1. Introduction
As is widely recognized, numerous effective approaches have been proposed for studying soliton solutions like the Hirota bilinear method [1], Darboux transform [2, 3], physics-informed neural networks [4, 5] 1 inverse scattering transform (IST) [6] and so on. Among these, the IST plays a role in resolving soliton solutions of completely integrable equations. The Riemann–Hilbert (RH) method is a newer form of IST that is garnering increasing interest and investigation [7]. Particularly, when dealing with high-order spectral issues, the RH approach is more convenient. Therefore, the application of the RH method to solve the high-order poles of some integrable equations has drawn increasing attention [8–11].
The crucial factor to solving nonlinear integrable equations is the pole of the transmission coefficient. The simple and the high-order pole are present in the transmission coefficient, but the RH problem is harder to solve in the case of the high-order pole than the one-order pole. Wadati combined distinct simple poles to research the third-order pole solitons of the modified KdV equation [12]. However, this approach is not practical for solutions involving multiple high-order poles and cannot ensure the regularity of the resulting pole solitons. Recently, Zhang and colleagues have shown an effective approach for obtaining multiple high-order pole solutions and ensuring their regularity, which is the improved RH method by using algebraic computation and matrix computation [13].
The AB system was suggested by Pedlosky in [14] and it can be transformed as follows [15]1.1 ), the following normalizing condition is satisfied
$\begin{eqnarray}\begin{array}{rcl}{A}_{{xt}} & = & {AB},\\ {B}_{x} & = & -\displaystyle \frac{1}{2}{\left({\left|A\right|}^{2}\right)}_{t},\end{array}\end{eqnarray}$
where A = A(x, t) is a complex function and B = B(x, t) is a real function. The two variables A and B represent the baroclinic wave packet's amplitude and basic flow correction which is caused by the baroclinic wave's self-rectification. According to equation ( $\begin{eqnarray}{\left({\left|A\right|}_{t}\right)}^{2}+{B}^{2}=1.\end{eqnarray}$
The AB equation's envelope periodic waves and solitary waves are used as models to describe ultra-short pulses in nonlinear optics and marginally unstable baroclinic wave packets in geophysical fluids [16–19]. Moreover, the AB system will reduce to the sinh-Gordon and sine-Gordon equations when A is a real function [20], which are crucial to both quantum field theory and nonlinear dynamics [21–23].Since its extremely momentous physical significance, the study on the AB equation is ongoing. For the multi-component AB system, Yu et al have used the Hirota bilinear method to report bright-dark soliton solutions [24, 25]. Multi-dark-dark soliton solutions, semirational solutions, rogue wave solutions and breather solutions have been studied via Darboux transformation [20, 26–29]. Geng and his collaborators have investigated bright and dark solitons with zero and nonzero backgrounds using the RH method [30, 31]. The nonlinear steepest decent approach has been used to study the long-time asymptotic behavior of solutions [32]. In this paper, we investigate the multiple high-order pole solutions of equation (1.1 ) according to the RH problem with zero boundary conditions. The primary objective is to establish the exact solution when the scattering data contains arbitrary order zeros. As far as we know, the following aspects of the AB system are still worth studying.
| • | It remains unclear whether the system permits a solution with mixed simple pole and high-order pole, even though it has been shown that the system has several special single-pole solutions [30, 31]. |
| • | Previously, the RH problem of the AB equation was solved using residue conditions instead of the Laurent expansion. |
| • | When the spectral parameters are pure imaginary, we obtain dark W-type soliton solutions, dark W-type double-pole solutions, and mixed dark W-type and double-pole solutions. The W-type solitons have been studied using modulation instability and Darboux transformation [33–35]. However, the latter two types of solitons are new and have not been reported in other literature. |
| • | The RH approach has not been used to study multiple high-order pole solutions for the AB system. |
This paper is organized in the following parts. First and foremost, in section 2 , we give the spectral problem and the corresponding RH problem is constructed with zero boundary conditions. In section 3 , the formula of the single high-order pole solutions is derived via Laurent expansion. Next, we generalize the soliton solutions for multiple high-order poles and analyze the dynamic behavior by the same method. Finally, section 4 will provide a succinct summary.
2. The Riemann–Hilbert problem
In this section, we carry out detailed spectral analysis from the Lax pair to prepare for constructing the RH problems.
