1. Introduction
Since nonlinear waves can describe many physical phenomena, nonlinear evolution equations and their exact solutions play an extremely important role in physics, biology, atmosphere and oceanography [1–4]. So far, scholars have developed many effective methods for solving nonlinear partial differential equations, such as inverse scattering transformation method [5], Riemann–Hilbert (RH) method [6, 7], Hirota bilinear method [8, 9], Darboux transformation method [10, 11] and $\bar{\partial }$-dressing method [12–14].
The RH method uses the spectral parameters in the Lax pair and adds many analytical tools in the complex analysis to solve the problems that need to be solved. It transforms the problem of solving the initial value of the partial differential equation into the RH problem of finding an analytic function with a specific jump form on a given curve. Next, the reconstruction formula is established to connect the solution of the equation with the solution of the RH problem, so as to solve the initial value problem of the differential equation by solving the RH problem. In addition, the RH method can also solve the high-order spectral problem, so it is very meaningful to study the RH method to solve the initial value problem of the integrable system. In general, when using the RH method to construct the RH problem, the initial value is first required to approach zero at infinity, but then Biondini and Kovai improved and proposed the RH problem under nonzero boundary conditions [15]. Scholars have also extended this method to other integrable equations, such as the nonlocal Hirota equation [16], derivative nonlinear Schrödinger equation [17], the generalized mixed nonlinear Schrödinger equation [18], the Gerdjikov–Ivanov equation [19] and the modified Landau–Lifshitz equation [20].
In this paper, we plan to consider the Lakshmanan–Porsezian–Daniel (LPD) equation with nonzero boundary conditions (also called fourth-order nonlinear Schrödinger equation) [21]1 ) can be given by the AKNS method as follows
$\begin{eqnarray}\begin{array}{c}{\rm{i}}{q}_{t}+{q}_{{xx}}+2{\left|q\right|}^{2}q+\beta {q}_{{xxxx}}+6{q}^{* }{q}_{x}^{2}\\ \qquad +\,4{\left|{q}_{x}\right|}^{2}q+8{\left|q\right|}^{2}{q}_{{xx}}+2{q}^{2}{q}_{{xx}}^{* }+6{\left|q\right|}^{4}q=0,\\ \qquad \qquad \qquad q\left(x,t\right)\to {q}_{\pm }{{\rm{e}}}^{{\rm{i}}\left(6\beta {q}_{0}^{4}+2{q}_{0}^{2}\right)t},x\to \pm \infty ,\end{array}\end{eqnarray}$
where q± is unrelated to t and $\left|{q}_{\pm }\right|={q}_{0}\ne 0$. The Lax pair of equation ( $\begin{eqnarray}{\phi }_{x}=X\phi ,\quad \ {\phi }_{t}=T\phi ,\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{ccl}X & = & -{\rm{i}}k{\sigma }_{3}+Q,\quad Q=\left(\begin{array}{cc}0 & q\\ -{q}^{* } & 0\end{array}\right),\\ {\sigma }_{3} & = & \left(\begin{array}{cc}1 & 0\\ 0 & -1\end{array}\right),\\ T & = & \left[3{\rm{i}}\beta {Q}^{4}-{\rm{i}}{Q}^{2}+{\rm{i}}\beta \left({Q}_{x}^{2}-{Q}_{{xx}}Q-{{QQ}}_{{xx}}\right)\right.\\ & & \left.-2k\beta \left({Q}_{x}Q-{{QQ}}_{x}\right)-2{\rm{i}}{k}^{2}\left(1-2\beta {Q}^{2}\right)+8{\rm{i}}\beta {k}^{4}\right]{\sigma }_{3}\\ & & -\,4{\rm{i}}\beta {k}^{2}{\sigma }_{3}{Q}_{x}-8\beta {k}^{3}Q+6{\rm{i}}\beta {Q}^{2}{Q}_{x}{\sigma }_{3}+{\rm{i}}{\sigma }_{3}{Q}_{x}\\ & & +\,{\rm{i}}\beta {\sigma }_{3}{Q}_{{xxx}}+2k\left(Q+\beta {Q}_{{xx}}-2\beta {Q}^{3}\right),\end{array}\end{eqnarray}$
and q = q(x, t) represents the complex envelope, φ is the eigenfunction, k is the spectral parameter, β is a real parameter, which stands for the strength of the higher-order linear and nonlinear effects, x and t represent the coordinates of time evolution and spatial distribution respectively, the subscript represents the partial derivative, * represents the complex conjugate.Equation (1 ) has a higher-order dispersion term, which can be used to describe the propagation of ultrashort pulses in light, and [22] found that equation (1 ) describes the amplitude modulation of the fundamental harmonic of the Stokes wave at different depths of the fluid level. The higher-order dispersion term of the equation describes the third-order dispersion effect and nonlinear dispersion effect. Since equation (1 ) has important applications in physical science, it has attracted extensive attention in recent years, and many scholars have studied this model. For example, [23] obtained rich bright, dark, singular, kink and periodic optical soliton solutions of equation (1 ) for the nonlinear Kerr law. In [24], the improved finite band integral method is used to find the periodic solution of equation (1 ). Based on the analysis of the characteristic spectral lines, [25] deduced the breather molecules composed of two, three and four different breathing atoms, and adjusted the value of the phase parameters to generate the synthesis mode of the molecules. In [20], the soliton solutions of the equation (1 ) were obtained by the modified generalized Darboux transformation. In [26], the formula of the general N soliton solutions of the integrable LPD equation was obtained by RH method and the dynamic behaviors of the soliton solutions were analyzed graphically. In [27], the soliton solutions and rogue wave solutions of the equation (1 ) were obtained by RH method. In general, the Plemelj formula and the residue condition can be used to solve the RH problem. However, for the case of multi-higher-order poles, the application of the residue condition to study the residue of each order is very complicated, and for the case of multi-high-order poles, the residue condition is not suitable for solving this problem. In this paper, our purpose is to obtain the soliton solutions with a higher-order pole and multi higher-order poles of equation (1 ) under nonzero boundary conditions, and to analyze the interactions between the soliton solutions.
The rest of the paper is arranged as follows. In section 2 , based on nonzero boundary conditions, we perform spectral analysis on equation (1 ) and construct a suitable RH problem for equation (1 ). In section 3 , we assume that the analytical scattering coefficient has a high-order zero point or multi-higher-order zero points, which leads to the high-order poles of the reflection coefficient in RH problem. By using the Laurent expansion and Taylor series expansion, the soliton solutions with a higher-order pole or multi-higher-order poles of the equation (1 ) can be derived. At the same time, the interactions of soliton solutions can be shown. In section 4 , we give the summary.
2. Riemann–Hilbert problem: nonzero boundary condition
2.1. Riemann surface and single-branched coordinate
By the transformation1 ) and boundary conditions are reduced to1 ) is the compatibility condition of the Lax pair
$\begin{eqnarray}q\to q{{\rm{e}}}^{{\rm{i}}\left(6\beta {q}_{0}^{4}+2{q}_{0}^{2}\right)t},\quad \ \phi \to {{\rm{e}}}^{{\rm{i}}\left(3\beta {q}_{0}^{4}+{q}_{0}^{2}\right)t{\sigma }_{3}}\phi ,\end{eqnarray}$
then equation ( $\begin{eqnarray}\begin{array}{l}{\rm{i}}{q}_{t}+{q}_{{xx}}+2{\left|q\right|}^{2}q+\beta {q}_{{xxxx}}+6{q}^{* }{q}_{x}^{2}\\ \qquad +4{\left|{q}_{x}\right|}^{2}q+8{\left|q\right|}^{2}{q}_{{xx}}+2{q}^{2}{q}_{{xx}}^{* }+6{\left|q\right|}^{4}q-6{q}_{0}^{4}q-2{q}_{0}^{2}q=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}q\left(x,t\right)\to {q}_{\pm },x\to \pm \infty .\end{eqnarray}$
