \ E'_3 = \!-E_1+E_5, \ E'_4 = \varepsilon E_1+\varepsilon E_5, \ E'_5 = \varepsilon E_2-\!\varepsilon E_6, \\ & & E'_6 = -E_2-E_6, \ E'_7 = \varepsilon E_3-\varepsilon E_4-\varepsilon E_8+\varepsilon E_9, \ E'_8 = -E_3+E_4-E_8+E_9, \\ & & E'_9 = -\varepsilon E_3-\varepsilon E_4+\varepsilon E_8+\varepsilon E_9, \ E'_{10} = E_3+E_4+E_8+E_9, \end{eqnarray}$

(6.2)

where $E_i, i = 1, 2, \cdots, 10$ are defined as (6.1), and satisfy the following commutative relations:

$\begin{eqnarray*} & & [E'_1, E'_2] = 0, \ [E'_1, E'_3] = E'_3, \ [E'_1, E'_4] = -E'_4, \ [E'_1, E'_5] = E'_5, \ [E'_1, E'_6] = -E'_6, \\ & & [E'_1, E'_7] = 2E'_7, \ [E'_1, E'_8] = -2E'_8, \ [E'_1, E'_9] = [E'_1, E'_{10}] = 0, \ [E'_2, E'_3] = -E'_3, \\ & & [E'_2, E'_4] = E'_4, \ [E'_2, E'_5] = E'_5, [E'_2, E'_6] = -E'_6, [E'_2, E'_7] = [E'_2, E'_8] = 0, [E'_2, E'_9] = 2E'_9, \\ & & [E'_2, E'_{10}]\! = \!-2E'_{10}, \ [E'_3, E'_4]\! = \!\varepsilon E'_1\!-\!\varepsilon E'_2, \ [E'_3, E'_5]\! = \!E'_7, \ [E'_3, E'_6]\! = \!-E'_{10}, \ [E'_3, E'_7]\! = \!0, \\ & & [E'_3, E'_8] = -2E'_6, \ [E'_3, E'_9] = 2E'_5, \ [E'_3, E'_{10}] = 0, \ [E'_4, E'_5] = \varepsilon E'_9, \ [E'_4, E'_6] = -\varepsilon E'_8, \\ & & [E'_4, E'_7] = 2\varepsilon E'_5, \ [E'_4, E'_8] = [E'_4, E'_9] = 0, \ [E'_4, E'_{10}] = -2\varepsilon E'_6, \ [E'_5, E'_6] = \varepsilon E'_1+\varepsilon E'_2, \\ & & [E'_5, E'_7] = 0, \ [E'_5, E'_8] = 2E'_4, \ [E'_5, E'_9] = 0, \ [E'_5, E'_{10}] = 2\varepsilon E'_3, \ [E'_6, E'_7] = -2\varepsilon E'_3, \\ & & [E'_6, E'_8] = [E'_6, E'_{10}] = 0, \ [E'_6, E'_9] = -2E'_4, \ [E'_7, E'_8] = 4\varepsilon E'_1, \ [E'_7, E'_9] = [E'_7, E'_{10}] = 0, \\ & &[E'_8, E'_9] = [E'_8, E'_{10}] = 0, \ [E'_9, E'_{10}] = 4\varepsilon E'_2. \end{eqnarray*}$

Consider the nonisospectral problem

$\begin{eqnarray*} \left\{ \begin{array}{ll} \varphi_x = U_3\varphi, \\ \varphi_t = V_3\varphi, \\ \lambda_t = \sum\limits_{i\geq 0}k_i(t)\lambda^{-i}, \\ \end{array} \right. \end{eqnarray*}$

where

$\begin{eqnarray} & & U_3 = \lambda(E'_1+E'_2)+\sqrt{2}u_1E'_5+\frac{\sqrt{2}}{2}u_2E'_6+u_3E'_7+\frac{u_4}{4}E'_8+\frac{u_5}{2}E'_9+\frac{u_6}{2}E'_{10}, \\ & & V_3 = aE'_1+bE'_2+\sqrt{2}cE'_3+\frac{\sqrt{2}}{2}dE'_4+\sqrt{2}eE'_5+\frac{\sqrt{2}}{2}fE'_6+gE'_7+\frac{h}{4}E'_8+\frac{p}{2}E'_9+\frac{q}{2}E'_{10}, \end{eqnarray}$