2.1. Spectral analysis
The Lax pair of equation (1.1 ) reads1.2 ), we can obtain2.5 ) into (2.1 ), we arrive at the equivalent spectrum problem as follows
$\begin{eqnarray}{{\rm{\Omega }}}_{x}=U{\rm{\Omega }},\quad {{\rm{\Omega }}}_{t}=V{\rm{\Omega }},\end{eqnarray}$
where $\begin{eqnarray}U=-{\rm{i}}\lambda {\sigma }_{3}+Q,\end{eqnarray}$
$\begin{eqnarray}V=-\displaystyle \frac{1}{4{\rm{i}}\lambda }{\sigma }_{3}B+\displaystyle \frac{1}{4{\rm{i}}\lambda }\widetilde{Q},\end{eqnarray}$
λ is spectrum parameter and $\begin{eqnarray*}Q=\left(\begin{array}{cc}0 & \displaystyle \frac{1}{2}A\\ -\displaystyle \frac{1}{2}{A}^{* } & 0\end{array}\right),\quad \widetilde{Q}=\left(\begin{array}{cc}0 & {\rm{i}}{A}_{t}\\ {\rm{i}}{A}_{t}^{* } & 0\end{array}\right),\quad {\sigma }_{3}=\left(\begin{array}{cc}1 & 0\\ 0 & -1\end{array}\right).\end{eqnarray*}$
Considering the zero boundary value condition and the normalization condition ( $\begin{eqnarray}A(x,t)\to 0,\quad B(x,t)\to 1\quad {\rm{as}}\quad x\to \pm \infty .\end{eqnarray}$
Then the modified Jost solutions will be given $\begin{eqnarray}{\omega }_{\pm }(x,t,\lambda )={{\rm{\Omega }}}_{\pm }(x,t,\lambda ){{\rm{e}}}^{{\rm{i}}\lambda x+\displaystyle \frac{1}{4{\rm{i}}\lambda }t}.\end{eqnarray}$
Thus ω±(x, t, λ) satisfies the following asymptotic property $\begin{eqnarray}{\omega }_{\pm }(x,t,\lambda )\to {\mathbb{I}},\quad x\to \pm \infty ,\end{eqnarray}$
where ${\mathbb{I}}$ is a 2 × 2 identity matrix. Then substituting ( $\begin{eqnarray}{\omega }_{\pm ,x}=-{\rm{i}}\lambda [{\sigma }_{3},{\omega }_{\pm }]+Q{\omega }_{\pm },\end{eqnarray}$
$\begin{eqnarray}{\omega }_{\pm ,t}=-\displaystyle \frac{1}{4{\rm{i}}\lambda }\left(-{\omega }_{\pm }{\sigma }_{3}+B{\sigma }_{3}{\omega }_{\pm }\right)+\displaystyle \frac{1}{4{\rm{i}}\lambda }\widetilde{Q}{\omega }_{\pm },\end{eqnarray}$
where [σ3, ω±] = σ3ω± − ω±σ3.Conveniently, for the sake of supporting the study of spectral analysis, the eigenfunctions ω±(x, λ) are accurately described
$\begin{eqnarray}{\omega }_{-}(x,\lambda )={\mathbb{I}}+{\int }_{-\infty }^{x}{{\rm{e}}}^{{\rm{i}}\lambda {\sigma }_{3}(y-x)}Q(y){\omega }_{-}(y,\lambda ){{\rm{e}}}^{-{\rm{i}}\lambda {\sigma }_{3}(y-x)}{\rm{d}}{y},\end{eqnarray}$
$\begin{eqnarray}{\omega }_{+}(x,\lambda )={\mathbb{I}}+{\int }_{+\infty }^{x}{{\rm{e}}}^{{\rm{i}}\lambda {\sigma }_{3}(y-x)}Q(y){\omega }_{+}(y,\lambda ){{\rm{e}}}^{-{\rm{i}}\lambda {\sigma }_{3}(y-x)}{\rm{d}}{y}.\end{eqnarray}$
Because Ω± are the linear associated matrix solutions of equation (2.1 ), there is a spectral matrix S(λ) which we obtain from equation (2.5 )
where ${\omega }_{\pm ,j}(x,t,\lambda )(j=1,2)$ denote the jth column of ${\omega }_{\pm }(x,t,\lambda )$, ${\mathbb{R}}$ is the real axis, ${C}^{\pm }$ mean the upper and lower half complex plane.
$\begin{eqnarray}{\omega }_{-}(x,t,\lambda ){{\rm{e}}}^{-{\rm{i}}\lambda x-\displaystyle \frac{1}{4{\rm{i}}\lambda }t}={\omega }_{+}(x,t,\lambda ){{\rm{e}}}^{-{\rm{i}}\lambda x-\displaystyle \frac{1}{4{\rm{i}}\lambda }t}S(\lambda ),\end{eqnarray}$
where $\begin{eqnarray*}S(\lambda )=\left(\begin{array}{cc}{s}_{11}(\lambda ) & {s}_{12}(\lambda )\\ {s}_{21}(\lambda ) & {s}_{22}(\lambda )\end{array}\right),\end{eqnarray*}$
and sij(i, j = 1, 2) are typically referred to as scattering data.${\omega }_{\pm }(x,t,\lambda )$ possess the following analytic properties:
| • | ${\omega }_{+,2}$, ${\omega }_{-,1}$ and s11 are continuous for ${C}^{+}\cup {\mathbb{R}}$ and can be analytically extended to ${C}^{+}$, |
| • | ${\omega }_{-,2}$, ${\omega }_{+,1}$ and s22 are continuous for ${C}^{-}\cup {\mathbb{R}}$ and can be analytically extended to ${C}^{-}$, |
The asymptotic expansion can be written as2.7 ) and the corresponding coefficients of λ are compared, one can obtain
$\begin{eqnarray}{\omega }_{\pm }(x,t,\lambda )={\omega }_{\pm }^{(0)}+\displaystyle \frac{{\omega }_{\pm }^{(1)}}{\lambda }+\displaystyle \frac{{\omega }_{\pm }^{(2)}}{{\lambda }^{2}}+...,\quad \lambda \to \infty ,\end{eqnarray}$
