1. Introduction
It is well established that the theory of solitons and integrable systems forms a crucial component of both mathematical physics and applied mathematics. Its core lies in investigating the integrability of nonlinear equations. A large class of nonlinear equations is integrable and exhibits several key integrable properties. For instance, the integrable equations can be derived from a linear spectral problem (Lax pairs) [1]. The Cauchy problem associated with such equations is amenable to investigation via the inverse scattering method [2, 3]. Soliton solutions of these equations can be obtained through the Darboux transformation method [4, 5]. Moreover, they have rich mathematical structures (infinite conservation laws, infinite symmetries, and bi-Hamiltonian structures).
Constructing the multi-component extensions of nonlinear integrable equations is also a hot topic in this field [6–21]. Integrable couplings refer to coupled systems that incorporate the given integrable equations as their subsystems. From a mathematical perspective, when considering a given integrable equation expressed as $u_t = K(u) = K(x,t,u,u_x,u_{xx},{\ldots})$, its integrable coupling takes the form of an enlarged triangular integrable system presented as follows
$\begin{align}u_t = K\left(u\right),~~v_t = S\left(u,v\right).\end{align}$
Once a generating scheme associated with a non-semisimple Lie algebra is established, integrable couplings can be constructed. The simplest non-semisimple Lie algebra, denoted by $\mathfrak{g}$, has the following triangular block matrix form $\begin{align*} \overline{\boldsymbol{U}} = \left( \begin{array}{cccc} \boldsymbol{U} & 0 \\ \boldsymbol{U}_1& \boldsymbol{U} \end{array} \right),\end{align*}$
where $\boldsymbol{U}$ and $\boldsymbol{U}_1$ are two arbitrary square matrices of the equal order. Many examples of integrable coupling hierarchies have been derived on the basis of the non-semisimple Lie algebra $\mathfrak{g}$. These include the Korteweg–de Vries (KdV) equation, the mKdV equation, the Toda lattice equation, the AKNS equation, and the Kaup–Newell equation. In 2018, Shen, Li, et al [22] studied the generalization of the AKNS integrable coupling with the form as $\begin{align}u_t = \widetilde{K}\left(u,v\right),~~v_t = \widetilde{S}\left(u,v\right),\end{align}$
by adding the perturbation term as $\begin{align*} \boldsymbol{U}_1 &= \left( \begin{array}{cccc} \lambda+\epsilon\left(ps+qr\right) & p \\q & -\lambda-\epsilon\left(ps+qr\right) \end{array} \right),\nonumber\\ \boldsymbol{U}& = \left( \begin{array}{cc}0 & 1 \\u-\lambda & 0 \end{array} \right).\end{align*}$
Subsequently, Wang and Zhang [23, 24] generalize the non-semisimple Lie algebra $\mathfrak{g}$ to a non-semisimple Lie algebra $\widetilde{\mathfrak{g}}$ which consists of square matrix of the following form: $\begin{align*} \overline{\boldsymbol{U}} = \left( \begin{array}{cccc} \boldsymbol{U} & \varepsilon \boldsymbol{U}_1 \\ \boldsymbol{U}_1&\boldsymbol{U} \end{array} \right),\end{align*}$
where $\varepsilon$ is an arbitrary constant. In 2024, Ma [25] considers a Liouville integrable Hamiltonian hierarchy by introducing the $4\times4$ matrix eigenvalue problem as $\begin{align*} \overline{\boldsymbol{U}} = \left( \begin{array}{cccc} \boldsymbol{U}_1 & \boldsymbol{U}_2 \\ \boldsymbol{T}\boldsymbol{U}_2\boldsymbol{T}^{-1} & \boldsymbol{T}\boldsymbol{U}_1\boldsymbol{T}^{-1} \end{array} \right),\end{align*}$
with $\begin{align*} \boldsymbol{T} = \left( \begin{array}{cccc}0 & -1 \\1 & 0 \end{array} \right)~\text{or}~\left( \begin{array}{cccc}0 & 1 \\-1 & 0 \end{array} \right).\end{align*}$
Inspired by the aforementioned literature, we investigate a new multi-component generalized KdV equation hierarchy, which is related with the linear spectral problem as follows $\begin{align} \boldsymbol{\Psi}_x = \overline{\boldsymbol{U}}\boldsymbol{\Psi},~~\overline{\boldsymbol{U}} = \left( \begin{array}{cccc} \boldsymbol{U} & \boldsymbol{U}_1 \\ \boldsymbol{U}_2 & \boldsymbol{U} \end{array} \right), \boldsymbol{U} = \left( \begin{array}{cc}0 & 1 \\u-\lambda & 0 \end{array} \right),\end{align}$
where $\boldsymbol{U}_1, \boldsymbol{U}_2$ are two square matrices of the same structure and $\boldsymbol{U}_1\neq \boldsymbol{U}_2$. To the best of our knowledge, investigations on the spectral problem of this special structure is still relatively scarce. The procedure for building the nonlinear integrable hierarchy of the present paper can be described as follows.Step 1: One ought to select an appropriate spectral matrix $\boldsymbol{U}$ to form a spectral problem $\boldsymbol{\psi}_x = \overline{\boldsymbol{U}}\boldsymbol{\psi}$, where the form of the matrix $\overline{\boldsymbol{U}}$ is defined as in (3 ).
