Li-Feng Chan^,1,?, Bao-Qiang Xia^,2,?, Ru-Guang Zhou^,2,ξ1 Kewen College, Jiangsu Normal University, Xuzhou 221116, China
2 School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Corresponding authors: ? E-mail:1018436266@qq.com? E-mail:xiabaoqiang@126.comξ E-mail:zhouruguang@jsnu.edu.cn

Received:2019-05-30Online:2019-12-1

Fund supported:

*Supported by National Natural Science Foundation of China under Grant. Nos. 11771186 Supported by National Natural Science Foundation of China under Grant. Nos. 1671177

Abstract
We present an integrable sl(2)-matrix Camassa-Holm (CH) equation.The integrability means that the equation possesses zero-curvature representation and infinitely many conservation laws.This equation includes two undetermined functions, which satisfy a system of constraint conditions and may be reduced to a lot of known multi-component peakon equations. We find a method to construct constraint condition and thus obtain many novel matrix CH equations. For the trivial reduction matrix CH equation we construct its N-peakon solutions.
Keywords： Camassa-Holm equation;zero-curvature representation;conservation laws;$N$-Peakon


PDF (105KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite
Cite this article
Li-Feng Chan, Bao-Qiang Xia, Ru-Guang Zhou. An Integrable Matrix Camassa-Holm Equation*. [J], 2019, 71(12): 1399-1404 doi:10.1088/0253-6102/71/12/1399

1 Introduction

In 1993, Camassa and Holm derived the following shallow water wave model^[1]

(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

(2) $$ m_t\!=\!F\!+\!F_x\!-\!\frac{1}{4}\left[(u\!-\!u_{x})(v\!+\!v_{x})m\!+\!m(v\!+\!v_{x})(u\!-\!u_{x})\right], \\ n_t\!=\!-G\!+\!G_x\!+\!\frac{1}{4}\left[(v\!+\!v_{x})(u\!-\!u_{x})n\!+\!n(u\!-\!u_{x})(v\!+\!v_{x})\right], \\ m=\!u\!-\!u_{xx},\\ n\!=\!v\!-\!v_{xx}, $$
where $u$, $v$, $m$, and $n$ are $sl(2)$-valued matrices

(3) $$ u=\left(\begin{array}{cc} u_{11}& u_{12}\\ u_{21}& -u_{11} \end{array} \right), \quad v=\left(\begin{array}{cc} v_{11}& v_{12}\\ v_{21}& -v_{11} \end{array} \right), m=\left(\begin{array}{cc} m_{11} & m_{12}\\ m_{21} & -m_{11} \end{array} \right),\quad n=\left(\begin{array}{cc} n_{11} & n_{12}\\ n_{21} & -n_{11} \end{array}\right), $$
$F$ 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:

(4) $$ mG=Fn, \quad Gm=nF. $$
Equation (2) is actually a matrix extension of a family of two-component peakon equations proposed in Ref. ^[20] (see Eq. (7) in Ref. ^[20]).

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),$

then Eq. (2) is reduced to the modified CH equation presented in Refs. ^[8--10].If we choose

(5) $$ u_{12}=v_{12}=u_{21}=v_{21}=0, \quad F=mH, \quad G=nH, $$
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:

(6a) $ 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),$
(6b) $V=-\frac{1}{2}\left( \begin{array}{cc} \lambda^{-2}I_2+E & -\lambda^{-1}(u-u_x)-\lambda F \\ \lambda^{-1}(v+v_x)+\lambda G & -\lambda^{-2}I_2-H \\ \end{array} \right) \equiv \left(\begin{array}{cccc}V_{11} & V_{12}\\V_{21} & V_{22}\end{array}\right),$
where $\lambda$ is a spectral parameter, $I_{2}$ is the $2\times2$ identity matrix, and the matrix functions $E$, $H$ are given by

(7a) $ E=\frac{1}{2}\left(uv+uv_x-u_xv-u_xv_x\right)$$
(7b) $ H=\frac{1}{2}\left(vu+v_xu-vu_x-v_xu_x\right).$
Consider a pair of linear spectral problems

(8) $$ \psi_x=U\psi, \quad \psi_t=V\psi, $$
where $U$ and $V$ are defined by Eqs. (6a) and (6b).