And equation ( $\begin{eqnarray}{\phi }_{x}=X\phi ,\quad \ {\phi }_{t}=T\phi ,\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{ccl}X & = & -{\rm{i}}k{\sigma }_{3}+Q,\\ T & = & \left[3{\rm{i}}\beta {Q}^{4}-{\rm{i}}{Q}^{2}+{\rm{i}}\beta \left({Q}_{x}^{2}-{Q}_{{xx}}Q-{{QQ}}_{{xx}}\right)\right.\\ & & -\,2k\beta \left({Q}_{x}Q-{{QQ}}_{x}\right)-2{\rm{i}}{k}^{2}\left(1-2\beta {Q}^{2}\right)\\ & & +\,\left.8{\rm{i}}\beta {k}^{4}-3{\rm{i}}\beta {q}_{0}^{4}-{\rm{i}}{q}_{0}^{2}\right]{\sigma }_{3}\\ & & -\,4{\rm{i}}\beta {k}^{2}{\sigma }_{3}{Q}_{x}-8\beta {k}^{3}Q+6{\rm{i}}\beta {Q}^{2}{Q}_{x}{\sigma }_{3}+{\rm{i}}{\sigma }_{3}{Q}_{x}\\ & & +\,{\rm{i}}\beta {\sigma }_{3}{Q}_{{xxx}}+2k\left(Q+\beta {Q}_{{xx}}-2\beta {Q}^{3}\right).\end{array}\end{eqnarray}$
Let x → ±∞ , and using boundary conditions equation (7 ), we can obtain that the asymptotic spectral problem is13 ), we obtain two single-valued functions
$\begin{eqnarray}{\psi }_{x}={X}_{\pm }\psi ,\quad \ {\psi }_{t}={T}_{\pm }\psi ,\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{cl}{X}_{\pm }\,=\, & -{\rm{i}}k{\sigma }_{3}+{Q}_{\pm },\quad \\ & {T}_{\pm }=\,\left(2k+4\beta {{kq}}_{0}^{2}-8\beta {k}^{3}\right){X}_{\pm },\quad {Q}_{\pm }=\left(\begin{array}{cc}0 & {q}_{\pm }\\ -{q}_{\pm }^{* } & 0\end{array}\right).\end{array}\end{eqnarray}$
It can be seen from the calculation that the eigenvalue of matrix X± are ±iλ, where $\begin{eqnarray}{\lambda }^{2}={k}^{2}+{q}_{0}^{2}=\left(k+{\rm{i}}{q}_{0}\right)\left(k-{\rm{i}}{q}_{0}\right).\end{eqnarray}$
The Riemann surface determined by the equation is composed of two complex planes S1 and S2 separated by the branch secant $\left[-{{\rm{i}}{q}}_{0},{{\rm{i}}{q}}_{0}\right]$, and the branch points are k = ±iq0. Therefore, $\lambda \left(k\right)$ is a single-valued function of k and the polar coordinates can be introduced to have $\begin{eqnarray}k+{\rm{i}}{q}_{0}={r}_{1}{{\rm{e}}}^{{\rm{i}}{\theta }_{1}},\quad k-{\rm{i}}{q}_{0}={r}_{2}{{\rm{e}}}^{{\rm{i}}{\theta }_{2}},\end{eqnarray}$
where $-\tfrac{\pi }{2}\ \lt \ {\theta }_{1},{\theta }_{2}\lt \tfrac{\pi }{2}.$ Then we can get two single analytic functions on Riemann surface $\begin{eqnarray}\lambda \left(k\right)=\left\{\begin{array}{c}{\left({r}_{1}{r}_{2}\right)}^{\tfrac{1}{2}}{{\rm{e}}}^{\tfrac{{\rm{i}}}{2}\left({\theta }_{1}+{\theta }_{2}\right)},\mathrm{on}\ {S}_{1},\\ -{\left({r}_{1}{r}_{2}\right)}^{\tfrac{1}{2}}{{\rm{e}}}^{\tfrac{{\rm{i}}}{2}\left({\theta }_{1}+{\theta }_{2}\right)},\mathrm{on}\ {S}_{2}.\end{array}\right.\end{eqnarray}$
In order to avoid the multiplicity of function and the complexity of Riemann surface, we introduce the affine parameter z and discuss this problem on the complex plane of z. We define single-valued variable $\begin{eqnarray}z=k+\lambda .\end{eqnarray}$
Then based on equation ( $\begin{eqnarray}k\left(z\right)=\displaystyle \frac{1}{2}\left(z-\displaystyle \frac{{q}_{0}^{2}}{z}\right),\quad \lambda \left(z\right)=\displaystyle \frac{1}{2}\left(z+\displaystyle \frac{{q}_{0}^{2}}{z}\right).\end{eqnarray}$
By directly calculating, we can get $\begin{eqnarray}\mathrm{Im}\left(\lambda \left(z\right)\right)=\mathrm{Im}\displaystyle \frac{{z}^{2}+{q}_{0}^{2}}{2z}=\displaystyle \frac{\left({\left|z\right|}^{2}-{q}_{0}^{2}\right)\mathrm{Im}z}{2{\left|z\right|}^{2}}.\end{eqnarray}$
On the standard z-plane, we define $\begin{eqnarray}\begin{array}{rcl}{D}^{+} & = & \left\{z\in C:\left({\left|z\right|}^{2}-{q}_{0}^{2}\right)\mathrm{Im}z\gt 0\right\},\\ {D}^{-} & = & \left\{z\in C:\left({\left|z\right|}^{2}-{q}_{0}^{2}\right)\mathrm{Im}z\lt 0\right\},\\ \sum & = & \left\{\left|z\right|={q}_{0},z\in C\right\}.\end{array}\end{eqnarray}$
It is easy to get $\begin{eqnarray}\left({\left|z\right|}^{2}-{q}_{0}^{2}\right)\mathrm{Im}z\left\{\begin{array}{c}=0,\ \mathrm{as}\,{\rm{z}}\in \sum ,\\ \gt 0,\ \mathrm{as}\,{\rm{z}}\in {D}^{+},\\ \lt 0,\ \mathrm{as}\ {\rm{z}}\in {D}^{-}.\end{array}\right.\end{eqnarray}$
2.2. The analytic properties of jost functions and scattering matrix
Since the eigenvalues of matrix X± are ±iλ, T± are $\pm \left(2k+4\beta {{kq}}_{0}^{2}-8\beta {k}^{3}\right){\rm{i}}\lambda $, and X± and T± can be diagonalized by the same eigenvector matrix satisfying19 ) into equation (9 ) to get9 ) is
$\begin{eqnarray}\begin{array}{ccl}{X}_{\pm } & = & {Y}_{\pm }(-{\rm{i}}\lambda {\sigma }_{3}){Y}_{\pm }^{-1},\quad \ \\ & & {{T}}_{\pm }={Y}_{\pm }\,\left(-\left(2k+4\beta {{kq}}_{0}^{2}-8\beta {k}^{3}\right){\rm{i}}\lambda {\sigma }_{3}\right){Y}_{\pm }^{-1},\end{array}\end{eqnarray}$
where $\begin{eqnarray}{Y}_{\pm }=\left(\begin{array}{cc}1 & -\displaystyle \frac{{{\rm{i}}{q}}_{\pm }}{z}\\ -\displaystyle \frac{{{\rm{i}}{q}}_{\pm }^{* }}{z} & 1\end{array}\right)=I-\displaystyle \frac{{\rm{i}}}{z}{\sigma }_{3}{Q}_{\pm }.\end{eqnarray}$
By directly calculating, we have $\begin{eqnarray}\det \left({Y}_{\pm }\right)=1+\displaystyle \frac{{q}_{0}^{2}}{{z}^{2}}=\eta ,\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{Y}_{\pm }^{-1} & = & \displaystyle \frac{1}{\eta }\left(\begin{array}{cc}1 & \tfrac{{{\rm{i}}{q}}_{\pm }}{z}\\ \tfrac{{{\rm{i}}{q}}_{\pm }^{* }}{z} & 1\end{array}\right)=\displaystyle \frac{1}{\eta }\left(I+\displaystyle \frac{{\rm{i}}}{z}{\sigma }_{3}{Q}_{\pm }\right)\\ & = & \displaystyle \frac{1}{\eta }\left(I-\displaystyle \frac{{\rm{i}}}{z}{Q}_{\pm }{\sigma }_{3}\right),z\ne \pm {{\rm{i}}{q}}_{0}.\end{array}\end{eqnarray}$
Then bringing equation ( $\begin{eqnarray}\begin{array}{rcl}{\left({Y}_{\pm }^{-1}\psi \right)}_{x} & = & -{\rm{i}}\lambda {\sigma }_{3}\left({Y}_{\pm }^{-1}\psi \right),\quad \ {\left({Y}_{\pm }^{-1}\psi \right)}_{t}\\ & = & -\left(2k+4\beta {{kq}}_{0}^{2}-8\beta {k}^{3}\right){\rm{i}}\lambda {\sigma }_{3}\left({Y}_{\pm }^{-1}\psi \right),\ z\ne \pm {{\rm{i}}{q}}_{0},\end{array}\end{eqnarray}$
and the solution of the asymptotic spectral problem equation ( $\begin{eqnarray}\psi \left(x,t,z\right)={Y}_{\pm }{{\rm{e}}}^{{\rm{i}}\theta \left(z\right){\sigma }_{3}},\end{eqnarray}$
where $\begin{eqnarray}\theta \left(z\right)=-\lambda \left(z\right)\left(x+\left(2k\left(z\right)+4\beta k\left(z\right){q}_{0}^{2}-8\beta k{\left(z\right)}^{3}\right)t\right).\end{eqnarray}$
Now, we define the Jost functions φ± as the solutions of the Lax pair equation (7 )
$\begin{eqnarray}{\phi }_{\pm }\sim {Y}_{\pm }{{\rm{e}}}^{{\rm{i}}\theta \left(z\right){\sigma }_{3}},x\to \pm \infty .\end{eqnarray}$
Furthermore, we introduce the modified eigenfunctions to eliminate exponential oscillations $\begin{eqnarray}{\mu }_{\pm }\left(x,t,z\right)={\phi }_{\pm }\left(x,t,z\right){{\rm{e}}}^{-{\rm{i}}\theta \left(z\right){\sigma }_{3}},\end{eqnarray}$
then we have $\begin{eqnarray}{\mu }_{\pm }\left(x,t,z\right)\sim {Y}_{\pm }\left(z\right),x\to \pm \infty .\end{eqnarray}$