where ${\omega }_{\pm }^{(i)}(i=0,1,2,\ldots \,)$ are independent of λ. When the expansion is substituted into the equation (x-part:
$\begin{eqnarray*}\begin{array}{l}\,O(\lambda ):\qquad 0=-{\rm{i}}[{\sigma }_{3},{\omega }_{\pm }^{(0)}],\\ \,O({\lambda }^{0}):\qquad {\omega }_{x}^{(0)}=-{\rm{i}}[{\sigma }_{3},{\omega }_{\pm }^{(1)}]+Q{\omega }_{\pm }^{(0)},\\ O({\lambda }^{-1}):\qquad {\omega }_{\pm ,x}^{(1)}=-{\rm{i}}[{\sigma }_{3},{\omega }_{\pm }^{(2)}]+Q{\omega }_{\pm }^{(1)}.\end{array}\end{eqnarray*}$
t-part:
$\begin{eqnarray*}\begin{array}{l}O(\lambda ):\qquad {\omega }_{\pm ,t}^{(0)}=0,\\ \Space{0ex}{0.04em}{0ex}O({\lambda }^{-1}):\qquad {\omega }_{\pm ,t}^{(1)}=-\displaystyle \frac{1}{4{\rm{i}}}(-{\omega }_{\pm }^{(0)}{\sigma }_{3}+B{\sigma }_{3}{\omega }_{\pm }^{(0)})\\ \Space{0ex}{0.04em}{0ex}+\displaystyle \frac{1}{4{\rm{i}}}\widetilde{Q}{\omega }_{\pm }^{(0)}.\end{array}\end{eqnarray*}$
From the above, we can deduce $\begin{eqnarray}{\omega }_{\pm }^{(0)}={\mathbb{I}},\quad A=4{\rm{i}}{\omega }_{\pm ,12}^{(1)},\quad B=-4{\rm{i}}{\left({\omega }_{\pm ,11}^{(1)}\right)}_{t}+1,\end{eqnarray}$
where the subscript ij represents the element in the j column and the i row of the matrix ω.2.2. Symmetries
For the purpose of making the following symmetry analysis easier, we define the reflection coefficients ϱ(λ) and $\tilde{\varrho }(\lambda )$:2.9 )
$\begin{eqnarray}\varrho (\lambda )=\displaystyle \frac{{s}_{21}}{{s}_{11}},\quad \tilde{\varrho }(\lambda )=\displaystyle \frac{{s}_{12}}{{s}_{22}},\quad \lambda \in {\mathbb{R}}.\end{eqnarray}$
Owing to the symmetry relation Q† = − Q, one can obtain $\begin{eqnarray}{\omega }_{\pm }^{\dagger }(x,t,{\lambda }^{* })={\omega }_{\pm }^{-1}(x,t,\lambda ),\end{eqnarray}$
where † means the Hermitian conjugation. Then the symmetry of matrix S can be inferred as follows by using equation ( $\begin{eqnarray}{S}^{\dagger }({\lambda }^{* })={S}^{-1}(\lambda ),\end{eqnarray}$
or $\begin{eqnarray}{s}_{22}^{* }({\lambda }^{* })={s}_{11}(\lambda ),\quad {s}_{21}^{* }({\lambda }^{* })=-{s}_{12}(\lambda ).\end{eqnarray}$
According to the above relationship and the definition of ϱ(λ) and $\tilde{\varrho }(\lambda )$, one can achieve $\begin{eqnarray}{\varrho }^{* }({\lambda }^{* })=-\tilde{\varrho }(\lambda ).\end{eqnarray}$
2.3. Riemann–Hilbert problem
We can propose a piecewise meromorphic function based on the analytic features of ω±(x, t, λ). Defining P(x, t, λ) as
$\begin{eqnarray}P(x,t,\lambda )=\left\{\begin{array}{ll}{P}^{+}=\left(\begin{array}{cc}\displaystyle \frac{{\omega }_{-,1}}{{s}_{11}} & {\omega }_{+,2}\end{array}\right), & \lambda \in {C}^{+},\\ {P}^{-}=\left(\begin{array}{cc}{\omega }_{+,1} & \displaystyle \frac{{\omega }_{-,2}}{{s}_{22}}\end{array}\right), & \lambda \in {C}^{-}.\end{array}\right.\end{eqnarray}$
Then the properties of P(x, t, λ) are listed.A multiplicative matrix RH problem is proposed:
| • | Analyticity : $P(x,t,\lambda )$ is analytic in ${\mathbb{C}}/{\mathbb{R}}$, |
| • | Jump condition: ${P}^{-}(x,t,\lambda )={P}^{+}(x,t,\lambda )J(x,t,\lambda )$, $\lambda \in {\mathbb{R}}$, |
| • | Asymptotic behaviors : $P(x,t,\lambda )\sim {\mathbb{I}}+O(\tfrac{1}{\lambda }),\quad \lambda \to \infty $, |
$\begin{eqnarray}J(x,t,\lambda )=\left(\begin{array}{cc}1 & {{\rm{e}}}^{2{\rm{\Delta }}}\tilde{\varrho }(\lambda )\\ -{{\rm{e}}}^{-2{\rm{\Delta }}}\varrho (\lambda ) & 1+\tilde{\varrho }(\lambda )\varrho (\lambda )\end{array}\right)\end{eqnarray}$
and ${\rm{\Delta }}=-{\rm{i}}\lambda x-\tfrac{1}{4{\rm{i}}\lambda }t$.Moreover, it is possible to write the solution as
$\begin{eqnarray}A(x,t)=4{\rm{i}}{P}_{12}^{(1)},\quad B(x,t)=-4{\rm{i}}{\left({P}_{11}^{(1)}\right)}_{t}+1.\end{eqnarray}$
3. RH problem with high-order poles
In this section we consider the reflectionless condition and start from a single high-order pole. Then the more general situation will be investigated.