The compatibility condition of Eq. (8), namely the zero-curvature representation generates

(9) $$ U_t-V_x+[U,V]=0. $$
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

(10a) $ \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),$
(10b) $ \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

(11) $$ (\ln \Phi_1)_x=-\frac{1}{2}I_{2}+\frac{1}{2}\lambda m\omega, $$
(12)$$ (\ln \Phi_1)_t=-\frac{1}{2}\Big[\lambda^{-2} -\lambda^{-1}(u-u_x)\omega\\ \quad\quad\quad\quad\,\,\,\,+\frac{1}{2}(u-u_x)(v+v_x)-\lambda F\omega\Big].$$
Equations (11) and (12) yield the following conservation law of Eq. (2):

(13) $$ \rho_t=A_x,$$
where

(14) $$ \rho=m\omega, $$
(15) $$ A=-\frac{1}{2}\lambda^{-1}(u\!-\!u_x)(v\!+\!v_x)\!+\!\lambda^{-2}(u\!-\!u_x)\omega\!+\!F\omega. $$
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 $\lambda$,

(16) $$ \omega=\sum_{j=0}^{\infty}\omega_j\lambda^{j}. $$
Substituting Eq. (16) into Eq. (11) and comparing powers of $\lambda$, we find

(17) $$ \omega_{0}=0, \quad \omega_{1}=\frac{1}{2}(v+v_x), \quad \omega_{2j}=0, \\ \omega_{2j+1}\!=\!\frac{1}{2}\!\left(1\!-\!\partial_{x}\right)^{-1}\!\!\left(\sum_{i\!+\!k\!=\!j\!+\!1, 1\leq i,k\leq j}\omega_i m\omega_{k}\!\right)\!,\quad j\!\geq\! 1. $$
By inserting Eqs. (16) and (17) into Eqs. (14) and (15), we finally obtain the following infinitely many

conserved densities and the associated fluxes

(18) $$ \rho_{0}=tr(m\omega_{0})=0, \quad A_0=0,\\ \rho_{1}=tr(m\omega_{1})=\frac{1}{2}(2m_{11}(v_{11}+v_{11}{x})+m_{12}(v_{21}+v_{21}{x})\\ \quad\quad+m_{21}(v_{12}+v_{12}{x})),\\ A_1=tr(u-u_x)\omega_{3}+F\omega_{1},\\ \rho_{3}=tr(m\omega_{3}), \quad A_3=tr[(u-u_x)\omega_5+F\omega_3],\\ \rho_{2j}=0, \quad A_{2j}=0,\\ \rho_{2j+1}=tr(m\omega_{2j+1}),\\ A_{2j+1}=tr[(u-u_x)\omega_{2j+3}+F\omega_{2j+1}],\quad j\geq 2. $$

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:

(19) $$ F=mf(u^{2}, v^{2}, u^{2}_{x}, v^{2}_{x}, u^{2}_{xx}, v^{2}_{xx}, \ldots),\\ G=nf(u^{2}, v^{2}, u^{2}_{x}, v^{2}_{x}, u^{2}_{xx}, v^{2}_{xx}, \ldots), $$
or

(20) $$ F\!=\!mf(uv\!+\!vu, u_{x}v_{x}\!+\!v_{x}u_{x}, uv_{x}\!+\!v_{x}u, u_{x}v\!+\!vu_{x}),\\ \\ G\!=\!nf(uv\!+\!vu, u_{x}v_{x}\!+\!v_{x}u_{x}, uv_{x}\!+\!v_{x}u, u_{x}v\!+\!vu_{x}), $$
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:

(21) $$ m_t=-\frac{1}{4}\left[\left(uv+uv_{x}-u_{x}v-u_{x}v_{x}\right)m+m\left(vu+v_{x}u-vu_{x}-v_{x}u_{x}\right)\right],\\ n_t=\frac{1}{4}\left[\left(vu+v_{x}u-vu_{x}-v_{x}u_{x}\right)n+n\left(uv+uv_{x}-u_{x}v-u_{x}v_{x}\right)\right],\quad m=u-u_{xx}, \quad n=v-v_{xx}, $$
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,

