2.1. Spectral analysis

It is well-known that cCH equation (Equation 1.3) admits the following Lax pair

(2.1)\begin{align} \phi_{x}=U\phi,\quad \phi_{t}=V\phi, \end{align}

where $U=U(x,t;\lambda)$ and $V=V(x,t;\lambda)$ are $2\times2$ matrices defined by

(2.2)\begin{align} U=\left(\begin{array}{cc} -\frac{Q}{2} & \frac{\lambda m}{2} \\ -\frac{\lambda m}{2} & \frac{Q}{2} \end{array}\right),\quad V=\left(\begin{array}{cc} \frac{Q}{\lambda^{2}}+\frac{QR}{2} & -\frac{u-Qu_x}{\lambda}-\frac{\lambda Rm}{2} \\ \frac{u+Qu_x}{\lambda}+\frac{\lambda Rm}{2} & -\frac{Q}{\lambda^{2}}-\frac{QR}{2} \end{array}\right), \end{align}

with

(2.3)\begin{equation} Q=Q(\nu,\lambda)=\sqrt{1-\frac{\nu\lambda^2}{2}},\ \ \ \ \ \ \ R=u^2-u_x^2. \end{equation}

It can be seen that under the following transformations

(2.4)\begin{equation} x=\tilde{x}, \quad t=\frac{2}{\nu}\tilde{t}, \quad u(x,t)=\sqrt{\frac{\nu}{2}}\tilde{u}(\tilde{x},\tilde{t}), \end{equation}

Eq. (Equation 1.3) turns to

(2.5)\begin{equation} \tilde{m}_{\tilde{t}}+(\tilde{m} \tilde{R})_{\tilde{x}}+2\tilde{u}_{\tilde{x}}=0, \quad \tilde{m}=\tilde{u}-\tilde{u}_{\tilde{x} \tilde{x}}, \quad \tilde{R}=\tilde{u}^2-\tilde{u}_{\tilde{x}}^2. \end{equation}

Without loss of generality, we pick ν = 2. To get rid of the multi-value problem of square roots, we introduce a new transformation

(2.6)\begin{equation} Q=\frac{i}{2}(\mu-\frac{1}{\mu}), \quad \lambda=\frac{1}{2}(\mu+\frac{1}{\mu}). \end{equation}

After doing a gauge transformation

(2.7)\begin{equation} \psi=P^{-1}\phi e^{J}, \end{equation}

with

(2.8)\begin{equation} P=P(x,t)=\sqrt{\frac{q+1}{2q}} \left(\begin{array}{cc} 1 & -\frac{im}{q+1} \\ -\frac{im}{q+1} & 1 \end{array}\right), \end{equation}

(2.9)\begin{equation} \delta=x-\int_{x}^{+\infty}(q-1)dy-\frac{2t}{\lambda^2}, \quad q=\sqrt{m^2+1}, \end{equation}

where we define $J=\frac{1}{2}Q\delta\sigma_3$. The asymptotic condition is shown as follows

\begin{align*} \psi(x,t,\mu)\sim I,\quad x\rightarrow\pm\infty. \end{align*}

Equation (Equation 1.3) has the following form Lax pair

(2.10a)\begin{align} &\psi_x+[J_x,\psi]=X\psi, \end{align}

(2.10b)\begin{align} &\psi_t+[J_t,\psi]=T\psi, \end{align}

where

\begin{equation*} X=\frac{im_x}{2q^2}\sigma_1+\frac{m}{2\mu q} \left(\begin{array}{cc} -im & 1 \\ -1 & im \end{array}\right),\nonumber \end{equation*}

\begin{equation*} \begin{split} T=&\frac{im_t}{2q^2}\sigma_1-\frac{mR}{2\mu q} \left(\begin{array}{cc} -im & 1 \\ -1 & im \end{array}\right) +\frac{(\mu^2-1)u_x}{\mu^2+1}\sigma_1-\frac{2 \mu u}{(\mu^2+1)q} \left(\begin{array}{cc} -im & 1 \\ -1 & im \end{array}\right) \\ &+\frac{2i\mu(\mu^2-1)}{(\mu^2+1)^2} \left(\begin{array}{cc} \frac{1}{q}-1 & -\frac{im}{q} \\ \frac{im}{q} & 1-\frac{1}{q} \end{array}\right),\nonumber \end{split} \end{equation*}

there $[A,B]=AB-BA$ and the notations $J=J(x,t,\mu)$, $X=X(x,t,\mu)$, $T=T(x,t,\mu)$ are $2\times2$ matrices. The σ ₁ is Pauli matrix and the Pauli matrices are that

\begin{equation*} \sigma_1= \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right),\quad \sigma_2= \left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right),\quad \sigma_3= \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right). \nonumber \end{equation*}

The two Jost solutions of (Equation 2.10a) are completely determined by Volterra integrable equations

(2.11)\begin{equation} \psi_\pm(x,t,\mu)=I+\int_{\pm\infty}^{x}e^{-\frac{i}{4}(\mu-\frac{1}{\mu}) \int_{y}^{x}qd\tau \hat{\sigma}_3}\left[X(y,t,\mu)\psi_\pm(y,t,\mu)\right]dy, \end{equation}

where $e^{A\hat{\sigma}_{3}}B=e^{A\sigma_{3}}Be^{-A\sigma_{3}}$. They have the analytic properties, symmetry properties, and asymptotic properties shown in the following three propositions.

Proposition 2.2. The Jost eigenfunctions $\psi_{\pm}(x,t,\mu)$ have the following symmetry conditions

(2.12)\begin{equation} \psi_{\pm}(\mu)=\sigma_1\psi_{\pm}^{*} (-\frac{1}{\mu^{*}})\sigma_1=\sigma_2\psi_{\pm}^{*} (\mu^{*})\sigma_2=\frac{1}{P^{2}}\sigma_2\psi_{\pm}(-\frac{1}{\mu})\sigma_2. \end{equation}

Proposition 2.3. The Jost eigenfunctions $\psi_{\pm}(x,t,\mu)$ possess asymptotic behaviour in the µ-plane

(2.13)\begin{equation} \psi_{\pm}(x,t,\mu)=I+\mathcal {O}(\frac{1}{\mu}),\quad \mu\rightarrow\pm\infty, \end{equation}

and according to the symmetry relations of $\psi_{\pm}(x,t,\mu)$, there are the same relations when $\mu\rightarrow 0$.

It can be seen that $\phi_{\pm}(x,t,\mu)$ are two fundamental matrix solutions of Lax pair (Equation 2.1). As a result, there exists a matrix $S(\mu)$ which independent of variables x and t satisfies

(2.14)\begin{equation} \phi_-(x,t,\mu)=\phi_+(x,t,\mu)S(\mu), \end{equation}

then on the basis of transformation (Equation 2.7), the relation (Equation 2.14) can be written in the following form

(2.15)\begin{equation} \psi_-(x,t,\mu)e^{-J(x,t,\mu)}=\psi_+(x,t,\mu)e^{-J(x,t,\mu)}S(\mu), \end{equation}

where $S(\mu)$ is a $2\times2$ matrix. We can derive the symmetries of $S(\mu)$ from proposition 2.2 as follows

(2.16)\begin{equation} S(\mu)=S^{*}(\frac{1}{\mu^{*}})=\sigma_3S(-\frac{1}{\mu})\sigma_3 =\sigma_{2}S^{*}(\mu^{*})\sigma_{2}. \end{equation}

According to symmetry relation (Equation 2.12), we can define the matrix $S(\mu)$ as the following form

(2.17)\begin{equation} S(\mu)=\left(\begin{array}{cc} a(\mu) & -b^{*}(\mu^{*}) \\ b(\mu) & a^{*}(\mu^{*}) \end{array}\right). \end{equation}

Because $tr(U)=0$, based on Abel theorem, we have $\det(\psi_{\pm})=1$, then $\det(S(\mu))=1$. By directly calculating Eq. (Equation 2.15), $a(\mu)$ and $b(\mu)$ can be expressed as

(2.18)\begin{equation} a(\mu)=Wr(\psi_{-1},\psi_{+2}), \quad b(\mu)=Wr(\psi_{+1},\psi_{-1}). \end{equation}

We define the reflection coefficient

(2.19)\begin{equation} \rho(\mu)=\frac{b(\mu)}{a(\mu)}. \end{equation}

Taking into account the properties of the Jost eigenfunctions $\psi_{\pm}(x,t,\mu)$, we obtain the corresponding properties to the elements $a(\mu)$ and $b(\mu)$ of the scattering matrix $S(\mu)$ as follows.

It is noticed that, since the branch cut points $\mu=\pm i$ exist in the extended complex µ-plane, the asymptotic behaviour of eigenfunctions $\psi_{\pm}(x,t,\mu)$ ought to be considered as $\mu\rightarrow \pm i$. Introducing a new Jost function

(2.21)\begin{equation} \tilde{\psi}_{\pm}(x,t,\mu)=\phi_{\pm}(x,t,\mu) e^{(\frac{Q}{2}x-\frac{Q}{\lambda^{2}}t)\sigma_{3}}, \end{equation}

then we have

(2.22)\begin{equation} \tilde{\psi}_{\pm}(x,t,\mu)\sim I,\quad x\rightarrow\pm \infty. \end{equation}

The Lax pair (Equation 2.1) can be converted to

(2.23a)\begin{align} &(\tilde{\psi}_{\pm})_{x}=-\frac{Q}{2}[\sigma_{3},\tilde{\psi}_{\pm}] +\tilde{X}\tilde{\psi}_{\pm}, \end{align}

(2.23b)\begin{align} &(\tilde{\psi}_{\pm})_{t}=\frac{Q}{\lambda^{2}}[\sigma_{3},\tilde{\psi}_{\pm}] +\tilde{T}\tilde{\psi}_{\pm}, \end{align}

with

\begin{align*} &\tilde{X}= \left(\begin{array}{cc} 0 & \frac{\lambda m}{2} \\ -\frac{\lambda m}{2} & 0 \end{array}\right),\\ &\tilde{T}=\frac{R}{2} \left(\begin{array}{cc} Q & -\lambda m \\ \lambda m & -Q \end{array}\right)+\frac{u-Qu_{x}}{\lambda} \left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right). \end{align*}

Taking into account the asymptotic expansion of $\tilde{\psi}$ at $\mu\rightarrow i$, we have

(2.24)\begin{equation} \tilde{\psi}=I+\tilde{\psi}_{1}(\mu-i)+\mathcal {O}((\mu-i)^{2}), \quad \mu\rightarrow i, \end{equation}

with

(2.25)\begin{equation} \tilde{\psi}_{1}= \left(\begin{array}{cc} 0 & -\frac{1}{2}(u+u_{x}) \\ -\frac{1}{2}(u-u_{x}) & 0 \end{array}\right). \end{equation}

According to the transformations (Equation 2.7) and (Equation 2.21), we have

(2.26)\begin{equation} \psi_{\pm}(x,t,\mu)=P^{-1}(x,t)\tilde{\psi}_{\pm}(x,t,\mu) e^{\frac{1}{2}Q(\mu)\int_{\pm\infty}^{x}(q-1)dy}. \end{equation}

Taking the limits of (Equation 2.26) as $\mu \rightarrow i $, we obtain the asymptotic of $a(\mu)$

(2.27)\begin{equation} a(\mu)=e^{-\frac{1}{2}\int_{-\infty}^{+\infty}(q-1)dx}(1+\mathcal {O}((\mu-i)^{2})), \quad \mu\rightarrow i. \end{equation}

The analytic properties of $\psi_{\pm}(x,t,\mu)$ stated above allow rewriting the relation (Equation 2.15) as a sectionally meromorphic matrix.

2.2. RH problem

Define a piecewise meromorphic $2\times2$-matrix valued function

(2.28)\begin{align} M(\mu)=M(x,t;\mu):=\left\{\begin{array}{ll} \left( \frac{\psi_{-,1}(x,t;\mu)}{a(\mu)}, \psi_{+,2}(x,t;\mu)\right), &\text{as } \mu\in \mathbb{C}^+,\\[12pt] \left( \psi_{+,1}(x,t;\mu),\frac{\psi_{-,2}(x,t;\mu)}{a^{*}(\mu^{*})}\right) , &\text{as }\mu\in \mathbb{C}^-. \end{array}\right. \end{align}

Since the function $\delta(x,t,\mu)$ has integral term, which directly lead to the difficulty to solve Eq. (Equation 1.3), a new space variable is introduced as follows

(2.33)\begin{equation} y(x,t)=x-\int_{x}^{+\infty}(q(s,t)-1)ds. \end{equation}

Defining the function $\tilde{M}(\mu)$ and phase function $\theta(\mu)$ on the new scale $y(x,t)$

(2.34)\begin{equation} \tilde{M}(\mu)=\tilde{M}(y,t,\mu)=M(x(y,t),t,\mu), \end{equation}

(2.35)\begin{equation} \theta(\mu)=\frac{1}{2}iQ(\mu)\left(\frac{y}{t}-\frac{2}{\lambda^{2}(\mu)}\right), \end{equation}

then, the RH problem 2.5 is transformed as follows.

Using the asymptotic property of function $\tilde{M}(\mu)$ as $\mu\rightarrow i$, the relationship between the solution of cCH equation and the RH problem is obtained

(2.40)\begin{equation} u(x,t)=\lim_{\mu \to i}\frac{1}{\mu-i}\left(1-\frac{(\tilde{M}_{11}(\mu)+\tilde{M}_{21}(\mu)) (\tilde{M}_{12}(\mu)+\tilde{M}_{22}(\mu))} {(\tilde{M}_{11}(i)+\tilde{M}_{21}(i))(\tilde{M}_{12}(i)+\tilde{M}_{22}(i))}\right), \end{equation}

with

(2.41)\begin{equation} x(y,t)= y-\ln\left(\frac{\tilde{M}_{12}(i)+\tilde{M}_{22}(i)}{\tilde{M}_{11}(i)+\tilde{M}_{21}(i)}\right), \end{equation}

where $\tilde{M}_{ij}(\mu)$ $(i,j=1,2)$ represents the element in raw i and column j of $\tilde{M}(\mu)$.

2.3. Single high-order pole solutions

In order to obtain a pure soliton solution, this section we will consider the reflectionless situation, i.e., $b(\mu)=0$. Then $V(\mu)=0$ for $\mu\in\mathbb{R}$. We assume that $\mu_{0}\in\mathbb{C}^{+}$ is the only one N-order zero point of the scattering data $a(\mu)$, then the N-order zero points $\left\{-\mu^{*}_{0}, -\frac{1}{\mu_{0}}, \frac{1}{\mu^{*}_{0}}\right\}\in \mathbb{C}^{+}$ are also obtained from the symmetry of $a(\mu)$. Additionally, $\left\{\mu^{*}_{0}, -\mu_{0}, -\frac{1}{\mu^{*}_{0}}, \frac{1}{\mu_{0}}\right\}\in\mathbb{C} ^{-}$ are the N-order zero points of the scattering data $a^{*}(\mu^{*})$.

The discrete spectrum is the set $\mathcal{Z}=\left\{\pm\mu_{0}, \pm\mu^{*}_{0}, \pm\frac{1}{\mu_{0}}, \pm\frac{1}{\mu^{*}_{0}}\right\}$, which can be shown in Figure 1. After taking $v_{1}=\mu_{0}$, $v_{2}=\frac{1}{\mu^{*}_{0}}$, the scattering data $a(\mu)$ can be expressed as

(2.42)\begin{equation} a(\mu)=\prod_{j=1}^{2}(\mu-v_{j})^{N}(\mu+v^{*}_{j})^{N}a_{0}(\mu), \end{equation}

Figure 1.

Distribution of the discrete spectrum $\mathcal{\mu}$.

where $a_{0}(\mu)\neq0$ for all $\mu\in\mathbb{C}^{+}$. Therefore, from the definition of $\tilde{M}(y,t,\mu)$, it can be seen that $\mu=v_{j}$ and $\mu=-v^{*}_{j}$ $(j=1,2)$ are N-order pole points of $\tilde{M}_{11}$. Simultaneously, $\mu=v^{*}_{j}$ and $\mu=-v_{j}$ $(j=1,2)$ are N-order pole points of $\tilde{M}_{12}$.

Based on the definitions of (Equation 2.14) and (Equation 2.15), under the new scale $y(x,t)$, we define $\breve{\phi}_{\pm}(y,t,\mu)=\phi_{\pm}(x,t,\mu)$ and $\breve{\psi}_{\pm}(y,t,\mu)=\psi_{\pm}(x,t,\mu)$ and obtain the following relationships

(2.43)\begin{equation} \breve{\phi}_-(y,t,\mu)=\breve{\phi}_+(y,t,\mu)S(\mu), \end{equation}

and

(2.44)\begin{equation} \breve{\psi}_-(y,t,\mu)e^{it\theta(\mu)\sigma_{3}}=\breve{\psi}_+(y,t,\mu)e^{it\theta(\mu)\sigma_{3}}S(\mu), \end{equation}

where $\theta(\mu)$ is defined in (Equation 2.35). Under the assumption that $\mu=v_{j}$ $(j=1,2)$ are N-order zero points of $a(\mu)$, from the following two relations, $a(\mu)$ can be represented as a Wronskian of analytic Jost solutions

(2.45)\begin{equation} a(\mu)=Wr(\breve{\phi}_{-1},\breve{\phi}_{+2}), \end{equation}

then at zero points v_j $(j=1,2)$, $a(\mu)$ vanishes to N-order, there exist complex constants $b_{j,s}$ $(j=1,2)$ $(s=1,2,\cdots,N)$ satisfy the follows

(2.46)\begin{equation} \frac{\partial^{m}[\breve{\phi}_+(y,t,v_{j})]_{2}}{\partial\mu^{m}}=\sum^{m}_{l=0}\left(\begin{array}{c} m \\ l \end{array}\right) b_{j,m-l+1}\frac{\partial^{l}[\breve{\phi}_-(y,t,v_{j})]_{1}}{\partial\mu^{l}}, \end{equation}

and

(2.47)\begin{equation} \frac{\partial^{m}[\breve{\psi}_+(y,t,v_{j})]_{2}}{\partial\mu^{m}}=\sum^{m}_{l=0}\left(\begin{array}{c} m \\ l \end{array}\right) b_{j,m-l+1}\frac{\partial^{l}[\breve{\psi}_-(y,t,v_{j})]_{1}e^{2it\theta(v_{j})}}{\partial\mu^{l}}, \end{equation}

with $m=0,1,\cdots,N-1$.