Step 2: Construct a particular Laurent series solution $\boldsymbol{\Gamma}$ that satisfies the stationary zero curvature equation $\boldsymbol{\Gamma}_x = [\overline{\boldsymbol{U}},\boldsymbol{\Gamma}]$.
Step 3: With the solution $\boldsymbol{\Gamma}$, introduce the temporal spectral problem $\boldsymbol{\psi}_{t_m} = \boldsymbol{V}_m\boldsymbol{\psi}$ to the zero curvature equation $\overline{\boldsymbol{U}}_t-\boldsymbol{V}_{m,x}+[\overline{\boldsymbol{U}},\boldsymbol{V}] = 0$, so that one can obtain an integrable equation.
Step 4: For some integrable equations, one could construct the bi-Hamiltonian structure $u_{t_m} = K_m(u) = J\frac{\delta}{\delta u}H_m,~m\unicode{x2A7E}0$ via the trace identity [26].
Introducing the temporal spectral matrices
$\begin{align*} \boldsymbol{V}_N = \left(\lambda^{N+1}\boldsymbol{\Gamma}\right)_++\boldsymbol{\Delta}_N\triangleq\left( \begin{array}{cccc} \boldsymbol{V}_1 & \boldsymbol{V}_2 \\ \boldsymbol{V}_3 & \boldsymbol{V}_4 \end{array} \right),~~\boldsymbol{\Gamma} = \sum_{k = 0}^{\infty}\lambda^{-k}\boldsymbol{\Gamma}_k,\end{align*}$
where $\boldsymbol{\Gamma}$ is a Laurent series matrix satisfying the stationary zero-curvature equation $\boldsymbol{\Gamma}_x = [\overline{\boldsymbol{U}},\boldsymbol{\Gamma}]$, and $\boldsymbol{\Delta}_N$ is a matrix independent of the spectral parameter $\lambda$. According to the zero-curvature equation $\begin{equation*} \overline{\boldsymbol{U}}_t-\boldsymbol{V}_{N,x}+\left[\overline{\boldsymbol{U}},\boldsymbol{V}_N\right] = 0,\end{equation*}$
i.e. $\begin{equation} \begin{aligned} &\boldsymbol{U}_t-\boldsymbol{V}_{1,x}+\left[\boldsymbol{U},\boldsymbol{V}_1\right]+\boldsymbol{U}_1\boldsymbol{V}_3-\boldsymbol{V}_2\boldsymbol{U}_2 = 0,\\ &\boldsymbol{U}_{1,t}-\boldsymbol{V}_{2,x}+\left[\boldsymbol{U},\boldsymbol{V}_2\right]+\boldsymbol{U}_1\boldsymbol{V}_4-\boldsymbol{V}_1\boldsymbol{U}_1 = 0,\\ &\boldsymbol{U}_{2,t}-\boldsymbol{V}_{3,x}+\left[\boldsymbol{U},\boldsymbol{V}_3\right]+\boldsymbol{U}_2\boldsymbol{V}_1-\boldsymbol{V}_4\boldsymbol{U}_2 = 0, \end{aligned}\end{equation}$
with the condition $\begin{align*} \left(\boldsymbol{V}_1-\boldsymbol{V}_4\right)_x+\left[\boldsymbol{U},\boldsymbol{V}_4-\boldsymbol{V}_1\right]+\boldsymbol{U}_2\boldsymbol{V}_2+\boldsymbol{V}_2\boldsymbol{U}_2-\boldsymbol{U}_1\boldsymbol{V}_3-\boldsymbol{V}_3\boldsymbol{U}_1 = 0,\end{align*}$
