1 Introduction
In 1993, Camassa and Holm derived the following shallow water wave model[1]
$$ m_{t}=2mu_{x}+m_{x}u\,, \quad m=u-u_{xx}+k\,, $$
where $k$ is an arbitrary constant. This equation was referred to as the Camassa-Holm (CH) equation nowadays and it has attracted much attention in recent years.The CH equation was first implicitly included in the work of Fuchssteiner and Fokas[2] as a very special case.Since the work of Camassa and Holm,[1] various studies on this equation have remarkably been developed; see, for example, Refs. [3--6] and references therein.The most attractive feature of the CH equation is that, in the case $k=0$,it admits peaked soliton (peakon) solutions.Such solutions are of great research interest and have been extensively studied from different points of view.In addition to the CH equation, other integrable models that admit peakon solutions have been found,such as the Degasperis-Procesi (DP) equation,[7] the modified CH equation,[8-10] the Novikov's cubic nonlinear equation,[11] and some multi-component extensions of CH equation.[12-21]
In this paper, we propose the following matrix CH-type equation
$$ m_t\!=\!F\!+\!F_x\!-\!\frac{1}{4}\left[(u\!-\!u_{x})(v\!+\!v_{x})m\!+\!m(v\!+\!v_{x})(u\!-\!u_{x})\right], \\ n_t\!=\!-G\!+\!G_x\!+\!\frac{1}{4}\left[(v\!+\!v_{x})(u\!-\!u_{x})n\!+\!n(u\!-\!u_{x})(v\!+\!v_{x})\right], \\ m=\!u\!-\!u_{xx},\\ n\!=\!v\!-\!v_{xx}, $$
where $u$, $v$, $m$, and $n$ are $sl(2)$-valued matrices
$$ u=\left(\begin{array}{cc} u_{11}& u_{12}\\ u_{21}& -u_{11} \end{array} \right), \quad v=\left(\begin{array}{cc} v_{11}& v_{12}\\ v_{21}& -v_{11} \end{array} \right), m=\left(\begin{array}{cc} m_{11} & m_{12}\\ m_{21} & -m_{11} \end{array} \right),\quad n=\left(\begin{array}{cc} n_{11} & n_{12}\\ n_{21} & -n_{11} \end{array}\right), $$
$F$ and $G$ are $sl(2)$ matrix valued functions whose entries are smooth functions of $u_{jk}$, $v_{jk}$, $j,k=1,2$ and their dervatives, and they satisfy the following constraints:
$$ mG=Fn, \quad Gm=nF. $$
By choosing appropriate $F$ and $G$ that satisfy Eq. (4), we can recover several known integrable peakon models as well as some new peakon models as special cases of Eq. (2).
For example, if we choose $u_{11}=v_{11}=0$, $u_{21}=v_{21}=2$, $u_{12}=v_{12}$, and
$ F=G=\left(\begin{array}{cc} 0& m_{12}u_{12} \\ 2u_{12} & 0\\\end{array} \right), $
then Eq. (2) is reduced to the celebrated CH Eq. (1).If we choose $u_{12}=v_{12}=u_{21}=v_{21}=0, u_{11}=v_{11}$, and
$ F=G=\frac{1}{2}\left(\begin{array}{cc} m_{11}(u^{2}_{11}-u^{2}_{11,x})& 0 \\ 0 & -m_{11}(u^{2}_{11}-u^{2}_{11,x}) \\ \end{array} \right),$
$$ u_{12}=v_{12}=u_{21}=v_{21}=0, \quad F=mH, \quad G=nH, $$
with $H$ being an arbitrary function of $u_{11}$, $v_{11}$ and their derivatives,then Eq. (2) is reduced to a family of two-component peakon equations proposed in Ref. [20] (see Eq. (7) in Ref. [20]).Furthermore, we recover the $U(1)$-invariant peakon equations presented in Refs. [22--23] from Eq. (2) by imposing the complex conjugate reduction $v_{11}=u^*_{11}$ on the reduction (5) with special choices of $H$ (see Refs. [19--20] for details).