(6.3)

where $u_1, u_2, \cdots, u_6$ and $a, b, c, d, e, f, g, h, p, q$ are the same as (2.2).

By solving the stationary zero curvature representation

$\begin{eqnarray} V_{3x} = \frac{\partial U_3}{\partial \lambda}\lambda_t+[U_3, V_3], \end{eqnarray}$

(6.4)

we can obtain the same equation as (2.4). This means that integrable hierarchies obtained from the linear nonisospectral problems $\left\{ \begin{array}{ll} \varphi_x = U_3\varphi, \\ \varphi_t = V_3\varphi, \\ \lambda_t = \sum_{i\geq 0}k_i(t)\lambda^{-i}\\ \end{array} \right.$ are the same as (2.5).

If we consider the spectral matrix $U_1$ and time spectral matrix $V_1$ in Lie algebra $G\mathfrak{sp}(4)$ , and choose spectral matrices $U_3$ and $V_3$ in Lie algebra $G\mathfrak{so}(3, 2)$ , then from zero curvature equations $V_{1x} = \frac{\partial U_1}{\partial \lambda}\lambda_t+[U_1, V_1]$ and $V_{3x} = \frac{\partial U_3}{\partial \lambda}\lambda_t+[U_3, V_3]$ , we can obtain the same nonisospectral integrable hierarchies. So, based on $\mathfrak{sp}(4)\cong \mathfrak{so}(3, 2)$ , as long as we select the corresponding spectral problem between Lie algebras $G\mathfrak{sp}(4)$ and $G\mathfrak{so}(3, 2)$ , and we can obtain the same hierarchies.

6.2. The nonisospectral integrable couplings

In this section, we will construct the nonisospectral integrable coupling hierarchies associated with Lie algebra $\mathfrak{so}(3, 2)$ . We consider the nonisospectral problem

$\begin{eqnarray*} \left\{ \begin{array}{ll} \varphi_x = \bar{U}_3\varphi, \\ \varphi_t = \bar{V}_3\varphi, \\ \lambda_t = \sum\nolimits_{i\geq 0}k_i(t)\lambda^{-i}, \\ \end{array} \right. \end{eqnarray*}$

where

$\begin{eqnarray} \bar{U}_3 = \left( \begin{array}{cc} U'_5 & \varepsilon U'_6 \\ U'_6 & U'_5 \\ \end{array} \right), \ \ \bar{V}_3 = \left( \begin{array}{cc} V'_5 & \varepsilon V'_6 \\ V'_6 & V'_5 \\ \end{array} \right), \end{eqnarray}$

(6.5)