we obtain a new three-component generalized KdV hierarchy. The main task of the present paper is constructing a hierarchy of three-component generalized KdV equation with the special linear spectral problem. Note that the last term of this equation can take arbitrary functions.2. Construction of the three-component generalized KdV hierarchy
In this paper, through the construction of a temporal auxiliary problem, we present the three-component generalized KdV hierarchy that is associated with the spatial spectral problem
$\begin{equation} \boldsymbol{\psi}_x = \overline{\boldsymbol{U}}\boldsymbol{\psi},\end{equation}$
where $\begin{equation*} \overline{\boldsymbol{U}} = \left( \begin{array}{cccc}0 & 1 & 0 & 1 \\u-\lambda&0&q&0\\0& 1 &0& 1 \\p&0&u-\lambda&0 \end{array} \right).\end{equation*}$
Assume that $\boldsymbol{\Gamma}$ has the form as
$\begin{equation*} \boldsymbol{\Gamma} = \left( \begin{array}{cccc}v_{11} & v_{12} & -v_{14} & v_{12} \\v_{13} & v_{14} & v_{23} & v_{14}\\v_{11} & v_{12} & -v_{14} & v_{12} \\v_{33} & -v_{11} & v_{13} & -v_{11} \end{array} \right).\end{equation*}$
Substituting the matrix $\boldsymbol{\Gamma}$ to the stationary zero-curvature equation $\boldsymbol{\Gamma}_x = [\overline{\boldsymbol{U}},\boldsymbol{\Gamma}]$, which yields that the following equations $\begin{align} &v_{11,x} = \left(\lambda-p-u\right)v_{12}+v_{13}+v_{33},~v_{12,x} = 2\left(v_{14}-v_{11}\right),\nonumber\\ &v_{13,x} = \left(q+u-\lambda\right)v_{11}-\left(p+u-\lambda\right)v_{14},\nonumber\\ &v_{14,x} = \left(q+u-\lambda\right)v_{12}-v_{13}-v_{23},\nonumber\\ &v_{23,x} = 2\left(\lambda-q-u\right)v_{14},~v_{33,x} = 2\left(p+u-\lambda\right)v_{11}. \end{align}$
Taking the Laurent series expansions $\begin{equation} \begin{aligned} &v_{11} = \sum_{j = 0}^{\infty}a_j\lambda^{-j-1},~v_{12} = \sum_{j = 0}^{\infty}b_j\lambda^{-j-1},~v_{13} = \sum_{j = 0}^{\infty}c_j\lambda^{-j},~\\ &v_{14} = \sum_{j = 0}^{\infty}d_j\lambda^{-j-1},~v_{23} = \sum_{j = 0}^{\infty}e_j\lambda^{-j},~v_{33} = \sum_{j = 0}^{\infty}f_j\lambda^{-j}, \end{aligned}\end{equation}$
to equation ( $\begin{equation} \begin{aligned} \lambda^0:~&b_0+c_0+f_0 = 0,~c_{0,x} = d_0-a_0,~b_0+c_0+e_0 = 0,~e_{0,x} = 2d_0,~f_{0,x} = -2a_0,\\ \lambda^{-k-1}:~&a_{k,x} = \left(-p-u\right)b_k+b_{k+1}+c_{k+1}+f_{k+1},~b_{k,x} = 2\left(d_k-a_k\right),~\\ &c_{k+1,x} = \left(q+u\right)a_k-\left(p+u\right)d_k+d_{k+1}-a_{k+1},~\\ &d_{k,x} = \left(q+u\right)b_k-b_{k+1}-c_{k+1}-e_{k+1},~\\ &e_{k+1,x} = -2\left(q+u\right)d_k+2d_{k+1},~f_{k+1,x} = 2\left(p+u\right)a_k-2a_{k+1}, ~k\unicode{x2A7E}0. \end{aligned}\end{equation}$