In this paper, we show that Eq. (2) admits Lax representation and possesses infinitely many conservation laws.We also derive the $N$-peakon solutions for a new matrix CH-type equation containing in the matrix CH equation (2) as special cases.
The whole paper is organized as follows. In Sec. 2, we present the Lax pair and conservation laws for Eq. (2).In Sec. 3, we discuss the $N$-peakon solutions for a new matrix CH-type equations containing in the matrix CH Eq. (2).
2 Zero-Curvature Representation and Conservation Laws
Let the $sl(2)$ valued matrices $m$, $n$, $u$, $v$ be defined by Eq. (3).
Let $F$ and $G$ be $2\times 2$ matrix valued functions such that their entries are smooth functions of $u_{jk}$, $v_{jk}$, $j,k=1,2$, and their dervatives, and they subject to the constraints (4). We introduce the matrices $U$ and $V$ as follows:
$ U=\frac{1}{2} \left(\begin{array}{cccc} -I_2 & \lambda m \\ - \lambda n & I_2 \\ \end{array}\right)\equiv\left(\begin{array}{cc}U_{11} & U_{12}\\U_{21} & U_{22}\end{array}\right),$
$V=-\frac{1}{2}\left( \begin{array}{cc} \lambda^{-2}I_2+E & -\lambda^{-1}(u-u_x)-\lambda F \\ \lambda^{-1}(v+v_x)+\lambda G & -\lambda^{-2}I_2-H \\ \end{array} \right) \equiv \left(\begin{array}{cccc}V_{11} & V_{12}\\V_{21} & V_{22}\end{array}\right),$
where $\lambda$ is a spectral parameter, $I_{2}$ is the $2\times2$ identity matrix, and the matrix functions $E$, $H$ are given by
$ E=\frac{1}{2}\left(uv+uv_x-u_xv-u_xv_x\right)$$
$ H=\frac{1}{2}\left(vu+v_xu-vu_x-v_xu_x\right).$
Consider a pair of linear spectral problems
$$ \psi_x=U\psi, \quad \psi_t=V\psi, $$
where $U$ and $V$ are defined by Eqs. (6a) and (6b).
The compatibility condition of Eq. (8), namely the zero-curvature representation generates
$$ U_t-V_x+[U,V]=0. $$
Substituting the expressions of $U$ and $V$ into Eq. (9) and recalling that $F$ and $G$ satisfy Eq. (4), we find that (9) is nothing but the matrix equation (2). Hence, Eq. (8) exactly gives the $4\times4$ matrix Lax pair of Eq. (2).
Next, let us construct conservation laws of Eq. (2).
We write Eq. (8) in the form
$ \left( \begin{array}{c} \Phi_1 \\ \Phi_2 \\ \end{array} \right)_x=\left(\begin{array}{cc} U_{11} & U_{12}\\ U_{21} & U_{22} \end{array} \right)\left( \begin{array}{c} \Phi_1\\ \Phi_2 \\ \end{array} \right),$
$ \left( \begin{array}{c} \Phi_1\\ \Phi_2 \\ \end{array} \right)_t=\left(\begin{array}{cc}V_{11} & V_{12}\\V_{21} & V_{22}\end{array}\right)\left( \begin{array}{c} \Phi_1\\ \Phi_2 \\ \end{array} \right),$
where $\Phi_1$, $\Phi_2$, $U_{ij}$ and $V_{ij}$, $1\leq i, j\leq 2$, are $2\times 2$ matrices.