and

$\begin{eqnarray} U'_5 & = & 2\lambda E_{10}+\sqrt{2}(u_1-\frac{u_2}{2})E_2-\sqrt{2}(u_1+\frac{u_2}{2})E_6+(u_3-\frac{u_4}{4}-\frac{u_5}{2}+\frac{u_6}{2})E_3 \\ & &+(-u_3+\frac{u_4}{4}-\frac{u_5}{2}+\frac{u_6}{2})E_4+(-u_3-\frac{u_4}{4}+\frac{u_5}{2}+\frac{u_6}{2})E_8+(u_3+\frac{u_4}{4}+\frac{u_5}{2}+\frac{u_6}{2})E_9, \\ U'_6 & = & \sqrt{2}(u'_1-\frac{u'_2}{2})E_2-\sqrt{2}(u'_1+\frac{u'_2}{2})E_6+(u'_3-\frac{u'_4}{4}-\frac{u'_5}{2}+\frac{u'_6}{2})E_3 \\ & &+(-u'_3+\frac{u'_4}{4}-\frac{u'_5}{2}+\frac{u'_6}{2})E_4+(-u'_3-\frac{u'_4}{4}+\frac{u'_5}{2}+\frac{u'_6}{2})E_8+(u'_3+\frac{u'_4}{4}+\frac{u'_5}{2}+\frac{u'_6}{2})E_9, \\ V'_5 & = & \sqrt{2}(\frac{d}{2}-c)E_1+\sqrt{2}(e-\frac{f}{2})E_2+(g-\frac{h}{4}-\frac{p}{2}+\frac{q}{2})E_3+(-g+\frac{h}{4}-\frac{p}{2}+\frac{q}{2})E_4 \\ & &+\sqrt{2}(c+\frac{d}{2})E_5-\sqrt{2}(e+\frac{f}{2})E_6+(a-b)E_7+(-g-\frac{h}{4}+\frac{p}{2}+\frac{q}{2})E_8 \\ & & +(g+\frac{h}{4}+\frac{p}{2}+\frac{q}{2})E_9+(a+b)E_{10}, \\ V'_6& = & \sqrt{2}(\frac{d'}{2}-c')E_1+\sqrt{2}(e'-\frac{f'}{2})E_2+(g'-\frac{h'}{4}-\frac{p'}{2}+\frac{q'}{2})E_3+(-g'+\frac{h'}{4}-\frac{p'}{2}+\frac{q'}{2})E_4 \\ & &+\sqrt{2}(c'+\frac{d'}{2})E_5-\sqrt{2}(e'+\frac{f'}{2})E_6+(a'-b')E_7+(-g'-\frac{h'}{4}+\frac{p'}{2}+\frac{q'}{2})E_8 \\ & & +(g'+\frac{h'}{4}+\frac{p'}{2}+\frac{q'}{2})E_9+(a'+b')E_{10}, \end{eqnarray}$

where $u_1, u_2, \cdots, u_6, u'_1, u'_2, \cdots, u'_6$ , and $a, b, c, d, e, f, g, h, p, q, a', b', c', d', e', f', g', h', p', q'$ are the same as (3.3).

Solving the stationary zero curvature equation

$\begin{eqnarray} \bar{V}_{3x} = \frac{\partial \bar{U}_3}{\partial \lambda}\lambda_t+[\bar{U}_3, \bar{V}_3], \end{eqnarray}$

(6.6)

we can obtain the same equations as (3.5) and (3.6). This means that integrable coupling systems obtained from the linear nonisospectral problems $\left\{ \begin{array}{ll} \varphi_x = \bar{U}_3\varphi, \\ \varphi_t = \bar{V}_3\varphi, \\ \lambda_t = \sum_{i\geq 0}k_i(t)\lambda^{-i}\\ \end{array} \right.$ are the same as (3.7).

If we consider the spectral matrix $\bar{U}_1$ and time spectral matrix $\bar{V}_1$ in Lie algebra $\mathfrak{sp}(4)$ , and choose spectral matrices $\bar{U}_3$ and $\bar{V}_3$ in Lie algebra $\mathfrak{so}(3, 2)$ , then from zero curvature equations $\bar{V}_{1x} = \frac{\partial \bar{U}_1}{\partial \lambda}\lambda_t+[\bar{U}_1, \bar{V}_1]$ and $\bar{V}_{3x} = \frac{\partial \bar{U}_3}{\partial \lambda}\lambda_t+[\bar{U}_3, \bar{V}_3]$ , we can obtain the same nonisospectral integrable coupling systems. So, based on $\mathfrak{sp}(4)\cong \mathfrak{so}(3, 2)$ , as long as we select the corresponding spectral problem between Lie algebras $\mathfrak{sp}(4)$ and $\mathfrak{so}(3, 2)$ , we can obtain the same integrable couplings.