On the basis of the definition of $M(\mu)$ defined by (Equation 2.28) and the asymptotic behaviour in RH problem 2.6, we set

(2.48a)\begin{align} &\tilde{M}_{11}(y,t,\mu)=1+\sum_{j=1}^{2}\sum_{s=1}^{N} \left(\frac{1}{(\mu-v_{j})^{s}}+\frac{(-1)^{s}}{(\mu+v^{*}_{j})^{s}}\right)F_{j,s}(y,t), \end{align}

(2.48b)\begin{align} &\tilde{M}_{12}(y,t,\mu)=\sum_{j=1}^{2}\sum_{s=1}^{N} \left(\frac{1}{(\mu-v^{*}_{j})^{s}}+\frac{(-1)^{s+1}}{(\mu+v_{j})^{s}}\right)G_{j,s}(y,t), \end{align}

where $F_{j,s}(y,t)$ and $G_{j,s}(y,t)$ are undetermined functions. Making use of Taylor series expansion, one can obtain the following relationships

(2.49)\begin{align} &e^{-2it\theta(\mu)}=\sum_{l=0}^{+\infty}\zeta_{j,l}(y,t)(\mu-v_{j})^{l},\quad e^{2it\theta(\mu)}=\sum_{l=0}^{+\infty}\zeta^{*}_{j,l}(y,t)(\mu-v^{*}_{j})^{l}, \end{align}

where the element $\zeta_{j,l}(y,t)$ is

\begin{align*} \zeta_{j,l}(y,t)=\lim_{\mu\rightarrow v_{j}}\frac{1}{l!}\frac{\partial^{l}}{\partial \mu^{l}}e^{-2it\theta(\mu)},\quad \zeta^{*}_{j,l}(y,t)=\lim_{\mu\rightarrow v^{*}_{j}}\frac{1}{l!}\frac{\partial^{l}}{\partial \mu^{l}}e^{2it\theta(\mu)}. \end{align*}

It can be obtained that the coefficient of item $(\mu-v_{j})^{-s}$ of function $\tilde{M}_{11}(y,t,\mu)$ is $F_{j,s}(y,t)$. Now we extend the residue theorem by combining (Equation 2.44) and (Equation 2.47) and obtain the following relations

(2.50)\begin{align} \begin{split} F_{j,s}&=\sum_{l=s}^{N}\sum_{m=0}^{l-s}\sum_{p=1}^{2}\sum_{q=1}^{N} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right) \times \\ &r_{j,l}\zeta_{j,l-s-m} \left\{\frac{(-1)^{m}}{(v_{j}-v^{*}_{p})^{q+m}} +\frac{(-1)^{m+q+1}}{(v_{j}+v_{p})^{q+m}}\right\}G_{p,q}, \end{split} \end{align}

where

(2.51)\begin{align} r_{j,l}=\lim_{\mu\rightarrow v_{j}}\frac{b_{j,l}}{(N-l)!}\frac{\partial^{N-l}}{\partial(\mu-v_{j})^{N-l}} \frac{(\mu-v_{j})^{N}}{a(\mu)}. \end{align}

Likewise, the coefficient of item $(\mu-v^{*}_{j})^{-s}$ of function $\tilde{M}_{12}(y,t,\mu)$ is $G_{j,s}(y,t)$. Through the same method, we have

(2.52)\begin{align} \begin{split} G_{j,s}&=-\sum_{l=s}^{N}r^{*}_{j,l}\zeta^{*}_{j,l-s} -\sum_{l=s}^{N}\sum_{m=0}^{l-s}\sum_{p=1}^{2}\sum_{q=1}^{N} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right)\times\\ &r^{*}_{j,l}\zeta^{*}_{j,l-s-m}\left\{\frac{(-1)^{m}} {(v^{*}_{j}-v_{p})^{q+m}} +\frac{(-1)^{m+q}}{(v^{*}_{j}+v^{*}_{p})^{q+m}}\right\}F_{p,q}. \end{split} \end{align}

Introducing the new notations for $j,p=1,2$

(2.53)\begin{align} |\eta_{j}\rangle&=(\eta_{j1},\cdots,\eta_{jN})^{T},\quad \eta_{js}=-\sum_{l=s}^{N}r^{*}_{j,l}\zeta^{*}_{j,l-s}(y,t), \end{align}

then defining the N × N matrices $\vartheta_{j,p}=[\vartheta_{j,p}]_{s,q}$ and $\tilde{\vartheta}_{j,p}=[\tilde{\vartheta}_{j,p}]_{s,q}$ for $s,q=1,2,\cdots,N$ as follows

(2.54)\begin{align} \begin{split} \vartheta_{j,p}=\sum_{l=s}^{N}\sum_{m=0}^{l-s} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right) r_{j,l}\zeta_{j,l-s-m}\left\{\frac{(-1)^{m}}{(v_{j}-v^{*}_{p})^{q+m}} +\frac{(-1)^{m+q+1}}{(v_{j}+v_{p})^{q+m}}\right\}, \end{split} \end{align}

(2.55)\begin{align} \begin{split} \tilde{\vartheta}_{j,p}=\sum_{l=s}^{N}\sum_{m=0}^{l-s} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right) r^{*}_{j,l}\zeta^{*}_{j,l-s-m}\left\{\frac{(-1)^{m}} {(v^{*}_{j}-v_{p})^{q+m}} +\frac{(-1)^{m+q}}{(v^{*}_{j}+v^{*}_{p})^{q+m}}\right\}, \end{split} \end{align}

(2.56)\begin{align} |F_{p}\rangle=(F_{p,1},\cdots,F_{p,N})^{T},\quad |G_{p}\rangle=(G_{p,1},\cdots,G_{p,N})^{T}, \end{align}

(2.57)\begin{align} \vartheta= \left(\begin{array}{cc} \vartheta_{1,1} & \vartheta_{1,2} \\ \vartheta_{2,1} & \vartheta_{2,2} \end{array}\right),\quad \tilde{\vartheta}= \left(\begin{array}{cc} \tilde{\vartheta}_{1,1} & \tilde{\vartheta}_{1,2} \\ \tilde{\vartheta}_{2,1} & \tilde{\vartheta}_{2,2} \end{array}\right),\quad I_{\sigma}= \left(\begin{array}{cc} I & 0 \\ 0 & I \end{array}\right), \end{align}

where I is a N × N identity matrix. Equations (Equation 2.50)–(Equation 2.52) can be written as a system of linear equations

(2.58)\begin{equation} \begin{aligned} &I|F_{p}\rangle-\vartheta_{j,p}|G_{p}\rangle=0,\\ &\tilde{\vartheta}_{j,p}|F_{p}\rangle+ I|G_{p}\rangle=|\eta_{j}\rangle. \end{aligned} \end{equation}

Then taking

(2.59)\begin{align} |F\rangle=(|F_{1}\rangle,|F_{2}\rangle)^{T},\quad |G\rangle=(|G_{1}\rangle,|G_{2}\rangle)^{T}, \end{align}

(2.60)\begin{align} |\eta\rangle=(|\eta_{1}\rangle,|\eta_{2}\rangle)^{T}. \end{align}

The system (Equation 2.58) has the following solution

(2.61)\begin{align} |F\rangle=(I_{\sigma}+\vartheta\tilde{\vartheta})^{-1}\vartheta|\eta\rangle,\quad |G\rangle=(I_{\sigma}+\tilde{\vartheta}\vartheta)^{-1}|\eta\rangle. \end{align}

The new expression of (Equation 2.48a) is derived as

(2.62)\begin{align} \tilde{M}_{11}(y,t,\mu)&=\frac{\det(I_{\sigma}+\vartheta\tilde{\vartheta}+|\eta\rangle\langle\Upsilon(\mu)|\vartheta)} {\det(I_{\sigma}+\vartheta\tilde{\vartheta})}, \end{align}

(2.63)\begin{align} \tilde{M}_{12}(y,t,\mu)&=\frac{\det(I_{\sigma}+\tilde{\vartheta}\vartheta+|\eta\rangle\langle\tilde{\Upsilon}(\mu)|)} {\det(I_{\sigma}+\tilde{\vartheta}\vartheta)}-1, \end{align}

with

(2.64)\begin{equation} \begin{split} &\langle\Upsilon(\mu)|=(\langle\Upsilon_{1}(\mu)|,\langle\Upsilon_{2}(\mu)|),\quad \langle\Upsilon_{j}(\mu)|=(\Upsilon_{j1},\ldots,\Upsilon_{jN}),\\ &\langle\tilde{\Upsilon}(\mu)| =(\langle\tilde{\Upsilon}_{1}(\mu)|,\langle\tilde{\Upsilon}_{2}(\mu)|),\quad \langle\tilde{\Upsilon}_{j}(\mu)|=(\tilde{\Upsilon}_{j1},\ldots,\tilde{\Upsilon}_{jN}),\\ &\langle\Upsilon^{\prime}(\mu)|= (\langle\Upsilon^{\prime}_{1}(\mu)|,\langle\Upsilon^{\prime}_{2}(\mu)|),\quad \langle\Upsilon^{\prime}_{j}(\mu)|=(\Upsilon^{\prime}_{j1},\ldots,\Upsilon^{\prime}_{jN}),\\ &\langle\tilde{\Upsilon}^{\prime}(\mu)| =(\langle\tilde{\Upsilon}^{\prime}_{1}(\mu)|,\langle\tilde{\Upsilon}^{\prime}_{2}(\mu)|),\quad \langle\tilde{\Upsilon}^{\prime}_{j}(\mu)|=(\tilde{\Upsilon}^{\prime}_{j1},\ldots,\tilde{\Upsilon}^{\prime}_{jN}), \end{split} \end{equation}

\begin{align*} &\Upsilon_{js}=\frac{1}{(\mu-v_{j})^{s}}+\frac{(-1)^{s}}{(\mu+v^{*}_{j})^{s}},\quad \tilde{\Upsilon}_{js}=\frac{1}{(\mu-v^{*}_{j})^{s}}+\frac{(-1)^{s+1}}{(\mu+v_{j})^{s}},\\ &\Upsilon^{\prime}_{js}=\frac{-s}{(\mu-v_{j})^{s+1}}+\frac{s(-1)^{s+1}}{(\mu+v^{*}_{j})^{s+1}},\quad \tilde{\Upsilon}^{\prime}_{js}=\frac{-s}{(\mu-v^{*}_{j})^{s+1}}+\frac{s(-1)^{s}}{(\mu+v_{j})^{s+1}}, \end{align*}

where we differentiate the components of vector $\langle\Upsilon(\mu)|$ from variable µ to obtain vector $\langle\Upsilon^{\prime}(\mu)|$, the vector $\langle\tilde{\Upsilon}^{\prime}(\mu)|$ can also be gained from the same way. In the following content, we will continue to use this definition. On the basis of the symmetry of $\tilde{M}(\mu)=\sigma_2\tilde{M}^{*}(\mu^{*})\sigma_2$, we have

(2.65)\begin{align} \tilde{M}_{22}(y,t,\mu)&=\frac{\det(I_{\sigma}+\tilde{\vartheta}^{*}\vartheta^{*}+|\eta^{*}\rangle\langle\Upsilon^{*}(\mu^{*})|\vartheta^{*})} {\det(I_{\sigma}+\tilde{\vartheta}^{*}\vartheta^{*})}, \end{align}

(2.66)\begin{align} \tilde{M}_{21}(y,t,\mu)&=-\frac{\det(I_{\sigma}+\vartheta^{*}\tilde{\vartheta}^{*}+|\eta^{*}\rangle\langle\tilde{\Upsilon}^{*}(\mu^{*})|)} {\det(I_{\sigma}+\vartheta^{*}\tilde{\vartheta}^{*})}+1. \end{align}

Hence the precise expression formula for the solution of cCH equation with single high-order pole under the ZBCs can be obtained in theorem 2.7.

Theorem 2.7

The solution of cCH equation with single high-order pole under the ZBCs can be obtained as follows

(2.67)\begin{align} u(x,t)=-\left( \frac{\tilde{M}^{\prime}_{11}(i)+\tilde{M}^{\prime}_{21}(i)}{\tilde{M}_{11}(i)+\tilde{M}_{21}(i)}+ \frac{\tilde{M}^{\prime}_{12}(i)+\tilde{M}^{\prime}_{22}(i)}{\tilde{M}_{12}(i)+\tilde{M}_{22}(i)} \right), \end{align}

with

\begin{align*} &\tilde{M}_{11}(i)=\frac{\det(I_{\sigma}+\vartheta\tilde{\vartheta}+|\eta\rangle\langle\Upsilon(i)|\vartheta)} {\det(I_{\sigma}+\vartheta\tilde{\vartheta})},\\ &\tilde{M}_{21}(i)=-\frac{\det(I_{\sigma}+\vartheta^{*}\tilde{\vartheta}^{*}+|\eta^{*}\rangle\langle\tilde{\Upsilon}^{*}(-i)|)} {\det(I_{\sigma}+\vartheta^{*}\tilde{\vartheta}^{*})}+1,\\ &\tilde{M}_{12}(i)=\frac{\det(I_{\sigma}+\tilde{\vartheta}\vartheta+|\eta\rangle\langle\tilde{\Upsilon}(i)|)} {\det(I_{\sigma}+\tilde{\vartheta}\vartheta)}-1\\ &\tilde{M}_{22}(i)=\frac{\det(I_{\sigma}+\tilde{\vartheta}^{*}\vartheta^{*}+|\eta^{*}\rangle\langle\Upsilon^{*}(-i)|\vartheta^{*})} {\det(I_{\sigma}+\tilde{\vartheta}^{*}\vartheta^{*})},\\ &\tilde{M}^{\prime}_{11}(i)=\frac{\det(I_{\sigma}+\vartheta\tilde{\vartheta}+|\eta\rangle\langle\Upsilon^{\prime}(i)|\vartheta)} {\det(I_{\sigma}+\vartheta\tilde{\vartheta})}-1,\\ &\tilde{M}^{\prime}_{21}(i)=-\frac{\det(I_{\sigma}+\vartheta^{*}\tilde{\vartheta}^{*}+|\eta^{*}\rangle\langle\tilde{\Upsilon}^{\prime *}(-i)|)} {\det(I_{\sigma}+\vartheta^{*}\tilde{\vartheta}^{*})}+1,\\ &\tilde{M}^{\prime}_{12}(i)=\frac{\det(I_{\sigma}+\tilde{\vartheta}\vartheta+|\eta\rangle\langle\tilde{\Upsilon}^{\prime}(i)|)} {\det(I_{\sigma}+\tilde{\vartheta}\vartheta)}-1,\\ &\tilde{M}^{\prime}_{22}(i)=\frac{\det(I_{\sigma}+\tilde{\vartheta}^{*}\vartheta^{*}+|\eta^{*}\rangle\langle\Upsilon^{\prime *}(-i)|\vartheta^{*})} {\det(I_{\sigma}+\tilde{\vartheta}^{*}\vartheta^{*})}-1, \end{align*}

where $x=x(y,t)$ is defined in (Equation 2.41), and the elements $|\eta\rangle$, I_σ, ϑ, $\tilde{\vartheta}$, $\langle\Upsilon(i)|$, $\langle\tilde{\Upsilon}(i)|$, $\langle\Upsilon^{\prime}(i)|$, and $\langle\tilde{\Upsilon}^{\prime}(i)|$ are respectively defined in (Equation 2.53), (Equation 2.57), and (Equation 2.64).

2.3.1. One-soliton solution

Next, we construct the simple one-soliton solution and compare the expressions of the solutions obtained by RH approach and Hirota’s bilinear method. In this case, the zero points of scattering data are one-order, and there is no need to calculate the high-order derivative formula of the Jost eigenfunctions and the complex Taylor expansions, which degenerates into the residue conditions of RH matrix. In order to compare with the solution obtained by bilinear method, we need to process the RH matrix and use the above construction process to obtain the simplest expression of solution.

For one-soliton solution, we have N = 1. Supposing $v_{1}=\mu_{0}$ is a first-order zero point of the scattering data $a(\mu)$, then the symmetrical relationships indicate that $\left\{-\mu^{*}_{0}, -\frac{1}{\mu_{0}}, \frac{1}{\mu^{*}_{0}}\right\}\in\mathbb{C}^{+}$ also are zero points of $a(\mu)$. Then from Eqs. (Equation 2.44), (Equation 2.45), and (Equation 2.47), we use the symmetry relationships in RH problem 2.6 to calculate the following residue conditions

(2.68)\begin{align} \begin{split} \mathop{\operatorname{Res}}_{\mu=\mu_{0}}\tilde{M}^{(1)}(y,t,\mu)&=c_{0}e^{-2it\theta(\mu_{0})}\tilde{M}^{(2)}(y,t,\mu_{0}),\\ \mathop{\operatorname{Res}}_{\mu=\frac{1}{\mu^{*}_{0}}}\tilde{M}^{(1)}(y,t,\mu)&=-\frac{c^{*}_{0}} {\mu^{*2}_{0}}e^{2it\theta^{*}(\mu_{0})}\tilde{M}^{(2)}(y,t,\frac{1}{\mu^{*}_{0}}),\\ \mathop{\operatorname{Res}}_{\mu=-\mu^{*}_{0}}\tilde{M}^{(1)}(y,t,\mu)&=-\frac{c_{0}} {\mu^{2}_{0}}e^{2it\theta^{*}(\mu_{0})}\tilde{M}^{(2)}(y,t,-\mu^{*}_{0}),\\ \mathop{\operatorname{Res}}_{\mu=-\frac{1}{\mu_{0}}}\tilde{M}^{(1)}(y,t,\mu)&=-\frac{c_{0}}{\mu^{2}_{0}} e^{-2it\theta(\mu_{0})}\tilde{M}^{(2)}(y,t,-\frac{1}{\mu_{0}}). \end{split} \end{align}

To simplify the calculation process, we redefine the RH matrix as

(2.69)\begin{align} \tilde{N}_{1}(\mu)=\tilde{M}_{11}(\mu)+\tilde{M}_{21}(\mu),\quad \tilde{N}_{2}(\mu)=\tilde{M}_{12}(\mu)+\tilde{M}_{22}(\mu). \end{align}

The characteristic function satisfies $\psi_{\pm}(\mu)=\sigma_1\psi_{\pm}(-\mu)\sigma_1$, which means that $\tilde{M}(\mu)=\sigma_1\tilde{M}(-\mu)\sigma_1$, then $\tilde{N}_{1}(\mu)=\tilde{N}_{2}(-\mu)$. Calculating from formulas (Equation 2.48a) and combining with the symmetries of matrix $\tilde{M}(\mu)$, we obtain

(2.70)\begin{align} \begin{split} \tilde{N}_{1}(\mu)=&1+\frac{c_{0}e^{-2it\theta(\mu_{0})}}{\mu-\mu_{0}}\tilde{N}_{1}(-\mu_{0})+ \frac{c^{*}_{0}e^{2it\theta^{*}(\mu_{0})}}{\mu+\mu^{*}_{0}}\tilde{N}_{1}(\mu^{*}_{0})+\\ &\frac{-\frac{c_{0}}{\mu^{2}_{0}}e^{-2it\theta(\mu_{0})}}{\mu+\frac{1}{\mu_{0}}}\tilde{N}_{1}(\frac{1}{\mu_{0}})+ \frac{-\frac{c^{*}_{0}}{\mu^{*2}_{0}}e^{2it\theta^{*}(\mu_{0})}}{\mu-\frac{1}{\mu^{*}_{0}}}\tilde{N}_{1}(-\frac{1}{\mu^{*}_{0}}), \end{split} \end{align}

there $c_{0}=\frac{b_{0}}{\dot{a}(\mu)}$ is some complex number, we can set $c_{0}=|c_{0}|e^{i\phi}$ $(\phi\in(0,\pi))$. The linear equation system shown in formula (Equation 2.58) can be simplified as

(2.71)\begin{align} \begin{split} \tilde{N}_{1}(-\mu_{0})=&1+\frac{c_{0}e^{-2it\theta(\mu_{0})}}{-\mu_{0}-\mu_{0}}\tilde{N}_{1}(-\mu_{0})+ \frac{c^{*}_{0}e^{2it\theta^{*}(\mu_{0})}}{-\mu_{0}+\mu^{*}_{0}}\tilde{N}_{1}(\mu^{*}_{0})+\\ &\frac{-\frac{c_{0}}{\mu^{2}_{0}}e^{-2it\theta(\mu_{0})}}{-\mu_{0}+\frac{1}{\mu_{0}}}\tilde{N}_{1}(\frac{1}{\mu_{0}})+ \frac{-\frac{c^{*}_{0}}{\mu^{*2}_{0}}e^{2it\theta^{*}(\mu_{0})}}{-\mu_{0}-\frac{1}{\mu^{*}_{0}}}\tilde{N}_{1}(-\frac{1}{\mu^{*}_{0}}), \end{split} \end{align}

(2.72)\begin{align} \begin{split} \tilde{N}_{1}(\mu^{*}_{0})=&1+\frac{c_{0}e^{-2it\theta(\mu_{0})}}{\mu^{*}_{0}-\mu_{0}}\tilde{N}_{1}(-\mu_{0})+ \frac{c^{*}_{0}e^{2it\theta^{*}(\mu_{0})}}{\mu^{*}_{0}+\mu^{*}_{0}}\tilde{N}_{1}(\mu^{*}_{0})+\\ &\frac{-\frac{c_{0}}{\mu^{2}_{0}}e^{-2it\theta(\mu_{0})}}{\mu^{*}_{0}+\frac{1}{\mu_{0}}}\tilde{N}_{1}(\frac{1}{\mu_{0}})+ \frac{-\frac{c^{*}_{0}}{\mu^{*2}_{0}}e^{2it\theta^{*}(\mu_{0})}}{\mu^{*}_{0}-\frac{1}{\mu^{*}_{0}}}\tilde{N}_{1}(-\frac{1}{\mu^{*}_{0}}), \end{split} \end{align}

(2.73)\begin{align} \begin{split} \tilde{N}_{1}(\frac{1}{\mu_{0}})=&1+\frac{c_{0}e^{-2it\theta(\mu_{0})}}{\frac{1}{\mu_{0}}-\mu_{0}}\tilde{N}_{1}(-\mu_{0})+ \frac{c^{*}_{0}e^{2it\theta^{*}(\mu_{0})}}{\frac{1}{\mu_{0}}+\mu^{*}_{0}}\tilde{N}_{1}(\mu^{*}_{0})+\\ &\frac{-\frac{c_{0}}{\mu^{2}_{0}}e^{-2it\theta(\mu_{0})}}{\frac{1}{\mu_{0}}+\frac{1}{\mu_{0}}}\tilde{N}_{1}(\frac{1}{\mu_{0}})+ \frac{-\frac{c^{*}_{0}}{\mu^{*2}_{0}}e^{2it\theta^{*}(\mu_{0})}}{\frac{1}{\mu_{0}}-\frac{1}{\mu^{*}_{0}}}\tilde{N}_{1}(-\frac{1}{\mu^{*}_{0}}), \end{split} \end{align}

(2.74)\begin{align} \begin{split} \tilde{N}_{1}(-\frac{1}{\mu^{*}_{0}})=&1+\frac{c_{0}e^{-2it\theta(\mu_{0})}}{-\frac{1}{\mu^{*}_{0}}-\mu_{0}}\tilde{N}_{1}(-\mu_{0})+ \frac{c^{*}_{0}e^{2it\theta^{*}(\mu_{0})}}{-\frac{1}{\mu^{*}_{0}}+\mu^{*}_{0}}\tilde{N}_{1}(\mu^{*}_{0})+\\ &\frac{-\frac{c_{0}}{\mu^{2}_{0}}e^{-2it\theta(\mu_{0})}}{-\frac{1}{\mu^{*}_{0}}+\frac{1}{\mu_{0}}}\tilde{N}_{1}(\frac{1}{\mu_{0}})+ \frac{-\frac{c^{*}_{0}}{\mu^{*2}_{0}}e^{2it\theta^{*}(\mu_{0})}}{-\frac{1}{\mu^{*}_{0}}-\frac{1}{\mu^{*}_{0}}}\tilde{N}_{1}(-\frac{1}{\mu^{*}_{0}}). \end{split} \end{align}

After solving this linear system, inputting the results into formula (Equation 2.70), we can obtain the solution of $\tilde{N}_{1}(\mu)$, then substituting it into the following formula

(2.75)\begin{align} u(x,t)=-\left( \frac{\tilde{N}^{\prime}_{1}(i)}{\tilde{N}_{1}(i)}+ \frac{\tilde{N}^{\prime}_{1}(-i)}{\tilde{N}_{1}(-i)} \right), \end{align}

where

(2.76)\begin{align} x(y,t)= y-\ln\left(\frac{\tilde{N}_{1}(-i)}{\tilde{N}_{1}(i)}\right), \end{align}

we can obtain the solution of the cCH equation.