Let $\omega=\Phi_2\Phi_1^{-1}$. From Eq. (10a), we find $\omega$ satisfies the following matrix Riccati equation
$ \omega_x=-\frac{1}{2}\lambda n+\omega-\frac{1}{2}\lambda\omega m \omega . $
From Eqs. (6a), (6b), and (10), we find
$$ (\ln \Phi_1)_x=-\frac{1}{2}I_{2}+\frac{1}{2}\lambda m\omega, $$
$$ (\ln \Phi_1)_t=-\frac{1}{2}\Big[\lambda^{-2} -\lambda^{-1}(u-u_x)\omega\\ \quad\quad\quad\quad\,\,\,\,+\frac{1}{2}(u-u_x)(v+v_x)-\lambda F\omega\Big].$$
Equations (11) and (12) yield the following conservation law of Eq. (2):
$$ \rho_t=A_x,$$
where
$$ \rho=m\omega, $$
$$ A=-\frac{1}{2}\lambda^{-1}(u\!-\!u_x)(v\!+\!v_x)\!+\!\lambda^{-2}(u\!-\!u_x)\omega\!+\!F\omega. $$
Usually, $\rho$ and $A$ are called a conserved density and an associated flux, respectively.
Equation (14) implies that $m\omega$ is a generating function of the conserved densities. We may derive explicit forms of the conserved densities by expanding $\omega$ in the positive powers of $\lambda$,
$$ \omega=\sum_{j=0}^{\infty}\omega_j\lambda^{j}. $$
Substituting Eq. (16) into Eq. (11) and comparing powers of $\lambda$, we find
$$ \omega_{0}=0, \quad \omega_{1}=\frac{1}{2}(v+v_x), \quad \omega_{2j}=0, \\ \omega_{2j+1}\!=\!\frac{1}{2}\!\left(1\!-\!\partial_{x}\right)^{-1}\!\!\left(\sum_{i\!+\!k\!=\!j\!+\!1, 1\leq i,k\leq j}\omega_i m\omega_{k}\!\right)\!,\quad j\!\geq\! 1. $$
By inserting Eqs. (16) and (17) into Eqs. (14) and (15), we finally obtain the following infinitely many
conserved densities and the associated fluxes
$$ \rho_{0}=tr(m\omega_{0})=0, \quad A_0=0,\\ \rho_{1}=tr(m\omega_{1})=\frac{1}{2}(2m_{11}(v_{11}+v_{11}{x})+m_{12}(v_{21}+v_{21}{x})\\ \quad\quad+m_{21}(v_{12}+v_{12}{x})),\\ A_1=tr(u-u_x)\omega_{3}+F\omega_{1},\\ \rho_{3}=tr(m\omega_{3}), \quad A_3=tr[(u-u_x)\omega_5+F\omega_3],\\ \rho_{2j}=0, \quad A_{2j}=0,\\ \rho_{2j+1}=tr(m\omega_{2j+1}),\\ A_{2j+1}=tr[(u-u_x)\omega_{2j+3}+F\omega_{2j+1}],\quad j\geq 2. $$
3 $N$-Peakon Solutions
As shown before, once having selected two undetermined matrices $F$ and $G$ satisfying Eq. (4),we may obtain a matrix CH-type equation that possesses a zero-curvature representation and infinitely many conservation laws.The following technical lemma is useful for us to select such two matrices.
Lemma 1 $If $A,B$ $\in$ $sl(2)$, then $AB+BA$ is a scalar matrix.
The lemma can be proved by a direct calculation.
Lemma 1 gives us a method to select functions $F$ and $G$. For example, we choose $F$ and $G$ as follows:
$$ F=mf(u^{2}, v^{2}, u^{2}_{x}, v^{2}_{x}, u^{2}_{xx}, v^{2}_{xx}, \ldots),\\ G=nf(u^{2}, v^{2}, u^{2}_{x}, v^{2}_{x}, u^{2}_{xx}, v^{2}_{xx}, \ldots), $$
or
$$ F\!=\!mf(uv\!+\!vu, u_{x}v_{x}\!+\!v_{x}u_{x}, uv_{x}\!+\!v_{x}u, u_{x}v\!+\!vu_{x}),\\ \\ G\!=\!nf(uv\!+\!vu, u_{x}v_{x}\!+\!v_{x}u_{x}, uv_{x}\!+\!v_{x}u, u_{x}v\!+\!vu_{x}), $$
where $f$ is an arbitrary polynomial about the matrices $u^{2}$, $v^{2}$, $u^{2}_{x}$, $v^{2}_{x}$, $\ldots$, $uv+vu$, $u_{x}v_{x}+v_{x}u_{x}$, $uv_{x}+v_{x}u$, $u_{x}v+vu_{x}$, or their linear combinations. Actually, from Lemma 1, we immediately conclude that $f$ is a scalar matrix, which in turn implies that $F$ and $G$ satisfy Eq. (4).Here we give the following two examples.