3.1. RH problem with single high-order pole
Suppose λ0 ∈ C+ is a high-order pole, according to the symmetric relations (2.14 ), obviously, ${\lambda }_{0}^{* }$ is the zero of s22, and the discrete spectrum reads:
$\begin{eqnarray}\{{\lambda }_{0},{\lambda }_{0}^{* }\}.\end{eqnarray}$
Let3.5 ), it yields3.6 ), we have
$\begin{eqnarray}{s}_{11}(\lambda )={\left(\lambda -{\lambda }_{0}\right)}^{H}{s}_{0}(\lambda ),\end{eqnarray}$
in which s0(λ) ≠ 0 in D+. Then ϱ(λ) and ϱ*(λ*) are expanded as $\begin{eqnarray}\begin{array}{rcl}\varrho (\lambda ) & = & {\varrho }_{0}({\lambda }_{0})+\displaystyle \sum _{m=1}^{H}\displaystyle \frac{{\varrho }_{m}}{{\left(\lambda -{\lambda }_{0}\right)}^{m}}\qquad \mathrm{and}\\ {\varrho }^{* }({\lambda }^{* }) & = & {\varrho }_{0}^{* }({\lambda }_{0}^{* })+\displaystyle \sum _{m=1}^{H}\displaystyle \frac{{\varrho }_{m}^{* }}{{\left(\lambda -{\lambda }_{0}^{* }\right)}^{m}},\end{array}\end{eqnarray}$
where ϱm are defined by $\begin{eqnarray}\begin{array}{rcl}{\varrho }_{m} & = & \mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{0}}\displaystyle \frac{1}{(H-m)!}\displaystyle \frac{{\partial }^{H-m}}{\partial {\lambda }^{H-m}}[{\left(\lambda -{\lambda }_{0}\right)}^{H}\varrho (\lambda )],\\ m & = & 1,2,3,\ldots ,H.\end{array}\end{eqnarray}$
Recalling the definition of matrix P, one can obtain that λ = λ0 is the Hth-order pole of P11(x, t, λ) while $\lambda ={\lambda }_{0}^{* }$ is the Hth-order pole of P12(x, t, λ). Then P11(x, t, λ) and P12(x, t, λ) are expanded via the Laurent series expansion: $\begin{eqnarray}{P}_{11}=1+\displaystyle \sum _{r=1}^{H}\left\{\displaystyle \frac{{L}_{r}(x,t)}{{\left(\lambda -{\lambda }_{0}\right)}^{r}}\right\},\end{eqnarray}$
$\begin{eqnarray}{P}_{12}=\displaystyle \sum _{r=1}^{H}\left\{\displaystyle \frac{{R}_{r}(x,t)}{{\left(\lambda -{\lambda }_{0}^{* }\right)}^{r}}\right\},\end{eqnarray}$
Lr(x, t) and Rr(x, t) are the undetermined function. Once they are solved, we can derive P(x, t, λ). Now we can work out Lr(x, t) and Rr(x, t). To this end, we expand these series $\begin{eqnarray}\begin{array}{rcl}{{\rm{e}}}^{-2{\rm{\Delta }}(\lambda )} & = & \displaystyle \sum _{d=0}^{+\infty }{g}_{d}(x,t){\left(\lambda -{\lambda }_{0}\right)}^{d},\\ {{\rm{e}}}^{2{\rm{\Delta }}(\lambda )} & = & \displaystyle \sum _{d=0}^{+\infty }{g}_{d}^{* }(x,t){\left(\lambda -{\lambda }_{0}^{* }\right)}^{d},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{P}_{11}(x,t,\lambda ) & = & \displaystyle \sum _{d=0}^{H}{\mu }_{d}(x,t){\left(\lambda -{\lambda }_{0}^{* }\right)}^{d},\\ {P}_{12}(x,t,\lambda ) & = & \displaystyle \sum _{d=0}^{H}{\nu }_{d}(x,t){\left(\lambda -{\lambda }_{0}\right)}^{d},\end{array}\end{eqnarray}$
where $\begin{eqnarray}{g}_{d}(x,t)=\mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{0}}\displaystyle \frac{1}{d!}\displaystyle \frac{{\partial }^{d}}{\partial {\lambda }^{d}}{{\rm{e}}}^{-2{\rm{\Delta }}},\end{eqnarray}$
$\begin{eqnarray}{\mu }_{d}(x,t)=\mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{0}^{* }}\displaystyle \frac{1}{d!}\displaystyle \frac{{\partial }^{d}}{\partial {\lambda }^{d}}{P}_{11}(x,t,\lambda ),\end{eqnarray}$
$\begin{eqnarray}{\nu }_{d}(x,t)=\mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{0}}\displaystyle \frac{1}{d!}\displaystyle \frac{{\partial }^{d}}{\partial {\lambda }^{d}}{P}_{12}(x,t,\lambda ).\end{eqnarray}$
The expansions in λ = λ0 can be obtained when λ ∈ C+ is as follows $\begin{eqnarray}\begin{array}{rcl}{P}_{11}(x,t,\lambda ) & = & \displaystyle \frac{{\omega }_{-,11}}{{s}_{11}}={\omega }_{+,11}+\varrho (\lambda ){{\rm{e}}}^{-2{\rm{\Delta }}}{\omega }_{+,12},\\ {P}_{12}(x,t,\lambda ) & = & {\omega }_{+,12}.\end{array}\end{eqnarray}$
Comparing the coefficients of ${\left(\lambda -{\lambda }_{0}\right)}^{-r}$ with equation ( $\begin{eqnarray}{L}_{r}(x,t)=\displaystyle \sum _{z=r}^{H}\displaystyle \sum _{d=0}^{z-r}{\varrho }_{z}{g}_{z-r-d}(x,t){\nu }_{d}(x,t).\end{eqnarray}$
Taking $\lambda ={\lambda }_{0}^{* }$ and λ ∈ C−, one can obtain $\begin{eqnarray}\begin{array}{rcl}{P}_{11}(x,t,\lambda ) & = & {\omega }_{+,11},\\ {P}_{12}(x,t,\lambda ) & = & \displaystyle \frac{{\omega }_{-,12}}{{s}_{22}}={\omega }_{+,12}+\tilde{\varrho }(\lambda ){{\rm{e}}}^{2{\rm{\Delta }}}{\omega }_{+,11}.\end{array}\end{eqnarray}$