To avoid all elements of matrix $\Gamma$ being constants, it is necessary here that the condition $b_0\neq0$ is satisfied. With the coefficients of $\lambda^0$ and $b_{0,x} = 2(d_0-a_0)$, we obtain the initial values $a_0 = 0, d_0 = 0, e_0 = f_0, b_{0,x} = 0, c_{0,x} = 0, f_{0,x} = 0.$ To ensure the uniqueness of the parameter, we set $c_0 = 1,e_0 = f_0 = 1, b_0 = -2$. Then we get the recursion relation of the Laurent series solution with $k = 0,1,2,\cdots,$ as follows: $\begin{equation*} \begin{aligned}a_{k+1}& = \frac{3}{4}\left(q-p\right)d_k+\left(\frac{5}{4}p-\frac{1}{4}q+u\right)a_k-\left(\frac{5}{16}p_x-\frac{3}{16}q_x+\frac{1}{8}u_x\right)b_k-\frac{3}{16}d_{k,xx}-\frac{5}{16}a_{k,xx},~~~~~~~~~\\b_{k+1}& = \frac{1}{4}\left(p+q+2u\right)b_k-\frac{1}{8}b_{k,x}+\frac{1}{4}\partial_x^{-1}\left(p+q+2u\right)b_{k,x},\\c_{k+1}& = \frac{1}{8}\left(p+q+2u\right)b_k-\frac{1}{16}b_{k,xx}+\partial_x^{-1}\left(\left(q+u\right)a_k-\left(p+u\right)d_k+\frac{1}{8}\left(p+q+2u\right)b_{k,x}\right),\\d_{k+1}& = \frac{3}{4}\left(p-q\right)a_k+\left(\frac{5}{4}q-\frac{1}{4}p+u\right)d_k+\left(\frac{5}{16}q_x-\frac{3}{16}p_x+\frac{1}{8}u_x\right)b_k-\frac{3}{16}a_{k,xx}-\frac{5}{16}d_{k,xx},~~~~~~~~~\\e_{k+1}& = \frac{3}{16}b_{k,xx}-d_{k,x}+\left(\frac{5}{8}q-\frac{3}{8}p+\frac{1}{4}u\right)b_k+\partial_x^{-1}\left(\left(\frac{3}{4}p-\frac{1}{4}q+\frac{1}{2}u\right)a_k-\left(\frac{3}{4}q-\frac{1}{4}p+\frac{1}{2}u\right)d_k\right),\\f_{k+1}& = \frac{3}{16}b_{k,xx}+a_{k,x}+\left(\frac{5}{8}p-\frac{3}{8}q+\frac{1}{4}u\right)b_k+\partial_x^{-1}\left(\left(\frac{3}{4}p-\frac{1}{4}q+\frac{1}{2}u\right)a_k-\left(\frac{3}{4}q-\frac{1}{4}p+\frac{1}{2}u\right)d_k\right). \end{aligned}\end{equation*}$
The first two sets of local functions are listed as follows $\begin{align}a_1 &= \frac{1}{8}\left(5p_x-3q_x+2u_x\right),~b_1 = -\frac{1}{2}\left(p+q+2u\right),~c_1 = -\frac{1}{4}\left(p+q+2u\right),~\nonumber\\d_1 &= \frac{1}{8}\left(3p_x-5q_x-2u_x\right),~e_1 = \frac{1}{4}\left(3p-5q-2u\right),~f_1 = \frac{1}{4}\left(3q-5p-2u\right),\nonumber\\a_2 &= \frac{1}{64}\left(15q-17p-2u\right)_{xxx}+\frac{3}{32}\left(p_x+q_x+2u_x\right)\left(3p-q+2u\right)+\frac{3}{8}\left(p_x-q_x\right)\left(p+q+2u\right),\nonumber\\b_2 &= \frac{1}{16}\left(p+q+2u\right)_{xx}-\frac{3}{16}\left(p+q+2u\right)^2,~~c_2 = \frac{1}{32}\left(p+q+2u\right)_{xx}-\frac{1}{4}\left(p-q\right)^2-\frac{1}{32}\left(p+q+2u\right)^2,\nonumber\\e_2 &= \frac{1}{32}\left(17q-15p+2u\right)_{xx}+\frac{1}{4}\left(p-q\right)^2-\frac{1}{32}\left(p+q+2u\right)\left(9q-7p+2u\right),\nonumber\\ &\quad