Example 1 $f=O$ (here $O$ denotes zero matrix).In this case, $F=G=O$, then Eq. (2) is reduced to the following matrix CH-type equation:
$$ m_t=-\frac{1}{4}\left[\left(uv+uv_{x}-u_{x}v-u_{x}v_{x}\right)m+m\left(vu+v_{x}u-vu_{x}-v_{x}u_{x}\right)\right],\\ n_t=\frac{1}{4}\left[\left(vu+v_{x}u-vu_{x}-v_{x}u_{x}\right)n+n\left(uv+uv_{x}-u_{x}v-u_{x}v_{x}\right)\right],\quad m=u-u_{xx}, \quad n=v-v_{xx}, $$
where $m$, $n$, $u$, $v$ are $sl(2)$ valued matrices defined by Eq. (3).
Example 2 $f=({1}/{4})(uv+vu-u_{x}v_{x}-v_{x}u_{x}).$
In this case,
$$ F=\frac{1}{4}m(uv+vu-u_{x}v_{x}-v_{x}u_{x}), \quad G=\frac{1}{4}n(uv+vu-u_{x}v_{x}-v_{x}u_{x}). $$
Then equation (2) becomes the following matrix equation:
$$ m_{t}=\frac{1}{4}[m(uv+vu-u_{x}v_{x}-v_{x}u_{x})]_{x}+\frac{1}{4}m(uv-u_{x}v_{x}-v_{x}u+vu_{x})-\frac{1}{4}(uv+uv_{x}-u_{x}v-u_{x}v_{x})m, \\ n_{t}=\frac{1}{4}[n(uv+vu-u_{x}v_{x}-v_{x}u_{x})]_{x}-\frac{1}{4}n(vu-v_{x}u_{x}-uv_{x}+u_{x}v)+\frac{1}{4}(vu+v_{x}u-vu_{x}-v_{x}u_{x})n, \\ m=u-u_{xx}, n=v-v_{xx}, $$
where $m$, $n$, $u$, $v$ are defined by Eq. (3).
In the following, we show that Eq. (21) has $N$-peakon solutions. To this end,we suppose the $N$-peakon solution of Eq. (21) in the following form
$$ u_{11}=\sum_{j=1}^N p_j(t)e^{-\mid x-q_j(t)\mid},\quad u_{12}=\sum_{j=1}^N r_j(t)e^{-\mid x-q_j(t)\mid}, \quad u_{21}=\sum_{j=1}^N s_j(t)e^{-\mid x-q_j(t)\mid},\\ v_{11}=\sum_{j=1}^N f_j(t)e^{-\mid x-q_j(t)\mid}, \quad v_{12}=\sum_{j=1}^N g_j(t)e^{-\mid x-q_j(t)\mid}, \quad v_{21}=\sum_{j=1}^N h_j(t)e^{-\mid x-q_j(t)\mid}.$$
In the distribution sense, we have
$$ u_{11,x}=-\sum_{j=1}^N p_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad m_{11}=2\sum_{j=1}^N p_j\delta(x-q_j),\\u_{12,x}=-\sum_{j=1}^N r_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad m_{12}=2\sum_{j=1}^N r_j\delta(x-q_j),\\u_{21,x}=-\sum_{j=1}^N s_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad m_{21}=2\sum_{j=1}^N s_j\delta(x-q_j),\\v_{11,x}=-\sum_{j=1}^N f_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad n_{11}=2\sum_{j=1}^N f_j\delta(x-q_j),\\v_{12,x}=-\sum_{j=1}^N g_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad n_{12}=2\sum_{j=1}^N g_j\delta(x-q_j),\\v_{21,x}=-\sum_{j=1}^N h_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad n_{21}=2\sum_{j=1}^N h_j\delta(x-q_j), $$
where sgn$(x)$ is the sign function.