By calculating, the equivalent Lax pair is obtained $\begin{eqnarray}{\left({Y}_{\pm }^{-1}{\mu }_{\pm }\right)}_{x}-{\rm{i}}\lambda \left[{Y}_{\pm }^{-1}{\mu }_{\pm },\ {\sigma }_{3}\right]={Y}_{\pm }^{-1}{\rm{\Delta }}{Q}_{\pm }{\mu }_{\pm },\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\left({Y}_{\pm }^{-1}{\mu }_{\pm }\right)}_{t}-{\rm{i}}\lambda \left(2k+4\beta {{kq}}_{0}^{2}-8\beta {k}^{3}\right)\\ \qquad \times \,\left[{Y}_{\pm }^{-1}{\mu }_{\pm },\ {\sigma }_{3}\right]={Y}_{\pm }^{-1}{\rm{\Delta }}{T}_{\pm }{\mu }_{\pm },\end{array}\end{eqnarray}$
where ΔQ± = Q − Q± and ${\rm{\Delta }}{T}_{\pm }\,=\,T\,-\,{T}_{\pm }\,=\left(2k+4\beta {{kq}}_{0}^{2}-8\beta {k}^{3}\right){\rm{\Delta }}{Q}_{\pm }$. The Lax pair can be written in the form of total differential $\begin{eqnarray}{\rm{d}}\left({{\rm{e}}}^{-{\rm{i}}\theta \left(z\right)\mathop{{\sigma }_{3}}\limits^{\wedge }}{Y}_{\pm }^{-1}{\mu }_{\pm }\right)={{\rm{e}}}^{-{\rm{i}}\theta \left(z\right)\mathop{{\sigma }_{3}}\limits^{\wedge }}\left[{Y}_{\pm }^{-1}\left({\rm{\Delta }}{Q}_{\pm }{\rm{d}}{x}+{\rm{\Delta }}{T}_{\pm }{\rm{d}}{t}\right){\mu }_{\pm }\right].\end{eqnarray}$
By integrating $\left(-\infty ,t\right)\to \left(x,t\right)$ and $\left(+\infty ,t\right)\to \left(x,t\right)$ along two special paths, two Jost functions solutions can be obtained as follows $\begin{eqnarray}\begin{array}{rcl}{\mu }_{-}\left(x,t,z\right) & = & {Y}_{-}\\ & & +{\displaystyle \int }_{-\infty }^{x}{Y}_{-}{{\rm{e}}}^{-{\rm{i}}\lambda \left(x-y\right)\mathop{{\sigma }_{3}}\limits^{\wedge }}\,\left[{Y}_{-}^{-1}{Q}_{1}\left(y,t,z\right){\mu }_{-}\left(y,t,z\right)\right]{\rm{d}}{y},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\mu }_{+}\left(x,t,z\right) & = & {Y}_{+}\\ & & +{\displaystyle \int }_{+\infty }^{x}{Y}_{+}{{\rm{e}}}^{-{\rm{i}}\lambda \left(x-y\right)\mathop{{\sigma }_{3}}\limits^{\wedge }}\,\left[{Y}_{+}^{-1}{Q}_{1}\left(y,t,z\right){\mu }_{+}\left(y,t,z\right)\right]{\rm{d}}{y},\end{array}\end{eqnarray}$
where ${{\rm{e}}}^{{\rm{i}}{k}\lambda \left(x-y\right){\sigma }_{3}^{\wedge }}A={{\rm{e}}}^{{\rm{i}}{k}\lambda \left(x-y\right){\sigma }_{3}}{{A}{\rm{e}}}^{-{\rm{i}}{k}\lambda \left(x-y\right){\sigma }_{3}},$ for a matrix A.For $q\left(x\right)-{q}_{\pm }\in {L}^{1}\left({R}^{\pm }\right)$, $t\in {R}^{+}$, the solutions of the equations (
| • | ${\mu }_{-,1}$, $\ {\mu }_{+,2}$ are analytic in ${D}^{+}$ and continuous in ${D}^{+}\cup \sum $, |
| • | ${\mu }_{-,2}$, $\ {\mu }_{+,1}$ are analytic in ${D}^{-}$ and continuous in ${D}^{-}\cup \sum $. |
Define ${\mu }_{\pm }=\left({\mu }_{\pm ,1},{\mu }_{\pm ,2}\right)$, $\ W\left(x,z\right)={Y}_{-}^{-1}{\mu }_{-}$, then the first column w of W is
$\begin{eqnarray}\begin{array}{ccl}w\left(x,t,z\right) & = & \left(\begin{array}{c}1\\ 0\end{array}\right)+\,{\int }_{-\infty }^{x}G\left(x-y,z\right){\rm{\Delta }}{Q}_{-}\left(y\right){Y}_{-}\left(y\right)w\left(y,z\right){\rm{d}}y,\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{ccc}G\left(x-y,z\right) & = & \mathrm{diag}\left(1,{{\rm{e}}}^{2{\rm{i}}\lambda \left(x-y\right)}\right){Y}_{-}^{-1}\left(z\right)\\ & = & \frac{1}{\eta }\left(\begin{array}{cc}1 & \tfrac{{{\rm{i}}{q}}_{-}}{z}\\ \tfrac{{{\rm{i}}{q}}_{-}^{* }}{z}{{\rm{e}}}^{2{\rm{i}}\lambda \left(x-y\right)} & {{\rm{e}}}^{2{\rm{i}}\lambda \left(x-y\right)}\end{array}\right),\end{array}\end{eqnarray}$
because ${{\rm{e}}}^{2{\rm{i}}\lambda \left(x-y\right)}={{\rm{e}}}^{2{\rm{i}}\left(x-y\right)\mathrm{Re}\left(\lambda \right)}{{\rm{e}}}^{-2\left(x-y\right)\mathrm{Im}\left(\lambda \right)}$, $y\lt x$, and $\mathrm{Im}\left(\lambda \right)\gt 0$, ${\mu }_{-,1}$ is analytic in ${D}^{+}$. Similarly, we can prove that ${\mu }_{+,2}$ is analytic in ${D}^{+}$, ${\mu }_{-,2}$ and ${\mu }_{+,1}$ are analytic in ${D}^{-}$. $\square $For $q\left(x\right)-{q}_{\pm }\in {L}^{1}\left(R\right),t\in {R}^{+}$, the Jost functions solutions of the equation (
| • | ${\phi }_{-,1}$, $\ {\phi }_{+,2}$ are analytic in ${D}^{+}$ and continuous in ${D}^{+}\cup \sum $, |
| • | ${\phi }_{-,2}$, $\ {\phi }_{+,1}$ are analytic in ${D}^{-}$ and continuous in ${D}^{-}\cup \sum $. |
The functions φ± are both fundamental matrix solutions of the Lax pair equation (8 ), thus there exists a matrix $S\left(z\right)$ that only depends on z, such that
$\begin{eqnarray}{\phi }_{+}\left(x,t,z\right)={\phi }_{-}\left(x,t,z\right)S\left(z\right),\ x,t\in R,\ z\in \sum ,\end{eqnarray}$
where $S\left(z\right)$ is called scattering matrix. Letting $S\left(z\right)=\left({s}_{{ij}}\right)$, after expanding the above equation by the column of the matrix, there are $\begin{eqnarray}{\phi }_{+,1}={s}_{11}{\phi }_{-,1}+{s}_{21}{\phi }_{-,2},\quad \ {\phi }_{+,2}={s}_{12}{\phi }_{-,1}+{s}_{22}{\phi }_{-,2}.\end{eqnarray}$
From Lax pair equation (8 ), we know that27 ), we have28 ), we can obtain37 ), we can get
$\begin{eqnarray}{\rm{tr}}\left(Q\right)={\rm{tr}}\left(T\right)=0,\end{eqnarray}$
and then by using Abel formula, we can get $\begin{eqnarray}{\left(\det {\phi }_{\pm }\right)}_{x}={\left(\det {\phi }_{\pm }\right)}_{t}=0.\end{eqnarray}$
Based on equation ( $\begin{eqnarray}\det \left({\mu }_{\pm }\right)=\det \left({\phi }_{\pm }{{\rm{e}}}^{-{\rm{i}}\theta \left(z\right){\sigma }_{3}}\right)=\det \left({\phi }_{\pm }\right).\end{eqnarray}$
Therefore, it is easy to get ${\left(\det {\mu }_{\pm }\right)}_{x}={\left(\det {\mu }_{\pm }\right)}_{t}=0$, which means $\det \left({\mu }_{\pm }\right)$ is independent with x, t. From equation ( $\begin{eqnarray}\det \left({\mu }_{\pm }\right)=\mathop{\mathrm{lim}}\limits_{x\to \pm \infty }\det \left({\mu }_{\pm }\right)=\det \left({Y}_{\pm }\right)=\eta \ne 0,\end{eqnarray}$
so μ± is invertible. Bringing the above equation into equation ( $\begin{eqnarray}{s}_{11}\left(z\right)=\displaystyle \frac{{\rm{Wr}}\left({\phi }_{+,1},{\phi }_{-,2}\right)}{\eta },\quad \ {s}_{12}\left(z\right)=\displaystyle \frac{{\rm{Wr}}\left({\phi }_{+,2},{\phi }_{-,2}\right)}{\eta },\ \end{eqnarray}$
$\begin{eqnarray}{s}_{21}\left(z\right)=\displaystyle \frac{{\rm{Wr}}\left({\phi }_{-,1},{\phi }_{+,1}\right)}{\eta },\quad \ {s}_{22}\left(z\right)=\displaystyle \frac{{\rm{Wr}}\left({\phi }_{-,1},{\phi }_{+,2}\right)}{\eta },\ \end{eqnarray}$
where $\mathrm{Wr}\left(\cdot ,\cdot \right)$ denotes Wronskian determinant.Based on the above conclusions and corollary 1 , there is the proposition 2.2 .
For $q\left(x\right)-{q}_{\pm }\in {L}^{1}\left(R\right),t\in {R}^{+}$, the scattering coefficients have the following analytic properties:
| • | ${s}_{11}\left(z\right)$ is analytic in ${D}^{-}$ and continuous in ${D}^{-}\cup \sum ,$ |
| • | ${s}_{22}\left(z\right)$ is analytic in ${D}^{+}$ and continuous in ${D}^{+}\cup \sum ,$ |
| • | ${s}_{12}\left(z\right)$ and ${s}_{21}\left(z\right)$ are continuous in $\sum .$ |
2.3. The symmetry properties of jost functions and scattering matrix
Since the Riemann–Hilbert (RH) problem with nonzero boundary conditions not only needs to deal with the map k ↦ k*, but also needs to deal with the sheets of the Riemann surface, and after removing the asymptotic oscillation, the Jost functions no longer tend to the identity matrix, which causes the complexity of the problem. From the Riemann surface, the transformation z ↦ z* implies $\left(k,\ \lambda \right)\mapsto \left({k}^{* },\ {\lambda }^{* }\right)$ and $z\mapsto -\tfrac{{q}_{0}^{2}}{z}$ implies $\left(k,\ \lambda \right)\mapsto \left(k,\ -\lambda \right)$ can be obtained.