d_2\left(p,q,u\right) = -a_2\left(q,p,u\right), f_2\left(p,q,u\right) = e_2\left(q,p,u\right),\nonumber\\a_3 &= \frac{15}{256}\left(p+q+2u\right)\left(\left(9p+q+10u\right)p_x+\left(p-7q-6u\right)q_x+2\left(5p-3q+2u\right)u_x\right)\nonumber\\ &\quad +\frac{5}{128}\left(\left(p_x+21q_x+22u_x\right)q_{xx}-\left(23p_x+3q_x+26u_x\right)p_{xx}+2\left(5p-3q+2u\right)u_{xx}\right)\nonumber\\ &\quad -\frac{5}{256}\left(\left(23p+11q+34u\right)p_{xxx}-3\left(3p+7q+10u\right)q_{xxx}+2\left(7p-5q+2u\right)u_{xxx}\right)\nonumber\\ &\quad +\frac{1}{512}\left(65p-63q+2u\right)_{xxxx},~~d_3\left(p,q,u\right) = -a_3\left(q,p,u\right),\nonumber\\b_3 &= \frac{-5}{64}\left(p+q+2u\right)^3+\frac{5}{128}\left(p_x+q_x+2u_x\right)^2+\frac{5}{64}\left(p+q+2u\right)\left(p+q+2u\right)_{xx}+\frac{1}{128}\left(p+q+2u\right)_{xxxx},\\c_3 &= \frac{1}{128}\left(p+q+2u\right)\left(6u_{xx}-25\left(p-q\right)^2-4(p+u\right)(q+u))-\frac{25}{256}(p_x-q_x)^2\nonumber\\ &\quad +\frac{1}{128}((35p-29q+6u)p_{xx}+(35q-29p+6u)q_{xx}+14(p_x+u_x)(q_x+u_x))-\frac{1}{256}(p+q+2u)_{xxxx},\nonumber\\e_3 &= \frac{1}{128}\left((p+q+2u)(60(p-q)^2-(7q-5p+2u)^2)+\frac{25}{2}(p_x-q_x)^2+79(q_x+u_x)^2-65(p_x+u_x)\right)\nonumber\\ &\quad+\frac{1}{128}\left((47q+55p+102u)q_{xx}+15(q-7p-6u)p_{xx}+2(31q-25p+6u)u_{xx}\right)\nonumber\\ &\quad+\frac{1}{256}(63p-65q-2u)_{xxxx},~~f_3(p,q,u) = e_3(q,p,u). \end{align}$
The functions $\{(a_k,b_k,c_k,d_k)\mid k\unicode{x2A7E}0\}$ must be all local. We need to solve these problems in future work.Taking the temporal spectral matrices as
$\begin{equation*} \boldsymbol{V}_N = \left(\lambda^{N+1}\boldsymbol{\Gamma}\right)_++\left( \begin{array}{cccc}0 &0 & 0& 0\\w_{21}^{\left(N\right)} & 0 & w_{23}^{\left(N\right)} & 0\\0 & 0 &0 & 0 \\w_{41}^{\left(N\right)} & 0 & w_{21}^{\left(N\right)} & 0 \end{array} \right),~~N\unicode{x2A7E}1,\end{equation*}$
where the subscript ‘+’ denotes the terms with non-negative power of $\lambda$. According to the zero-curvature equation $\overline{\boldsymbol{U}}_t-\boldsymbol{V}_{N,x}+[\overline{\boldsymbol{U}},\boldsymbol{V}_N] = 0$, we obtain the three-component generalized KdV hierarchy $\begin{align} &u_{t_N} = -\left(q+u\right)a_{N}+\left(p+u\right)d_{N}+c_{N+1,x}+w_{21,x}^{\left(N\right)},\nonumber\\ &q_{t_N} = 2\left(q+u\right)d_{N}+e_{N+1,x}+w_{23,x}^{\left(N\right)},\nonumber\\ &p_{t_N} = -2\left(p+u\right)a_{N}+f_{N+1,x}+w_{41,x}^{\left(N\right)}, \end{align}$