Substituting these expressions into Eq. (21) and integrating against test functions with compact support,we arrive at the $N$-peakon dynamical system as follows:
$ q_{j,t}=0, \\ p_{j,t}=-\frac{1}{4}\Big\{[p_j\sum_{i,k=1}^N (2p_if_k+r_ih_k+s_ig_k)+s_j\sum_{i,k=1}^N(p_ig_k-r_if_k)+r_j\sum_{i,k=1}^N(p_ih_k-s_if_k)] \\ \times[1+{\rm{}sgn}(q_j-q_k)-{\rm{}sgn}(q_j-q_i)+{\rm{}sgn}(q_j-q_i){\rm{}sgn}(q_j-q_k)] e^{ -\mid q_j-q_i\mid-\mid q_j-q_k\mid} \\ +\Big[\frac{2}{3}p^{2}_jg_j-\frac{4}{3}p_jr_jf_j-\frac{2}{3}r^{2}_jh_j\Big]\Big\},\\ r_{j,t}=-\frac{1}{4}\Big\{[p_j\sum_{i,k=1}^N (2r_if_k-2p_ig_k)+r_j\sum_{i,k=1}^N(2p_if_k+2r_ih_k)] \\ \times[1+{\rm{}sgn}(q_j-q_k)-{\rm{}sgn}(q_j-q_i)+{\rm{}sgn}(q_j-q_i){\rm{}sgn}(q_j-q_k)] e^{ -\mid q_j-q_i\mid-\mid q_j-q_k\mid} \\ +\Big[\frac{2}{3}p^{2}_jg_j-\frac{4}{3}p_jr_jf_j-\frac{2}{3}r^{2}_jh_j\Big]\Big\}, s_{j,t}=-\frac{1}{4}\Big\{[p_j\sum_{i,k=1}^N (2s_if_k-2p_ih_k)+s_j\sum_{i,k=1}^N(2p_if_k+2s_ig_k)] \\ \times[1+{\rm{}sgn}(q_j-q_k)-{\rm{}sgn}(q_j-q_i)+{\rm{}sgn}(q_j-q_i){\rm{}sgn}(q_j-q_k)] e^{ -\mid q_j-q_i\mid-\mid q_j-q_k\mid} \\ + \Big[ \frac{2}{3} p_j^2 h_j -\frac{4}{3}p_js_jf_j-\frac{2}{3}s^{2}_jg_j\Big]\Big\}, \\ f_{j,t}=\frac{1}{4}\Big\{[f_j\sum_{i,k=1}^N (2p_if_k+s_ig_k+r_ih_k)+g_j\sum_{i,k=1}^N(s_if_k-p_ih_k)+h_j\sum_{i,k=1}^N(r_if_k-p_ig_k)] \\ \times[1+{\rm{}sgn}(q_j-q_k)-{\rm{}sgn}(q_j-q_i)+{\rm{}sgn}(q_j-q_i){\rm{}sgn}(q_j-q_k)] e^{ -\mid q_j-q_i\mid-\mid q_j-q_k\mid}\\ +\frac{2}{3}[-f^{2}_jp_j-f_jr_jh_j-f_js_jg_j+g_jp_jh_j]\Big\},\\ g_{j,t}=\frac{1}{4}\Big\{[g_j\sum_{i,k=1}^N (2p_if_k+2s_ig_k)+f_j\sum_{i,k=1}^N(2p_ig_k-2r_if_k)] \\ \times[1+{\rm{}sgn}(q_j-q_k)-{\rm{}sgn}(q_j-q_i)+{\rm{}sgn}(q_j-q_i){\rm{}sgn}(q_j-q_k)] e^{ -\mid q_j-q_i\mid-\mid q_j-q_k\mid}\\ +\Big[\frac{2}{3}f^{2}_jr_j-\frac{4}{3}p_jf_jg_j-\frac{2}{3}g^{2}_js_j\Big]\Big\},\\ h_{j,t}=\frac{1}{4}\Big\{[f_j\sum_{i,k=1}^N (2p_ih_k-2s_if_k)+h_j\sum_{i,k=1}^N(2p_if_k+2r_ih_k)]e^{ -\mid q_j-q_i\mid-\mid q_j-q_k\mid} \\ \\ \times[1+{\rm{}sgn}(q_j-q_k)-{\rm{}sgn}(q_j-q_i)+{\rm{}sgn}(q_j-q_i){\rm{}sgn}(q_j-q_k)] e^{ -\mid q_j-q_i\mid-\mid q_j-q_k\mid}\\ +\Big[\frac{2}{3}f^{2}_js_j-\frac{4}{3}p_jh_jf_j-\frac{2}{3}h^{2}_jr_j\Big]\Big\}.$
Equations $q_{j,t}=0$ imply that the peakons are stationary and the solution is in the form of separation of variables.
We now discuss the single peakon solution of Eq. (21). For $N=1$, the above $N$-peakon dynamical system becomes
$$ q_{1,t}(t)=0,\\ p_{1,t}(t)=-\frac{1}{3}p_{1}(t)(p_{1}(t)f_{1}(t)+r_{1}(t)h_{1}(t))+\frac{1}{3}s_{1}(t)(f_{1}(t)r_{1}(t)-p_{1}(t)g_{1}(t)),\\ r_{1,t}(t)=-\frac{1}{3}p_{1}(t)(f_{1}(t)r_{1}(t)-p_{1}(t)g_{1}(t))-\frac{1}{3}r_{1}(t)(p_{1}(t)f_{1}(t)+r_{1}(t)h_{1}(t)),\\s_{1,t}(t)=-\frac{1}{3}p_{1}(t)(f_{1}(t)s_{1}(t)-p_{1}(t)h_{1}(t))-\frac{1}{3}s_{1}(t)(p_{1}(t)f_{1}(t)+s_{1}(t)g_{1}(t)),\\f_{1,t}(t)=\frac{1}{3}f_{1}(t)(p_{1}(t)f_{1}(t)+g_{1}(t)s_{1}(t))+\frac{1}{3}h_{1}(t)(r_{1}(t)f_{1}(t)-p_{1}(t)g_{1}(t)),\\g_{1,t}(t)= \frac{1}{3}g_{1}(t)(p_{1}(t)f_{1}(t)+g_{1}(t)s_{1}(t))+\frac{1}{3}f_{1}(t)(p_{1}(t)g_{1}(t)-r_{1}(t)f_{1}(t)),\\h_{1,t}(t)=\frac{1}{3}h_{1}(t)(p_{1}(t)f_{1}(t)+h_{1}(t)r_{1}(t))+\frac{1}{3}f_{1}(t)(p_{1}(t)h_{1}(t)-f_{1}(t)s_{1}(t)). $$
From Eq. (26), we can take
$$ q_{1}(t)=0, \\ p_{1}(t)f_{1}(t)+h_{1}(t)r_{1}(t)=A_{1},\\ f_{1}(t)r_{1}(t)-p_{1}(t)g_{1}(t)=A_{2},\\ p_{1}(t)f_{1}(t)+s_{1}(t)g_{1}(t)=A_{3},\\ f_{1}(t)s_{1}(t)-p_{1}(t)h_{1}(t)=A_{4}, $$
where $A_{1}, A_{2}, A_{3}, A_{4}$ are arbitrary constants.Let us specify the following cases:
Case 1 $(A_{1}-A_{3})^{2}-4A_{2}A_{4}>0$.