Next, we discuss the solution when scattering data $a(\mu)$ has only two zero points through two examples, where the discrete spectrum has four points.

Case 1: Supposing $v_{1}=\mu_{0}=e^{i\tau}\in\mathbb{C}^{+}$ $(\tau\in(0,\pi))$, $v_{2}=\frac{1}{\mu^{*}_{0}}=v_{1}$. Then $|v_{1}|=1$, and we further assume that $\mu_{0}=r_{1}+ir_{2}$. It can be obtained that $\theta(e^{i\tau})=\theta(-e^{-i\tau})$. Next we define $\chi_{1}\triangleq\theta(e^{i\tau})=-\frac{i\sin\tau y}{2t}+\frac{i\sin\tau}{\cos^{2}\tau}$, hence $e^{-2it\chi_{1}}$ is a real number about variables y and t. Thus algebraic system (Equation 2.70)–(Equation 2.74) have the following forms

(2.77)\begin{align} \tilde{N}_{1}(\mu)=1+\frac{|c_{0}|e^{i\phi}e^{\chi_{1}}}{\mu-e^{i\tau}}\tilde{N}_{1}(-e^{i\tau})+ \frac{|c_{0}|e^{-i\phi}e^{\chi_{1}}}{\mu+e^{-i\tau}}\tilde{N}_{1}(e^{-i\tau}), \end{align}

(2.78)\begin{align} \begin{split} \tilde{N}_{1}(-e^{i\tau})=&1+\frac{|c_{0}|e^{i\phi}e^{\chi_{1}}}{-2e^{i\tau}}\tilde{N}_{1}(-e^{i\tau})+ \frac{|c_{0}|e^{-i\phi}e^{\chi_{1}}}{e^{-i\tau}-e^{i\tau}}\tilde{N}_{1}(e^{-i\tau}),\\ \tilde{N}_{1}(e^{-i\tau})=&1+\frac{|c_{0}|e^{i\phi}e^{\chi_{1}}}{e^{-i\tau}-e^{i\tau}}\tilde{N}_{1}(-e^{i\tau})+ \frac{|c_{0}|e^{-i\phi}e^{\chi_{1}}}{2e^{-i\tau}}\tilde{N}_{1}(e^{-i\tau}), \end{split} \end{align}

then we have the solutions

(2.79)\begin{align} \begin{split} \tilde{N}_{1}(-e^{i\tau})=&\frac{1}{H_{1}}(1-\frac{r_{1}}{2ir_{2}}|c_{0}|e^{\chi_{1}-i\phi+i\tau}),\\ \tilde{N}_{1}(e^{-i\tau})=&\frac{1}{H_{1}}(1-\frac{r_{1}}{2ir_{2}}|c_{0}|e^{\chi_{1}+i\phi-i\tau}), \end{split} \end{align}

where

(2.80)\begin{align} H_{1}=1+\frac{r^{2}_{1}}{4r^{2}_{2}}|c_{0}|e^{2\chi_{1}}+i|c_{0}|e^{\chi_{1}}\sin(\phi-\tau), \end{align}

then we have

(2.81)\begin{align} \begin{split} \tilde{N}_{1}(\mu)=1+\frac{1}{H_{1}}\left(\frac{|c_{0}|e^{\phi+\chi_{1}}- \frac{r_{1}|c_{0}|^{2}e^{2\chi_{1}+\phi}}{2ir_{2}e^{-i(\tau-\phi)}}}{\mu-e^{i\tau}}+ \frac{|c_{0}|e^{\chi_{1}-\phi}-\frac{r_{1}|c_{0}|^{2}e^{2\chi_{1}-\phi}}{2ir_{2}e^{i\tau}}}{\mu+e^{i(\tau-\phi)}}\right), \end{split} \end{align}

after assuming the parameter values are $|c_{0}|=\frac{2r_{2}\,\mathrm{sgn}\,(r_{1})}{r_{1}}$ and $\phi-\tau=\frac{\pi}{2}$, the expression of solution can be obtained as follows

(2.82)\begin{align} u(x,t)=-\frac{4r_{2}}{r^{3}_{1}}\,\mathrm{sgn}\,(r_{1})\frac{\cosh(\chi_{1})}{cosh(2\chi_{1})+\frac{1+r^{2}_{2}}{r^{2}_{1}}}, \end{align}

with

(2.83)\begin{align} x=x(y,t)=y-\ln\left(\frac{1+\frac{1+r_{2}}{1-r_{2}}e^{2\chi_{1}}}{1+\frac{1-r_{2}}{1+r_{2}}e^{2\chi_{1}}}\right). \end{align}

Remark 2.8. By comparing with the soliton solution of cCH equation obtained in reference [Reference Matsuno31], it can be seen that the solution constructed by the RH method is exactly the same as the solution obtained by Hirota’s bilinear method after special parameter selection. Both of the above methods are used to study the construction of soliton solution for cCH equation under ZBCs. The difference is that [Reference Matsuno31] constructs multiple soliton solutions, while our article constructs arbitrary-order pole solutions. Similarly, when the orders of the zero points degenerate to one, the expressions of solutions obtained in these two articles are the same for multiple zero points case.

Remark 2.9. According to the detailed analysis of literature [Reference Matsuno31], it can be seen that when the parameters take different values, we can obtain smooth soliton solutions and symmetric singular soliton solutions. When the selected zero point µ ₀ is complex and its modulus is not equal to zero, the breather solutions can be obtained.

From expression (Equation 2.83), it can be seen that when $x\rightarrow\pm\infty$, there exists $y\rightarrow\pm\infty$. The solution (Equation 2.82) indicates that $r_{1}=0$ $(\tau=\frac{\pi}{2})$ is a singularity point and $cosh(2\chi_{1})+\frac{1+r^{2}_{2}}{r^{2}_{1}}$ has no zero point; otherwise, the following relationship will presence

(2.84)\begin{align} r^{2}_{1}(e^{-2yr_{2}+\frac{2tr_{2}}{r^{2}_{1}}}+e^{2yr_{2}-\frac{2tr_{2}}{r^{2}_{1}}})=-2(1+r^{2}_{2}), \end{align}

since the point $\mu_{0}\in\mathbb{C}^{+}$, the right-hand side of the equation is always negative and the left-hand side is always positive, which leads to a contradiction. According to our analysis of constructing the RH problem, the points $\mu=0,\pm i$ are singularities which corresponding to the obtained solution. There $\mu_{0}\in\mathbb{C}^{+}$ and it has only one singularity point $\mu_{0}=i$.

To demonstrate the characteristics of solitons, we present wave propagations including images of local structure, density, and intensity distribution in figures 2–4 after selecting different parameters. It can be seen that the single smooth soliton solutions are obtained at these values. Specifically, figure 2(a) shows a Hump soliton solution. figure 3(a) is a critical situation where the soliton exhibits a top horizontal characteristic. Finally, figure 4(a) shows the M-shape soliton solution. From these figures, it can be seen that as the values of τ increasing, the peak of the soliton first appears in a horizontal state and then sinks inward.

Figure 2.

(a)–(c) describe the local structure, density, and intensity profiles with different times of one-soliton solution $|u|$. Parameters $r_{1}=\frac{\sqrt{3}}{2}$, $r_{2}=\frac{1}{2}$, $\tau=\frac{\pi}{6}$.

Figure 2 Long description

Three panels labeled (a), (b) and (c), each representing a different view of a one-soliton solution. Panel (a) is a 3D surface plot. The x-axis is labeled x, representing the spatial coordinate and ranges from approximately minus 10 to 10. The t-axis, representing time, ranges from approximately minus 5 to 5. The vertical z-axis represents the amplitude of the soliton field. The surface shows a single smooth localized peak rising sharply from a flat base, with the peak centered near x equals 0 and t equals 0. The peak amplitude reaches a maximum value visible at the apex of the surface and the surface slopes symmetrically downward in all directions from this central maximum. Panel (b) is a 2D contour density plot. The x-axis is labeled x and ranges from approximately minus 10 to 10. The y-axis is labeled t and ranges from approximately minus 5 to 5. The density plot shows a diagonal band running from the lower-left region to the upper-right region of the plot area, indicating that the soliton peak travels linearly across the spatial domain as time progresses. The band contains a sequence of contour regions reflecting high amplitude at the center and lower amplitude at the outer edges. Panel (c) is a line plot showing intensity profiles. The x-axis is labeled x and ranges from approximately minus 10 to 10. The y-axis represents amplitude or intensity magnitude. Three curves are plotted, each corresponding to a distinct parameter case as identified in the legend. The first curve, drawn with a solid line, shows the narrowest and tallest peak, centered near x equals 0, with the highest amplitude among the three curves. The second curve, drawn with a dashed line, shows a moderately wide peak of intermediate height, also centered near x equals 0. The third curve, drawn with a dash-dot line, shows the broadest and shortest peak among the three, centered near x equals 0. All three curves are symmetric about x equals 0 and each decays to near zero amplitude at the outer edges of the x range.

Figure 3.

(a)–(c) describe the local structure, density, and intensity profiles with different times of one-soliton solution $|u|$. Parameters $r_{1}=\frac{\sqrt{2}}{2}$, $r_{2}=\frac{\sqrt{2}}{2}$, $\tau=\frac{\pi}{4}$.

Figure 3 Long description

Three panels labeled (a), (b) and (c) present the local structure, density and intensity profiles of a one-soliton solution under different parameter conditions. Panel (a) is a three-dimensional surface plot. The horizontal axes are labeled x and t and the vertical axis represents the amplitude or intensity of the soliton solution. The surface shows a single smooth peak rising sharply from a near-flat base. The peak is located approximately at x equals 0 and t equals 0, reaching a maximum height of approximately 1.5 units on the vertical axis. The surface slopes symmetrically away from the peak in all directions, forming a bell-shaped mound. The base of the surface extends from approximately negative 4 to positive 4 along the x-axis and from negative 4 to positive 4 along the t-axis. Panel (b) is a two-dimensional density or intensity map. The horizontal axis is labeled x and the vertical axis is labeled t. Both axes appear to range from approximately negative 4 to positive 4. A narrow high-intensity band runs diagonally across the map from the lower-left region toward the upper-right region, indicating the trajectory of the soliton across the x-t plane. The surrounding region shows uniformly low intensity. The high-intensity band is narrow and well-defined, with the peak intensity concentrated along a central ridge. A colorbar is present alongside the plot, with lower values at one end and higher values at the other end, corresponding to the intensity range across the map. Panel (c) contains three overlapping line profiles plotted on the same axes. The horizontal axis represents x, ranging from approximately negative 10 to positive 10. The vertical axis represents intensity, ranging from 0 to approximately 1.5. Three curves are shown, each corresponding to a different parameter value as indicated in the legend, which lists three entries distinguished by different line styles. The first curve is centered near x equals negative 2, reaching a peak of approximately 1.4. The second curve is centered near x equals 0, reaching a peak of approximately 1.4. The third curve is centered near x equals positive 2, reaching a peak of approximately 1.4. All three curves have similar heights and widths, but their horizontal positions differ, indicating a lateral shift in the soliton position as the parameter changes. Each curve has a symmetric bell shape with a full width at approximately half maximum of roughly 2 units.

Figure 4.

(a)–(c) describe the local structure, density, and intensity profiles with different times of one-soliton solution $|u|$. Parameters $r_{1}=\frac{1}{2}$, $r_{2}=\frac{\sqrt{3}}{2}$, $\tau=\frac{\pi}{3}$.

Figure 4 Long description

Plot (a) is a three-dimensional surface plot. The horizontal axes represent spatial coordinates and the vertical axis represents amplitude. The surface shows a single narrow peak rising sharply from a flat background near zero. The peak is smooth and symmetric, with the surrounding region remaining uniformly low. Plot (b) is a two-dimensional density or intensity map. Both axes represent spatial coordinates. A single diagonal high-intensity stripe runs across the plot from one corner toward the opposite side. The surrounding area shows uniformly low intensity. The stripe is narrow and well-defined. Plot (c) shows line graphs of intensity profiles. The horizontal axis represents spatial position and the vertical axis represents intensity. Three curves are plotted, each corresponding to a different time value as indicated by the legend. Each curve shows a single peak. The three peaks are centered at different positions along the horizontal axis, indicating a shift in peak location across the three time values. The peaks appear similar in height and width. The legend identifies three separate entries corresponding to the three curves shown.

Case 2: Supposing $v_{1}=\mu_{0}=ir\in i\mathbb{R}$, $v_{2}=\frac{1}{\mu^{*}_{0}}$. It can be obtained that $\theta(\mu_{0})=\theta(\frac{1}{\mu^{*}_{0}})$. Next we define $\chi_{2}\triangleq\theta(\mu_{0})=-\frac{i}{4}\frac{r^{2}+1}{r}\left(\frac{y}{t}+\frac{8r^{2}}{(r^{2}-1)^{2}}\right)$, hence $e^{-2it\chi_{2}}$ is a real number about variables y and t. Thus algebraic system (Equation 2.70)–(Equation 2.74) have the following forms

(2.85)\begin{align} \tilde{N}_{1}(\mu)=1+\frac{|c_{0}|e^{i\phi}e^{\chi_{2}}}{\mu-ir}\tilde{N}_{1}(-ir)+ \frac{\frac{|c_{0}|}{r^{2}}e^{-i\phi}e^{\chi_{2}}}{\mu+\frac{1}{ir}}\tilde{N}_{1}(\frac{1}{ir}), \end{align}

(2.86)\begin{align} \begin{split} \tilde{N}_{1}(-ir)=&1+\frac{|c_{0}|e^{i\phi}e^{\chi_{2}}}{-2ir}\tilde{N}_{1}(-ir)+ \frac{\frac{|c_{0}|}{r^{2}}e^{-i\phi}e^{\chi_{2}}}{-ir+\frac{1}{ir}}\tilde{N}_{1}(\frac{1}{ir}),\\ \tilde{N}_{1}(\frac{1}{ir})=&1+\frac{|c_{0}|e^{i\phi}e^{\chi_{2}}}{\frac{1}{ir}-ir}\tilde{N}_{1}(-ir)+ \frac{\frac{|c_{0}|}{r^{2}}e^{-i\phi}e^{\chi_{2}}}{\frac{2}{ir}}\tilde{N}_{1}(\frac{1}{ir}), \end{split} \end{align}

the solutions are

(2.87)\begin{align} \begin{split} \tilde{N}_{1}(-ir)=&\frac{1}{H_{2}}(1-\frac{1-r^2}{2ir(1+r^2)}|c_{0}|e^{\chi_{2}+\phi}),\\ \tilde{N}_{1}(\frac{1}{ir})=&\frac{1}{H_{2}}(1+\frac{1-r^2}{2ir(1+r^2)}|c_{0}|e^{\chi_{2}+\phi}), \end{split} \end{align}

with

(2.88)\begin{align} H_{2}=1-\frac{(r^{2}-1)^{2}}{4r^{2}(r^{2}+1)^{2}}|c_{0}|^{2}e^{2\chi_{2}}-\frac{i|c_{0}|e^{\chi_{2}+\phi}}{r}. \end{align}

Substitute formula (Equation 2.87) into (Equation 2.85) and take an appropriate value for c ₀ to calculate the solution as follows

(2.89)\begin{align} u(x,t)=\frac{16r^2(1+r^2)}{(1-r^2)^3}\frac{\sinh(\chi_{2})}{cosh(2\chi_{2})+\frac{(1+r^{2})^2+4r^2}{(1-r^2)^2}}, \end{align}

with

(2.90)\begin{align} x=x(y,t)=y-\ln\left(\frac{1+\frac{(1+r)^2}{(1-r)^2}e^{2\chi_{2}}}{1+\frac{(1-r)^2}{(1+r)^2}e^{2\chi_{2}}}\right). \end{align}

The solution (Equation 2.89) indicates that r = 1 is a singularity point and $cosh(2\chi_{2})+\frac{(1+r^{2})^2+4r^2}{(1-r^2)^2}$ has no zero point; otherwise, the following relationship will presence

(2.91)\begin{align} (1-r^2)^2(e^{2\chi_{2}}+e^{-2\chi_{2}})=-2(1+r^2)^2-8r^4, \end{align}

this is clearly impossible.

Remark 2.10. After comparison, when selecting appropriate parameters, the expression of solution (Equation 2.89) solved by RH approach is the same as the solution (3.9a) solved by Hirota’s bilinear method in reference [Reference Matsuno31]. Then the antisymmetric singular soliton can be obtained.

2.3.2. Two-order pole solution

For the N = 2 case, it corresponds to the solution with a two-order pole. The expression of $u(x,t)$ defined in theorem 2.7.

Case 3: Assuming that $v_{1}=\mu_{0}=e^{i\tau}$ be one two-order zero point of the scattering data $a(\mu)$, then $v_{2}=e^{i\tau}=v_{1}$ is also the two-order zero point of $a(\mu)$. The discrete spectrum is the set $\mathcal{Z}=\left\{e^{i\tau}, -e^{-i\tau}, e^{-i\tau}, -e^{i\tau}\right\}$. We choose some parameters to simulate the solution, then the different dynamic models can be obtained in Figures (5)–(6).

Figure 5.