with $w_{21}^{(N)} = w_{21}^{(N)}(p,q,u,p_x,q_x,u_x,{\ldots})$ being an arbitrary function, and $\begin{align}w_{23}^{\left(N\right)}& = \left(q+u\right)b_{N}-d_{N,x}-c_{N+1}-e_{N+1}-w_{21}^{\left(N\right)},\nonumber\\w_{41}^{\left(N\right)} & = \left(p+u\right)b_{N}+a_{N,x}-c_{N+1}-f_{N+1}-w_{21}^{\left(N\right)},\end{align}$
i.e. the three-component generalized KdV hierarchy is $\begin{align}u_{t_N}&= -\left(q+u\right)a_{N}+\left(p+u\right)d_{N}+c_{N+1,x}+w_{21,x}^{\left(N\right)},\\q_{t_N}&= 2\left(q+u\right)\left(2d_{N}-a_{N}\right)+\left(q_x+u_x\right)b_{N}-d_{N,xx}\nonumber\\ &\quad -c_{N+1,x}-w_{21,x}^{\left(N\right)},\nonumber\\p_{t_N}&= 2\left(p+u\right)\left(d_{N}-2a_{N}\right)+\left(p_x+u_x\right)b_{N}+a_{N,xx}\nonumber\\ &\quad -c_{N+1,x}-w_{21,x}^{\left(N\right)}. \end{align}$
We note that here $w_{21}^{(N)}$ can be an arbitrary function, and in this paper, we are more interested in the case where $w_{21}^{(N)}\neq0$.When $N = 1$, we have 9 ).
$\begin{equation*} \boldsymbol{V}_1 = \left( \begin{array}{cccc}a_1 & b_1-2\lambda & -d_1 & b_1-2\lambda \\ \lambda^2+c_1\lambda+c_2+w_{21}^{\left(1\right)} & d_1 & \lambda^2+e_1\lambda+\left(q+u\right)b_{1}-d_{1,x}-c_2-w_{21}^{\left(1\right)} & d_1\\a_1 & b_1-2\lambda & -d_1 & b_1-2\lambda \\ \lambda^2+f_1\lambda+\left(p+u\right)b_{1}+a_{1,x}-c_2-w_{21}^{\left(1\right)} & -a_1 & \lambda^2+c_1\lambda+c_2+w_{21}^{\left(1\right)} & -a_1 \end{array} \right),\end{equation*}$
where $a_1,b_1,c_1,d_1,e_1,f_1$ are defined by equation (The first member of this hierarchy is the following three-component generalized KdV equation:
$\begin{align}u_{t_1} &= \frac{1}{32}\left(p+q+2u\right)_{xxx}-\frac{3}{16}\left(p+q+2u\right)\left(p_x+q_x+2u_x\right)+w_{21,x}^{\left(1\right)},\\q_{t_1} &= \frac{1}{32}\left(19q-13p+6u\right)_{xxx}+\frac{3}{16}\left(\left(3p-q+2u\right)p_x\right.\nonumber\\ &\quad \left.-\left(5p+9q+14u\right)q_x-2\left(p+5q+6u\right)u_x\right)-w_{21,x}^{\left(1\right)},\\p_{t_1} &= \frac{1}{32}\left(19p-13q+6u\right)_{xxx}+\frac{3}{16}\left(\left(3q-p+2u\right)q_x\right.\nonumber\\ &\quad \left.-\left(5q+9p+14u\right)p_x-2\left(q+5p+6u\right)u_x\right)-w_{21,x}^{\left(1\right)}. \end{align}$
Note that both $q$ and $p$ must be either zero or non-zero simultaneously here.When $w_{21} = \frac{3}{16}u_{xx}-\frac{9}{8}u^2$, $p = q = 0$, equation (13 ) reduces to the KdV equation
$\begin{equation} \widetilde{u}_{t}+6\widetilde{u}\widetilde{u}_{x}+\widetilde{u}_{xxx} = 0,\end{equation}$
with ${t_1}\rightarrow-4t$, $u\rightarrow-\frac{1}{2}\widetilde{u}$.When $N = 2$, we have 9 ). The second member of this three-component generalized KdV hierarchy is 15 ) reduces to the fifth order KdV equation