In this case, a solution of Eq. (26) is given by
$$ p_{1}(t)\!=\!C_{1}e^{\lambda_{1}t}+C_{2}e^{\lambda_{2}t}, \quad f_{1}(t)=C^{'}_{1}e^{\lambda_{1}t}+C^{'}_{2}e^{\lambda_{2}t}. r_{1}(t)\!=\!C_{0}e^{-({1}/{3})A_{1}t}-\frac{A_{2}C_{3}}{3\lambda_{1}+A_{1}}e^{\lambda_{1}t}-\frac{C_{4}A_{2}}{3\lambda_{2}+A_{1}} e^{\lambda_{2}t}, s_{1}(t)\!=\!C_{5}e^{-({1}/{3})A_{3}t}-\frac{A_{4}C_{6}}{3\lambda_{1}+A_{3}}e^{\lambda_{1}t}-\frac{C_{7}A_{4}}{3\lambda_{2}+A_{3}} e^{\lambda_{2}t}, g_{1}(t)\!=\!C^{'}_{0}e^{({1}/{3})A_{3}t}-\frac{A_{2}C^{'}_{3}}{3\lambda_{1}+A_{3}}e^{\lambda_{1}t}-\frac{C^{'}_{4}A_{2}}{3\lambda_{2}-A_{1}} e^{\lambda_{2}t}, h_{1}(t)\!=\!C^{'}_{5}e^{({1}/{3})A_{1}t}-\frac{A_{4}C^{'}_{6}}{3\lambda_{1}-A_{1}}e^{\lambda_{1}t}-\frac{C^{'}_{7}A_{4}}{3\lambda_{2}-A_{1}} e^{\lambda_{2}t}, $$
where $C_{j}$, $C'_{j}$, $j=0,1,\ldots,7$, are arbitrary constants, and $\lambda_{1,2}=-\frac{A_1+A_3}{6}\pm \frac{\sqrt{(A_1-A_3)^2-4A_2A_4}}{6}.$
Case 2 $(A_{1}-A_{3})^{2}-4A_{2}A_{4}=0$.
In this case, a solution of Eq. (26) is given by
$$ p_{1}(t)=C_{1}e^{\lambda_{1}t}+C_{2}te^{\lambda_{1}t}, f_{1}(t)=C^{'}_{1}e^{\lambda_{1}t}+C^{'}_{2}te^{\lambda_{1}t}, \\ r_{1}(t)=C_{0}e^{-({1}/{3})A_{1}t}-\frac{A_{2}C_{3}}{3\lambda_{1}+A_{1}}e^{\lambda_{1}t}+\frac{A_{2}C_{4}}{3(\lambda_{1}+\frac{1}{3}A_{1})^{2}} e^{\lambda_{1}t}-A_{2}C_{4}\frac{t}{3\lambda_{1}+A_{1}}e^{\lambda_{1}t}, \\ s_{1}(t)=C_{5}e^{-({1}/{3})A_{3}t}-\frac{A_{4}C_{6}}{3\lambda_{1}+A_{3}}e^{\lambda_{1}t}+\frac{A_{4}C_{7}}{3(\lambda_{1}+\frac{1}{3}A_{3})^{2}} e^{\lambda_{1}t}-A_{4}C_{7}\frac{t}{3\lambda_{1}+A_{3}}e^{\lambda_{1}t}, \\ g_{1}(t)=C^{'}_{0}e^{({1}/{3})A_{3}t}-\frac{A_{2}C^{'}_{3}}{3\lambda_{1}-A_{3}}e^{\lambda_{1}t}+\frac{A_{2}C^{'}_{4}}{3(\lambda_{1}-\frac{1}{3}A_{3})^{2}} e^{\lambda_{1}t}-A_{2}C^{'}_{4}\frac{t}{3\lambda_{1}-A_{3}}e^{\lambda_{1}t}, \\ h_{1}(t)=C^{'}_{5}e^{-({1}/{3})A_{1}t}-\frac{A_{4}C^{'}_{6}}{3\lambda_{1}-A_{1}}e^{\lambda_{1}t}+\frac{A_{4}C^{'}_{7}}{3(\lambda_{1}-\frac{1}{1}A_{3})^{2}} e^{\lambda_{1}t}-A_{4}C^{'}_{7}\frac{t}{3\lambda_{1}-A_{1}}e^{\lambda_{1}t}, $$
where
$\lambda_{1,2}=-\frac{A_1+A_3}{6}.$
Case 3 $(A_{1}-A_{3})^{2}-4A_{2}A_{4}＜0$.