(a)–(c) describe the local structure, density, and intensity profiles with different times of the soliton solutions $|u|$ with one two-order pole. Parameters $r_{1,1}=2$, $r_{1,2}=1$, $r_{2,1}=1$, $r_{2,2}=2$, $\tau=\frac{\pi}{6}$.

Figure 5 Long description

(a) A 3D surface plot showing the local structure of soliton solutions over time. The axes are labeled x, t and z, representing spatial and temporal dimensions. A diagonal ridge indicates a significant feature in the data. (b) A heatmap displaying density profiles with axes labeled from negative 20 to positive 20. The color scale represents varying density levels, transitioning from red to blue. (c) A line graph illustrating intensity profiles with the horizontal axis from negative 20 to positive 20 and the vertical axis from 0 to 40. The legend shows different colored lines representing distinct intensity levels. Peaks and troughs indicate changes in intensity over time. Together, these graphs demonstrate the relationship between structure, density and intensity in soliton solutions, highlighting key features and transitions.

Figure 6.

(a)–(c) describe the local structure, density, and intensity profiles with different times of the soliton solutions $|u|$ with one two-order pole. Parameters $r_{1,1}=2$, $r_{1,2}=1$, $r_{2,1}=1$, $r_{2,2}=2$, $\tau=\frac{3\pi}{4}$.

Figure 6 Long description

A) Three-dimensional surface plot representing the local structure profile of a soliton solution. The vertical axis represents amplitude or wave quantity and the two horizontal axes represent spatial and temporal coordinates. The surface forms a largely flat plane with a single sharp, narrow ridge running diagonally across it, indicating a localized elevation concentrated along one axis direction. The ridge rises steeply from the flat base and drops back sharply, consistent with a single soliton structure. Axis labels and numeric tick values are not legible at this resolution. B) Two-dimensional density plot representing the spatial and temporal evolution of the soliton solution. The horizontal axis represents one coordinate, spanning approximately minus 15 to 15 and the vertical axis represents the other coordinate, spanning approximately minus 10 to 10. The plot is divided into two broad regions separated by a narrow, sharply curved transition band running vertically near the left portion of the domain. A vertical color scale bar is included alongside the plot. The transition band is narrow and well-defined, indicating a steep gradient in the plotted quantity across that boundary. The two surrounding regions appear uniform in value, with the transition being the dominant feature. C) Two-dimensional intensity profile plot. The horizontal axis represents a spatial or temporal coordinate and the vertical axis represents intensity or magnitude of the soliton solution. Multiple overlaid traces are present, with at least three distinguishable curves. All traces remain near the zero baseline across most of the domain. Near one end of the horizontal axis, several narrow, tall peaks appear in close proximity. One peak is visibly taller than the others. The remaining peaks are smaller but still sharply defined. The peaks are clustered within a narrow region of the domain. A vertical reference line is also drawn near the peak cluster. The traces overlap closely in the flat baseline region and diverge only at the peak locations.

Setting $\tau=\frac{\pi}{6}$ and $\left\{r_{1,1}=2, r_{1,2}=1, r_{2,1}=1, r_{2,2}=2\right\}$. On the basis of the construction process of single high-order pole solutions, the following relationships are calculated.

\begin{align*} \zeta_{1,0}=\zeta_{2,0}=e^{\frac{4t}{3}-\frac{y}{2}},\quad \zeta_{1,1}=\zeta_{2,1}=\frac{-8i(40t-9y)e^{\frac{4t}{3}-\frac{y}{2}}(i\sqrt{3}-1)} {(i\sqrt{3}+3)^3(\sqrt{3}+i)^2}, \end{align*}

\begin{align*} \vartheta_{1,1}=\vartheta_{1,3}=\frac{5\sqrt{3}e^{\frac{4t}{3}-\frac{y}{2}}}{9} \left(\sqrt{3}\left(\frac{21+27y}{40}-\frac{3i}{2}-3t\right)+it-\frac{9i(y-1)}{40}+\frac{9}{10}\right), \end{align*}

\begin{align*} \vartheta_{1,2}=\vartheta_{1,4}=-\frac{2880e^{\frac{4t}{3}-\frac{y}{2}}}{(i\sqrt{3}+3)^3(\sqrt{3}+i)^4} \left(\sqrt{3}\left(\frac{6-y}{40}-\frac{i}{10}+\frac{t}{9}\right)+it-\frac{9i(y+1)}{40}-\frac{1}{2}\right), \end{align*}

\begin{align*} \vartheta_{3,1}=\vartheta_{3,3}=\frac{10\sqrt{3}e^{\frac{4t}{3}-\frac{y}{2}}}{9} \left(\sqrt{3}\left(\frac{21+27y}{40}-\frac{3i}{8}-3t\right)+it-\frac{9i(y-1)}{40}+\frac{9}{40}\right), \end{align*}

\begin{align*} \vartheta_{3,2}=\vartheta_{3,4}=-\frac{5760e^{\frac{4t}{3}-\frac{y}{2}}}{(i\sqrt{3}+3)^3(\sqrt{3}+i)^4} \left(\sqrt{3}\left(\frac{6-y}{40}-\frac{i}{40}+\frac{t}{9}\right)+it-\frac{9i(y+2)}{40}-\frac{1}{8}\right), \end{align*}

\begin{align*} \vartheta_{2,1}=\vartheta_{2,3}=-\frac{e^{\frac{4t}{3}-\frac{y}{2}}(i\sqrt{3}-2)}{\sqrt{3}+i},\quad \vartheta_{2,2}=\vartheta_{2,4}=-\frac{e^{\frac{4t}{3}-\frac{y}{2}}(2i\sqrt{3}+3)}{(\sqrt{3}+i)^{2}}, \end{align*}

\begin{align*} \vartheta_{4,1}=\vartheta_{4,3}=-\frac{e^{\frac{4t}{3}-\frac{y}{2}}(2i\sqrt{3}-4)}{\sqrt{3}+i},\quad \vartheta_{4,2}=\vartheta_{4,4}=-\frac{e^{\frac{4t}{3}-\frac{y}{2}}(4i\sqrt{3}+6)}{(\sqrt{3}+i)^{2}}, \end{align*}

\begin{align*} \eta_{11}=-2e^{\frac{4t}{3}-\frac{y}{2}}-\frac{8ie^{\frac{4t}{3}-\frac{y}{2}}(1+i\sqrt{3})(40t-9y)}{(i\sqrt{3}-3)^3(i-\sqrt{3})^2},\quad \eta_{12}=-e^{\frac{4t}{3}-\frac{y}{2}}, \end{align*}

\begin{align*} \eta_{21}=-e^{\frac{4t}{3}-\frac{y}{2}}-\frac{16ie^{\frac{4t}{3}-\frac{y}{2}}(1+i\sqrt{3})(40t-9y)}{(i\sqrt{3}-3)^3(i-\sqrt{3})^2},\quad \eta_{22}=-2e^{\frac{4t}{3}-\frac{y}{2}}. \end{align*}

Substitute the above results into the expression of solution (Equation 2.67) to obtain the two-order pole solution, and the propagation phenomenon of soliton is shown in figure 5. It can be seen that the energies of two waves for solution decrease after collision; thus, it is not an elastic collision that occurs. According to the previous analysis, the singularity of the solution is $\mu_{0}=i$.

Using a similar analysis, when the value is set to $\mu_{0}=\frac{3\pi}{4}$, the soliton dynamics behaviour shown in figure 6. figure 6(b) presents the density image of the wave propagation process, while figure 6(c) shows the intensity distribution at different times.

2.4. Multiple high-order pole solutions

Now the general circumstance that $a(\mu)$ has N high-order zero points $\mu_{1},\mu_{2},\ldots,\mu_{N}$, $\mu_{k}\in\mathbb{C}^{+}$ for $k=1,2,\ldots,N$ and the corresponding powers are $n_{1},n_{2},\ldots,n_{N}$, respectively. Assume that $v^{k}_{1}=\mu_{k}$ and $v^{k}_{2}=\frac{1}{\mu^{*}_{k}}$. Similar to the case of one high-order pole, from the definition of $\tilde{M}(y,t,\mu)$, it can be seen that $\mu=v^{k}_{j}$ and $\mu=-v^{k*}_{j}$ $(j=1,2)$ are n_k-order pole points of $\tilde{M}_{11}$. Simultaneously, $\mu=v^{k*}_{j}$ and $\mu=-v^{k}_{j}$ $(j=1,2)$ are n_k-order pole points of $\tilde{M}_{12}$.

Under the assumption that $\mu=v^{k}_{j}$ $(j=1,2)$ are n_k-order zero points of $a(\mu)$ for $k=1,2,\ldots,N$, utilizing the Wronskian $a(\mu)=Wr(\breve{\phi}_{-1},\breve{\phi}_{+2})$, there exist complex constants $b^{k}_{j,s}$ $(s=1,2,\cdots,n_N)$ satisfy the follows

(2.92)\begin{equation} \frac{\partial^{m}[\tilde{\phi}_+(y,t,v^{k}_{j})]_{2}}{\partial\mu^{m}}=\sum^{m}_{l=0}\left(\begin{array}{c} m \\ l \end{array}\right) b^{k}_{j,m-l+1}\frac{\partial^{l}[\tilde{\phi}_-(y,t,v^{k}_{j})]_{1}}{\partial\mu^{l}}, \end{equation}

and

(2.93)\begin{equation} \frac{\partial^{m}[\tilde{\psi}_+(y,t,v^{k}_{j})]_{2}}{\partial\mu^{m}}=\sum^{m}_{l=0}\left(\begin{array}{c} m \\ l \end{array}\right) b^{k}_{j,m-l+1}\frac{\partial^{l}[\tilde{\psi}_-(y,t,v^{k}_{j})]_{1}e^{2it\theta(v^{k}_{j})}}{\partial\mu^{l}}, \end{equation}

with $m=0,1,\ldots,n_{N-1}$. Utilizing the parallel approach in above, we define the $n_k\times n_k$ matrices $\Theta^{j,p}_{kd}=[\Theta^{j,p}_{kd}]_{s,q}$ and $\tilde{\Theta}^{j,p}_{kd}=[\tilde{\Theta}^{j,p}_{kd}]_{s,q}$ for $s,q=1,2,\cdots,n_{k}$ and introduce the following notations for $j,p=1,2$

(2.94)\begin{align} |\eta^{k}_{j}\rangle=(\eta^{k}_{j1},\cdots,\eta^{k}_{jn_{k}})^{T},\quad \eta^{k}_{js}=-\sum_{l=s}^{n_{k}}r^{k*}_{j,l}\zeta^{k*}_{j,l-s}(y,t), \end{align}

(2.95)\begin{align} r^{k}_{j,l}=\lim_{\mu\rightarrow v^{k}_{j}}\frac{b^{k}_{j,l}}{(n_{k}-l)!}\frac{\partial^{n_{k}-l}}{\partial(\mu-v^{k}_{j})^{n_{k}-l}} \frac{(\mu-v^{k}_{j})^{n_{k}}}{a(\mu)}, \end{align}

(2.96)\begin{align} |\Gamma\rangle=(|\Gamma^{1}\rangle,\cdots,|\Gamma^{N}\rangle)^{T},\quad |\Gamma^{k}\rangle=(|\eta^{k}_{1}\rangle,|\eta^{k}_{2}\rangle)^{T}, \end{align}

(2.97)\begin{align} \begin{split} \Theta^{j,p}_{kd}=\sum_{l=s}^{n_k}\sum_{m=0}^{l-s} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right) r^{k}_{j,l}\zeta^{k}_{j,l-s-m}\left\{\frac{(-1)^{m}}{(v^{k}_{j}-v^{k*}_{p})^{q+m}} +\frac{(-1)^{m+q+1}}{(v^{k}_{j}+v^{k}_{p})^{q+m}}\right\}, \end{split} \end{align}

(2.98)\begin{align} \begin{split} \tilde{\Theta}^{j,p}_{kd}=\sum_{l=s}^{n_k}\sum_{m=0}^{l-s} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right) r^{k*}_{j,l}\zeta^{k*}_{j,l-s-m}\left\{\frac{(-1)^{m}} {(v^{k*}_{j}-v^{k}_{p})^{q+m}} +\frac{(-1)^{m+q}}{(v^{k*}_{j}+v^{k*}_{p})^{q+m}}\right\}, \end{split} \end{align}

(2.99)\begin{align} \Theta= \begin{pmatrix} [\Theta^{j,p}_{11}] & \cdots & [\Theta^{j,p}_{1N}] \\ \vdots & \ddots & \vdots \\ [\Theta^{j,p}_{N1}] & \cdots & [\Theta^{j,p}_{NN}] \\ \end{pmatrix},\quad \tilde{\Theta}= \begin{pmatrix} [\tilde{\Theta}^{j,p}_{11}] & \cdots & [\tilde{\Theta}^{j,p}_{1N}] \\ \vdots & \ddots & \vdots \\ [\tilde{\Theta}^{j,p}_{N1}] & \cdots & [\tilde{\Theta}^{j,p}_{NN}] \\ \end{pmatrix}, \end{align}

(2.100)\begin{align} I_{\epsilon}= \begin{pmatrix} I_{\epsilon_{1}} & & \\ & \ddots & \\ & & I_{\epsilon_{N}} \\ \end{pmatrix},\quad I_{\epsilon_{k}}= \begin{pmatrix} I & \\ & I \\ \end{pmatrix}_{2n_{k}\times2n_{k}} k=1,\cdots,N, \end{align}

(2.101)\begin{equation} \begin{split} &\langle Y(\mu)|=(\langle Y^{1}_{1}(\mu)|,\langle Y^{1}_{2}(\mu)|,\cdots,\langle Y^{N}_{1}(\mu)|,\langle Y^{N}_{2}(\mu)|),\\ &\langle\tilde{Y}(\mu)| =(\langle\tilde{Y}^{1}_{1}(\mu)|,\langle\tilde{Y}^{1}_{2}(\mu)|,\cdots, \langle\tilde{Y}^{N}_{1}(\mu)|,\langle\tilde{Y}^{N}_{2}(\mu)|),\\ &\langle Y^{\prime}(\mu)|= (\langle Y^{1\prime}_{1}(\mu)|,\langle Y^{1\prime}_{2}(\mu)|,\cdots, \langle Y^{N\prime}_{1}(\mu)|,\langle Y^{N\prime}_{2}(\mu)|),\\ &\langle\tilde{Y}^{\prime}(\mu)| =(\langle\tilde{Y}^{1\prime}_{1}(\mu)|,\langle\tilde{Y}^{1\prime}_{2}(\mu)|,\cdots, \langle\tilde{Y}^{N\prime}_{1}(\mu)|,\langle\tilde{Y}^{N\prime}_{2}(\mu)|), \end{split} \end{equation}

\begin{align*} &\langle Y^{k}_{j}(\mu)|=(Y^{k}_{j1},\cdots,Y^{k}_{jn_k}),\quad Y^{k}_{js}=\frac{1}{(\mu-v^{k}_{j})^{s}}+\frac{(-1)^{s}}{(\mu+v^{k*}_{j})^{s}},\\ &\langle\tilde{Y}^{k}_{j}(\mu)|=(\tilde{Y}^{k}_{j1},\cdots,\tilde{Y}^{k}_{jn_k}),\quad \tilde{Y}^{k}_{js}=\frac{1}{(\mu-v^{k*}_{j})^{s}}+\frac{(-1)^{s+1}}{(\mu+v^{k}_{j})^{s}},\\ &\langle Y^{k\prime}_{j}(\mu)|=(Y^{k\prime}_{j1},\cdots,Y^{k\prime}_{jn_k}),\quad Y^{k\prime}_{js}=\frac{-s}{(\mu-v^{k}_{j})^{s+1}}+\frac{s(-1)^{s+1}}{(\mu+v^{k*}_{j})^{s+1}},\\ &\langle\tilde{Y}^{k\prime}_{j}(\mu)|=(\tilde{Y}^{k\prime}_{j1},\cdots,\tilde{Y}^{k\prime}_{jn_k}),\quad \tilde{Y}^{k\prime}_{js}=\frac{-s}{(\mu-v^{k*}_{j})^{s+1}}+\frac{s(-1)^{s}}{(\mu+v^{k}_{j})^{s+1}}. \end{align*}

Similar to the theorem 2.7, we give the solution of cCH equation with multiple high-order poles as follows.

Theorem 2.11

The solution of cCH equation with multiple high-order poles under the ZBCs can be obtained as follows

(2.102)\begin{align} u(x,t)=-\left( \frac{\tilde{M}^{\prime}_{11}(i)+\tilde{M}^{\prime}_{21}(i)}{\tilde{M}_{11}(i)+\tilde{M}_{21}(i)}+ \frac{\tilde{M}^{\prime}_{12}(i)+\tilde{M}^{\prime}_{22}(i)}{\tilde{M}_{12}(i)+\tilde{M}_{22}(i)} \right), \end{align}

with

\begin{align*} &\tilde{M}_{11}(i)=\frac{\det(I_{\epsilon}+\Theta\tilde{\Theta}+|\Gamma\rangle\langle Y(i)|\Theta)} {\det(I_{\epsilon}+\Theta\tilde{\Theta})},\\ &\tilde{M}_{21}(i)=-\frac{\det(I_{\epsilon}+\Theta^{*}\tilde{\Theta}^{*}+|\Gamma^{*}\rangle\langle\tilde{Y}^{*}(-i)|)} {\det(I_{\epsilon}+\Theta^{*}\tilde{\Theta}^{*})}+1,\\ &\tilde{M}_{12}(i)=\frac{\det(I_{\epsilon}+\tilde{\Theta}\Theta+|\Gamma\rangle\langle\tilde{Y}(i)|)} {\det(I_{\epsilon}+\tilde{\Theta}\Theta)}-1,\\ &\tilde{M}_{22}(i)=\frac{\det(I_{\epsilon}+\tilde{\Theta}^{*}\Theta^{*}+|\Gamma^{*}\rangle\langle Y^{*}(-i)|\Theta^{*})} {\det(I_{\epsilon}+\tilde{\Theta}^{*}\Theta^{*})},\\ &\tilde{M}^{\prime}_{11}(i)=\frac{\det(I_{\epsilon}+\Theta\tilde{\Theta}+|\Gamma\rangle\langle Y^{\prime}(i)|\Theta)} {\det(I_{\epsilon}+\Theta\tilde{\Theta})}-1,\\ &\tilde{M}^{\prime}_{21}(i)=-\frac{\det(I_{\epsilon}+\Theta^{*}\tilde{\Theta}^{*}+|\Gamma^{*}\rangle\langle\tilde{Y}^{\prime *}(-i)|)} {\det(I_{\epsilon}+\Theta^{*}\tilde{\Theta}^{*})}+1,\\ &\tilde{M}^{\prime}_{12}(i)=\frac{\det(I_{\epsilon}+\tilde{\Theta}\Theta+|\Gamma\rangle\langle\tilde{Y}^{\prime}(i)|)} {\det(I_{\epsilon}+\tilde{\Theta}\Theta)}-1,\\ &\tilde{M}^{\prime}_{22}(i)=\frac{\det(I_{\epsilon}+\tilde{\Theta}^{*}\Theta^{*}+|\Gamma^{*}\rangle\langle Y^{\prime *}(-i)|\Theta^{*})} {\det(I_{\epsilon}+\tilde{\Theta}^{*}\Theta^{*})}-1, \end{align*}

and $x=x(y,t)$ can be obtained by (Equation 2.41), the elements $|\Gamma\rangle$, I_ϵ, Θ, $\tilde{\Theta}$, $\langle Y(i)|$, $\langle\tilde{Y}(i)|$, $\langle Y^{\prime}(i)|$ and $\langle \tilde{Y}^{\prime}(i)|$ are defined in (Equation 2.96), (Equation 2.99), (Equation 2.100), and (Equation 2.101).