(22) $$ F=\frac{1}{4}m(uv+vu-u_{x}v_{x}-v_{x}u_{x}), \quad G=\frac{1}{4}n(uv+vu-u_{x}v_{x}-v_{x}u_{x}). $$
Then equation (2) becomes the following matrix equation:

(23) $$ m_{t}=\frac{1}{4}[m(uv+vu-u_{x}v_{x}-v_{x}u_{x})]_{x}+\frac{1}{4}m(uv-u_{x}v_{x}-v_{x}u+vu_{x})-\frac{1}{4}(uv+uv_{x}-u_{x}v-u_{x}v_{x})m, \\ n_{t}=\frac{1}{4}[n(uv+vu-u_{x}v_{x}-v_{x}u_{x})]_{x}-\frac{1}{4}n(vu-v_{x}u_{x}-uv_{x}+u_{x}v)+\frac{1}{4}(vu+v_{x}u-vu_{x}-v_{x}u_{x})n, \\ m=u-u_{xx}, n=v-v_{xx}, $$
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

(24) $$ u_{11}=\sum_{j=1}^N p_j(t)e^{-\mid x-q_j(t)\mid},\quad u_{12}=\sum_{j=1}^N r_j(t)e^{-\mid x-q_j(t)\mid}, \quad u_{21}=\sum_{j=1}^N s_j(t)e^{-\mid x-q_j(t)\mid},\\ v_{11}=\sum_{j=1}^N f_j(t)e^{-\mid x-q_j(t)\mid}, \quad v_{12}=\sum_{j=1}^N g_j(t)e^{-\mid x-q_j(t)\mid}, \quad v_{21}=\sum_{j=1}^N h_j(t)e^{-\mid x-q_j(t)\mid}.$$
In the distribution sense, we have

(25) $$ u_{11,x}=-\sum_{j=1}^N p_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad m_{11}=2\sum_{j=1}^N p_j\delta(x-q_j),\\u_{12,x}=-\sum_{j=1}^N r_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad m_{12}=2\sum_{j=1}^N r_j\delta(x-q_j),\\u_{21,x}=-\sum_{j=1}^N s_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad m_{21}=2\sum_{j=1}^N s_j\delta(x-q_j),\\v_{11,x}=-\sum_{j=1}^N f_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad n_{11}=2\sum_{j=1}^N f_j\delta(x-q_j),\\v_{12,x}=-\sum_{j=1}^N g_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad n_{12}=2\sum_{j=1}^N g_j\delta(x-q_j),\\v_{21,x}=-\sum_{j=1}^N h_j{\rm{}sgn}(x-q_j)e^{-\mid x-q_j\mid}, \quad n_{21}=2\sum_{j=1}^N h_j\delta(x-q_j), $$
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

(26) $$ q_{1,t}(t)=0,\\ p_{1,t}(t)=-\frac{1}{3}p_{1}(t)(p_{1}(t)f_{1}(t)+r_{1}(t)h_{1}(t))+\frac{1}{3}s_{1}(t)(f_{1}(t)r_{1}(t)-p_{1}(t)g_{1}(t)),\\ r_{1,t}(t)=-\frac{1}{3}p_{1}(t)(f_{1}(t)r_{1}(t)-p_{1}(t)g_{1}(t))-\frac{1}{3}r_{1}(t)(p_{1}(t)f_{1}(t)+r_{1}(t)h_{1}(t)),\\s_{1,t}(t)=-\frac{1}{3}p_{1}(t)(f_{1}(t)s_{1}(t)-p_{1}(t)h_{1}(t))-\frac{1}{3}s_{1}(t)(p_{1}(t)f_{1}(t)+s_{1}(t)g_{1}(t)),\\f_{1,t}(t)=\frac{1}{3}f_{1}(t)(p_{1}(t)f_{1}(t)+g_{1}(t)s_{1}(t))+\frac{1}{3}h_{1}(t)(r_{1}(t)f_{1}(t)-p_{1}(t)g_{1}(t)),\\g_{1,t}(t)= \frac{1}{3}g_{1}(t)(p_{1}(t)f_{1}(t)+g_{1}(t)s_{1}(t))+\frac{1}{3}f_{1}(t)(p_{1}(t)g_{1}(t)-r_{1}(t)f_{1}(t)),\\h_{1,t}(t)=\frac{1}{3}h_{1}(t)(p_{1}(t)f_{1}(t)+h_{1}(t)r_{1}(t))+\frac{1}{3}f_{1}(t)(p_{1}(t)h_{1}(t)-f_{1}(t)s_{1}(t)). $$
From Eq. (26), we can take