The symmetry properties of Jost function, scattering matrix and scattering coefficients can be proved
| ${\phi }_{\pm }\left(x,t,z\right)={\sigma }_{2}{\phi }_{\pm }^{* }\left(x,t,{z}^{* }\right){\sigma }_{2}$, ${S}\left(z\right)={\sigma }_{2}{S}^{* }\left({z}^{* }\right){\sigma }_{2}$, $\ \rho \left(z\right)\,=-{\tilde{\rho }}^{* }\left({z}^{* }\right)$, | |
| ${\phi }_{\pm }\left(x,t,z\right)=-\tfrac{{\rm{i}}}{z}{\phi }_{\pm }\left(x,t,-\tfrac{{q}_{0}^{2}}{z}\right){\sigma }_{3}{Q}_{\pm }$, ${S}\left(z\right)={\left({\sigma }_{3}{Q}_{-}\right)}^{-1}S\left(-\tfrac{{q}_{0}^{2}}{z}\right){\sigma }_{3}{Q}_{+}$, $\ \rho \left(z\right)=\tfrac{{q}_{-}}{{q}_{-}^{* }}\tilde{\rho }\left(-\tfrac{{q}_{0}^{2}}{z}\right),$ |
Let φ be a solution of scattering problem equation (
$\begin{eqnarray}{\phi }_{x}=\left(-{\rm{i}}\lambda {\sigma }_{3}+Q\right)\phi ,\end{eqnarray}$
which leads to $\begin{eqnarray}{\left[{\sigma }_{2}{\phi }^{* }\left({z}^{* }\right){\sigma }_{2}\right]}_{x}={\sigma }_{2}\left[{\rm{i}}{\lambda }^{* }\left({z}^{* }\right){\sigma }_{3}+{Q}^{* }\right]{\phi }^{* }\left({z}^{* }\right){\sigma }_{2}.\end{eqnarray}$
Since ${\sigma }_{2}{\sigma }_{3}{\sigma }_{2}=-{\sigma }_{3}$, $\ {\sigma }_{2}{Q}^{* }{\sigma }_{2}=Q$, $\ {k}^{* }\left({z}^{* }\right)=k\left(z\right)$, we have $\begin{eqnarray}{\left[{\sigma }_{2}{\phi }^{* }\left({z}^{* }\right){\sigma }_{2}\right]}_{x}=\left[-{\rm{i}}\lambda \left(z\right){\sigma }_{3}+Q\right]\left({\sigma }_{2}{\phi }^{* }\left({z}^{* }\right){\sigma }_{2}\right).\end{eqnarray}$
Therefore, ${\sigma }_{2}{\phi }^{* }\left({z}^{* }\right){\sigma }_{2}$ is also the solution of equation ( $\begin{eqnarray}{\sigma }_{2}{Y}_{\pm }^{* }\left({z}^{* }\right){\sigma }_{2}={Y}_{\pm }\left(z\right),\quad \ {\sigma }_{2}{{\rm{e}}}^{-{\rm{i}}{\theta }^{* }\left({z}^{* }\right){\sigma }_{3}}{\sigma }_{2}={{\rm{e}}}^{{\rm{i}}\theta \left(z\right){\sigma }_{3}},\end{eqnarray}$
we can get $\begin{eqnarray}{\sigma }_{2}{Y}_{\pm }^{* }\left({z}^{* }\right){\sigma }_{2}={Y}_{\pm }\left(z\right){{\rm{e}}}^{{\rm{i}}\theta \left(z\right){\sigma }_{3}}+O\left(1\right),\ x\to \pm \infty .\end{eqnarray}$
From the uniqueness of the solution of the scattering problem, we can obtain $\begin{eqnarray}{\phi }_{\pm }\left(x,t,z\right)={\sigma }_{2}{\phi }^{* }\left(x,t,{z}^{* }\right){\sigma }_{2}.\end{eqnarray}$
By directly calculating, we have $\begin{eqnarray}-\displaystyle \frac{{\rm{i}}}{z}{Y}_{\pm }\left(-\displaystyle \frac{{q}_{0}^{2}}{z}\right){{\rm{e}}}^{-{\rm{i}}\theta \left(z\right){\sigma }_{3}}{\sigma }_{3}{Q}_{\pm }={Y}_{\pm }\left(z\right){{\rm{e}}}^{{\rm{i}}\theta \left(z\right){\sigma }_{3}},\end{eqnarray}$
then $\begin{eqnarray}{\phi }_{\pm }\left(x,t,z\right)=-\frac{{\rm{i}}}{z}{\phi }_{\pm }\left(x,t,-\frac{{q}_{0}^{2}}{z}\right){\sigma }_{3}{Q}_{\pm }.\end{eqnarray}$
Next, we prove the symmetry properties of the scattering matrix. Bringing equation ( $\begin{eqnarray}{S}\left(z\right)={\sigma }_{2}{S}^{* }\left({z}^{* }\right){\sigma }_{2},\end{eqnarray}$
and ${s}_{11}\left(z\right)={s}_{22}^{* }\left({z}^{* }\right),\ {s}_{12}\left(z\right)=-{s}_{21}^{* }\left({z}^{* }\right)$ after expansion, so $\begin{eqnarray}\rho \left(z\right)=-{\tilde{\rho }}^{* }\left({z}^{* }\right).\end{eqnarray}$
Similarly, bringing equation ( $\begin{eqnarray}\ {S}\left(z\right)={\left({\sigma }_{3}{Q}_{-}\right)}^{-1}S\left(-\displaystyle \frac{{q}_{0}^{2}}{z}\right){\sigma }_{3}{Q}_{+}.\end{eqnarray}$
After expansion, we have $\begin{eqnarray}\begin{array}{rcl}{s}_{11}\left(z\right) & = & \displaystyle \frac{{q}_{+}^{* }}{{q}_{-}^{* }}{s}_{22}\left(-\displaystyle \frac{{q}_{0}^{2}}{z}\right),\quad \ {s}_{12}\left(z\right)=\displaystyle \frac{{q}_{+}}{{q}_{-}^{* }}{s}_{21}\left(-\displaystyle \frac{{q}_{0}^{2}}{z}\right),\\ {s}_{21}\left(z\right) & = & \displaystyle \frac{{q}_{+}^{* }}{{q}_{-}}{s}_{12}\left(-\displaystyle \frac{{q}_{0}^{2}}{z}\right),\quad \ {s}_{22}\left(z\right)=\displaystyle \frac{{q}_{+}}{{q}_{-}}{s}_{11}\left(-\displaystyle \frac{{q}_{0}^{2}}{z}\right),\end{array}\end{eqnarray}$
then $\begin{eqnarray}\rho \left(z\right)=\displaystyle \frac{{q}_{-}}{{q}_{-}^{* }}\tilde{\rho }\left(-\displaystyle \frac{{q}_{0}^{2}}{z}\right).\end{eqnarray}$
2.4. The asymptotic properties of jost functions and scattering matrix
It is necessary to discuss the asymptotic properties of Jost functions and scattering matrix in order to solve the RH problem in the next section.
It can be proved by equations (
| • | ${\mu }_{\pm }=I+O\left({z}^{-1}\right)$, $z\to \infty ,$ |
| • | ${\mu }_{\pm }=-\tfrac{{\rm{i}}}{z}{\sigma }_{3}{Q}_{\pm }+O\left(1\right)$, $z\to 0.$ |
| • | $S\left(z\right)=I+O\left({z}^{-1}\right)$, $\ z\to \infty ,$ |
| • | $S\left(z\right)=\mathrm{diag}\left(\tfrac{{q}_{-}}{{q}_{+}},\tfrac{{q}_{+}}{{q}_{-}}\right)+O\left(z\right)$, $\ z\to 0.$ |
Asymptotic expansion is made at $z=\infty $ as follows
$\begin{eqnarray}{\mu }_{\pm }={\mu }_{0}^{\pm }+\displaystyle \frac{{\mu }_{1}^{\pm }}{z}+\displaystyle \frac{{\mu }_{2}^{\pm }}{{z}^{2}}+\cdots .\end{eqnarray}$
Bringing equations ( $\begin{eqnarray}\begin{array}{l}{\left[\left(I-\displaystyle \frac{{\rm{i}}}{z}{Q}_{\pm }{\sigma }_{3}\right)\left({\mu }_{0}^{\pm }+\displaystyle \frac{{\mu }_{1}^{\pm }}{z}+\cdots \right)\right]}_{x}\\ \qquad -{\rm{i}}\lambda \left[\left(I-\displaystyle \frac{{\rm{i}}}{z}{Q}_{\pm }{\sigma }_{3}\right)\left({\mu }_{0}^{\pm }+\displaystyle \frac{{\mu }_{1}^{\pm }}{z}+\cdots \right),\,{\sigma }_{3}\right]\\ \,=\,\left(I-\displaystyle \frac{{\rm{i}}}{z}{Q}_{\pm }{\sigma }_{3}\right){\rm{\Delta }}{Q}_{\pm }\left({\mu }_{0}^{\pm }+\displaystyle \frac{{\mu }_{1}^{\pm }}{z}+\cdots \right),\\ {\,\left[\left(I-\displaystyle \frac{{\rm{i}}}{z}{Q}_{\pm }{\sigma }_{3}\right)\left({\mu }_{0}^{\pm }+\displaystyle \frac{{\mu }_{1}^{\pm }}{z}+\cdots \right)\right]}_{t}\\ \qquad -\,{\rm{i}}\left(2k+4\beta {{kq}}_{0}^{2}-8\beta {k}^{3}\right)\\ \,\times \,\left[\left(I-\displaystyle \frac{{\rm{i}}}{z}{Q}_{\pm }{\sigma }_{3}\right)\left({\mu }_{0}^{\pm }+\displaystyle \frac{{\mu }_{1}^{\pm }}{z}+\cdots \right),\,{\sigma }_{3}\right]\\ \,=\,\left(I-\displaystyle \frac{{\rm{i}}}{z}{Q}_{\pm }{\sigma }_{3}\right){\rm{\Delta }}{T}_{\pm }\left({\mu }_{0}^{\pm }+\displaystyle \frac{{\mu }_{1}^{\pm }}{z}+\cdots \right).\end{array}\end{eqnarray}$
Combining with equation ( $\begin{eqnarray}-\frac{{\rm{i}}}{2}\left[{\mu }_{0}^{\pm },{\sigma }_{3}\right]=0,\end{eqnarray}$
$\begin{eqnarray}{\mu }_{0,x}^{\pm }-\displaystyle \frac{{\rm{i}}}{2}\left[{\mu }_{1}^{\pm }-{{\rm{i}}{Q}}_{\pm }{\sigma }_{3}{\mu }_{0}^{\pm },{\sigma }_{3}\right]={\rm{\Delta }}{Q}_{\pm }{\mu }_{0}^{\pm },\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{\rm{i}}\beta }{2}\left[{\mu }_{1}^{\pm }-{{\rm{i}}{Q}}_{\pm }{\sigma }_{3}{\mu }_{0}^{\pm },{\sigma }_{3}\right]=-\beta {\rm{\Delta }}{Q}_{\pm }{\mu }_{0}^{\pm }.\end{eqnarray}$
From equation ( $\begin{eqnarray}\mathop{\mathrm{lim}}\limits_{z\to \infty }{Y}_{\pm }=\mathop{\mathrm{lim}}\limits_{z\to \infty }\mathop{\mathrm{lim}}\limits_{x\to \pm \infty }\mu =\mathop{\mathrm{lim}}\limits_{x\to \pm \infty }\mathop{\mathrm{lim}}\limits_{z\to \infty }\left({\mu }_{0}^{\pm }+O\left({z}^{-1}\right)\right).\end{eqnarray}$
Therefore, we have ${\mu }_{0}^{\pm }=I.$ By taking this into equation ( $\begin{eqnarray}{\mu }_{1}^{\pm }=-{\rm{i}}{\sigma }_{3}Q,\end{eqnarray}$
then $\begin{eqnarray}\begin{array}{ccc}{\mu }_{\pm } & = & I+O\left({z}^{-1}\right),\ z\to \infty ,\\ {\mu }_{\pm } & = & -\frac{{\rm{i}}}{z}{\sigma }_{3}{Q}_{\pm }+O\left(1\right),\ z\to 0.\end{array}\end{eqnarray}$
For the scattering matrix $S\left(z\right)$, when $z\to \infty $, combining with equations ( $\begin{eqnarray}S\left(z\right)=I+O\left({z}^{-1}\right).\end{eqnarray}$