3.1. Spectral analysis

After doing the transformations $u(x,t)=\hat{u}(x-t,t)+1$, $\hat{m}=\hat{u}-\hat{u}_{xx}+1$ and $\hat{R}=\hat{u}^{2}-\hat{u}^{2}_{x}+2\hat{u}$, the Lax pair (Equation 2.1) admits the forms

(3.1)\begin{align} \hat{\phi}_{x}=\hat{U}\hat{\phi},\quad \hat{\phi}_{t}=\hat{V}\hat{\phi}, \end{align}

(3.2)\begin{align} \hat{U}=\left(\begin{array}{cc} -\frac{Q}{2} & \frac{\lambda \hat{m}}{2} \\ -\frac{\lambda \hat{m}}{2} & \frac{Q}{2} \end{array}\right), \hat{V}=\left(\begin{array}{cc} \frac{Q}{\lambda^{2}}+\frac{Q\hat{R}}{2} & -\frac{\hat{u}-Q\hat{u}_x+1}{\lambda}-\frac{\lambda \hat{R}\hat{m}}{2} \\ \frac{\hat{u}+Q\hat{u}_x+1}{\lambda}+\frac{\lambda \hat{R}\hat{m}}{2} & -\frac{Q}{\lambda^{2}}-\frac{Q\hat{R}}{2} \end{array}\right). \end{align}

Using the similar approach with ZBCs, we have the gauge transformation

(3.3)\begin{align} \hat{\psi}=F^{-1}\hat{\phi} e^{\hat{J}}, \end{align}

with

\begin{align*} F=\left(\begin{array}{cc} 1 & \frac{\lambda}{\sqrt{Q^{2}-\lambda^{2}}+Q} \\ \frac{\lambda}{\sqrt{Q^{2}-\lambda^{2}}+Q} & 1 \end{array}\right), \end{align*}

where we define $\hat{J}=p\sigma_{3}$. $\hat{\psi}$ satisfies the asymptotic conditions

\begin{align*} \hat{\psi}(x,t,\lambda)\sim I,\quad x\rightarrow\pm\infty. \end{align*}

The parameter $p(x,t,\lambda)$ is defined as follows

(3.4)\begin{align} p=\frac{\sqrt{1-2\lambda^{2}}}{2}\left(x-\int_{x}^{+\infty}(\hat{m}(\xi,t)-1)d\xi -\frac{2}{\lambda^{2}}t\right). \end{align}

Equation (Equation 1.3) admits the following form Lax pair

(3.5a)\begin{align} &\hat{\psi}_x+[\hat{J}_x,\hat{\psi}]=\hat{X}\hat{\psi}, \end{align}

(3.5b)\begin{align} &\hat{\psi}_t+[\hat{J}_t,\hat{\psi}]=\hat{T}\hat{\psi}, \end{align}

with

\begin{align*} \hat{X}=\frac{\lambda(\hat{m}-1)Q}{2\sqrt{1-2\lambda^{2}}} \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right)+ \frac{(\hat{m}-1)Q^{2}}{2\sqrt{1-2\lambda^{2}}} \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right), \end{align*}

\begin{align*} \hat{T}=&\frac{1}{2\sqrt{1-2\lambda^{2}}}\left(\lambda\hat{R}(\hat{m}-1)+\frac{2\hat{u}}{\lambda}\right) Q\left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right)+ \frac{\hat{u}_{x}Q}{\lambda} \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \\ &-\frac{1}{\sqrt{1-2\lambda^{2}}}\left(\hat{u}+\frac{1}{2}((\hat{m}-1)\hat{R}+2\hat{u})Q^{2}-\lambda^{2}\hat{u}\right)\sigma_{3}, \end{align*}

where $\hat{J}=\hat{J}(x,t,\lambda)$, $\hat{X}=\hat{X}(x,t,\lambda)$, $\hat{T}=\hat{T}(x,t,\lambda)$ are $2\times2$ matrices.

A new spectral parameter k satisfied $2\lambda^{2}=4k^{2}+1$ is introduced, and to avoid multi-value problem, we define

(3.6)\begin{equation} \lambda=-\frac{1}{2\sqrt{2}}\left(\mu+\frac{1}{\mu}\right),\quad k=\frac{1}{4}\left(\mu-\frac{1}{\mu}\right), \end{equation}

then we have

(3.7)\begin{equation} p=-\frac{i(\mu^{2}-1)}{4\mu}\left(\int_{x}^{+\infty}(\hat{m}(\xi,t)-1)d\xi-x+ \frac{16\mu^{2}}{(\mu^{2}+1)^{2}}t\right), \end{equation}

and

\begin{align*} \hat{X}(x,t,\mu)&=\frac{i((\mu^{2}+1)^{2}-8\mu^{2})(\hat{m}-1)}{8\mu(\mu^{2}-1)}\sigma_{3}\\ &+\frac{i(\mu^{2}+1)\sqrt{8\mu^{2}-(\mu^{2}+1)^{2}}(\hat{m}-1)}{8|\mu|(\mu^{2}-1)} \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right), \end{align*}

the two Jost solutions of (Equation 3.5a) with the parameter µ are completely determined by Volterra integrable equations

(3.8)\begin{align} \hat{\psi}_{\pm}(x,t,\mu)=I+\int_{\pm\infty}^{x}e^{\frac{i(\mu^{2}-1)}{4\mu}\int_{x}^{\xi}\hat{m}(\tau,t)d\tau\hat{\sigma}_{3}} \left[\hat{X}(\xi,t,\mu)\hat{\psi}_{\pm}(\xi,t,\mu)\right]d\xi. \end{align}

The Jost eigenfunctions $\hat{\psi}_{\pm}(x,t,\mu)$ have the analytic properties, symmetry properties, and asymptotic properties shown in the following three propositions.

Proposition 3.2. The Jost eigenfunctions $\hat{\psi}_{\pm}(x,t,\mu)$ have the following symmetry conditions

(3.9)\begin{align} \hat{\psi}_{\pm}(\mu)=\sigma_{1}\hat{\psi}^{*}_{\pm}(\mu^{*})\sigma_{1}= \sigma_{2}\hat{\psi}_{\pm}(-\mu)\sigma_{2}=\sigma_{1}\hat{\psi}_{\pm}(\frac{1}{\mu})\sigma_{1}. \end{align}

Proposition 3.3. The Jost eigenfunctions $\hat{\psi}_{\pm}(x,t,\mu)$ possess asymptotic behaviour in the µ-plane

(3.10)\begin{align} \begin{split} \hat{\psi}_{\pm}(x,t,\mu)&=I+\mathcal {O}(\frac{1}{\mu}),\quad \mu\rightarrow\infty,\\ \hat{\psi}_{\pm}(x,t,\mu)&=I+\mathcal {O}(\mu),\quad \mu\rightarrow0,\\ \hat{\psi}_{\pm}(x,t,\mu)&= \frac{i}{4(\mu-1)}\alpha_{\pm}(x,t)\Sigma_{1}+\mathcal {O}(1),\quad \mu\rightarrow1,\\ \hat{\psi}_{\pm}(x,t,\mu)&= -\frac{i}{4(\mu+1)}\alpha_{\pm}(x,t)\Sigma_{2}+\mathcal {O}(1),\quad \mu\rightarrow-1, \end{split} \end{align}

there $\alpha_{\pm}(x,t)\in\mathbb{R}$, $$\Sigma_{1}=\left(\begin{array}{cc} -1 & 1 \\ -1 & 1 \end{array}\right)$$ and $$\Sigma_{2}=\left(\begin{array}{cc} 1 & 1 \\ -1 & -1 \end{array}\right)$$.

Because $\hat{\phi}_{\pm}(x,t,\mu)$ are two fundamental matrix solutions of the Lax pair (Equation 3.1), there exists a linear relation between them.

(3.11)\begin{align} \hat{\phi}_{-}(x,t,\mu)=\hat{\phi}_{+}(x,t,\mu)S(\mu), \end{align}

using the transformation (Equation 3.3), matrix $S(\mu)$ can be introduced as follows

(3.12)\begin{align} \hat{\psi}_{-}(x,t,\mu)=\hat{\psi}_{+}(x,t,\mu)e^{-p(x,t,\mu)\sigma_{3}}S(\mu)e^{p(x,t,\mu)\sigma_{3}}, \end{align}

there $\mu\in\mathbb{R},\mu\neq\pm1$. On the basis of symmetry conditions of $\hat{\psi}_{\pm}(x,t,\mu)$, the matrix $S(\mu)$ can be expressed as

(3.13)\begin{align} S(\mu)=\left(\begin{array}{cc} a(\mu) & b^{*}(\mu^{*}) \\ b(\mu) & a^{*}(\mu^{*}) \end{array}\right). \end{align}

Taking into account the properties of the Jost eigenfunctions $\hat{\psi}_{\pm}(x,t,\mu)$, we obtain the corresponding properties to the elements $a(\mu)$ and $b(\mu)$ of the scattering matrix $S(\mu)$ as follows.

The analytic properties of $\hat{\psi}_{\pm}(x,t,\mu)$ stated above allow rewriting the relation (Equation 3.12) as a sectionally meromorphic matrix.

3.2. RH problem

A piecewise meromorphic $2\times2$-matrix valued function $M(x,t,\mu)$ is introduced as follows

(3.14)\begin{align} M(x,t;\mu)=\left\{\begin{array}{ll} \left( \frac{\hat{\psi}_{-,1}(x,t;\mu)}{a(\mu)}, \hat{\psi}_{+,2}(x,t;\mu)\right), &\text{as } Im\mu \gt 0,\\[12pt] \left( \hat{\psi}_{+,1}(x,t;\mu),\frac{\hat{\psi}_{-,2}(x,t;\mu)}{a^{*}(\mu^{*})}\right) , &\text{as } Im\mu \lt 0.\\ \end{array}\right. \end{align}

At the same time, we define the reflection coefficient

(3.15)\begin{align} r(\mu)=\frac{b(\mu)}{a(\mu)},\quad \mu\in\mathbb{R}. \end{align}

We substitute Eqs. (Equation 3.6) into Lax pair (Equation 3.5a) and take the new transformation

(3.23)\begin{align} \hat{\psi}_{0}=\hat{\varphi}e^{\hat{J}_{0}}, \end{align}

with

(3.24)\begin{align} p_{0}=\frac{i(\mu^{2}-1)}{4\mu}x-\frac{4i(\mu^{2}-1)\mu}{(\mu^{2}+1)^{2}}t, \end{align}

there $\hat{J}_{0}=p_{0}\sigma_{3}$, the Lax pair can be converted to

(3.25a)\begin{align} &\hat{\psi}_{0x}+[\hat{J}_{0x},\hat{\psi}_{0}]=\hat{X}_{0}\hat{\psi}_{0}, \end{align}

(3.25b)\begin{align} &\hat{\psi}_{0t}+[\hat{J}_{0t},\hat{\psi}_{0}]=\hat{T}_{0}\hat{\psi}_{0}, \end{align}

with

\begin{align*} \hat{X_{0}}(x,t,\mu)&=-\frac{(\mu^{2}+1)\sqrt{8\mu^{2}-(\mu^{2}+1)^{2}}(\hat{m}-1)}{8|\mu|(\mu^{2}-1)}\sigma_{2}\\ &+\left(\frac{i((\mu^{2}+1)^{2}-8\mu^{2})(\hat{m}-1)}{8\mu(\mu^{2}-1)}-\frac{i(\hat{m}-1)(\mu^{2}-1)}{4\mu} \right)\sigma_{3}, \end{align*}

\begin{align*} \hat{T_{0}}(x,t,\mu)=\frac{i(\mu^{2}-1)}{4\mu}\hat{R}\hat{m}\sigma_{3} +\frac{i(\mu^{2}-1)}{2\mu}\hat{u}\sigma_{3}+\hat{T}(x,t,\mu). \end{align*}

Then the Jost solutions of Equation 3.25a are expressed as

(3.26)\begin{align} \hat{\psi}_{0\pm}(x,t,\mu)=I+\int_{\pm\infty}^{x}e^{-\frac{i(\mu^{2}-1)}{4\mu}(x-\xi)\hat{\sigma}_{3}} \left[\hat{X}_{0}(\xi,t,\mu)\hat{\psi}_{0\pm}(\xi,t,\mu)\right]d\xi. \end{align}

Utilizing the transformation (Equation 3.23), there exist matrices $r_{\pm}(\mu)$ satisfying

(3.27)\begin{align} \hat{\psi}_{\pm}(x,t,\mu)=\hat{\psi}_{0\pm}(x,t,\mu)e^{-\hat{J}_{0}}r_{\pm}(\mu)e^{\hat{J}}. \end{align}

Thus $r_{\pm}(\mu)=e^{\hat{J}_{0}(\pm\infty,t,\mu)-\hat{J}(\pm\infty,t,\mu)}$. On the basis of (Equation 3.7) and (Equation 3.24),

(3.28)\begin{align} r_{+}(\mu)=I, \quad r_{-}(\mu)=e^{\frac{i(\mu^{2}-1)}{4\mu}\int_{-\infty}^{+\infty}(\hat{m}(\xi,t)-1)d\xi\sigma_{3}}. \end{align}

Because $\hat{X}_{0}(x,t,\pm i)\equiv0$, making use of (Equation 3.26) and $a(\mu)=\det(\hat{\psi}_{-,1},\hat{\psi}_{+,2})$, we have

(3.29)\begin{align} M(x,t,i)=\left(\begin{array}{cc} e^{\frac{1}{2}\int_{x}^{+\infty}(\hat{m}(\xi,t)-1)d\xi} & 0 \\ 0 & e^{-\frac{1}{2}\int_{x}^{+\infty}(\hat{m}(\xi,t)-1)d\xi} \end{array}\right), \end{align}

with symmetry of $M(x,t,\mu)$, we can obtain $M(x,t,-i)$. Besides, the symmetry conditions (Equation 3.22) indicate M(i) is a diagonal matrix with real entries, taking into account $\det M\equiv1$, there exists $f(x,t)\in\mathbb{R}$ such that $M_{11}(x,t,i)=f(x,t)$ and $M_{22}(x,t,i)=\frac{1}{f(x,t)}$.

Based on the new space variable $y(x,t)=x-\int_{x}^{+\infty}(\hat{m}(\xi,t)-1)d\xi$ and $\tilde{M}(y,t,\mu)=M(x,t,\mu)$, we have a new RH problem.

The implicit expression of the potential function u under NZBCs is constructed by using the solution of RH problem.

Theorem 3.9

Consider that $\tilde{M}(y,t,\mu)$ is a solution of RH problem 3.8, according to the definition of $\tilde{M}(y,t,\mu)$ at $\mu=i$, we make $\hat{\Phi}_{1}(x(y,t),t)=\tilde{\Phi}_{1}(y,t)=\tilde{M}_{11}(y,t,i)+\tilde{M}_{21}(y,t,i)$ and $\hat{\Phi}_{2}(x(y,t),t)=\tilde{\Phi}_{2}(y,t)=\tilde{M}_{12}(y,t,i)+\tilde{M}_{22}(y,t,i)$, then the solution $\hat{u}(x,t)$ of Cauchy problem (Equation 1.3) has the following representation

(3.36)\begin{align} \left(\hat{R}+\frac{2\hat{u}}{\hat{m}}\right)(\hat{m}+1)=\partial_{t}\ln\frac{\tilde{\Phi}_{1}(y,t)}{\tilde{\Phi}_{2}(y,t)}, \end{align}

with

\begin{align*} \hat{R}=\hat{u}^{2}-\hat{u}^{2}_{x}+2\hat{u},\quad \hat{m}=\hat{u}-\hat{u}_{xx}+1. \end{align*}

Proof. Considering the new notations $\tilde{R}(y,t)=\hat{R}(x(y,t),t)$, $\tilde{m}(y,t)=\hat{m}(x(y,t),t)$, $\tilde{u}(y,t)=\hat{u}(x(y,t),t)$ and $\tilde{u}_{x}(y,t)=\hat{u}_{x}(x(y,t),t)$, with regard to $x(y(x,t),t)=x$, we have

(3.37)\begin{align} \partial_{t}(x(y(x,t),t))=x_{y}(y,t)y_{t}(x,t)+x_{t}(y,t)=0. \end{align}

Since $y(x,t)=x-\int_{x}^{+\infty}(\hat{m}(\xi,t)-1)d\xi$, then

(3.38)\begin{align} x_{y}(y,t)=\frac{1}{\tilde{m}(y,t)}. \end{align}

Based on $\hat{m}_{t}=-(\hat{m} \hat{R})_{x}-2\hat{u}_{x}$ and unite (Equation 3.37)–(Equation 3.38), we have

(3.39)\begin{align} x_{t}(y,t)=\tilde{R}(y,t)+\frac{2\tilde{u}(y,t)}{\tilde{m}(y,t)}. \end{align}

By (Equation 3.29), we have

(3.40)\begin{align} \hat{\Phi}_{1}(x,t)=e^{\frac{1}{2}\int_{x}^{+\infty}(\tilde{m}(\xi,t)-1)d\xi},\quad \hat{\Phi}_{2}(x,t)=e^{-\frac{1}{2}\int_{x}^{+\infty}(\tilde{m}(\xi,t)-1)d\xi}, \end{align}

then

(3.41)\begin{align} x=y(x,t)+\ln\frac{\tilde{\Phi}_{1}(y,t)}{\tilde{\Phi}_{2}(y,t)}. \end{align}

Differential equation (Equation 3.41) with respect to t,

(3.42)\begin{align} x_{t}=-(\tilde{R}(y,t)\tilde{m}(y,t)+2\tilde{u}(y,t))+\partial_{t}\ln\frac{\tilde{\Phi}_{1}(y,t)}{\tilde{\Phi}_{2}(y,t)}, \end{align}

making further calculation, we can get (Equation 3.36), then the theorem 3.9 can be proved.

Next, in order to obtain the solution of Lax pair (Equation 3.1) represented by the new variable $y(x,t)$, using the gauge transformation $\hat{\varphi}=F^{-1}\hat{\phi}$, there we have $\tilde{\varphi}(y,t,\lambda)=\hat{\varphi}(x,t,\lambda)$. Taking (Equation 3.38) and (Equation 3.39) into consideration, the Lax pair (Equation 3.1) turns into

(3.43)\begin{align} \tilde{\varphi}_{y}=\tilde{U}\tilde{\varphi},\quad \tilde{\varphi}_{t}=\tilde{V}\tilde{\varphi}, \end{align}

there $\tilde{U}=\tilde{X}-ik\sigma_{3}$ and $\tilde{V}=\tilde{T}+\frac{2ik}{\lambda^{2}}\sigma_{3}$. Then we change the parameters to µ and expanding $\tilde{X}$ and $\tilde{T}$ at the singularities $\mu=\pm1, \pm i$

(3.44a)\begin{align} &\tilde{\varphi}_{y}+\frac{i(\mu^{2}-1)}{4\mu}\sigma_{3}\tilde{\varphi}=\tilde{X}_{1}\tilde{\varphi}, \end{align}

(3.44b)\begin{align} &\tilde{\varphi}_{t}-\frac{4i(\mu^{2}-1)\mu}{(\mu^{2}+1)^{2}}\sigma_{3}\tilde{\varphi}=\tilde{T}_{1}\tilde{\varphi}, \end{align}

with

\begin{align*} \tilde{X}_{1}=-\frac{iv_{1}}{\mu-1}\Sigma_{1}+\frac{iv_{1}}{\mu+1}\Sigma_{2}+v_{1}\sigma_{2}, \end{align*}

\begin{align*} \tilde{T}_{1}=-\frac{iv_{2}}{\mu-1}\Sigma_{1}+\frac{iv_{2}}{\mu+1}\Sigma_{2} +\frac{v_{3}\Delta_{1}+v_{4}\Delta_{2}}{\mu-i}+ \frac{v_{3}\Delta_{2}+v_{4}\Delta_{1}}{\mu+i}, \end{align*}

where $$\Delta_{1}=\left(\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right)$$, $$\Delta_{2}=\left(\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}\right)$$, Σ₁ and Σ₂ are shown in proposition 3.3. Besides, we also have $v_{1}(y,t)=-\frac{\tilde{m}-1}{2\tilde{m}}$, $v_{2}(y,t)=\tilde{u}$, $v_{3}(y,t)=-\tilde{u}-\tilde{u}_{x}$, and $v_{4}(y,t)=\tilde{u}-\tilde{u}_{x}$.

Proposition 3.10. Assume $\tilde{M}(y,t,\mu)$ is a solution of RH problem 3.8, then

(3.45)\begin{align} \tilde{\varphi}(y,t,\mu)=\tilde{M}(y,t,\mu)e^{-\tilde{p}(y,t,\mu)\sigma_{3}} \end{align}

satisfies the Lax pair (Equation 3.44a), there $\tilde{p}(y,t,\mu)$ is shown in (Equation 3.32).