(27) $$ q_{1}(t)=0, \\ p_{1}(t)f_{1}(t)+h_{1}(t)r_{1}(t)=A_{1},\\ f_{1}(t)r_{1}(t)-p_{1}(t)g_{1}(t)=A_{2},\\ p_{1}(t)f_{1}(t)+s_{1}(t)g_{1}(t)=A_{3},\\ f_{1}(t)s_{1}(t)-p_{1}(t)h_{1}(t)=A_{4}, $$
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

(28) $$ p_{1}(t)\!=\!C_{1}e^{\lambda_{1}t}+C_{2}e^{\lambda_{2}t}, \quad f_{1}(t)=C^{'}_{1}e^{\lambda_{1}t}+C^{'}_{2}e^{\lambda_{2}t}. r_{1}(t)\!=\!C_{0}e^{-({1}/{3})A_{1}t}-\frac{A_{2}C_{3}}{3\lambda_{1}+A_{1}}e^{\lambda_{1}t}-\frac{C_{4}A_{2}}{3\lambda_{2}+A_{1}} e^{\lambda_{2}t}, s_{1}(t)\!=\!C_{5}e^{-({1}/{3})A_{3}t}-\frac{A_{4}C_{6}}{3\lambda_{1}+A_{3}}e^{\lambda_{1}t}-\frac{C_{7}A_{4}}{3\lambda_{2}+A_{3}} e^{\lambda_{2}t}, g_{1}(t)\!=\!C^{'}_{0}e^{({1}/{3})A_{3}t}-\frac{A_{2}C^{'}_{3}}{3\lambda_{1}+A_{3}}e^{\lambda_{1}t}-\frac{C^{'}_{4}A_{2}}{3\lambda_{2}-A_{1}} e^{\lambda_{2}t}, h_{1}(t)\!=\!C^{'}_{5}e^{({1}/{3})A_{1}t}-\frac{A_{4}C^{'}_{6}}{3\lambda_{1}-A_{1}}e^{\lambda_{1}t}-\frac{C^{'}_{7}A_{4}}{3\lambda_{2}-A_{1}} e^{\lambda_{2}t}, $$
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

(29) $$ p_{1}(t)=C_{1}e^{\lambda_{1}t}+C_{2}te^{\lambda_{1}t}, f_{1}(t)=C^{'}_{1}e^{\lambda_{1}t}+C^{'}_{2}te^{\lambda_{1}t}, \\ r_{1}(t)=C_{0}e^{-({1}/{3})A_{1}t}-\frac{A_{2}C_{3}}{3\lambda_{1}+A_{1}}e^{\lambda_{1}t}+\frac{A_{2}C_{4}}{3(\lambda_{1}+\frac{1}{3}A_{1})^{2}} e^{\lambda_{1}t}-A_{2}C_{4}\frac{t}{3\lambda_{1}+A_{1}}e^{\lambda_{1}t}, \\ s_{1}(t)=C_{5}e^{-({1}/{3})A_{3}t}-\frac{A_{4}C_{6}}{3\lambda_{1}+A_{3}}e^{\lambda_{1}t}+\frac{A_{4}C_{7}}{3(\lambda_{1}+\frac{1}{3}A_{3})^{2}} e^{\lambda_{1}t}-A_{4}C_{7}\frac{t}{3\lambda_{1}+A_{3}}e^{\lambda_{1}t}, \\ g_{1}(t)=C^{'}_{0}e^{({1}/{3})A_{3}t}-\frac{A_{2}C^{'}_{3}}{3\lambda_{1}-A_{3}}e^{\lambda_{1}t}+\frac{A_{2}C^{'}_{4}}{3(\lambda_{1}-\frac{1}{3}A_{3})^{2}} e^{\lambda_{1}t}-A_{2}C^{'}_{4}\frac{t}{3\lambda_{1}-A_{3}}e^{\lambda_{1}t}, \\ h_{1}(t)=C^{'}_{5}e^{-({1}/{3})A_{1}t}-\frac{A_{4}C^{'}_{6}}{3\lambda_{1}-A_{1}}e^{\lambda_{1}t}+\frac{A_{4}C^{'}_{7}}{3(\lambda_{1}-\frac{1}{1}A_{3})^{2}} e^{\lambda_{1}t}-A_{4}C^{'}_{7}\frac{t}{3\lambda_{1}-A_{1}}e^{\lambda_{1}t}, $$
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