Similarly, comparing ${\left(\lambda -{\lambda }_{0}^{* }\right)}^{-r}$'s coefficients with equation ( $\begin{eqnarray}{R}_{r}(x,t)=-\displaystyle \sum _{z=r}^{H}\displaystyle \sum _{d=0}^{z-r}{\varrho }_{z}^{* }{g}_{z-r-d}^{* }(x,t){\mu }_{d}(x,t).\end{eqnarray}$
In addition, μd(x, t) and νd(x, t) are expressed via direct calculation
$\begin{eqnarray}\begin{array}{l}{\mu }_{d}(x,t)\\ \quad =\left\{\begin{array}{ll}1+\displaystyle \sum _{r=1}^{H}\{\displaystyle \frac{{L}_{r}(x,t)}{{\left({\lambda }_{0}^{* }-{\lambda }_{0}\right)}^{r}}\}, & d=0,\\ \displaystyle \sum _{r=1}^{H}\left(\begin{array}{c}r+d-1\\ d\end{array}\right)\{\displaystyle \frac{{\left(-1\right)}^{d}{L}_{r}(x,t)}{{\left({\lambda }_{0}^{* }-{\lambda }_{0}\right)}^{r+d}}\}, & d=1,2,3,\ldots ,\end{array}\right.\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\nu }_{d}(x,t) & = & \displaystyle \sum _{r=1}^{H}\left(\begin{array}{c}r+d-1\\ d\end{array}\right)\left\{\displaystyle \frac{{\left(-1\right)}^{d}{R}_{r}(x,t)}{{\left({\lambda }_{0}-{\lambda }_{0}^{* }\right)}^{r+d}}\right\},\\ d & = & 0,1,2,\ldots ,\end{array}\end{eqnarray}$
then $\begin{eqnarray}\begin{array}{rcl}{L}_{r}(x,t) & = & \displaystyle \sum _{z=r}^{H}\displaystyle \sum _{d=0}^{z-r}\displaystyle \sum _{p=1}^{H}\left(\begin{array}{c}p+d-1\\ d\end{array}\right){\varrho }_{z}{g}_{z-r-d}(x,t)\\ & & \Space{0ex}{4.04em}{0ex}\times \left\{\displaystyle \frac{{\left(-1\right)}^{d}{R}_{p}(x,t)}{{\left({\lambda }_{0}-{\lambda }_{0}^{* }\right)}^{d+p}}\right\},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{R}_{r}(x,t) & = & -\displaystyle \sum _{z=r}^{H}{\varrho }_{z}^{* }{g}_{z-r-d}^{* }(x,t)\\ & & -\displaystyle \sum _{z=r}^{H}\displaystyle \sum _{d=0}^{z-r}\displaystyle \sum _{p=1}^{H}\left(\begin{array}{c}p+d-1\\ d\end{array}\right){\varrho }_{z}^{* }{g}_{z-r-d}^{* }(x,t)\\ & & \times \left\{\displaystyle \frac{{\left(-1\right)}^{d}{L}_{p}(x,t)}{{\left({\lambda }_{0}^{* }-{\lambda }_{0}\right)}^{d+p}}\right\}.\end{array}\end{eqnarray}$
To facilitate representation, we introduce notations $\begin{eqnarray}| \vartheta \rangle ={\left({\vartheta }_{1},{\vartheta }_{2},\ldots ,{\vartheta }_{H}\right)}^{{\rm{T}}},\quad {\vartheta }_{r}=-\displaystyle \sum _{z=r}^{H}{\varrho }_{z}^{* }{g}_{z-r-d}^{* }(x,t),\end{eqnarray}$
$\begin{eqnarray}| L\rangle ={\left({L}_{1},{L}_{2},\ldots ,{L}_{H}\right)}^{{\rm{T}}},\quad | R\rangle ={\left({R}_{1},{R}_{2},\ldots ,{R}_{H}\right)}^{{\rm{T}}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\rm{\Upsilon }} & = & {\left[{{\rm{\Upsilon }}}_{{rp}}\right]}_{H\times H}=\left[\displaystyle \sum _{z=r}^{H}\displaystyle \sum _{d=0}^{z-r}\left(\begin{array}{c}p+d-1\\ d\end{array}\right)\right.\\ & & \left.\times \displaystyle \frac{{\left(-1\right)}^{d}{\varrho }_{z}{g}_{z-r-d}(x,t)}{{\left({\lambda }_{0}-{\lambda }_{0}^{* }\right)}^{d+p}}\right].\end{array}\end{eqnarray}$
Therefore, the equations (3.18 )–(3.22 ) can be written as follows
$\begin{eqnarray}\left\{\begin{array}{l}{\rm{\Lambda }}| L\rangle -{\rm{\Upsilon }}| R\rangle =0,\\ {{\rm{\Upsilon }}}^{* }| L\rangle +{\rm{\Lambda }}| R\rangle =| \vartheta \rangle .\end{array}\right.\end{eqnarray}$
By calculation, ∣L⟩ and $\left|R\right\rangle $ are explicitly expressed as $\begin{eqnarray}| L\rangle ={\rm{\Upsilon }}{\left({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }}\right)}^{-1}| \vartheta \rangle ,\end{eqnarray}$
$\begin{eqnarray}| R\rangle ={\left({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }}\right)}^{-1}| \vartheta \rangle ,\end{eqnarray}$
where $\begin{eqnarray}{\rm{\Lambda }}={I}_{H\times H}.\end{eqnarray}$
So the expansions of P11 and P12 are given as $\begin{eqnarray}{P}_{11}(x,t,\lambda )=\displaystyle \frac{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }}+| \vartheta \rangle \langle K| {\rm{\Upsilon }})}{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }})},\end{eqnarray}$
$\begin{eqnarray}{P}_{12}(x,t,\lambda )=\displaystyle \frac{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }}+| \vartheta \rangle \langle {K}_{1}| }{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }})}-1,\end{eqnarray}$