For the sake of testifying the compatibility conditions $\tilde{U}_{t}-\tilde{V}_{y}+[\tilde{U},\tilde{V}]=0$ of Lax pair (Equation 3.43) can be deduced Eq. (Equation 1.3) with variables (y, t), we have the theorem 3.11 and theorem 3.12.

Theorem 3.11

Equation (Equation 1.3) with variables (y, t) can be shown as follows

(3.54)\begin{align} \tilde{m}^{-1}_{t}(y,t)=2\tilde{u}_{x}(y,t)-\frac{2\tilde{u}\tilde{m}_{x}}{\tilde{m}^{3}}(y,t) +\frac{2\tilde{u}_{x}}{\tilde{m}^{2}}(y,t), \end{align}

with

\begin{align*} \tilde{m}(y,t)=\tilde{u}(y,t)-\tilde{u}_{xx}(y,t)+1,\quad x_{y}(y,t)=\tilde{m}^{-1}(y,t). \end{align*}

Proof. According to equations $\hat{m}_{t}=-(\hat{m} \hat{R})_{x}-2\hat{u}_{x}$, $x_{t}(y,t)=\tilde{R}(y,t)+\frac{2\tilde{u}(y,t)}{\tilde{m}(y,t)}$ and $\tilde{R}_{x}=2\tilde{m}\tilde{u}_{x}$,

\begin{align*} \tilde{m}_{t}(y,t)&=\tilde{m}_{x}(x(y,t),t)x_{t}(y,t)+\tilde{m}_{t}(x(y,t),t)\\ &=\left(-\frac{2\tilde{u}\tilde{m}_{x}}{\tilde{m}^{3}}+\frac{2\tilde{u}_{x}}{\tilde{m}^{2}}\right) (-\tilde{m}^{2})+2\tilde{u}_{x}(-\tilde{m}^{2}), \end{align*}

then we have Eq. (Equation 3.54), and the theorem 3.11 can be proved.

Theorem 3.12

It can be demonstrable that the functions $\alpha_{1}(y,t)$, $\alpha_{2}(y,t)$, $g_{1}(y,t)$, and $g_{2}(y,t)$ of proposition 3.10 content the following algebraic system

(3.55a)\begin{align} &\alpha_{1t}+(g_{1}+g_{2})=0, \end{align}

(3.55b)\begin{align} &\alpha_{2}+(g_{1}-g_{2})=0, \end{align}

(3.55c)\begin{align} &(g_{1}-g_{2})_{y}-(1+2\alpha_{1})(g_{1}+g_{2})=0, \end{align}

(3.55d)\begin{align} &(g_{1}+g_{2})_{y}+2\alpha_{1}-(1+2\alpha_{1})(g_{1}-g_{2})=0, \end{align}

then the compatibility conditions of Lax pair (Equation 3.43) can be satisfied.

Proposition 3.13. In compliance with proposition 3.10 and theorem 3.11, the relations of $\tilde{m}$ and $\tilde{\mu}$ with notations $\left\{\alpha_{1}, \alpha_{2}, g_{1}, g_{2}\right\}$ can be available as follows

(3.56)\begin{align} \tilde{m}=\frac{1}{1+2\alpha_{1}},\quad \tilde{u}=\alpha_{2}, \end{align}

with the conditions $x_{y}=1+2\alpha_{1}$ and $(\alpha_{1}\alpha_{2})_{x}=-\frac{\alpha_{2x}}{2}$, we can derive Eq. (Equation 3.54) from (Equation 3.55a)–(Equation 3.55d).

Proof. Actually, (Equation 3.55a) can be obtained by (Equation 3.54), and from (Equation 3.55c) we have

\begin{align*} \tilde{u}_{x}=\tilde{u}_{y}x^{-1}_{y}=-(g_{1}+g_{2}). \end{align*}

According to

\begin{align*} \tilde{m}=\tilde{u}-(\tilde{u}_{x})_{y}\tilde{m}+1, \end{align*}

we have

\begin{align*} \frac{1}{1+2\alpha_{1}}=-(g_{1}-g_{2})+(g_{1}+g_{2})_{y}\frac{1}{1+2\alpha_{1}}+1, \end{align*}

which is Eq. (Equation 3.55d).

Combining Eqs. (Equation 3.56) and (Equation 3.55b), we can further obtain the relationship between the solution of cCH equation and the RH problem as shown in (Equation 3.82).

3.3. Single high-order pole solutions

In order to obtain a pure soliton solution, this section we will consider the reflectionless situation, i.e., $b(\mu)=0$ $(\mu\in\mathbb{R})$. Then $\tilde{G}(\mu)=0$ for $\mu\in\mathbb{R}$. Let $z_{0}\in\mathbb{C}^{+}$ be the N-order zero point of the scattering data $a(\mu)$, then $\left\{-z_{0}^{*}, -\frac{1}{z_{0}}, \frac{1}{z_{0}^{*}}\right\}\in\mathbb{C}^{+}$ are also the N-order zero points of the scattering data $a(\mu)$. Furthermore, $\left\{z_{0}^{*}, -z_{0}, -\frac{1}{z_{0}^{*}}, \frac{1}{z_{0}}\right\}\in\mathbb{C}^{-}$ are the N-order zero points of the scattering data $a^{*}(\mu^{*})$. The discrete spectrum is the set $\mathcal{X}=\left\{\pm z_{0}, \pm z^{*}_{0}, \pm\frac{1}{z_{0}}, \pm\frac{1}{z^{*}_{0}}\right\}$, which is same as the case in figure 1. Taking $w_{1}=z_{0}$, $w_{2}=\frac{1}{z^{*}_{0}}$, the scattering data $a(\mu)$ has the following form

(3.57)\begin{align} a(\mu)=\prod_{j=1}^{2}(\mu-w_{j})^{N}(\mu+w^{*}_{j})^{N}a_{0}(\mu), \end{align}

where $a_{0}(\mu)\neq0$ for all $\mu\in\mathbb{C}^{+}$. Therefore, from the definition of $\tilde{M}(y,t,\mu)$, it can be seen that $\mu=w_{j}$ and $\mu=-w^{*}_{j}$ $(j=1,2)$ are N-order pole points of $\tilde{M}_{11}$. Simultaneously, $\mu=w^{*}_{j}$ and $\mu=-w_{j}$ $(j=1,2)$ are N-order pole points of $\tilde{M}_{12}$. According to the definitions of (Equation 3.11) and (Equation 3.12), under the new scale $y(x,t)$, we define $\check{\phi}_{\pm}(y,t,\mu)=\hat{\phi}_{\pm}(x,t,\mu)$ and $\check{\psi}_{\pm}(y,t,\mu)=\hat{\psi}_{\pm}(x,t,\mu)$ and obtain the following relationships

(3.58)\begin{equation} \check{\phi}_-(y,t,\mu)=\check{\phi}_+(y,t,\mu)S(\mu), \end{equation}

and

(3.59)\begin{equation} \check{\psi}_-(y,t,\mu)e^{-it\theta(\mu)\sigma_{3}}=\check{\psi}_+(y,t,\mu)e^{-it\theta(\mu)\sigma_{3}}S(\mu), \end{equation}

there we redefine relation $\tilde{p}(\mu)=it\theta(\mu)$, $\tilde{p}(\mu)$ is shown in (Equation 3.32). Under the assumption that $\mu=w_{j}$ $(j=1,2)$ are N-order zero points of $a(\mu)$, from the following two relations, we have a Wronskian form

(3.60)\begin{equation} a(\mu)=Wr(\check{\phi}_{-1},\check{\phi}_{+2}), \end{equation}

then $a(\mu)$ vanishes to N-order at zero points w_j $(j=1,2)$, there exist complex constants $b_{j,s}$ $(j=1,2)$ $(s=1,2,\cdots,N)$ satisfy the follows

(3.61)\begin{equation} \frac{\partial^{m}[\check{\phi}_+(y,t,w_{j})]_{2}}{\partial\mu^{m}}=\sum^{m}_{l=0}\left(\begin{array}{c} m \\ l \end{array}\right) b_{j,m-l+1}\frac{\partial^{l}[\check{\phi}_-(y,t,w_{j})]_{1}}{\partial\mu^{l}}, \end{equation}

and

(3.62)\begin{equation} \frac{\partial^{m}[\check{\psi}_+(y,t,w_{j})]_{2}}{\partial\mu^{m}}=\sum^{m}_{l=0}\left(\begin{array}{c} m \\ l \end{array}\right) b_{j,m-l+1}\frac{\partial^{l}[\check{\psi}_-(y,t,w_{j})]_{1}e^{2it\theta(w_{j})}}{\partial\mu^{l}}, \end{equation}

with $m=0,1,\ldots,N-1$. From the normalization condition of RH problem 3.8, one can set

(3.63a)\begin{align} &\tilde{M}_{11}=1+\sum_{j=1}^{2}\sum_{s=1}^{N} \left(\frac{1}{(\mu-w_{j})^{s}}+\frac{(-1)^{s}}{(\mu+w^{*}_{j})^{s}}\right)F_{j,s} -\frac{i\tilde{\alpha}_{+}(y,t)}{4(\mu-1)}D-\frac{i\tilde{\alpha}_{+}(y,t)}{4(\mu+1)}D, \end{align}

(3.63b)\begin{align} &\tilde{M}_{12}=\sum_{j=1}^{2}\sum_{s=1}^{N} \left(\frac{1}{(\mu-w^{*}_{j})^{s}}+\frac{(-1)^{s+1}}{(\mu+w_{j})^{s}}\right)G_{j,s} +\frac{i\tilde{\alpha}_{+}(y,t)}{4(\mu-1)}-\frac{i\tilde{\alpha}_{+}(y,t)}{4(\mu+1)}, \end{align}

where D is shown in (Equation 3.21) and $\tilde{\alpha}_{+}(y,t)\in\mathbb{R}$. Besides, $F_{j,s}=F_{j,s}(y,t)$ and $G_{j,s}=G_{j,s}(y,t)$ are undetermined functions. Once these functions are solved, the solution of cCH equation will be acquired. To address this problem, making use of Taylor series expansion, one has

(3.64)\begin{align} &e^{2it\theta(\mu)}=\sum_{l=0}^{+\infty}\zeta_{j,l}(y,t)(\mu-w_{j})^{l}, \end{align}

(3.65)\begin{align} \zeta_{j,l}(y,t)=\lim_{\mu\rightarrow w_{j}}\frac{1}{l!}\frac{\partial^{l}}{\partial \mu^{l}}e^{2it\theta(\mu)}. \end{align}

It can be obtained that the coefficient of item $(\mu-w_{j})^{-s}$ of function $\tilde{M}_{11}(y,t,\mu)$ is $F_{j,s}(y,t)$. Now we extend the residue theorem by combining (Equation 3.59) and (Equation 3.62) and obtain the following relations

(3.66)\begin{align} \begin{split} F_{j,s}&=(-1)^{m}\frac{i\tilde{\alpha}_{+}(y,t)}{4}\sum_{l=s}^{N}\sum_{m=0}^{l-s} \left(\frac{r_{j,l}\zeta_{j,l-s-m}}{(w_{j}-1)^{m+1}}-\frac{r_{j,l}\zeta_{j,l-s-m}}{(w_{j}+1)^{m+1}}\right)\\ &+\sum_{l=s}^{N}\sum_{m=0}^{l-s}\sum_{p=1}^{2}\sum_{q=1}^{N} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right) r_{j,l}\zeta_{j,l-s-m}\times\\ &\left\{\frac{(-1)^{m}}{(w_{j}-w^{*}_{p})^{q+m}} +\frac{(-1)^{m+q+1}}{(w_{j}+w_{p})^{q+m}}\right\}G_{p,q}, \end{split} \end{align}

where

(3.67)\begin{align} r_{j,l}=\lim_{\mu\rightarrow v_{j}}\frac{b_{j,l}}{(N-l)!}\frac{\partial^{N-l}}{\partial(\mu-w_{j})^{N-l}} \frac{(\mu-w_{j})^{N}}{a(\mu)}. \end{align}

Likewise, the coefficient of item $(\mu-w^{*}_{j})^{-s}$ of function $\tilde{M}_{12}(y,t,\mu)$ is $G_{j,s}(y,t)$. Through the same method, we have

(3.68)\begin{align} \begin{split} G_{j,s}&=(-1)^{m+1}\frac{i\tilde{\alpha}_{+}(y,t)}{4}D\sum_{l=s}^{N}\sum_{m=0}^{l-s} \left(\frac{r^{*}_{j,l}\zeta^{*}_{j,l-s-m}}{(w^{*}_{j}-1)^{m+1}}+ \frac{r^{*}_{j,l}\zeta^{*}_{j,l-s-m}}{(w^{*}_{j}+1)^{m+1}}\right)\\ &+\sum_{l=s}^{N}\sum_{m=0}^{l-s} +\sum_{l=s}^{N}\sum_{m=0}^{l-s}\sum_{p=1}^{2}\sum_{q=1}^{N} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right)\times\\ &r^{*}_{j,l}\zeta^{*}_{j,l-s-m}\left\{\frac{(-1)^{m}} {(w^{*}_{j}-w_{p})^{q+m}} +\frac{(-1)^{m+q}}{(w^{*}_{j}+w^{*}_{p})^{q+m}}\right\}F_{p,q}, \end{split} \end{align}

The system (Equation 2.58) has the following solution For the sake of convenience, selecting the suitable notations for $j,p=1,2,$

(3.69)\begin{align} |\eta^{1}_{j}\rangle=(\eta^{1}_{j1},\ldots,\eta^{1}_{jN})^{T},\quad |\eta^{2}_{j}\rangle=(\eta^{2}_{j1},\ldots,\eta^{2}_{jN})^{T}, \end{align}

\begin{align*} \eta^{1}_{js}&=\frac{i\tilde{\alpha}_{+}(y,t)}{4}\sum_{l=s}^{N}\sum_{m=0}^{l-s} \left(\frac{(-1)^{m}}{(w_{j}-1)^{m+1}}-\frac{(-1)^{m}}{(w_{j}+1)^{m+1}} \right)r_{j,l}\zeta_{j,l-s-m},\\ \eta^{2}_{js}&=\sum_{l=s}^{N}\sum_{m=0}^{l-s}r^{*}_{j,l}\zeta^{*}_{j,l-s-m}\\ &-\frac{i\tilde{\alpha}_{+}(y,t)}{4}D\sum_{l=s}^{N}\sum_{m=0}^{l-s} \left(\frac{(-1)^{m}}{(w^{*}_{j}-1)^{m+1}}+\frac{(-1)^{m}}{(w^{*}_{j}+1)^{m+1}} \right)r^{*}_{j,l}\zeta^{*}_{j,l-s-m}, \end{align*}

then defining the N × N matrices $\Omega_{j,p}=[\Omega_{j,p}]_{s,q}$ and $\tilde{\Omega}_{j,p}=[\tilde{\Omega}_{j,p}]_{s,q}$ for $s,q=1,2,\cdots,N$ as follows

(3.70)\begin{align} \begin{split} \Omega_{j,p}=\sum_{l=s}^{N}\sum_{m=0}^{l-s} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right) r_{j,l}\zeta_{j,l-s-m}\left\{\frac{(-1)^{m}}{(w_{j}-w^{*}_{p})^{q+m}} +\frac{(-1)^{m+q+1}}{(w_{j}+w_{p})^{q+m}}\right\}, \end{split} \end{align}

(3.71)\begin{align} \begin{split} \tilde{\Omega}_{j,p}=\sum_{l=s}^{N}\sum_{m=0}^{l-s} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right) r^{*}_{j,l}\zeta^{*}_{j,l-s-m}\left\{\frac{(-1)^{m}} {(w^{*}_{j}-w_{p})^{q+m}} +\frac{(-1)^{m+q}}{(w^{*}_{j}+w^{*}_{p})^{q+m}}\right\}, \end{split} \end{align}

(3.72)\begin{align} |A_{1}\rangle=(|\eta^{1}_{1}\rangle, |\eta^{1}_{2}\rangle)^{T},\quad |A_{2}\rangle=(-|\eta^{2}_{1}\rangle, -|\eta^{2}_{2}\rangle)^{T}, \end{align}

(3.73)\begin{align} |F_{p}\rangle=(F_{p,1},\ldots,F_{p,N})^{T},\quad |G_{p}\rangle=(G_{p,1},\ldots,G_{p,N})^{T}, \end{align}

(3.74)\begin{align} \Omega= \left(\begin{array}{cc} \Omega_{1,1} & \Omega_{1,2} \\ \Omega_{2,1} & \Omega_{2,2} \end{array}\right),\quad \tilde{\Omega}= \left(\begin{array}{cc} \tilde{\Omega}_{1,1} & \tilde{\Omega}_{1,2} \\ \tilde{\Omega}_{2,1} & \tilde{\Omega}_{2,2} \end{array}\right),\quad I_{\varepsilon}= \left(\begin{array}{cc} -I & 0 \\ 0 & -I \end{array}\right), \end{align}

where I is a N × N identity matrix and the definition of Ω matrix and $\tilde{\Omega}$ matrix is the same. Combining functions (Equation 3.66)–(Equation 3.68), we have

(3.75)\begin{equation} \begin{aligned} &I|F_{p}\rangle-\Omega_{j,p}|G_{p}\rangle=|\eta^{1}_{j}\rangle,\\ &\tilde{\Omega}_{j,p}|F_{p}\rangle-I|G_{p}\rangle=-|\eta^{2}_{j}\rangle. \end{aligned} \end{equation}

After taking

(3.76)\begin{align} |K_1\rangle=(|F_{1}\rangle,|F_{2}\rangle)^{T},\quad |K_2\rangle=(|G_{1}\rangle,|G_{2}\rangle)^{T}, \end{align}

by straight calculation, we have

(3.77)\begin{align} \begin{split} &|K_{1}\rangle=(I_{\varepsilon}+\Omega\tilde{\Omega})^{-1}\Omega|A_{2}\rangle- (I_{\varepsilon}+\Omega\tilde{\Omega})^{-1}|A_{1}\rangle,\\ &|K_{2}\rangle=-(I_{\varepsilon}+\tilde{\Omega}\Omega)^{-1}\tilde{\Omega}|A_{1}\rangle+ (I_{\varepsilon}+\tilde{\Omega}\Omega)^{-1}|A_{2}\rangle. \end{split} \end{align}

The new expression of Equation 3.63a are derived as

(3.78)\begin{equation} \begin{split} \tilde{M}_{11} &=1-\frac{i\tilde{\alpha}_{+}(y,t)}{4}D\left(\frac{1}{\mu-1}+\frac{1}{\mu+1}\right)\\ &+\frac{\det(I_{\varepsilon}+\Omega\tilde{\Omega}+\Omega|A_{2}\rangle\langle E(\mu)|)- \det(I_{\varepsilon}+\Omega\tilde{\Omega}+|A_{1}\rangle\langle E(\mu)|)} {\det(I_{\varepsilon}+\Omega\tilde{\Omega})}, \end{split} \end{equation}

(3.79)\begin{equation} \begin{split} \tilde{M}_{12}&=\frac{i\tilde{\alpha}_{+}(y,t)}{4}\left(\frac{1}{\mu-1}-\frac{1}{\mu+1}\right)\\ &+\frac{\det(I_{\varepsilon}+\tilde{\Omega}\Omega+|A_{2}\rangle\langle\tilde{E}(\mu)|)- \det(I_{\varepsilon}+\tilde{\Omega}\Omega+\tilde{\Omega}|A_{1}\rangle\langle\tilde{E}(\mu)|)} {\det(I_{\varepsilon}+\tilde{\Omega}\Omega)}, \end{split} \end{equation}

with

(3.80)\begin{equation} \begin{split} &\langle E(\mu)|=(\langle E_{1}(\mu)|,\langle E_{2}(\mu)|),\quad \langle E_{j}(\mu)| =(E_{j1},\ldots,E_{jN}),\\ &\langle\tilde{E}(\mu)|=(\langle\tilde{E}_{1}(\mu)|,\langle\tilde{E}_{2}(\mu)|),\quad \langle\tilde{E}_{j}(\mu)| =(\tilde{E}_{j1},\ldots,\tilde{E}_{jN}),\\ &\langle E^{\prime}(\mu)|= (\langle E^{\prime}_{1}(\mu)|,\langle E^{\prime}_{2}(\mu)|),\quad \langle E^{\prime}_{j}(\mu)|= (E^{\prime}_{j1},\ldots,E^{\prime}_{jN}),\\ &\langle\tilde{E}^{\prime}(\mu)| =(\langle\tilde{E}^{\prime}_{1}(\mu)|,\langle\tilde{E}^{\prime}_{2}(\mu)|),\quad \langle\tilde{E}^{\prime}_{j}(\mu)|= (\tilde{E}^{\prime}_{j1},\ldots,\tilde{E}^{\prime}_{jN}), \end{split} \end{equation}

\begin{align*} &E_{js}=\frac{1}{(\mu-w_{j})^{s}}+\frac{(-1)^{s}}{(\mu+w^{*}_{j})^{s}},\quad \tilde{E}_{js}=\frac{1}{(\mu-w^{*}_{j})^{s}}+\frac{(-1)^{s+1}}{(\mu+w_{j})^{s}},\\ &E^{\prime}_{js}=\frac{-s}{(\mu-w_{j})^{s+1}}+\frac{s(-1)^{s+1}}{(\mu+w^{*}_{j})^{s+1}},\quad \tilde{E}^{\prime}_{js}=\frac{-s}{(\mu-w^{*}_{j})^{s+1}}+\frac{s(-1)^{s}}{(\mu+w_{j})^{s+1}}. \end{align*}

According to the symmetry conditions (Equation 3.22), we get that

(3.81)\begin{align} \begin{split} \tilde{M}_{21}&=-\frac{i\tilde{\alpha}_{+}(y,t)}{4}\left(\frac{1}{\mu-1}-\frac{1}{\mu+1}\right)\\ &+\frac{\det(I_{\varepsilon}+\Omega^{*}\tilde{\Omega}^{*}+|A^{*}_{2}\rangle\langle \tilde{E}^{*}(\mu^{*})|)- \det(I_{\varepsilon}+\Omega^{*}\tilde{\Omega}^{*}+\tilde{\Omega}^{*}|A^{*}_{1}\rangle\langle \tilde{E}^{*}(\mu^{*})|)} {\det(I_{\varepsilon}+\Omega^{*}\tilde{\Omega}^{*})}. \end{split} \end{align}

Hence the precise expression formula for the solution of cCH equation with single high-order pole under the NZBCs can be obtained in theorem 3.14.