When $z\to 0$, we can obtain $\begin{eqnarray}S\left(z\right)=\mathrm{diag}\left(\frac{{q}_{-}}{{q}_{+}},\frac{{q}_{+}}{{q}_{-}}\right)+O\left(z\right).\end{eqnarray}$
2.5. The Riemann–Hilbert problem
Expanding equation (36 ), we have
$\begin{eqnarray}\begin{array}{rcl}{\mu }_{+,1} & = & {s}_{11}{\mu }_{-,1}+{s}_{21}{{\rm{e}}}^{-2{\rm{i}}\theta \left(z\right)}{\mu }_{-,2},\\ {\mu }_{+,2} & = & {s}_{12}{{\rm{e}}}^{2{\rm{i}}\theta \left(z\right)}{\mu }_{-,1}+{s}_{22}{\mu }_{-,2},\end{array}\end{eqnarray}$
furthermore, we define piecewise analytic functions $\begin{eqnarray}M\left(x,t,z\right)=\left\{\begin{array}{l}{M}^{-}\left(x,t,z\right)=\left(\tfrac{{\mu }_{+,1}}{{s}_{11}},{\mu }_{-,2}\right),z\in {{\rm{D}}}^{-},\\ {M}^{+}\left(x,t,z\right)=\left({\mu }_{-,1},\tfrac{{\mu }_{+,2}}{{s}_{22}}\right),z\in {{\rm{D}}}^{+}.\end{array}\right.\end{eqnarray}$
Therefore, the following generalized RH problem can be obtained.$M\left(x,t,z\right)$ satisfies
| • | $M\left(x,t,z\right)$ is analytic in $C\setminus \sum .$ |
| • | ${M}^{-}\left(x,t,z\right)={M}^{+}\left(x,t,z\right)\left(I-G\left(x,t,z\right)\right)$, $\,z\in \sum ,$ where $G\left(x,t,z\right)=\left(\begin{array}{cc}\rho \left(z\right)\tilde{\rho }\left(z\right) & \rho \left(z\right){{\rm{e}}}^{2{\rm{i}}\theta \left(z\right)}\\ -\tilde{\rho }\left(z\right){{\rm{e}}}^{-2{\rm{i}}\theta \left(z\right)} & 0\end{array}\right)$. |
| • | $\left\{\begin{array}{c}M\left(x,t,z\right)=I+O\left({z}^{-1}\right),z\to \infty ,\\ M\left(x,t,z\right)=-\tfrac{{\rm{i}}}{z}{\sigma }_{3}{Q}_{-}+O\left(1\right),z\to 0.\end{array}\right.$ |
$\begin{eqnarray}q\left(x,t\right)={\rm{i}}\mathop{\mathrm{lim}}\limits_{z\to \infty }{\left({zM}\right)}_{12}.\end{eqnarray}$
2.6. Discrete spectrum
Suppose that z0 is an N-order zero point of ${s}_{22}\left(z\right)$ at ${D}^{+}\cap \left\{z\in C:\mathrm{Im}z\gt 0,\ \left|z\right|\gt {q}_{0}\right\}$. According to the symmetry conditions of proposition 2.3 , we know
$\begin{eqnarray}\begin{array}{rcl}{s}_{22}\left({z}_{0}\right) & = & 0\iff {s}_{11}^{* }\left({z}_{0}^{* }\right)\\ & = & 0\iff {s}_{11}\left(-\displaystyle \frac{{q}_{0}^{2}}{{z}_{0}}\right)=0\iff {s}_{22}\left(-\displaystyle \frac{{q}_{0}^{2}}{{z}_{0}^{* }}\right)=0.\end{array}\end{eqnarray}$
So, the discrete spectrum is the set $\begin{eqnarray}Z=\left\{{z}_{0},{z}_{0}^{* },-\displaystyle \frac{{q}_{0}^{2}}{{z}_{0}},-\displaystyle \frac{{q}_{0}^{2}}{{z}_{0}^{* }}\right\}.\end{eqnarray}$
3. Soliton solutions
3.1. The RH problem with a single higher-order pole
Let z0 ∈ D+ be the Nth-order pole, from the symmetry properties of proposition 2.3 , we know that −z0, $\pm \tfrac{{q}_{0}^{2}}{{z}_{0}^{* }}\in {D}^{+}$ is also the N-order zero point of ${s}_{22}\left(z\right)$. Then $\pm {z}_{0}^{* }$ and $\pm \tfrac{{q}_{0}^{2}}{{z}_{0}}$ are the N-order poles of ${s}_{11}\left(z\right)$. Taking $v$1 = z0 and ${\upsilon }_{2}=\tfrac{{q}_{0}^{2}}{{z}_{0}^{* }}$, then ${s}_{22}\left(z\right)$ can be expressed as1 ) can also be obtained from equation (75b ).
$\begin{eqnarray}{s}_{22}\left({z}^{2}\right)={\left(z-{\upsilon }_{1}^{2}\right)}^{N}{\left(z-{\upsilon }_{2}^{2}\right)}^{N}{s}_{0}\left(z\right),\end{eqnarray}$
where ${s}_{0}\left(z\right)\ne 0$ in D+. According to the Laurent series expansion in poles, the reflection coefficients $\rho \left(z\right)$ and ${\rho }^{* }\left({z}^{* }\right)$ can be expanded as $\begin{eqnarray}\begin{array}{r}\rho \left(z\right)={\rho }_{0}\left(z\right)+\displaystyle \sum _{j=1}^{N}\displaystyle \frac{{\rho }_{1,j}}{{\left(z-{\upsilon }_{1}\right)}^{j}},\,\\ \,\rho \left(z\right)={\tilde{\rho }}_{0}(z)+\displaystyle \sum _{j=1}^{N}\displaystyle \frac{{\left(-1\right)}^{j+1}{\rho }_{2,j}}{{\left(z-{\upsilon }_{2}\right)}^{j}},\,\\ {\rho }^{* }\left({z}^{* }\right)={\rho }_{0}^{* }\left({z}^{* }\right)+\displaystyle \sum _{j=1}^{N}\displaystyle \frac{{\rho }_{1,j}^{* }}{{\left(z-{\upsilon }_{1}^{* }\right)}^{j}},\,\\ {\rho }^{* }\left({z}^{* }\right)={\tilde{\rho }}_{0}^{* }\left({z}^{* }\right)+\displaystyle \sum _{j=1}^{N}\displaystyle \frac{{\left(-1\right)}^{j+1}{\rho }_{2,j}^{* }}{{\left(z-{\upsilon }_{2}^{* }\right)}^{j}},\,\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}{\rho }_{m,{m}_{j}} & = & \mathop{\mathrm{lim}}\limits_{z\to {\upsilon }_{m}}\displaystyle \frac{1}{\left(N-{m}_{j}\right)!}\displaystyle \frac{{\partial }^{N-{m}_{j}}}{\partial {z}^{N-{m}_{j}}}\left[{\left(z-{\upsilon }_{m}\right)}^{N}{\rho }_{m}\left(z\right)\right],\\ m & = & 1,2,\ {m}_{j}=1,2,\cdots N,\end{array}\end{eqnarray}$
and ${\rho }_{0},m\left(z\right)$ and ${\hat{\rho }}_{0},m\left(z\right)$ are analytic for all z ∈ D+. According to the definition of $M\left(x,t,z\right)$, we know that z = ±$v$m and $z=\pm {\upsilon }_{m}^{* }$ are N-order poles of ${M}_{12}\left(x,t,z\right)$ and ${M}_{11}\left(x,t,z\right)$, respectively. Therefore, $M\left(x,t,z\right)$ can be expanded as $\begin{eqnarray}{M}_{11}\left(x,t,z\right)=1+\displaystyle \sum _{m=1}^{2}\displaystyle \sum _{l=1}^{N}\left(\displaystyle \frac{{F}_{m.l}\left(x,t\right)}{{\left(z-{\upsilon }_{m}^{* }\right)}^{l}}+\displaystyle \frac{{G}_{m,l}\left(x,t\right)}{{\left(z+{\upsilon }_{m}^{* }\right)}^{l}}\right),\end{eqnarray}$
$\begin{eqnarray}{M}_{12}\left(x,t,z\right)=-\displaystyle \frac{{\rm{i}}}{z}{q}_{-}+\displaystyle \sum _{m=1}^{2}\displaystyle \sum _{l=1}^{N}\left(\displaystyle \frac{{H}_{m,l}\left(x,t\right)}{{\left(z-{\upsilon }_{m}\right)}^{l}}+\displaystyle \frac{{L}_{m,l}\left(x,t\right)}{{\left(z-{\upsilon }_{m}\right)}^{l}}\right),\end{eqnarray}$
where ${F}_{m,l}\left(x,t\right),\ {G}_{m,l}\left(x,t\right),\ {H}_{m,l}\left(x,t\right)$ and ${L}_{m,l}\left(x,t\right)\left(l=1,2,\cdots N\right)$ are unknown functions to be determined. Once these functions are solved, the solution of equation (In what follows, according to Taylor series expansion, we have75b ) and (78 ) into equation (79 ), and comparing the coefficients of ${\left(z-{\upsilon }_{m}\right)}^{-s}$ with equation (75b ), we can obtain77a ) and substituting equations (75b ) and (77c ) into them, we have80a ), we can obtain75b ) and (86a ), we have4 ), the N-order soliton solution of equation (1 ) is
$\begin{eqnarray}\begin{array}{l}{{\rm{e}}}^{2{\rm{i}}\theta \left(z\right)}=\displaystyle \sum _{s=0}^{+\infty }{f}_{1,s}\left(x,t\right){\left(z-{\upsilon }_{1}\right)}^{s},\ \\ {{\rm{e}}}^{2{\rm{i}}\theta \left(z\right)}=\displaystyle \sum _{s=0}^{+\infty }{\left(-1\right)}^{s}{f}_{2,s}\left(x,t\right){\left(z-{\upsilon }_{2}\right)}^{s},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{{\rm{e}}}^{-2{\rm{i}}\theta \left(z\right)}=\displaystyle \sum _{s=0}^{+\infty }{f}_{1,s}^{* }\left(x,t\right){\left(z-{\upsilon }_{1}^{* }\right)}^{s},\ \\ {{\rm{e}}}^{-2{\rm{i}}\theta \left(z\right)}=\displaystyle \sum _{s=0}^{+\infty }{\left(-1\right)}^{s}{f}_{2,s}^{* }\left(x,t\right){\left(z-{\upsilon }_{2}^{* }\right)}^{s},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{M}_{11}\left(x,t,z\right) & = & \displaystyle \sum _{s=0}^{+\infty }{g}_{1,s}\left(x,t\right){\left(z-{\upsilon }_{1}\right)}^{s},\ \\ {M}_{11}\left(x,t,z\right) & = & \displaystyle \sum _{s=0}^{+\infty }{\left(-1\right)}^{s}{g}_{2,s}\left(x,t\right){\left(z-{\upsilon }_{2}\right)}^{s},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{M}_{12}\left(x,t,z\right) & = & \displaystyle \sum _{s=0}^{+\infty }{h}_{1,s}\left(x,t\right){\left(z-{\upsilon }_{1}^{* }\right)}^{s},\\ {M}_{12}\left(x,t,z\right) & = & \displaystyle \sum _{s=0}^{+\infty }{\left(-1\right)}^{s}{h}_{2,s}\left(x,t\right){\left(z-{\upsilon }_{2}^{* }\right)}^{s},\end{array}\end{eqnarray}$
where $\begin{eqnarray}{f}_{m,s}\left(x,t\right)=\mathop{\mathrm{lim}}\limits_{z\to {\upsilon }_{m}}\displaystyle \frac{1}{s!}\displaystyle \frac{{\partial }^{s}}{\partial {z}^{s}}{{\rm{e}}}^{2{\rm{i}}\theta \left(z\right)},\end{eqnarray}$