$\begin{align*} &u_{t_2} = -\left(q+u\right)a_{2}+\left(p+u\right)d_{2}+c_{3,x}+w_{21,x}^{\left(2\right)},\\ &q_{t_2} = 2\left(q+u\right)\left(2d_{2}-a_{2}\right)+\left(q_x+u_x\right)b_{2}-d_{2,xx}-c_{3,x}-w_{21,x}^{\left(2\right)},~\\ &p_{t_2} = 2\left(p+u\right)\left(d_{2}-2a_{2}\right)+\left(p_x+u_x\right)b_{2}+a_{2,xx}-c_{3,x}-w_{21,x}^{\left(2\right)},\end{align*}$
where $a_k,b_k,c_k,d_k,e_k,f_k,c_3$ ($k = 1,2$) are defined by ( $\begin{align} &u_{t_2}+\frac{15}{128}\left(p+q+2u\right)^2\left(p+q+2u\right)_{x}-\frac{5}{64}\left(p+q+2u\right)_{x}\left(p+q+2u\right)_{xx}\nonumber\\ &\quad\times \frac{5}{128}\left(p+q+2u\right)\left(p+q+2u\right)_{xxx}+\frac{\left(p+q+2u\right)_{xxxxx}}{256}+w_{21,x}^{\left(2\right)} = 0,\nonumber\\ &p_{t_2}+\frac{15}{128}\left(p+q+2u\right)\left(\left(11p+3q+14u\right)p_x+\left(3p-5q-2u\right)q_x\right.\nonumber\\ &\quad\left.+2\left(7p-q+6u\right)u_x\right)-\frac{5}{64}\left(5\left(5p_x+q_x+6u_x\right)p_{xx}\right.\nonumber\\ &\quad\left.+\left(p_x-19q_x-18u_x\right)q_{xx}+2\left(13p_x-7q_x+6u_x\right)u_{xx}\right)\nonumber\\ &\quad -\frac{5}{128}\left(\left(25p+13q+38u\right)p_{xxx}-\left(7p+19q+26u\right)q_{xxx}\right.\nonumber\\ &\quad\left.+6\left(3p-q+2u\right)u_{xxx}\right)+\frac{1}{256}\left(67p-61q+6u\right)_{xxxxx}+w_{21,x}^{\left(2\right)} = 0,\nonumber\\ &q_{t_2}+\frac{15}{128}\left(p+q+2u\right)\left(\left(11q+3p+14u\right)q_x+\left(3q-5p-2u\right)p_x\right.\nonumber\\ &\quad\left. +2\left(7q-p+6u\right)u_x\right)-\frac{5}{64}\left(5\left(5q_x+p_x+6u_x\right)q_{xx}\right.\nonumber\\ &\quad\left. +\left(q_x-19p_x-18u_x\right)p_{xx} +2\left(13q_x-7p_x+6u_x\right)u_{xx}\right)\nonumber\\ &\quad-\frac{5}{128}\left(\left(25q+13p+38u\right)q_{xxx}+\left(7q+19p+26u\right)p_{xxx}\right.\nonumber\\ &\quad\left.-6\left(3q-p+2u\right)u_{xxx}\right)+\frac{1}{256}\left(67q-61p+6u\right)_{xxxxx}+w_{21,x}^{\left(2\right)} = 0. \end{align}$
When $w_{21}^{(2)} = -\frac{3}{128}(u_{xxxx}-20uu_{xx}-10u_x^2+40u^3)$, $p = q = 0$, equation ( $\begin{equation} \widetilde{u}_{t}+\widetilde{u}_{xxxxx}+10\widetilde{u}\widetilde{u}_{xxx}+20\widetilde{u}_x\widetilde{u}_{xx}+30\widetilde{u}^2\widetilde{u}_x = 0,\end{equation}$
with ${t_2}\rightarrow32t,u\rightarrow-\frac{1}{2}\widetilde{u}$.In the following, we present two specific three-component generalized KdV equations by choosing different $w_{21}$. One is to make $p$ and $q$ have the same status in $w_{21}^{(1)}$, and the other is to set $p$ and $q$ not have the same status in $w_{21}^{(1)}$.