Theorem 3.14

The solution of cCH equation with single high-order pole under the nonzero condition can be derived as the form

(3.82)\begin{align} \hat{u}(x,t)=-2(\tilde{M}^{\prime}_{12}(i)\tilde{M}_{11}(i) +\tilde{M}^{\prime}_{21}(i)\tilde{M}^{-1}_{11}(i)), \end{align}

\begin{align*} x=x(y,t)=y+2\ln\tilde{M}_{11}(i), \end{align*}

with

\begin{align*} \tilde{M}_{11}(i)&=1-\frac{\tilde{\alpha}_{+}(y,t)D}{4}+ \frac{\det(I_{\varepsilon}+\Omega\tilde{\Omega}+\Omega|A_{2}\rangle\langle E(i)|)- \det(I_{\varepsilon}+\Omega\tilde{\Omega}+|A_{1}\rangle\langle E(i)|)}{\det(I_{\varepsilon}+\Omega\tilde{\Omega})},\\ \tilde{M}^{\prime}_{12}(i)&=\frac{\tilde{\alpha}_{+}(y,t)}{4}+ \frac{\det(I_{\varepsilon}+\tilde{\Omega}\Omega+|A_{2}\rangle\langle\tilde{E}^{\prime}(i)|)- \det(I_{\varepsilon}+\tilde{\Omega}\Omega+\tilde{\Omega}|A_{1}\rangle\langle\tilde{E}^{\prime}(i)|)} {\det(I_{\varepsilon}+\tilde{\Omega}\Omega)},\\ \tilde{M}^{\prime}_{21}(i)& =-\frac{\tilde{\alpha}_{+}(y,t)}{4}\\ &\quad+\frac{\det(I_{\varepsilon}+\Omega^{*}\tilde{\Omega}^{*}+|A^{*}_{2}\rangle\langle \tilde{E}^{\prime*}(-i)|)- \det(I_{\varepsilon}+\Omega^{*}\tilde{\Omega}^{*}+\tilde{\Omega}^{*}|A^{*}_{1}\rangle\langle \tilde{E}^{\prime*}(-i)|)} {\det(I_{\varepsilon}+\Omega^{*}\tilde{\Omega}^{*})}, \end{align*}

where D is shown in (Equation 3.21), $\tilde{\alpha}_{+}(y,t)\in\mathbb{R}$, and the elements $|A_{1}\rangle$, $|A_{2}\rangle$, I_ɛ, Ω, $\tilde{\Omega}$, $\langle E(i)|$, $\langle\tilde{E}(i)|$, $\langle E^{\prime}(i)|$, $\langle \tilde{E}^{\prime}(i)|$ are defined in (Equation 3.72), (Equation 3.74), and (Equation 3.80).

3.3.1. One-soliton solution

Next, to construct the simple one-soliton solution of cCH equation under NZBCs, we consider the specific data as follows.

Case 4: For this case, we consider the situation N = 1, Assuming $w_{1}=z_{0}=e^{i\lambda}\in\mathbb{C}^{+}$, $(\lambda\in(0,\pi))$ is one-order pole of scattering data $a(\mu)$, then we have $w_{2}=\frac{1}{z^{*}_{0}}=w_{1}$. It can be obtained that $\theta(e^{i\lambda})=\theta(-e^{-i\lambda})$. Next we define $\theta_{1}\triangleq\theta(e^{i\lambda})=\frac{i\sin\lambda y}{2t}-\frac{2i\sin\lambda}{\cos^{2}\lambda}$, hence $e^{2it\theta_{2}}$ is real. Suppose that D = 1, $\tilde{\alpha}_{+}=1$, $r_{1,1}=2$, $r_{2,1}=1$. Then the parameters satisfy the following relationships, there $\Omega_{1,1}=\Omega_{1,2}=\Omega_{2,1}=\Omega_{2,2}$, $\tilde{\Omega}_{1,1}=\tilde{\Omega}_{1,2}=\tilde{\Omega}_{2,1}=\tilde{\Omega}_{2,2}$, $\eta^{1}_{21}=\frac{1}{2}\eta^{1}_{11}$, $\eta^{2}_{21}=\frac{1}{2}\eta^{2}_{11}$, $\zeta_{1,0}=\zeta_{2,0}$.

\begin{align*} \zeta_{1,0}=e^{-\frac{4\sin\lambda}{\cos^{2}\lambda e^{2i\lambda}}\left(\left(t+\frac{y}{8}\right)e^{2i\lambda}+\frac{y}{16}(e^{4i\lambda}+1)\right)}, \end{align*}

\begin{align*} \Omega_{1,1}=2e^{-\frac{4\sin\lambda}{\cos^{2}\lambda e^{2i\lambda}}\left(\left(t+\frac{y}{8}\right)e^{2i\lambda}+\frac{y}{16}(e^{4i\lambda}+1)\right)} \left(\frac{1}{e^{i\lambda}-e^{-i\lambda}+\frac{1}{2e^{i\lambda}}}\right), \end{align*}

\begin{align*} \tilde{\Omega}_{1,1}=2e^{-\frac{((16t+2y)e^{-2i\lambda}+y(e^{-4i\lambda}+1))\sin\lambda}{4\cos^{2}\lambda e^{2i\lambda}}}\left(\frac{1}{e^{-i\lambda}-e^{i\lambda}-\frac{1}{2e^{-i\lambda}}}\right), \end{align*}

\begin{align*} \eta^{1}_{11}=\frac{i}{2}\left(\frac{1}{e^{i\lambda}-1}-\frac{1}{e^{i\lambda}+1}\right) e^{-\frac{4\sin\lambda}{\cos^{2}\lambda e^{2i\lambda}}\left(\left(t+\frac{y}{8}\right)e^{2i\lambda}+\frac{y}{16}(e^{4i\lambda}+1)\right)}, \end{align*}

\begin{align*} \eta^{2}_{11}=\left(2-\frac{i}{2}\left(\frac{1}{e^{i\lambda}-1}+\frac{1}{e^{i\lambda}+1}\right)\right)e^{-\frac{4\sin\lambda}{\cos^{2}\lambda e^{2i\lambda}}\left(\left(t+\frac{y}{8}\right)e^{2i\lambda}+\frac{y}{16}(e^{4i\lambda}+1)\right)}, \end{align*}

substituting the above results into the formula (Equation 3.82) we can obtain the one-soliton solution of cCH equation under NZBCs.

Supposing $\lambda=\frac{\pi}{6}$, the expression for the solution is

(3.83)\begin{align} \hat{u}(x,t)=-2\left(T^{1}_{2}T^{1}_{1}+\frac{T^{1}_{3}}{T^{1}_{1}}\right), \end{align}

\begin{align*} x=x(y,t)=y+2\ln T^{1}_{1}, \end{align*}

\begin{align*} T^{1}_{1}=\frac{12ie^{-\frac{8t}{3}-\frac{y}{2}}+153i\sqrt{3}e^{-\frac{16t}{3}-y}+20i\sqrt{3}-246e^{-\frac{16t}{3}-y}-20} {(i\sqrt{3}-1)\left(27e^{-\frac{16t}{3}-\frac{y}{2}}+18ie^{-\frac{16t}{3}-\frac{y}{2}}\sqrt{3}+4\right)}, \end{align*}

\begin{align*} T^{1}_{2}=\frac{141ie^{-\frac{16t}{3}-\frac{y}{2}}\sqrt{3}-6e^{-\frac{8t}{3}-\frac{y}{2}}+216e^{-\frac{16t}{3}-\frac{y}{2}}+32} {27e^{-\frac{16t}{3}-\frac{y}{2}}+18ie^{-\frac{16t}{3}-\frac{y}{2}}\sqrt{3}+4}, \end{align*}

\begin{align*} T^{1}_{3}=-\frac{18(i\sqrt{3}-1)e^{-\frac{8t}{3}-\frac{y}{2}}-(351i\sqrt{3}+189)e^{-\frac{16t}{3}-\frac{y}{2}}-32i\sqrt{3}+32} {(i\sqrt{3}-1)\left(18ie^{-\frac{16t}{3}-\frac{y}{2}}-27e^{-\frac{16t}{3}-\frac{y}{2}}-4\right)}. \end{align*}

In accordance with the construction process of the RH problem, it can be known that there are singularities $\left\{0,\pm1,\pm i\right\}$ of the spectral parameter z. Since we choose $z_{0}\in\mathbb{C}^{+}$ here and $|z_{0}|=1$, $z_{0}=i$ is the singularity of solution $\hat{u}$. From the propagation of waves in figure 7, the bright soliton solution is the hump-kink interacted solution. Based on the intensity distribution figure captured at various time points, when the wave peaks and then drops a certain height, it continues to propagate but does not return to the same height as before.

Figure 7.

(a)–(c) describe the local structure, density, and intensity profiles with different times of one-soliton solution $|u|$. Parameters $r_{1,1}=2$, $r_{2,1}=1$, $\lambda=\frac{\pi}{6}$.

Figure 7 Long description

Three plots labeled (a), (b) and (c) show different aspects of a one-soliton solution at different times. Image (a) is a three-dimensional surface plot showing the local structure. The horizontal plane axes are labeled x and t and the vertical axis represents the amplitude of the solution. The surface shows a single sharp peak rising from a near-flat base, with the peak located near the center of the plotted domain. The x-axis and t-axis each span from approximately negative 5 to positive 5. Image (b) is a two-dimensional density plot. The horizontal axis represents x, spanning from approximately negative 10 to positive 10. The vertical axis represents t, also spanning from approximately negative 10 to positive 10. The density scale runs from low to high values. A narrow near-vertical bright band is visible near x equals 0, running along the full extent of the t-axis. The surrounding region shows uniformly low density values. This band indicates the spatial concentration of the soliton across all time values shown. Image (c) is a line plot showing intensity profiles. The horizontal axis represents x, spanning from approximately negative 10 to positive 10. The vertical axis represents intensity, spanning from 0 upward. Three curves are plotted, each corresponding to a different time value. All three curves show a single peak centered near x equals 0. The tallest and narrowest peak reaches approximately 1.0 on the vertical axis. The second curve shows a peak of moderate height, slightly lower than the first. The third curve shows the broadest and lowest peak among the three. The three peaks are all located near x equals 0, with no visible lateral shift between them.

Supposing $\lambda=\frac{\pi}{3}$, the solution can be obtained as follows

(3.84)\begin{align} \hat{u}(x,t)=-2\left(T^{2}_{2}T^{2}_{1}+\frac{T^{2}_{3}}{T^{2}_{1}}\right), \end{align}

\begin{align*} x=x(y,t)=y+2\ln T^{2}_{1}, \end{align*}

\begin{align*} T^{2}_{1}=&\frac{1}{\varpi^{2}_{1}}\left((5520-696i+(1080-816i)\sqrt{3})e^{(-16t-y)\sqrt{3}}\right)+\\ &\frac{1}{\varpi^{2}_{1}}\left(320e^{-\frac{\sqrt{3}(16t+y)}{2}}\sqrt{3}+64-96i+(64i-32)\sqrt{3}\right), \end{align*}

\begin{align*} T^{2}_{2}=&\frac{1}{\varpi^{2}_{2}}\left((-1056-1728i-(192+1248i)\sqrt{3})e^{(-16t-y)\sqrt{3}}+896-1536i+512\sqrt{3}\right)+\\ &\frac{1}{\varpi^{2}_{2}}\left((-15360+28800i+(-9600+15360i)\sqrt{3})e^{-\frac{\sqrt{3}(16t+y)}{2}}-896i\sqrt{3}\right), \end{align*}

\begin{align*} T^{2}_{3}=&\frac{1}{\varpi^{2}_{3}}\left((1056+1728i-(192+1248i)\sqrt{3})e^{(-16t-y)\sqrt{3}}-896+1536i+512\sqrt{3}\right)+\\ &\frac{1}{\varpi^{2}_{3}}\left((-7680+17280i+(5760-7680i)\sqrt{3})e^{-\frac{\sqrt{3}(16t+y)}{2}}-896i\sqrt{3}\right), \end{align*}

\begin{align*} \varpi^{2}_{1}&=80(1+i\sqrt{3})(2-\sqrt{3})((3+6i\sqrt{3})e^{(-16t-y)\sqrt{3}})+4,\\ \varpi^{2}_{2}&=-40(3+2\sqrt{3})(i\sqrt{3}-1)((3+6i)e^{-\frac{\sqrt{3}(16t+y)}{2}}+4),\\ \varpi^{2}_{3}&=-320(2-\sqrt{3})^{2}(i\sqrt{3}+1)((6i\sqrt{3}-3)e^{-\frac{\sqrt{3}(16t+y)}{2}}-4). \end{align*}

From figure 8, it can be seen that the peak of soliton does not immediately propagate smoothly when it falls from the highest point but rather fluctuates and appears as a smaller peak.

Figure 8.

(a)–(c) describe the local structure, density, and intensity profiles with different times of one-soliton solutions $|u|$. Parameters $r_{1,1}=2$, $r_{2,1}=1$, $\lambda=\frac{\pi}{3}$.

Figure 8 Long description

Three plots together show the local structure, density and intensity profiles of one-soliton solutions at different times. Plot (a) is a three-dimensional surface plot. The horizontal axes each range from negative 10 to positive 10. The vertical axis represents the amplitude or height of the soliton. A single narrow ridge runs diagonally across the surface, indicating a localized soliton peak. The ridge remains concentrated in a narrow band. Plot (b) is a two-dimensional density plot. The horizontal axis ranges from negative 4 to positive 4 and the vertical axis ranges from negative 4 to positive 4. A narrow vertical band of high density runs through the center of the plot. The surrounding regions show low density. A color scale on the right side of the plot ranges from low values at the bottom to high values at the top, encoding the density magnitude. Plot (c) is a line plot with three curves, each corresponding to a different time point as identified in the legend. The horizontal axis ranges from approximately negative 30 to positive 30. The vertical axis ranges from approximately negative 1 to positive 3. All three curves show a sharp peak near the center of the horizontal axis. The tallest peak reaches approximately 3 on the vertical axis and is narrow. A secondary smaller peak appears to the right of the main peak for at least one curve. The remaining portions of all three curves stay near zero. The three curves differ in their peak heights and the presence or position of the secondary peak, indicating how the soliton intensity profile changes over time.

3.3.2. Two-order pole solution

In an effort to construct the two-order pole solution of cCH equation under NZBCs, we consider the specific data as follows. This is for N = 2 case, and the expression of $u(x,t)$ defined in theorem 3.14.

Case 5: Assuming that $w_{1}=z_{0}=e^{i\lambda}\in\mathbb{C}^{+}$ be the two-order zero point of the scattering data $a(\mu)$, then $w_{2}=w_{1}$ is also the two-order zero point of $a(\mu)$. The discrete spectrum is the set $\mathcal{X}=\left\{e^{i\lambda}, -e^{-i\lambda}, e^{-i\lambda}, -e^{i\lambda}\right\}$. For the purpose of displaying characteristics of soliton propagations more clearly, we select some appropriate parameters to construct the two-order pole solutions of cCH equation. The dynamic behaviours of solitons are shown in Figures (9)–(10).

Figure 9.

(a)–(c) describe the local structure, density, and intensity profiles with different times of the soliton solutions $|u|$ with one two-order pole. Parameters $r_{1,1}=1$, $r_{1,2}=1$, $r_{2,1}=1$, $r_{2,2}=1$, $\lambda=\frac{\pi}{6}$.

Figure 9 Long description

Three panels labeled (a), (b) and (c) present soliton solution profiles with one two-order pole across spatial variable x and time variable t. Panel (a) is a 3D surface plot. The x-axis is labeled x, units not shown and the y-axis is labeled t, units not shown. The z-axis is labeled with an intensity-related quantity, units not shown. A narrow, sharply peaked ridge rises steeply along a diagonal path across the x-t plane. The ridge is concentrated within a narrow band, indicating a localized soliton structure. The surface is flat and near zero away from this ridge. Panel (b) is a 2D heatmap or colormap plot of density over the x-t plane. The x-axis is labeled x, units not shown, ranging approximately from minus 20 to 30. The y-axis is labeled t, units not shown, ranging approximately from minus 20 to 20. A dominant diagonal band runs from lower left to upper right across the plot, flanked by narrower secondary bands. The central band occupies a wider region compared to the surrounding bands. The density values transition sharply across band boundaries. Panel (c) is an overlaid line plot showing intensity profiles. The x-axis is labeled t, units not shown, ranging approximately from minus 3 to 3. The y-axis is labeled with an intensity quantity, units not shown. Multiple curves are plotted together, each corresponding to a different value of x or a different time slice. The curves share a common baseline near zero and each rises to a distinct sharp peak near t equals 0. The peaks are narrow and closely spaced. Individual curves are distinguishable by line style, though legend labels are not clearly readable at this resolution.

Figure 10.

(a)–(c) describe the local structure, density, and intensity profiles with different times of the soliton solutions $|u|$ with one two-order pole. Parameters $r_{1,1}=1$, $r_{1,2}=1$, $r_{2,1}=1$, $r_{2,2}=1$, $\lambda=\frac{5\pi}{6}$.