$\begin{eqnarray}{g}_{m,s}\left(x,t\right)=\mathop{\mathrm{lim}}\limits_{z\to {\upsilon }_{m}}\displaystyle \frac{1}{s!}\displaystyle \frac{{\partial }^{s}}{\partial {z}^{s}}{M}_{11}\left(x,t,z\right),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{h}_{m,s}\left(x,t\right) & = & \mathop{\mathrm{lim}}\limits_{z\to {\upsilon }_{m}^{* }}\displaystyle \frac{1}{s!}\displaystyle \frac{{\partial }^{s}}{\partial {z}^{s}}{M}_{12}\left(x,t,z\right),\\ m & = & 1,2,\ {\rm{s}}=0,1,2\cdots .\end{array}\end{eqnarray}$
When z ∈ D+, the function $M\left(z\right)$ has the expansions at $z={\upsilon }_{m}\left(m=1,2.\right),$ $\begin{eqnarray}{M}_{11}\left(x,t,z\right)={\mu }_{-,11}\left(z\right)=\displaystyle \sum _{s=0}^{+\infty }{g}_{m,s}\left(x,t\right){\left(z-{\upsilon }_{m}\right)}^{s},\end{eqnarray}$
$\begin{eqnarray}{M}_{12}\left(x,t,z\right)=\displaystyle \frac{{\mu }_{+,12}\left(z\right)}{{s}_{22}\left(z\right)}={\mu }_{-,12}\left(z\right)+\rho \left(z\right){{\rm{e}}}^{2{\rm{i}}\theta \left(z\right)}{\mu }_{-,11}\left(z\right).\end{eqnarray}$
Substituting equations ( $\begin{eqnarray}{H}_{l}\left(x,t\right)=\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}{\rho }_{1,j}{f}_{1,j-l-s}\left(x,t\right){g}_{1,s}\left(x,t\right),\end{eqnarray}$
$\begin{eqnarray}{L}_{l}\left(x,t\right)=\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}{\left(-1\right)}^{l+1}{\rho }_{2,j}{f}_{2,j-l-s}\left(x,t\right){g}_{2,s}\left(x,t\right).\end{eqnarray}$
Similarly, when z ∈ D−, $\begin{eqnarray}\begin{array}{ccl}{F}_{l}\left(x,t\right) & = & -\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}{\rho }_{1,j}^{\ast }{f}_{1,j-l-s}^{\ast }\left(x,t\right){h}_{1,s}\left(x,t\right),\\ {G}_{l}\left(x,t\right) & = & -\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}{\left(-1\right)}^{l+1}{\rho }_{2,j}^{\ast }{f}_{2,j-l-s}^{\ast }\left(x,t\right){h}_{2,s}\left(x,t\right),\end{array}\end{eqnarray}$
can be gotten by comparing the coefficients of ${\left(z-{\upsilon }_{m}^{* }\right)}^{-s}$. Actually, gm,s and hm,s can be expressed by these functions ${H}_{m,l}\left(x,t\right),\ {L}_{m,l}\left(x,t\right)$ and ${F}_{m,l}\left(x,t\right),\ {G}_{m,l}\left(x,t\right)$ respectively. Recalling equation ( $\begin{eqnarray}\begin{array}{l}{h}_{1,s}\left(x,t\right)=\frac{{\left(-1\right)}^{s+1}}{{\left({\upsilon }_{1}^{* }\right)}^{s+1}}{\rm{i}}{q}_{-}\\ \,+\displaystyle \sum _{l=1}^{N}\left(\begin{array}{l}l+s-1\\ s\end{array}\right)\left(\frac{{\left(-1\right)}^{s}{H}_{l}\left(x,t\right)}{{\left({\upsilon }_{1}^{* }-{\upsilon }_{1}\right)}^{s+l}}+\frac{{\left(-1\right)}^{s}{L}_{l}\left(x,t\right)}{{\left({\upsilon }_{1}^{* }-{\upsilon }_{2}\right)}^{s+l}}\right),\\ \,s=0,1,2\cdots ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{h}_{2,s}\left(x,t\right) & = & \displaystyle \frac{{\left(-1\right)}^{s+1}}{{\left({\upsilon }_{2}^{* }\right)}^{s+1}}{{\rm{i}}{q}}_{-}+\displaystyle \sum _{l=1}^{N}\left(\begin{array}{c}l+s-1\\ s\end{array}\right)\\ & & \times \,\left(\displaystyle \frac{{\left(-1\right)}^{s}{H}_{l}\left(x,t\right)}{{\left({\upsilon }_{2}^{* }-{\upsilon }_{1}\right)}^{s+l}}+\displaystyle \frac{{\left(-1\right)}^{s}{L}_{l}\left(x,t\right)}{{\left({\upsilon }_{2}^{* }-{\upsilon }_{2}\right)}^{s+l}}\right),\\ & & s=0,1,2\cdots ,\end{array}\end{eqnarray}$
$\begin{eqnarray}{g}_{1,s}\left(x,t\right)=\left\{\begin{array}{ll}1+\displaystyle \sum _{l=1}^{N}\left(\tfrac{{\left(-1\right)}^{s}{F}_{l}\left(x,t\right)}{{\left({\upsilon }_{1}-{\upsilon }_{1}^{* }\right)}^{s+l}}+\tfrac{{\left(-1\right)}^{s}{G}_{l}\left(x,t\right)}{{\left({\upsilon }_{1}-{\upsilon }_{2}^{* }\right)}^{s+l}}\right), & s=0,\\ \displaystyle \sum _{l=1}^{N}\left(\begin{array}{c}l+s-1\\ s\end{array}\right)\left(\tfrac{{\left(-1\right)}^{s}{F}_{l}\left(x,t\right)}{{\left({\upsilon }_{1}-{\upsilon }_{2}^{* }\right)}^{s+l}}+\tfrac{{\left(-1\right)}^{s}{L}_{l}\left(x,t\right)}{{\left({\upsilon }_{2}-{\upsilon }_{2}^{* }\right)}^{s+l}}\right), & s=0,1,2\cdots ,\end{array}\right.\end{eqnarray}$
$\begin{eqnarray}{g}_{2,s}\left(x,t\right)=\left\{\begin{array}{cc}1+\displaystyle \sum _{l=1}^{N}\left(\tfrac{{\left(-1\right)}^{s}{F}_{l}\left(x,t\right)}{{\left({\upsilon }_{2}-{\upsilon }_{1}^{* }\right)}^{s+l}}+\tfrac{{\left(-1\right)}^{s}{G}_{l}\left(x,t\right)}{{\left({\upsilon }_{2}-{\upsilon }_{2}^{* }\right)}^{s+l}}\right), & s=0,\\ \displaystyle \sum _{l=1}^{N}\left(\begin{array}{c}l+s-1\\ s\end{array}\right)\left(\displaystyle \frac{{\left(-1\right)}^{s}{F}_{l}\left(x,t\right)}{{\left({\upsilon }_{2}-{\upsilon }_{2}^{* }\right)}^{s+l}}+\displaystyle \frac{{\left(-1\right)}^{s}{L}_{l}\left(x,t\right)}{{\left({\upsilon }_{2}-{\upsilon }_{2}^{* }\right)}^{s+l}}\right), & s=0,1,2\cdots .\end{array}\right.\end{eqnarray}$
Then, substituting the above equations into equation ( $\begin{eqnarray}{F}_{l}\left(x,t\right)=-{\rm{i}}{q}_{-}\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}\frac{{\left(-1\right)}^{s+1}}{{\left({\upsilon }_{1}^{* }\right)}^{s+1}}{\rho }_{1,j}^{* }{f}_{1,j-l-s}^{* }\left(x,t\right)\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}\displaystyle \sum _{p=1}^{N}\left(\begin{array}{c}p+s-1\\ s\end{array}\right){\rho }_{1,j}^{* }{f}_{1,j-l-s}^{* }\left(x,t\right)\\ \qquad \times \,\left(\displaystyle \frac{{\left(-1\right)}^{s}{H}_{p}\left(x,t\right)}{{\left({\upsilon }_{1}^{* }-{\upsilon }_{1}\right)}^{s+p}}+\displaystyle \frac{{\left(-1\right)}^{s}{L}_{p}\left(x,t\right)}{{\left({\upsilon }_{1}^{* }-{\upsilon }_{2}\right)}^{s+p}}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}{G}_{l}\left(x,t\right)=-{\rm{i}}{q}_{-}\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}\frac{{\left(-1\right)}^{l+1}{\left(-1\right)}^{s+1}}{{\left({\upsilon }_{2}^{* }\right)}^{s+1}}{\rho }_{2,j}^{* }{f}_{2,j-l-s}^{* }\left(x,t\right)\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}\displaystyle \sum _{p=1}^{N}{\left(-1\right)}^{l+1}\left(\begin{array}{c}p+s-1\\ s\end{array}\right){\rho }_{2,j}^{* }{f}_{2,j-l-s}^{* }\left(x,t\right)\\ \qquad \times \,\left(\displaystyle \frac{{\left(-1\right)}^{s}{H}_{p}\left(x,t\right)}{{\left({\upsilon }_{2}^{* }-{\upsilon }_{1}\right)}^{s+p}}+\displaystyle \frac{{\left(-1\right)}^{s}{L}_{p}\left(x,t\right)}{{\left({\upsilon }_{2}^{* }-{\upsilon }_{2}\right)}^{s+p}}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}{H}_{l}\left(x,t\right)=\displaystyle \sum _{j=l}^{N}{\rho }_{1,j}{f}_{1,j-l}\left(x,t\right)\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\ +\ \displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}\displaystyle \sum _{p=1}^{N}\left(\begin{array}{c}p+s-1\\ s\end{array}\right){\rho }_{1,j}{f}_{1,j-l-s}\left(x,t\right)\\ \qquad \times \,\left(\displaystyle \frac{{\left(-1\right)}^{s}{F}_{p}\left(x,t\right)}{{\left({\upsilon }_{1}-{\upsilon }_{1}^{* }\right)}^{s+p}}+\displaystyle \frac{{\left(-1\right)}^{s}{G}_{p}\left(x,t\right)}{{\left({\upsilon }_{1}-{\upsilon }_{2}^{* }\right)}^{s+p}}\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}{L}_{l}\left(x,t\right)=\displaystyle \sum _{j=l}^{N}{\left(-1\right)}^{l+1}{\rho }_{2,j}{f}_{2,j-l}\left(x,t\right)\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}+\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}\displaystyle \sum _{p=1}^{N}{\left(-1\right)}^{l+1}\left(\begin{array}{c}p+s-1\\ s\end{array}\right){\rho }_{2,j}{f}_{2,j-l-s}\left(x,t\right)\\ \left(\displaystyle \frac{{\left(-1\right)}^{s}{F}_{p}\left(x,t\right)}{{\left({\upsilon }_{2}-{\upsilon }_{1}^{* }\right)}^{s+p}}+\displaystyle \frac{{\left(-1\right)}^{s}{G}_{p}\left(x,t\right)}{{\left({\upsilon }_{2}-{\upsilon }_{2}^{* }\right)}^{s+p}}\right).\end{array}\end{eqnarray}$