(30) $$ p_{1}(t)=C_{1}e^{at}\cos {bt}+C_{2}te^{at}\sin{bt},\\ f_{1}(t)=C^{'}_{1}e^{-at}\cos {bt}+C^{'}_{2}te^{-at}\sin{bt},\\ r_{1}(t)=C_{0}e^{-({1}/{3})A_{1}t}-\frac{1}{3}C_{3}A_{2}\Big[\frac{(\cos{bt}+b\sin{bt})(a+\frac{1}{3}A_{1})}{b^{2}+(a+\frac{1}{3}A_{1})^{2}}\Big]e^{at}-\frac{1}{3}C_{4}A_{2}\Big[\frac{\sin{bt}(a+\frac{1}{3}A_{1})-b\cos{bt}}{(a+\frac{1}{3}A_{1})^{2}+b^{2}}\Big]e^{at}, \\ \\ s_{1}(t)=C_{5}e^{-({1}/{3})A_{3}t}-\frac{1}{3}C_{6}A_{4}\Big[\frac{(\cos{bt}+b\sin{bt})(a+\frac{1}{3}A_{3})}{b^{2}+(a+\frac{1}{3}A_{3})^{2}}\Big]e^{at}-\frac{1}{3}C_{7}A_{4}\Big[\frac{\sin{bt}(a+\frac{1}{3}A_{3})-b\cos{bt}}{(a+\frac{1}{3}A_{3})^{2}+b^{2}}\Big]e^{at}, \\ \\ g_{1}(t)=C^{\prime}_{0}e^{({1}/{3})A_{3}t}-\frac{1}{3}C^{\prime}_{3}A_{2}\Big[\frac{(\cos{bt}+b\sin{bt})(-a-\frac{1}{3}A_{3})}{b^{2}+(-a-\frac{1}{3}A_{3})^{2}}\Big]e^{-at}-\frac{1}{3}C^{\prime}_{4}A_{2}\Big[\frac{\sin{bt}(-a-\frac{1}{3}A_{3})-b\cos{bt}}{(-a-\frac{1}{3}A_{3})^{2}+b^{2}}\Big]e^{-at}, \\ \\ h_{1}(t)=C^{\prime}_{5}e^{({1}/{3})A_{1}t}-\frac{1}{3}C^{\prime}_{6}A_{4}\Big[\frac{(\cos{bt}+b\sin{bt})(-a-\frac{1}{3}A_{1})}{b^{2}+(-a-\frac{1}{3}A_{1})^{2}}\Big]e^{-at}-\frac{1}{3}C^{\prime}_{7}A_{4}\Big[\frac{\sin{bt}(-a-\frac{1}{3}A_{1})-b\cos{bt}}{(-a-\frac{1}{3}A_{1})^{2}+b^{2}}\Big]e^{-at},$$
where

$ a=-\frac{A_{1}+A_{3}}{6}, \quad b=\frac{\sqrt{4A_{2}A_{4}-(A_{1}-A_{3})^{2}}}{6}.$

Reference View Option By original order
By published year
By cited within times
By Impact factor

[1] R. Camassa and D. Holm, Phys. Rev. Lett. 71(1993) 1661.
DOI:10.1103/PhysRevLett.71.1661URLPMID:10054466 [Cited within: 2]

[2] B. Fuchssteiner and A. Fokas , Physica D 4 ( 1981) 47.
[Cited within: 1]

[3] A. Constantin , Proc. R. Soc. Lond. A 457 ( 2001) 953.
[Cited within: 1]