In this case, a solution of Eq. (26) is given by
$$ p_{1}(t)=C_{1}e^{at}\cos {bt}+C_{2}te^{at}\sin{bt},\\ f_{1}(t)=C^{'}_{1}e^{-at}\cos {bt}+C^{'}_{2}te^{-at}\sin{bt},\\ r_{1}(t)=C_{0}e^{-({1}/{3})A_{1}t}-\frac{1}{3}C_{3}A_{2}\Big[\frac{(\cos{bt}+b\sin{bt})(a+\frac{1}{3}A_{1})}{b^{2}+(a+\frac{1}{3}A_{1})^{2}}\Big]e^{at}-\frac{1}{3}C_{4}A_{2}\Big[\frac{\sin{bt}(a+\frac{1}{3}A_{1})-b\cos{bt}}{(a+\frac{1}{3}A_{1})^{2}+b^{2}}\Big]e^{at}, \\ \\ s_{1}(t)=C_{5}e^{-({1}/{3})A_{3}t}-\frac{1}{3}C_{6}A_{4}\Big[\frac{(\cos{bt}+b\sin{bt})(a+\frac{1}{3}A_{3})}{b^{2}+(a+\frac{1}{3}A_{3})^{2}}\Big]e^{at}-\frac{1}{3}C_{7}A_{4}\Big[\frac{\sin{bt}(a+\frac{1}{3}A_{3})-b\cos{bt}}{(a+\frac{1}{3}A_{3})^{2}+b^{2}}\Big]e^{at}, \\ \\ g_{1}(t)=C^{\prime}_{0}e^{({1}/{3})A_{3}t}-\frac{1}{3}C^{\prime}_{3}A_{2}\Big[\frac{(\cos{bt}+b\sin{bt})(-a-\frac{1}{3}A_{3})}{b^{2}+(-a-\frac{1}{3}A_{3})^{2}}\Big]e^{-at}-\frac{1}{3}C^{\prime}_{4}A_{2}\Big[\frac{\sin{bt}(-a-\frac{1}{3}A_{3})-b\cos{bt}}{(-a-\frac{1}{3}A_{3})^{2}+b^{2}}\Big]e^{-at}, \\ \\ h_{1}(t)=C^{\prime}_{5}e^{({1}/{3})A_{1}t}-\frac{1}{3}C^{\prime}_{6}A_{4}\Big[\frac{(\cos{bt}+b\sin{bt})(-a-\frac{1}{3}A_{1})}{b^{2}+(-a-\frac{1}{3}A_{1})^{2}}\Big]e^{-at}-\frac{1}{3}C^{\prime}_{7}A_{4}\Big[\frac{\sin{bt}(-a-\frac{1}{3}A_{1})-b\cos{bt}}{(-a-\frac{1}{3}A_{1})^{2}+b^{2}}\Big]e^{-at},$$
where
$ a=-\frac{A_{1}+A_{3}}{6}, \quad b=\frac{\sqrt{4A_{2}A_{4}-(A_{1}-A_{3})^{2}}}{6}.$