Let us define $\begin{eqnarray*}\begin{array}{l}\left|{\eta }_{1}\right\rangle ={\left({\eta }_{\mathrm{1,1}},{\eta }_{\mathrm{1,2}},\cdots {\eta }_{1,N}\right)}^{{\rm{T}}},\\ {\eta }_{1,l}=-{\rm{i}}{q}_{-}\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}\frac{{\left(-1\right)}^{s+1}}{{\left({\upsilon }_{1}^{* }\right)}^{s+1}}{\rho }_{1,j}^{* }{f}_{1,j-l-s}^{* }\left(x,t\right),\\ \left|{\eta }_{2}\right\rangle ={\left({\eta }_{\mathrm{2,1}},{\eta }_{\mathrm{2,2}},\cdots {\eta }_{2,N}\right)}^{{\rm{T}}},\\ {\eta }_{2,l}=-{\rm{i}}{q}_{-}\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}{\left(-1\right)}^{l+1}\\ \frac{{\left(-1\right)}^{s+1}}{{\left({\upsilon }_{2}^{* }\right)}^{s+1}}{\rho }_{2,j}^{* }{f}_{2,j-l-s}^{* }\left(x,t\right),\\ \left|{\hat{\eta }}_{1}\right\rangle ={\left({\hat{\eta }}_{\mathrm{1,1}},{\hat{\eta }}_{\mathrm{1,2}},\cdots {\hat{\eta }}_{1,N}\right)}^{{\rm{T}}},\\ {\hat{\eta }}_{1,l}=\displaystyle \sum _{j=l}^{N}{\rho }_{1,j}{f}_{1,j-l}\left(x,t\right),\\ \left|{\hat{\eta }}_{2}\right\rangle ={\left({\hat{\eta }}_{\mathrm{2,1}},{\hat{\eta }}_{\mathrm{2,2}},\cdots {\hat{\eta }}_{2,N}\right)}^{{\rm{T}}},\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}{\hat{\eta }}_{2,l}=\displaystyle \sum _{j=l}^{N}{\left(-1\right)}^{l+1}{\rho }_{2,j}{f}_{2,j-l}\left(x,t\right),\\ {{\rm{\Omega }}}_{1}={\left[{{\rm{\Omega }}}_{1,{lp}}\right]}_{N\times N}=-\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}\left(\begin{array}{c}p+s-1\\ s\end{array}\right)\\ \,\times \,\frac{{\left(-1\right)}^{s}{\rho }_{1,j}^{* }{f}_{1,j-s-p}^{* }\left(x,t\right)}{{\left({\upsilon }_{1}^{* }-{\upsilon }_{1}\right)}^{s+p}},\\ {{\rm{\Omega }}}_{2}={\left[{{\rm{\Omega }}}_{2,{lp}}\right]}_{N\times N}=-\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}\left(\begin{array}{c}p+s-1\\ s\end{array}\right)\\ \,\times \,\frac{{\left(-1\right)}^{s}{\rho }_{1,j}^{* }{f}_{1,j-s-p}^{* }\left(x,t\right)}{{\left({\upsilon }_{1}^{* }-{\upsilon }_{2}\right)}^{s+p}},\\ {{\rm{\Omega }}}_{3}={\left[{{\rm{\Omega }}}_{3,{lp}}\right]}_{N\times N}=-\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}\left(\begin{array}{c}p+s-1\\ s\end{array}\right)\\ \,\times \,\frac{{\left(-1\right)}^{l+1}{\left(-1\right)}^{s}{\rho }_{2,j}^{* }{f}_{2,j-s-p}^{* }\left(x,t\right)}{{\left({\upsilon }_{2}^{* }-{\upsilon }_{1}\right)}^{s+p}},\\ {{\rm{\Omega }}}_{4}={\left[{{\rm{\Omega }}}_{1,{lp}}\right]}_{N\times N}=-\displaystyle \sum _{j=l}^{N}\displaystyle \sum _{s=0}^{j-l}\left(\begin{array}{c}p+s-1\\ s\end{array}\right)\\ \,\times \,\frac{{\left(-1\right)}^{l+1}{\left(-1\right)}^{s}{\rho }_{2,j}^{* }{f}_{2,j-s-p}^{* }\left(x,t\right)}{{\left({\upsilon }_{2}^{* }-{\upsilon }_{1}\right)}^{s+p}},\\ \left|F\right\rangle ={\left({F}_{1},\cdots ,{F}_{N}\right)}^{{\rm{T}}},\ \ \ \left|G\right\rangle ={\left({G}_{1},\cdots ,{G}_{N}\right)}^{{\rm{T}}},\\ \left|H\right\rangle ={\left({H}_{1},\cdots ,{H}_{N}\right)}^{{\rm{T}}},\quad \ \left|L\right\rangle ={\left({L}_{1},\cdots ,{L}_{N}\right)}^{{\rm{T}}}.\end{array}\end{eqnarray}$
Let $\begin{eqnarray}\begin{array}{l}{\rm{\Omega }}=\left(\begin{array}{cc}{{\rm{\Omega }}}_{1} & {{\rm{\Omega }}}_{2}\\ {{\rm{\Omega }}}_{3} & {{\rm{\Omega }}}_{4}\end{array}\right),\quad \ \left|{\alpha }_{1}\right\rangle ={\left(\left|{\eta }_{1}\right\rangle \left|{\eta }_{2}\right\rangle \right)}^{{\rm{T}}},\\ \left|{\alpha }_{2}\right\rangle ={\left(\left|{\hat{\eta }}_{1}\right\rangle \ \left|{\hat{\eta }}_{2}\right\rangle \right)}^{{\rm{T}}},\end{array}\end{eqnarray}$
so we have $\begin{eqnarray}\begin{array}{rcl}\left(\begin{array}{cc}\left|H\right\rangle & \left|L\right\rangle \end{array}\right) & = & -{\left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}\right)}^{-1}{{\rm{\Omega }}}^{* }\left|{\alpha }_{1}\right\rangle \\ & & +\,{\left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}\right)}^{-1}\left|{\alpha }_{2}\right\rangle ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\left(\begin{array}{cc}\left|F\right\rangle & \left|G\right\rangle \end{array}\right) & = & -{\rm{\Omega }}\left({\left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}\right)}^{-1}{{\rm{\Omega }}}^{* }\left|{\alpha }_{1}\right\rangle \right)\\ & & +\,{\left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}\right)}^{-1}\left|{\alpha }_{2}\right\rangle +\left|{\alpha }_{1}\right\rangle ,\end{array}\end{eqnarray}$
where $\begin{eqnarray}{I}_{\sigma }=\left(\begin{array}{cc}{I}_{N\times N} & 0\\ 0 & {I}_{N\times N}\end{array}\right).\end{eqnarray}$
Applying equations ( $\begin{eqnarray}\begin{array}{ccl}{M}_{12}\left(x,t,z\right) & = & -\frac{{\rm{i}}}{z}{q}_{-}\\ & & +\,\frac{\det \left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}+\left|{\alpha }_{2}\right\rangle \left\langle Y\right|\right)-\det \left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}+{{\rm{\Omega }}}^{* }\left|{\alpha }_{1}\right\rangle \left\langle Y\right|\right)}{\det \left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}\right)},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\left\langle Y\right|=\left(\displaystyle \frac{1}{z-{\upsilon }_{1}},\cdots ,\displaystyle \frac{1}{{\left(z-{\upsilon }_{1}\right)}^{N}},\displaystyle \frac{1}{z-{\upsilon }_{2}},\cdots ,\displaystyle \frac{1}{{\left(z-{\upsilon }_{2}\right)}^{N}}\right).\end{eqnarray}$
With the nonzero boundary conditions equation ( $\begin{eqnarray}\begin{array}{rcl}q\left(x,t\right) & = & {\rm{i}}\mathop{\mathrm{lim}}\limits_{z\to \infty }{\left({zM}\right)}_{12}\\ & = & {\rm{i}}\mathop{\mathrm{lim}}\limits_{z\to \infty }z\left(-\displaystyle \frac{{\rm{i}}}{z}{q}_{-}+\displaystyle \frac{\det \left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}+\left|{\alpha }_{2}\right\rangle \left\langle Y\right|\right)-\det \left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}+{{\rm{\Omega }}}^{* }\left|{\alpha }_{1}\right\rangle \left\langle Y\right|\right)}{\det \left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}\right)}\right)\\ & = & {q}_{-}+{\rm{i}}\displaystyle \frac{\det \left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}+\left|{\alpha }_{2}\right\rangle \left\langle {Y}_{0}\right|\right)-\det \left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}+{{\rm{\Omega }}}^{* }\left|{\alpha }_{1}\right\rangle \left\langle {Y}_{0}\right|\right)}{\det \left({I}_{\sigma }+{{\rm{\Omega }}}^{* }{\rm{\Omega }}\right)},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\left\langle {Y}_{0}\right|={\left(\mathrm{1,0},\cdots ,\mathrm{0,1,0},\cdots ,0\right)}_{1\times 2N}.\end{eqnarray}$
Figure 1. The soliton solution with one simple pole of the LPD equation with $v$1 = 2i. (a) The soliton solution. (b) The density plot. |
Figure 2. The soliton solution with one simple pole of the LPD equation with ${\upsilon }_{1}=\tfrac{1}{3}+2{\rm{i}}$. (a) The soliton solution. (b) The density plot. |
It can be found that when z is a pure imaginary number, the propagation of the solution is parallel to the t-axis, which is usually called the steady-state breather soliton solution. When the real part of z is not equal to zero, the propagation of the solution is not parallel to the x-axis and t-axis, it is usually called the unsteady breather soliton solution, and it can be found that the arrangement of the soliton solution is very close. Comparing with the focusing nonlinear Schrödinger equation [27], the solution will be arranged more closely, that is, the propagation speed will be accelerated due to the addition of β in equation (1 ).
Figure 3. The soliton solution with one second-order pole of the LPD equation with $v$1 = 2i. (a) The soliton solution. (b) The density plot. |
Figure 4. The soliton solution with one second-order pole of the LPD equation with ${\upsilon }_{1}=\tfrac{1}{3}+2{\rm{i}}$. (a) The soliton solution. (b) The density plot. |
It can be found that compared with the simple pole soliton solution, the second-order pole soliton solution has two columns of breather solutions moving and colliding, and the propagation speed of the solution is still very fast.