[4] Z. Qiao , Commun. Math. Phys. 239(2003) 309.
DOI:10.1007/s00220-003-0880-yURL
This paper shows that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We see that the well-known CH equation is included in the negative order CH hierarchy while the Dym type equation is included in the positive order CH hierarchy. Furthermore, under two constraint conditions between the potentials and the eigenfunctions, the CH spectral problem is cast in: 1. a new Neumann-like N-dimensional system when it is restricted into a symplectic submanifold of ^2N which is proven to be integrable by using the Dirac-Poisson bracket and the r-matrix process; and 2. a new Bargmann-like N-dimensional system when it is considered in the whole ^2N which is proven to be integrable by using the standard Poisson bracket and the r-matrix process.

[5] A. Himonas, G. Misiolek, G. Ponce, Y. Zhou , Comm. Math. Phys. 271(2007) 511.
DOI:10.1007/s00220-006-0172-4URL

[6] J. Lenells, J. Differential Eqns . 217(2005) 393.
DOI:10.1016/j.jde.2004.09.007URL [Cited within: 1]

[7] A. Degasperis, D. Holm, A. Hone , Theor. Math. Phys. 133(2002) 1463.
DOI:10.1023/A:1021186408422URL [Cited within: 1]

[8] A. Fokas , Physica D 87 ( 1995) 145.
[Cited within: 2]

[9] B. Fuchssteiner , Physica D 95 ( 1996) 229.


[10] Z. Qiao, J. Math. Phys . 47(2006) 112701.
[Cited within: 2]

[11] A. Hone and J. Wang, J. Phys. A: Math. Theor. 41(2008) 372002.
DOI:10.1088/1751-8113/41/37/372002URL [Cited within: 1]

[12] D. Holm and R. Ivanov, J. Phys A: Math. Theor. 43(2010) 492001.
DOI:10.1088/1751-8113/43/49/492001URL [Cited within: 1]

[13] M. Chen, S. Liu, Y. Zhang , Lett. Math. Phys. 75(2006) 1.
DOI:10.1007/s11005-005-0041-7URL
An explicit reciprocal transformation between a two-component generalization of the Camassa–Holm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established. This transformation enables one to obtain solutions of the 2-CH system from those of the first negative flow of the AKNS hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH system are presented

[14] X. Geng and B. Xue , Nonlinearity 22 ( 2009) 1847.


[15] X. Geng and B. Xue Adv. Math. 226(2011) 827.
DOI:10.1016/j.aim.2010.07.009URL
A three-component generalization of Camassa-Holm equation with peakon solutions is proposed, which is associated with a 3 x 3 matrix spectral problem with three potentials. With the aid of the zero-curvature equation, we derive a hierarchy of new nonlinear evolution equations and establish their Hamiltonian structures. The three-component generalization of Camassa-Holm equation is exactly a negative flow in the hierarchy and admits exact solutions with N-peakons and an infinite sequence of conserved quantities. (C) 2010 Elsevier Inc.

[16] J. Song, C. Qu, Z. Qiao, J. Math. Phys . 52(2011) 013503.


[17] C. Qu, J. Song, R. Yao , SIGMA Symmetry Integrability Geom Methods Appl. 9(2013) 001.


[18] N. Li, Q. Liu, Z. Popowicz, J. Geom.Phys . 85(2013) 29.


[19] B. Xia and Z. Qiao , Proc. R. Soc. A 471 ( 2015) 0750.
[Cited within: 1]

[20] B. Xia, Z. Qiao, R. Zhou , Stud. Appl. Math. 135(2015) 248.
DOI:10.1111/sapm.2015.135.issue-3URL [Cited within: 5]

[21] B. Xia, Z. Qiao, R. Zhou, J. Nonlinear Math Phys. 22(2015) 155.
DOI:10.1080/14029251.2015.996446URL [Cited within: 1]

[22] S.C. Anco and F. Mobasheramini , Physica D 355 ( 2017) 1.
[Cited within: 1]

[23] S.C. Anco, X. Chang, J. Szmigielski , Stud. Appl. Math. 141(2018) 680.
DOI:10.1111/sapm.2018.141.issue-4URL [Cited within: 1]