Figure 10 Long description

Three panels labeled a, b and c, each depicting soliton solutions with one two-order pole at different times. Panel a is a 3D surface plot. The x-axis is labeled x, units not shown and the y-axis is labeled t, units not shown. The z-axis represents the amplitude of the soliton solution, units not shown. The surface displays a sharp ridge running diagonally across the plot, indicating the soliton position shifts as time increases. The ridge rises steeply from a near-flat base to a prominent peak, with the surrounding surface remaining close to zero. Panel b is a 2D density plot. The x-axis is labeled x, ranging approximately from minus 30 to 30, units not shown. The y-axis is labeled t, ranging approximately from minus 30 to 30, units not shown. The plot shows color-coded intensity values across the x-t plane. A narrow diagonal band runs from the lower-left region toward the upper-right region, representing the soliton trajectory. The surrounding regions display near-uniform low-intensity values. Transition boundaries between regions are sharp and clearly defined. Panel c is a line plot showing intensity profiles at different values of time t. The x-axis is labeled x, units not shown. The y-axis is labeled intensity, units not shown. Multiple curves are present, each corresponding to a distinct value of t. A dominant narrow peak is visible near x equals 0, with the peak height substantially larger than the surrounding profile. Smaller secondary peaks appear symmetrically on either side of the central peak. The curves differ in peak height and width across the plotted time values. No legend labels are visible to individually identify each curve; multiple distinct curves are shown without textual labels.

Setting $\lambda=\frac{\pi}{6}$, $\left\{r_{1,1}=1, r_{1,2}=1, r_{2,1}=1, r_{2,2}=1\right\}$, D = 1, $\tilde{\alpha}_{+}=1$. Besides, $\Omega_{1,1}=\Omega_{1,3}=\Omega_{3,1}=\Omega_{3,3}$, $\Omega_{1,2}=\Omega_{1,4}=\Omega_{3,2}=\Omega_{3,4}$, $\Omega_{2,1}=\Omega_{2,3}=\Omega_{4,1}=\Omega_{4,3}$, $\Omega_{2,2}=\Omega_{2,4}=\Omega_{4,2}=\Omega_{4,4}$, $\tilde{\Omega}_{1,1}=\tilde{\Omega}_{1,3}=\tilde{\Omega}_{3,1}=\tilde{\Omega}_{3,3}$, $\tilde{\Omega}_{1,2}=\tilde{\Omega}_{1,4}=\tilde{\Omega}_{3,2}=\tilde{\Omega}_{3,4}$, $\tilde{\Omega}_{2,1}=\tilde{\Omega}_{2,3}=\tilde{\Omega}_{4,1}=\tilde{\Omega}_{4,3}$, $\tilde{\Omega}_{2,2}=\tilde{\Omega}_{2,4}=\tilde{\Omega}_{4,2}=\tilde{\Omega}_{4,4}$. In view of the construction process for simple high-order pole solutions under NZBCs and the parameter expressions of obtained solutions in theorem 3.14, we further obtain

\begin{align*} \zeta_{1,0}=\zeta_{2,0}=e^{-\frac{2t}{3}-\frac{y}{2}}, \end{align*}

\begin{align*} \zeta_{1,1}=\zeta_{2,1}=\frac{8i(20t+9y)e^{-\frac{2t}{3}-\frac{y}{2}}(i\sqrt{3}-1)} {(i\sqrt{3}+3)^3(\sqrt{3}+i)^2}, \end{align*}

\begin{align*} \Omega_{1,1}=\frac{5\sqrt{3}}{18}\left(\left(-\frac{21+27y}{20}+\frac{3i}{2}-3t\right)\sqrt{3} +it+\frac{9i(y-1)}{20}-\frac{9}{10}\right)e^{-\frac{2t}{3}-\frac{y}{2}}, \end{align*}

\begin{align*} \Omega_{1,2}=\frac{1440e^{-\frac{2t}{3}-\frac{y}{2}}}{(i\sqrt{3}+3)^3(\sqrt{3}+i)^4} \left(\sqrt{3}\left(\frac{y-6}{20}+\frac{i}{10}+\frac{t}{9}\right)+it+\frac{9i(y+1)}{20}+\frac{1}{2}\right), \end{align*}

\begin{align*} \Omega_{2,1}=-\frac{e^{-\frac{2t}{3}-\frac{y}{2}}(-2+i\sqrt{3})}{\sqrt{3}+i},\quad \Omega_{2,2}=\frac{e^{-\frac{2t}{3}-\frac{y}{2}}(-3-2i\sqrt{3})}{(\sqrt{3}+i)^2}, \end{align*}

\begin{align*} \tilde{\Omega}_{1,1}=\frac{1}{24}\left((-6+3i+20it+9iy)\sqrt{3}+27+18i+20t+9y\right) e^{-\frac{2t}{3}-\frac{y}{2}}, \end{align*}

\begin{align*} \tilde{\Omega}_{1,2}=\frac{1120\left(\left(\frac{3}{10}+\frac{3i}{14}+\frac{t}{7}+\frac{9y}{140}\right)\sqrt{3} +it+\frac{9iy}{20}-\frac{27}{70}+\frac{9i}{10}\right)e^{-\frac{2t}{3}-\frac{y}{2}}}{(i\sqrt{3}-3)^3(i-\sqrt{3})^4}, \end{align*}

\begin{align*} \tilde{\Omega}_{2,1}=\frac{-i\sqrt{3}e^{-\frac{2t}{3}-\frac{y}{2}}}{i-\sqrt{3}},\quad \tilde{\Omega}_{2,2}=\frac{(2i\sqrt{3}-1)e^{-\frac{2t}{3}-\frac{y}{2}}}{(i-\sqrt{3})^2}, \end{align*}

\begin{align*} \eta^{1}_{11}=\eta^{1}_{21}=\frac{-1280i\left(\left(-\frac{6}{5}+\frac{3i}{5}-t-\frac{9y}{20}\right) \sqrt{3}+it+\frac{9iy}{20}\right)e^{-\frac{2t}{3}-\frac{y}{2}}} {4(i\sqrt{3}-1)^2(i\sqrt{3}+3)^3(\sqrt{3}+i)^2}, \end{align*}

\begin{align*} \eta^{2}_{11}=\eta^{2}_{21}=\frac{(-1920i\sqrt{3}-7680it-3456iy+5760-2304i)e^{-\frac{2t}{3}-\frac{y}{2}}} {4(i\sqrt{3}+1)^2(i\sqrt{3}-3)^3(i-\sqrt{3})^2}, \end{align*}

\begin{align*} \eta^{1}_{12}=\eta^{1}_{22}=\frac{ie^{-\frac{2t}{3}-\frac{y}{2}}}{i\sqrt{3}-1},\quad \eta^{2}_{12}=\eta^{2}_{22}=-\frac{3e^{-\frac{2t}{3}-\frac{y}{2}}}{2}, \end{align*}

then substituting the above results into the expression of solution (Equation 3.82), we can obtain the two-order pole solution of cCH equation, Figure 9 shows the dynamic behaviour of soliton propagation, with three images presenting the local structure, density, and intensity profiles at different times of two-order pole solutions with the above parameters.

In addition, we select parameter $\lambda=\frac{5\pi}{6}$ and use the same process to obtain the dynamic propagation of the solution shown in Figure 10. From these two Figures 9–10, it can be seen that when the two waves collide at the centre, a larger peak is generated, and then they continue to propagate along the previous trajectory, but the energy decreases.

3.4. Multiple high-order pole solutions

The general circumstance that scattering data $a(\mu)$ has N high-order poles $z_{1},z_{2},\cdots,z_{N}$ will be considered, there $z_{k}\in\mathbb{C}^{+}$ for all $k=1,2,\ldots,N$, and their powers are $n_{1},n_{2},\ldots,n_{N}$, respectively. Let $w^{k}_{1}=z_{k}$ and $w^{k}_{2}=\frac{1}{z^{*}_{k}}$, $k=1,2,\ldots,N$, just like the case of one high-order pole, from the definition of $\tilde{M}(y,t,\mu)$, it can be seen that $\mu=w^{k}_{j}$ and $\mu=-w^{k*}_{j}$ $(j=1,2)$ are N-order pole points of $\tilde{M}_{11}$. Simultaneously, $\mu=w^{k*}_{j}$ and $\mu=-w^{k}_{j}$ $(j=1,2)$ are N-order pole points of $\tilde{M}_{12}$.

Under the assumption that $\mu=w^{k}_{j}$ $(j=1,2)$ are n_k-order zero points of $a(\mu)$ for $k=1,2,\ldots,N$, by the Wronskian $a(\mu)=Wr(\check{\phi}_{-1},\check{\phi}_{+2})$, there exist complex constants $b^{k}_{j,s}$ $(s=1,2,\ldots,n_N)$ satisfy the follows

(3.85)\begin{equation} \frac{\partial^{m}[\check{\phi}_+(y,t,w^{k}_{j})]_{2}}{\partial\mu^{m}}=\sum^{m}_{l=0}\left(\begin{array}{c} m \\ l \end{array}\right) b^{k}_{j,m-l+1}\frac{\partial^{l}[\check{\phi}_-(y,t,w^{k}_{j})]_{1}}{\partial\mu^{l}}, \end{equation}

and

(3.86)\begin{equation} \frac{\partial^{m}[\check{\psi}_+(y,t,w^{k}_{j})]_{2}}{\partial\mu^{m}}=\sum^{m}_{l=0}\left(\begin{array}{c} m \\ l \end{array}\right) b^{k}_{j,m-l+1}\frac{\partial^{l}[\check{\psi}_-(y,t,w^{k}_{j})]_{1}e^{2it\theta(w^{k}_{j})}}{\partial\mu^{l}}, \end{equation}

with $m=0,1,\ldots,n_{N-1}$. Utilizing the parallel approach in above, we define the $n_k\times n_k$ matrices $\Xi^{j,p}_{kd}=[\Xi^{j,p}_{kd}]_{s,q}$ and $\tilde{\Xi}^{j,p}_{kd}=[\tilde{\Xi}^{j,p}_{kd}]_{s,q}$ for $s,q=1,2,\ldots,n_{k}$ and introduce the following notations for $j,p=1,2$

(3.87)\begin{align} &|\eta^{1k}_{j}\rangle=(\eta^{1k}_{j1},\cdots,\eta^{1k}_{jN})^{T},\quad |\eta^{2k}_{j}\rangle=(\eta^{2k}_{j1},\cdots,\eta^{2k}_{jN})^{T}, \end{align}

\begin{align*} \eta^{1k}_{js}=\frac{i\tilde{\alpha}_{+}(y,t)}{4}\sum_{l=s}^{n_{k}}\sum_{m=0}^{l-s} \left(\frac{(-1)^{m}}{(w^{k}_{j}-1)^{m+1}}-\frac{(-1)^{m}}{(w^{k}_{j}+1)^{m+1}} \right)r^{k}_{j,l}\zeta^{k}_{j,l-s-m}, \end{align*}

\begin{align*} \eta^{2k}_{js}&=\sum_{l=s}^{n_{k}}\sum_{m=0}^{l-s}r^{k*}_{j,l}\zeta^{k*}_{j,l-s-m}\\ &-\frac{i\tilde{\alpha}_{+}(y,t)}{4}D\sum_{l=s}^{n_{k}}\sum_{m=0}^{l-s} \left(\frac{(-1)^{m}}{(w^{k*}_{j}-1)^{m+1}}+\frac{(-1)^{m}}{(w^{k*}_{j}+1)^{m+1}} \right)r^{k*}_{j,l}\zeta^{k*}_{j,l-s-m}, \end{align*}

(3.88)\begin{align} r^{k}_{j,l}=\lim_{\mu\rightarrow w^{k}_{j}}\frac{b^{k}_{j,l}}{(n_{k}-l)!}\frac{\partial^{n_{k}-l}}{\partial(\mu-w^{k}_{j})^{n_{k}-l}} \frac{(\mu-w^{k}_{j})^{n_{k}}}{a(\mu)}, \end{align}

(3.89)\begin{align} \begin{split} &|B_{1}\rangle=(|A^{1}_{1}\rangle, |A^{2}_{1}\rangle, \cdots, |A^{N}_{1}\rangle)^{T},\quad |A^{k}_{1}\rangle=(|\eta^{1k}_{1}\rangle, |\eta^{1k}_{2}\rangle)^{T},\\ &|B_{2}\rangle=(|A^{1}_{2}\rangle, |A^{2}_{2}\rangle, \cdots, |A^{N}_{2}\rangle)^{T},\quad |A^{k}_{2}\rangle=(-|\eta^{2k}_{1}\rangle, -|\eta^{2k}_{2}\rangle)^{T}, \end{split} \end{align}

(3.90)\begin{align} \begin{split} \Xi^{j,p}_{k,d}=\sum_{l=s}^{n_{k}}\sum_{m=0}^{l-s} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right) r^{k}_{j,l}\zeta^{k}_{j,l-s-m}\left\{\frac{(-1)^{m}}{(w^{k}_{j}-w^{k*}_{p})^{q+m}} +\frac{(-1)^{m+q+1}}{(w^{k}_{j}+w^{k}_{p})^{q+m}}\right\}, \end{split} \end{align}

(3.91)\begin{align} \begin{split} \tilde{\Xi}^{j,p}_{k,d}=\sum_{l=s}^{n_{k}}\sum_{m=0}^{l-s} \left( \begin{array}{c} q+m-1\\ m \\ \end{array} \right) r^{k*}_{j,l}\zeta^{k*}_{j,l-s-m}\left\{\frac{(-1)^{m}} {(w^{k*}_{j}-w^{k}_{p})^{q+m}} +\frac{(-1)^{m+q}}{(w^{k*}_{j}+w^{k*}_{p})^{q+m}}\right\}, \end{split} \end{align}

(3.92)\begin{align} \Xi= \begin{pmatrix} [\Xi^{j,p}_{11}] & \cdots & [\Xi^{j,p}_{1N}] \\ \vdots & \ddots & \vdots \\ [\Xi^{j,p}_{N1}] & \cdots & [\Xi^{j,p}_{NN}] \\ \end{pmatrix},\quad \tilde{\Xi}= \begin{pmatrix} [\tilde{\Xi}^{j,p}_{11}] & \cdots & [\tilde{\Xi}^{j,p}_{1N}] \\ \vdots & \ddots & \vdots \\ [\tilde{\Xi}^{j,p}_{N1}] & \cdots & [\tilde{\Xi}^{j,p}_{NN}] \\ \end{pmatrix}, \end{align}

(3.93)\begin{align} I_{\delta}= \begin{pmatrix} I_{\delta_{1}} & & \\ & \ddots & \\ & & I_{\delta_{N}} \\ \end{pmatrix},\quad I_{\delta_{k}}= \begin{pmatrix} -I & \\ & -I \\ \end{pmatrix}_{2n_{k}\times2n_{k}} k=1,\ldots,N, \end{align}

(3.94)\begin{equation} \begin{split} &\langle \Lambda(\mu)|=(\langle \Lambda^{1}_{1}(\mu)|,\langle \Lambda^{1}_{2}(\mu)|,\ldots,\langle \Lambda^{N}_{1}(\mu)|,\langle \Lambda^{N}_{2}(\mu)|),\\ &\langle \tilde{\Lambda}(\mu)|=(\langle \tilde{\Lambda}^{1}_{1}(\mu)|,\langle \tilde{\Lambda}^{1}_{2}(\mu)|,\ldots,\langle \tilde{\Lambda}^{N}_{1}(\mu)|,\langle \tilde{\Lambda}^{N}_{2}(\mu)|),\\ &\langle \Lambda^{\prime}(\mu)|=(\langle \Lambda^{1\prime}_{1}(\mu)|,\langle \Lambda^{1\prime}_{2}(\mu)|,\ldots,\langle \Lambda^{N\prime}_{1}(\mu)|,\langle \Lambda^{N\prime}_{2}(\mu)|),\\ &\langle \tilde{\Lambda}^{\prime}(\mu)|=(\langle \tilde{\Lambda}^{1\prime}_{1}(\mu)|,\langle \tilde{\Lambda}^{1\prime}_{2}(\mu)|,\ldots,\langle \tilde{\Lambda}^{N\prime}_{1}(\mu)|,\langle \tilde{\Lambda}^{N\prime}_{2}(\mu)|), \end{split} \end{equation}

with

\begin{align*} &\langle\Lambda^{k}_{j}(\mu)|=(\Lambda^{k}_{j1},\ldots,\Lambda^{k}_{jn_k}),\quad \Lambda^{k}_{js}=\frac{1}{(\mu-w^{k}_{j})^{s}}+\frac{(-1)^{s}}{(\mu+w^{k*}_{j})^{s}},\\ &\langle\tilde{\Lambda}^{k}_{j}(\mu)|=(\tilde{\Lambda}^{k}_{j1},\ldots,\tilde{\Lambda}^{k}_{jn_k}),\quad \tilde{\Lambda}^{k}_{js}=\frac{1}{(\mu-w^{k*}_{j})^{s}}+\frac{(-1)^{s+1}}{(\mu+w^{k}_{j})^{s}},\\ &\langle\Lambda^{k\prime}_{j}(\mu)|=(\Lambda^{k\prime}_{j1},\ldots,\Lambda^{k\prime}_{jn_k}),\quad \Lambda^{k\prime}_{js}=\frac{-s}{(\mu-w^{k}_{j})^{s+1}}+\frac{s(-1)^{s+1}}{(\mu+w^{k*}_{j})^{s+1}},\\ &\langle\tilde{\Lambda}^{k\prime}_{j}(\mu)|=(\tilde{\Lambda}^{k\prime}_{j1},\ldots,\tilde{\Lambda}^{k\prime}_{jn_k}),\quad \tilde{\Lambda}^{k\prime}_{js}=\frac{-s}{(\mu-w^{k*}_{j})^{s+1}}+\frac{s(-1)^{s}}{(\mu+w^{k}_{j})^{s+1}}. \end{align*}

Similar to theorem 3.14, we give the solution of cCH equation with multiple high-order poles as follows.

Theorem 3.15

The multiple high-order pole solutions of cCH equation under the nonzero condition can be obtained as follows

(3.95)\begin{align} \hat{u}(x,t)=-2(\tilde{M}^{\prime}_{12}(i)\tilde{M}_{11}(i) +\tilde{M}^{\prime}_{21}(i)\tilde{M}^{-1}_{11}(i)), \end{align}

\begin{align*} x=x(y,t)=y+2\ln\tilde{M}_{11}(i), \end{align*}

with

\begin{align*} \tilde{M}_{11}(i)& =1-\frac{\tilde{\alpha}_{+}(y,t)D}{4}+ \frac{\det(I_{\delta}+\Xi\tilde{\Xi}+\Xi|B_{2}\rangle\langle\Lambda(i)|)- \det(I_{\delta}+\Xi\tilde{\Xi}+|B_{1}\rangle\langle\Lambda(i)|)}{\det(I_{\delta}+\Xi\tilde{\Xi})},\\ \tilde{M}^{\prime}_{12}(i)& =\frac{\tilde{\alpha}_{+}(y,t)}{4}+ \frac{\det(I_{\delta}+\tilde{\Xi}\Xi+|B_{2}\rangle\langle\tilde{\Lambda}^{\prime}(i)|)- \det(I_{\delta}+\tilde{\Xi}\Xi+\tilde{\Xi}|B_{1}\rangle\langle\tilde{\Lambda}^{\prime}(i)|)} {\det(I_{\delta}+\tilde{\Xi}\Xi)},\\ \tilde{M}^{\prime}_{21}(i)& =-\frac{\tilde{\alpha}_{+}(y,t)}{4}\\ & \quad +\frac{\det(I_{\delta}+\Xi^{*}\tilde{\Xi}^{*}+|\Lambda^{*}_{2}\rangle\langle \tilde{\Lambda}^{\prime*}(-i)|)- \det(I_{\delta}+\Xi^{*}\tilde{\Xi}^{*}+\tilde{\Xi}^{*}|\Lambda^{*}_{1}\rangle\langle \tilde{\Lambda}^{\prime*}(-i)|)} {\det(I_{\delta}+\Xi^{*}\tilde{\Xi}^{*})}, \end{align*}

where D is shown in (Equation 3.21), $\tilde{\alpha}_{+}(y,t)\in\mathbb{R}$, the elements $|B_{1}\rangle$, $|B_{2}\rangle$, I_δ, Ξ, $\tilde{\Xi}$, $\langle\Lambda(i)|$, $\langle\tilde{\Lambda}(i)|$, $\langle \Lambda^{\prime}(i)|$, and $\langle \tilde{\Lambda}^{\prime}(i)|$ are defined in (Equation 3.89), (Equation 3.92), (Equation 3.93), and (Equation 3.94).