where $\begin{eqnarray}\langle K(\lambda )| =(\displaystyle \frac{1}{(\lambda -{\lambda }_{0})},\displaystyle \frac{1}{{\left(\lambda -{\lambda }_{0}\right)}^{2}},\ldots ,\displaystyle \frac{1}{{\left(\lambda -{\lambda }_{0}\right)}^{H}}),\end{eqnarray}$
$\begin{eqnarray}\langle {K}_{1}(\lambda )| =(\displaystyle \frac{1}{(\lambda -{\lambda }_{0}^{* })},\displaystyle \frac{1}{{\left(\lambda -{\lambda }_{0}^{* }\right)}^{2}},\ldots ,\displaystyle \frac{1}{{\left(\lambda -{\lambda }_{0}^{* }\right)}^{H}}).\end{eqnarray}$
With the rapidly decaying initial condition (
$\begin{eqnarray}\begin{array}{rcl}A(x,t) & = & 4{\rm{i}}\displaystyle \frac{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }}+| \vartheta \rangle \langle \widetilde{K}| )}{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }})},\\ B(x,t) & = & 1-4{\rm{i}}{\left(\displaystyle \frac{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }}+| \vartheta \rangle \langle \widetilde{K}| {\rm{\Upsilon }})}{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }})}\right)}_{t},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\langle \tilde{K}| \,=\,{\left(\mathrm{1,0,0},\ldots ,0\right)}_{1\times H}.\end{eqnarray}$
Case 1: Let H = 1, and one soliton can be obtained and shown in figure 1. B is a W-type dark soliton as λ0 is purely imaginary. If λ is not purely imaginary, B will be a dark soliton. The elements of equation (3.30 ) can be derived from
$\begin{eqnarray}{{\rm{\Upsilon }}}_{11}=\displaystyle \frac{{\varrho }_{1}{g}_{0}}{{\lambda }_{0}-{\lambda }_{0}^{* }},\quad {\vartheta }_{1}=-{\varrho }_{1}^{* }{g}_{0}^{* }.\end{eqnarray}$
Figure 1. The single solution with parameters (1) ϱ1 = 1, λ0 = i. (2) ϱ1 = 1, λ0 = 2 + i. |
Case 2: Considering a second-order pole soliton, we take H = 2 and get figure 2. As same as the first example, B has dark W-type double-pole soliton solutions, depending on whether λ0 is purely imaginary. The elements of equation (3.30 ) can be derived
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Upsilon }}}_{11} & = & \displaystyle \frac{{\varrho }_{1}{g}_{0}}{{\lambda }_{0}-{\lambda }_{0}^{* }}+\displaystyle \frac{{\varrho }_{2}{g}_{1}}{{\lambda }_{0}-{\lambda }_{0}^{* }}-\displaystyle \frac{{\varrho }_{2}{g}_{0}}{{\left({\lambda }_{0}-{\lambda }_{0}^{* }\right)}^{2}},\\ {{\rm{\Upsilon }}}_{12} & = & \displaystyle \frac{{\varrho }_{1}{g}_{0}}{{\left({\lambda }_{0}-{\lambda }_{0}^{* }\right)}^{2}}+\displaystyle \frac{{\varrho }_{2}{g}_{1}}{{\left({\lambda }_{0}-{\lambda }_{0}^{* }\right)}^{2}}-\displaystyle \frac{2{\varrho }_{2}{g}_{0}}{{\left({\lambda }_{0}-{\lambda }_{0}^{* }\right)}^{3}},\\ {{\rm{\Upsilon }}}_{21} & = & \displaystyle \frac{{\varrho }_{2}{g}_{0}}{{\lambda }_{0}-{\lambda }_{0}^{* }},\quad {{\rm{\Upsilon }}}_{22}=\displaystyle \frac{{\varrho }_{2}{g}_{0}}{{\left({\lambda }_{0}-{\lambda }_{0}^{* }\right)}^{2}},\\ \vartheta & = & -{\varrho }_{1}^{* }{g}_{0}^{* }-{\varrho }_{2}^{* }{g}_{1}^{* },\quad \vartheta =-{\varrho }_{2}^{* }{g}_{0}^{* }.\end{array}\end{eqnarray}$
Figure 2. The second-order pole solution with parameters ϱ1 = 1, ϱ2 = 2. (1) λ0 = i. (2) λ0 = 1 + i. |
3.2. RH problem with multiple high-order poles
Assume λ1, λ2, …, λH (λz ∈ D+, z = 1, 2,…,H) are H high-order zeros of s11, and the powers are h1, h2, …, hH, respectively. Based on the Laurent series, ϱz(λ) and ${\varrho }_{z}^{* }({\lambda }^{* })$ are expanded as
$\begin{eqnarray}\begin{array}{rcl}{\varrho }_{z}(\lambda ) & = & {\varrho }_{z,0}(\lambda )+\displaystyle \sum _{{m}_{z}=1}^{{h}_{z}}\displaystyle \frac{{\varrho }_{z,{m}_{z}}}{{\left(\lambda -{\lambda }_{z}\right)}^{{m}_{z}}},\\ {\varrho }_{z}^{* }({\lambda }^{* }) & = & {\varrho }_{z,0}^{* }({\lambda }^{* })+\displaystyle \sum _{{m}_{z}=1}^{{h}_{z}}\displaystyle \frac{{\varrho }_{z,{m}_{z}}}{{\left(\lambda -{\lambda }_{z}^{* }\right)}^{{m}_{z}}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\varrho }_{z,{m}_{z}}=\mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{z}}\displaystyle \frac{1}{({h}_{z}-{m}_{z})!}\displaystyle \frac{{\partial }^{{h}_{z}-{m}_{z}}}{\partial {\lambda }^{{h}_{z}-{m}_{z}}}[{\left(\lambda -{\lambda }_{z}\right)}^{{h}_{z}}\varrho (\lambda )],\end{eqnarray}$
and ϱz,0(λ)(z = 1,…,H) are analytic for all λ ∈ D+ .Then we obtain the multiple high-order solitons as follows.