7. Conclusions

By adding any real number $\varepsilon$ , we construct the generalized Lie algebras $G\mathfrak{sp}(4)$ , $G\mathfrak{so}(5)$ , and $G\mathfrak{so}(3, 2)$ . Based on these three Lie algebras, we introduce the spectral parameter $\lambda_t = \sum_{i\geq 0}k_i(t)\lambda^{-i}$ , and obtain the nonisospectral integrable hierarchies and their Hamiltonian structures of these three Lie algebras. Additionally, based on the semi-direct sum decomposition of Lie algebras, we derive the integrable coupling systems associated with Lie algebras $\mathfrak{sp}(4)$ , $\mathfrak{so}(5)$ , and $\mathfrak{so}(3, 2)$ . At the same time, we use $\mathfrak{sp}(4)\cong\mathfrak{so}(3, 2)$ , and further discuss the relationship between the integrable couplings systems corresponding to these two Lie algebras.

Author contributions

Baiying He: Conceived of the study, Completed the computations, Writing-original draft; Siyu Gao: Writing review and editing, Writing-original draft. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

The authors would like to thank the referee for valuable comments and suggestions on this paper, which have considerably improved its presentation and quality.

This work is partially supported by National Natural Science Foundation of China (Nos.12001467 and 12471024) and Nanhu Scholars Program for Young Scholars of XYNU.

Conflict of interest

The authors declare that they have no conflicts of interest.

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On periodic Ambrosetti-Prodi-type problems

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Abstract

1. Introduction

2. The nonisospectral integrable hierarchy associated with the generalized Lie algebra $G\mathfrak{sp}(4)$

3. The nonisospectral integrable coupling hierarchy associated with Lie algebra $\mathfrak{sp}(4)$

3.1. Preliminaries

3.2. The nonisospectral integrable couplings

4. The nonisospectral integrabl hierarchy of the generalized Lie algebra $G\mathfrak{so}(5)$

5. The nonisospectral integrable coupling hierarchy

6. The nonisospectral integrable hierarchy associated with Lie algebra $\mathfrak{so}(3, 2)$

6.1. The generalized Lie algebra $G\mathfrak{so}(3, 2)$

6.2. The nonisospectral integrable couplings

7. Conclusions

Author contributions

Acknowledgments

Conflict of interest

References

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AIMS Mathematics

On periodic Ambrosetti-Prodi-type problems

Related Papers:

Abstract

1. Introduction

2. The nonisospectral integrable hierarchy associated with the generalized Lie algebra Gsp(4) G\mathfrak{sp}(4)

3. The nonisospectral integrable coupling hierarchy associated with Lie algebra sp(4) \mathfrak{sp}(4)

3.1. Preliminaries

3.2. The nonisospectral integrable couplings

4. The nonisospectral integrabl hierarchy of the generalized Lie algebra Gso(5) G\mathfrak{so}(5)

5. The nonisospectral integrable coupling hierarchy

6. The nonisospectral integrable hierarchy associated with Lie algebra so(3,2) \mathfrak{so}(3, 2)

6.1. The generalized Lie algebra Gso(3,2) G\mathfrak{so}(3, 2)

6.2. The nonisospectral integrable couplings

7. Conclusions

Author contributions

Acknowledgments

Conflict of interest

References

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Catalog

2. The nonisospectral integrable hierarchy associated with the generalized Lie algebra $G\mathfrak{sp}(4)$

3. The nonisospectral integrable coupling hierarchy associated with Lie algebra $\mathfrak{sp}(4)$

4. The nonisospectral integrabl hierarchy of the generalized Lie algebra $G\mathfrak{so}(5)$

6. The nonisospectral integrable hierarchy associated with Lie algebra $\mathfrak{so}(3, 2)$

6.1. The generalized Lie algebra $G\mathfrak{so}(3, 2)$