(1) When $w_{21}^{(1)} = \frac{3}{16}u_{xx}$, equation (13 ) reduces to
$\begin{align} &u_t = \frac{1}{32}\left(p_{xxx}+q_{xxx}+8u_{xxx}-6\left(p+q+2u\right)\left(p_x+q_x+2u_x\right)\right),\nonumber\\ &q_t = \frac{1}{32}\left(19q_{xxx}-13p_{xxx}+6\left(3p-q+2u\right)p_x-6\left(5p+9q+14u\right)q_x\right.\nonumber\\ &\quad \left. -12\left(p+5q+6u\right)u_x\right),\nonumber\\ &p_t = \frac{1}{32}\left(19p_{xxx}-13q_{xxx}+6\left(3q-p+2u\right)q_x-6\left(5q+9p+14u\right)p_x\right.\nonumber\\ &\quad \left.-12\left(q+5p+6u\right)u_x\right). \end{align}$
(2) When $w_{21}^{(1)} = \frac{3}{16}u_{xx}-\frac{13}{32}p_{xx}-\frac{3}{8}u(p+q)$, equation (13 ) reduces to
$\begin{align}u_t &= \frac{1}{32}\left(8u_{xxx}-12p_{xxx}+q_{xxx}-6\left(p+q+4u\right)\left(p_x+q_x+4u_x\right)\right.\nonumber\\ &\quad \left.+72uu_x\right),\nonumber\\q_t& = \frac{1}{32}\left(19q_{xxx}+6\left(3p-q+4u\right)p_x-6\left(5p+9q+12u\right)q_x\right.\nonumber\\ &\quad\left. -24u_x\left(2q+3u\right)\right),\nonumber\\p_t& = \frac{1}{32}\left(32p_{xxx}-13q_{xxx}+6\left(3q-p+4u\right)q_x-6\left(5q+9p+12u\right)p_x\right.\nonumber\\ &\quad \left.-24u_x\left(2p+3u\right)\right). \end{align}$
3. Conclusions
Inspired by the relevant literature on the integrable coupling of nonlinear integrable equations [20–25], we adopt the non-semisimple Lie algebra form as
$\begin{equation*} \overline{\boldsymbol{U}} = \left( \begin{array}{cccc} \boldsymbol{U} & \boldsymbol{U}_1 \\ \boldsymbol{U}_2 & \boldsymbol{U} \end{array} \right),\end{equation*}$
where $\boldsymbol{U}_1\neq \boldsymbol{U}_2$, and then establish the hierarchy of three-component generalized KdV equation. We must emphasize that the last term of this equation can choose arbitrary functions.It is commonly known that exact solutions hold significant importance for nonlinear evolution equations as they can function as seeding solutions for a class of localized structures that exist within physical and engineering systems. For example, the KdV equation is a well-known nonlinear evolution equation that describes the propagation of shallow water waves, and its exact soliton solutions provide a fundamental understanding of the behavior of these waves. These solitons can exist in various physical systems like optical fibers and plasma physics. In future work, we will extend the $2$nd-order square matrices $\boldsymbol{U}_1$ and $\boldsymbol{U}_2$ associated with the linear spectral problem to the $n\,({\ldots})$th-order square matrices, and construct the corresponding nonlinear equation hierarchy. Inspired by the relevant literature [27], we will also investigate the negative flow of the three-component generalized KdV hierarchy, as well as its Hamiltonian structure, soliton solutions and infinite conservation laws.