3.2. The RH problem with multiple higher-order poles
In this section, we will discuss the case that ${s}_{22}\left(z\right)$ has N-order zero points z1, z2, ⋯, zn ∈ D+ and their powers are n1, n2, ⋯ nN, respectively. Let
$\begin{eqnarray}{\xi }_{j}=\left\{\begin{array}{cc}{z}_{j} & ,\ j=1,2,\cdots ,N,\\ -\tfrac{{q}_{0}^{2}}{{z}_{j-N}^{* }} & ,\ j=N+1,N+2,\cdots ,2N.\end{array}\right.\end{eqnarray}$
Then the reflection coefficient $\rho \left(z\right)$ can be expressed as follows $\begin{eqnarray}\begin{array}{rcl}\rho \left(z\right) & = & {\rho }_{0}\left(z\right)+\displaystyle \sum _{{m}_{j}=1}^{{n}_{j}}\displaystyle \frac{{\rho }_{j,{m}_{j}}}{{\left(z-{\xi }_{j}\right)}^{{m}_{j}}},j=1,2,\cdots ,N,\\ \rho \left(z\right) & = & {\tilde{\rho }}_{0}\left(z\right)+\displaystyle \sum _{{m}_{j}=1}^{{n}_{j}}\displaystyle \frac{{\left(-1\right)}^{{m}_{j}+1}{\rho }_{j,{m}_{j}}}{{\left(z-{\xi }_{j}\right)}^{{m}_{j}}},\\ j & = & N+1,N+2,\cdots ,2N,\end{array}\end{eqnarray}$
where ${\rho }_{0}\left(z\right)$ and ${\tilde{\rho }}_{0}\left(z\right)$ are analytic for all z ∈ D+ and $\begin{eqnarray}{\rho }_{j,{m}_{j}}=\mathop{\mathrm{lim}}\limits_{z\to {\xi }_{j}}\displaystyle \frac{1}{\left({n}_{j}-{m}_{j}\right)!}\displaystyle \frac{{\partial }^{{n}_{j}-{m}_{j}}}{\partial {z}^{{n}_{j}-{m}_{j}}}\left[{\left(z-{\xi }_{m}\right)}^{{n}_{j}}\rho \left(z\right)\right].\end{eqnarray}$
Like the case of one higher-order pole discussed in above, the solution of equation (1 ) with multiple higher-order poles can be obtained as follows
$\begin{eqnarray}\begin{array}{ccl}q\left(x,t\right) & = & {q}_{-}\\ & & +\,{\rm{i}}\frac{\det \left({I}_{\sigma }+{{K}}^{* }{K}+\left|{X}\right\rangle \langle {Y}_{0}| \right)-\det \left({I}_{\sigma }+{{K}}^{* }{K}+{{K}}^{* }\left|{\rm{\Gamma }}\right\rangle \langle {Y}_{0}| \right)}{\det \left({I}_{\sigma }+{{K}}^{* }{K}\right)},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{cll}\left|{\Gamma }\right\rangle & = & {\left({\tau }_{1},{\tau }_{2},\cdots ,{\tau }_{2N}\right)}^{{\rm{T}}},\quad {\tau }_{j}={\left({\tau }_{j,1},{\tau }_{j,2},\cdots ,{\tau }_{j,{n}_{j}}\right)}^{{\rm{T}}},\\ \left|{X}\right\rangle & = & {\left({\chi }_{1},{\chi }_{2},\cdots ,{\chi }_{2N}\right)}^{{\rm{T}}},\quad {\chi }_{j}={\left({\chi }_{j,1},{\chi }_{j,2},\cdots ,{\chi }_{j,{n}_{j}}\right)}^{{\rm{T}}},\\ {\tau }_{j,l} & = & -{\rm{i}}{q}_{-}\displaystyle \sum _{{m}_{j}=l}^{N}\displaystyle \sum _{{s}_{j}=0}^{{m}_{j}-l}\frac{{\left(-1\right)}^{{s}_{j}+1}}{{\left({\xi }_{1}^{* }\right)}^{{s}_{j}+1}}\\ & & \times \,{\rho }_{j,{m}_{j}}^{* }{f}_{j,{m}_{j}-{s}_{j}-l}^{* }\left(x,t\right),j=1,2,\cdots N,\\ {\tau }_{j,l} & = & -{\rm{i}}{q}_{-}\displaystyle \sum _{{m}_{j}=l}^{N}\displaystyle \sum _{{s}_{j}=0}^{{m}_{j}-l}\frac{{\left(-1\right)}^{l+1}{\left(-1\right)}^{{s}_{j}+1}}{{\left({\xi }_{1}^{* }\right)}^{{s}_{j}+1}}\\ & & \times \,{\rho }_{j,{m}_{j}}^{* }{f}_{j,{m}_{j}-{s}_{j}-l}^{* }\left(x,t\right),j=N+1,N+2,\cdots 2N,\\ {\chi }_{j,l} & = & \displaystyle \sum _{{m}_{j}=l}^{{n}_{j}}{\rho }_{j,{m}_{j}}{f}_{j,{m}_{j}-l}\left(x,t\right),j=1,2,\cdots N,\\ {\chi }_{j,l} & = & \displaystyle \sum _{{m}_{j}=l}^{{n}_{j}}{\left(-1\right)}^{l+1}{\rho }_{j,{m}_{j}}{f}_{j,{m}_{j}-l}\left(x,t\right),\\ j & = & N+1,N+2,\cdots 2N,\\ {\boldsymbol{K}} & =\, & \left(\begin{array}{llll}\left({\kappa }_{11}\right) & \left({\kappa }_{12}\right) & \cdots & \left({\kappa }_{1\left(2N\right)}\right)\\ \left({\kappa }_{21}\right) & \left({\kappa }_{22}\right) & \cdots & \left({\kappa }_{2\left(2N\right)}\right)\\ \vdots & \vdots & & \vdots \\ \left({\kappa }_{\left(2N\right)1}\right) & \left({\kappa }_{\left(2N\right)2}\right) & \cdots & \left({\kappa }_{\left(2N\right)\left(2N\right)}\right)\end{array}\right),\\ \left({\theta }_{{js}}\right) & = & {\left({\left({\theta }_{{js}}\right)}_{{pq}}\right)}_{{n}_{j}\times {n}_{s}}\\ & = & -\displaystyle \sum _{{m}_{j}=p}^{{n}_{j}}\displaystyle \sum _{{s}_{j}=0}^{{m}_{j}-p}\left(\begin{array}{l}q+{s}_{j}-1\\ {s}_{j}\end{array}\right)\\ & & \times \,{\frac{{\left(-1\right)}^{{s}_{j}}{\rho }_{j,{m}_{j}}^{* }{f}_{j,{m}_{j}-{s}_{j}-p}^{* }\left(x,t\right)}{{\left({\xi }_{j}^{* }-{\xi }_{s}\right)}^{{s}_{j}+q}}}_{{n}_{j}\times {n}_{s}},\\ j & = & 1,2,\cdots N,\\ \left({\theta }_{{js}}\right) & = & {\left({\left({\theta }_{{js}}\right)}_{{pq}}\right)}_{{n}_{j}\times {n}_{s}}\\ & = & {\left(-\displaystyle \sum _{{m}_{j}=p}^{{n}_{j}}\displaystyle \sum _{{s}_{j}=0}^{{m}_{j}-p}{\left(-1\right)}^{l+1}\left(\begin{array}{l}q+{s}_{j}-1\\ {s}_{j}\end{array}\right)\frac{{\left(-1\right)}^{{s}_{j}}{\rho }_{j,{m}_{j}}^{* }{f}_{j,{m}_{j}-{s}_{j}-p}^{* }\left(x,t\right)}{{\left({\xi }_{j}^{* }-{\xi }_{s}\right)}^{{s}_{j}+q}}\right)}_{{n}_{j}\times {n}_{s}},\\ j & = & N+1,N+2,\cdots 2N,\\ \langle {Y}_{0}| & = & {\left(\mathrm{1,0},\cdots ,0\right)}_{1\times {n}_{1}},\ \ {\left(\mathrm{1,0},\cdots ,0\right)}_{1\times {n}_{2}},\,\cdots ,{\left(\mathrm{1,0},\cdots ,0\right)}_{1\times {n}_{j}},\\ {I}_{\sigma } & = & \left(\begin{array}{lccc}{I}_{{n}_{1}\times {n}_{1}} & & & \\ & {I}_{{n}_{2}\times {n}_{2}} & & \\ & & \ddots & \\ & & & {I}_{{n}_{j}\times {n}_{j}}\end{array}\right),\ \ \ j=1,2,\cdots ,2N.\end{array}\end{eqnarray}$
Here, we consider the propagation of solutions corresponding to a simple pole and a second-order pole. Let $\beta =\tfrac{1}{2}$, ρ11 = ρ12 = ρ21 = ρ31 = ρ32 = ρ41 = q0 = 1, the dynamic behaviors of the solution can be obtained as shown in figure 5.
Figure 5. The soliton solutions with one simple pole and a second-order pole of the LPD equation. (a) The soliton solutions corresponding to ${z}_{1}=2{\rm{i}},{z}_{2}=\tfrac{1}{3}{\rm{i}}$. (b) The soliton solutions corresponding to ${z}_{1}=\tfrac{1}{3}+2{\rm{i}},{z}_{2}=\tfrac{1}{3}{\rm{i}}$. (c) The soliton solutions corresponding to ${z}_{1}=\tfrac{1}{3}+2{\rm{i}},{z}_{2}=1-2{\rm{i}}$. |
It can be found that in figure 5(a), when the two poles are a pure imaginary number and a non-pure imaginary number, the propagation behavior of the solution is a nonlinear superposition of the breather soliton solution corresponding to the pole of a first-order pure imaginary number and the breather soliton solution corresponding to the second-order pole of a non-pure imaginary number. In figure 5(b), when both poles are pure imaginary numbers, there are three columns of breather soliton solutions parallel to the t-axis moving and colliding. In figure 5(c), when the two poles are non-pure imaginary numbers, the breather solution corresponding to a simple pole and the breather soliton solution corresponding to the second-order pole collide after motion, but the propagation direction of the solution does not change significantly.
4. Summary
In this paper, we consider the Lakshmanan–Porsezian–Daniel (LPD) equation with nonzero boundary conditions. By analyzing the analyticity, symmetry, asymptotic properties of the Jost functions and scattering matrix, we construct the related Riemann–Hilbert (RH) problem. Due to the complexity of the Plemelj formula and the residue condition, in order to obtain the soliton solutions of equation (1 ) with multi-higher-order poles under nonzero boundary conditions, we perform Laurent expansion and Taylor series expansion on the scattering data ${s}_{22}\left(z\right)$ at the zero points to solve the corresponding RH problem.
In addition, we also analyze the dynamic behaviors of the soliton solutions. For the case of a higher-order pole, such as a simple pole and a second-order pole, the propagation of the solutions is shown in figures 1–4. The shape of the solution is similar to that of the breather soliton solution, and the propagation speed of the solution is faster. For the case of multiple higher-order poles, for example, a first-order pole plus a second-order pole, the propagation of the solution is shown in figure 5. It can be found that the propagation behaviors of the solutions can be regarded as a nonlinear superposition of a simple pole and a second-order pole, which is a very interesting phenomenon. The idea used in this paper can be extended to other integrable evolution equations, and the results can be used to study the long-time asymptotic property of equation (1 ) under nonzero boundary conditions.