With the rapidly decaying initial condition (
$\begin{eqnarray}\begin{array}{rcl}A(x,t) & = & 4{\rm{i}}\displaystyle \frac{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }}+| \vartheta \rangle \langle \widetilde{K}| )}{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }})},\\ B(x,t) & = & 1-4{\rm{i}}{\left(\displaystyle \frac{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }}+| \vartheta \rangle \langle \widetilde{K}| {\rm{\Upsilon }})}{\det ({\rm{\Lambda }}+{{\rm{\Upsilon }}}^{* }{\rm{\Upsilon }})}\right)}_{t},\end{array}\end{eqnarray}$
where $\begin{eqnarray}| \vartheta \rangle ={\left[{\vartheta }_{1},{\vartheta }_{2},\ldots ,{\vartheta }_{H}\right]}^{{\rm{T}}},\quad | {\vartheta }_{z}\rangle =[{\vartheta }_{z,1},{\vartheta }_{z,2},\ldots ,{\vartheta }_{z,{h}_{z}}],\end{eqnarray}$
$\begin{eqnarray}{\vartheta }_{z,d}=-\displaystyle \sum _{{m}_{z}=d}^{{h}_{z}}{\varrho }_{z,{m}_{z}}^{* }{g}_{z,{m}_{z}-d}^{* }(x,t),\end{eqnarray}$
$\begin{eqnarray}\langle \tilde{K}| \,=\,[\langle {\tilde{K}}^{1}| ,\langle {\tilde{K}}^{2}| ,\ldots ,\langle {\tilde{K}}^{H}| ],\langle {\tilde{K}}^{z}| \,=\,{\left(\mathrm{1,0},\ldots ,0\right)}_{1\times {h}_{z}},\end{eqnarray}$
$\begin{eqnarray}{\rm{\Upsilon }}=\left(\begin{array}{cccc}\left[{{\rm{\Upsilon }}}_{11}\right] & [{{\rm{\Upsilon }}}_{12}] & ... & [{{\rm{\Upsilon }}}_{1H}]\\ \left[{{\rm{\Upsilon }}}_{21}\right] & [{{\rm{\Upsilon }}}_{22}] & ... & [{{\rm{\Upsilon }}}_{2H}]\\ \vdots & \vdots & \vdots & \vdots \\ \left[{{\rm{\Upsilon }}}_{H1}\right] & [{{\rm{\Upsilon }}}_{H2}] & ... & [{{\rm{\Upsilon }}}_{{HH}}]\end{array}\right),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\left[{{\rm{\Upsilon }}}_{{zd}}\right] & = & {\left({\left[{{\rm{\Upsilon }}}_{{zd}}\right]}_{p,q}\right)}_{{h}_{z}\times {h}_{d}}=\displaystyle \sum _{{m}_{z}=p}^{{h}_{z}}\displaystyle \sum _{{r}_{z}=0}^{{m}_{z}-p}\left(\begin{array}{c}p+{r}_{z}-1\\ {r}_{z}\end{array}\right)\\ & & \times \displaystyle \frac{{\left(-1\right)}^{{r}_{z}}{\varrho }_{z,{m}_{z}}{g}_{z,{m}_{z}-p-{r}_{z}}(x,t)}{{\left({\lambda }_{z}-{\lambda }_{d}^{* }\right)}^{{r}_{z}+q}},\end{array}\end{eqnarray}$
$\begin{eqnarray}{\rm{\Lambda }}=\left(\begin{array}{cccc}{I}_{{h}_{1}\times {h}_{1}} & & & \\ & {I}_{{h}_{2}\times {h}_{2}} & & \\ & & \ddots & \\ & & & {I}_{{h}_{H}\times {h}_{H}}\end{array}\right).\end{eqnarray}$
Case 3: Now, we take H = 2, h1 = h2 = 1, then the two-soliton solutions can be obtained in figure 3. Unlike chaos theory and inelastic collisions [36, 37], the observation clearly shows that the collision between two solitons is elastic, meaning that neither the amplitude nor the velocity direction is affected. In addition, if the spectral parameters are purely imaginary, W-type solitons will be generated. In this case, the elements can be given as
$\begin{eqnarray}\begin{array}{rcl}{\left[{{\rm{\Upsilon }}}_{11}\right]}_{11} & = & \displaystyle \frac{{\varrho }_{\mathrm{1,1}}{g}_{\mathrm{1,0}}}{{\lambda }_{1}-{\lambda }_{1}^{* }},\quad {\left[{{\rm{\Upsilon }}}_{12}\right]}_{11}=\displaystyle \frac{{\varrho }_{\mathrm{1,1}}{g}_{\mathrm{1,0}}}{{\lambda }_{1}-{\lambda }_{2}^{* }},\\ {\left[{{\rm{\Upsilon }}}_{21}\right]}_{11} & = & \displaystyle \frac{{\varrho }_{\mathrm{2,1}}{g}_{\mathrm{2,0}}}{{\lambda }_{2}-{\lambda }_{1}^{* }},\quad {\left[{{\rm{\Upsilon }}}_{22}\right]}_{11}=\displaystyle \frac{{\varrho }_{\mathrm{2,1}}{g}_{\mathrm{2,0}}}{{\lambda }_{2}-{\lambda }_{2}^{* }},\\ {\vartheta }_{\mathrm{1,1}} & = & -{\varrho }_{1,1}^{* }{g}_{1,0}^{* },\quad {\vartheta }_{\mathrm{2,1}}=-{\varrho }_{2,1}^{* }{g}_{2,0}^{* }.\end{array}\end{eqnarray}$
Figure 3. The two-soliton solution with parameters ϱ1,1 = ϱ2,1 = 1. (1) λ1 = i, λ2 = 2i. (2) λ1 = 2 + i, λ2 = 0.5i. |
Case 4: For one second-order pole and one simple pole, i.e., H = 2, h1 = 2, h2 = 1, the mixed solution of a soliton and a double-pole can be obtained and shown in figure 4. The relevant elements of solutions can be expressed as
$\begin{eqnarray}\begin{array}{rcl}{\left[{{\rm{\Upsilon }}}_{11}\right]}_{11} & = & \displaystyle \frac{{\varrho }_{\mathrm{1,1}}{g}_{\mathrm{1,0}}}{{\lambda }_{1}-{\lambda }_{1}^{* }}+\displaystyle \frac{{\varrho }_{\mathrm{1,2}}{g}_{\mathrm{1,1}}}{{\lambda }_{1}-{\lambda }_{1}^{* }}-\displaystyle \frac{{\varrho }_{\mathrm{1,2}}{g}_{\mathrm{1,0}}}{{\left({\lambda }_{1}-{\lambda }_{1}^{* }\right)}^{2}},\\ {\left[{{\rm{\Upsilon }}}_{11}\right]}_{12} & = & \displaystyle \frac{{\varrho }_{\mathrm{1,1}}{g}_{\mathrm{1,0}}}{{\left({\lambda }_{1}-{\lambda }_{1}^{* }\right)}^{2}}+\displaystyle \frac{{\varrho }_{\mathrm{1,2}}{g}_{\mathrm{1,1}}}{{\left({\lambda }_{1}-{\lambda }_{1}^{* }\right)}^{2}}-\displaystyle \frac{2{\varrho }_{\mathrm{1,2}}{g}_{\mathrm{1,0}}}{{\left({\lambda }_{1}-{\lambda }_{1}^{* }\right)}^{3}},\\ {\left[{{\rm{\Upsilon }}}_{11}\right]}_{21} & = & \displaystyle \frac{{\varrho }_{\mathrm{1,2}}{g}_{\mathrm{1,0}}}{{\lambda }_{1}-{\lambda }_{1}^{* }},\quad {\left[{{\rm{\Upsilon }}}_{11}\right]}_{22}=\displaystyle \frac{{\varrho }_{\mathrm{1,2}}{g}_{\mathrm{1,0}}}{{\left({\lambda }_{1}-{\lambda }_{1}^{* }\right)}^{2}},\\ {\left[{{\rm{\Upsilon }}}_{12}\right]}_{11} & = & \displaystyle \frac{{\varrho }_{\mathrm{1,1}}{g}_{\mathrm{1,0}}}{{\lambda }_{1}-{\lambda }_{2}^{* }}+\displaystyle \frac{{\varrho }_{\mathrm{1,2}}{g}_{\mathrm{1,1}}}{{\lambda }_{1}-{\lambda }_{2}^{* }}-\displaystyle \frac{{\varrho }_{\mathrm{1,2}}{g}_{\mathrm{1,0}}}{{\left({\lambda }_{1}-{\lambda }_{2}^{* }\right)}^{2}},\\ {\left[{{\rm{\Upsilon }}}_{12}\right]}_{21} & = & \displaystyle \frac{{\varrho }_{\mathrm{1,2}}{g}_{\mathrm{1,0}}}{{\lambda }_{1}-{\lambda }_{2}^{* }},\quad {\left[{{\rm{\Upsilon }}}_{21}\right]}_{11}=\displaystyle \frac{{\varrho }_{\mathrm{2,1}}{g}_{\mathrm{2,0}}}{{\lambda }_{2}-{\lambda }_{1}^{* }},\\ {\left[{{\rm{\Upsilon }}}_{21}\right]}_{12} & = & \displaystyle \frac{{\varrho }_{\mathrm{2,1}}{g}_{\mathrm{2,0}}}{{\left({\lambda }_{2}-{\lambda }_{1}^{* }\right)}^{2}},\quad {\left[{{\rm{\Upsilon }}}_{22}\right]}_{11}=\displaystyle \frac{{\varrho }_{\mathrm{2,1}}{g}_{\mathrm{2,0}}}{{\lambda }_{2}-{\lambda }_{2}^{* }},\\ {\vartheta }_{\mathrm{1,1}} & = & -{\varrho }_{1,1}^{* }{g}_{1,0}^{* }-{\varrho }_{1,2}^{* }{g}_{1,1}^{* },\\ {\vartheta }_{\mathrm{1,2}} & = & -{\varrho }_{1,2}^{* }{g}_{1,0}^{* },\quad {\vartheta }_{\mathrm{2,1}}=-{\varrho }_{2,1}^{* }{g}_{2,0}^{* }.\end{array}\end{eqnarray}$
Figure 4. A simple and a second-order pole solution with parameters ϱ1,1 = ϱ1,2 = ϱ2,1 = 1. (1) λ1 = 0.5i, λ2 = 2i. (2) λ1 = 0.5i, λ2 = 0.2 + 0.2i. (3) λ1 = 0.5 + 0.5i, λ2 = 0.2 + 0.2i. |
4. Conclusion
In this paper, multiple high-order poles of the AB system have been investigated using the RH approach. For this purpose, firstly, we construct the RH problem on the ground of the spectral analysis. Next, the expression of the soliton solution can be determined under the condition of no reflection. It is noteworthy that it is inconvenient to solve the soliton solutions by the residue condition. Therefore, we adopt the Laurent expansion and the determinant expression to construct the exact expression of the multiple high-order pole solutions. Finally, a few distinctive characteristics of these solutions are visualized and described in a clear and concise manner. Three types of soliton solutions are obtained, including single soliton solutions, double-pole solutions, and mixed solutions. All solutions of A are bright soliton solutions, while all solutions of B are dark soliton solutions. Furthermore, it is an interesting phenomenon that if the spectral parameters are purely imaginary, dark solitons of the W-type will appear. And the collisions between these solitons are all elastic. In the future, we will plan to extend this study to multi-components or the matrix for the AB system.