N-Triple-Pole Solitons in Matrix NLS Systems: Inverse Scattering Transform Under Nonzero Boundary Conditions

Abstract

This work presents the first systematic development of the inverse scattering transform for matrix nonlinear Schrödinger equations in the case where the discrete spectrum has triple poles, under nonzero boundary conditions at infinity. These systems arise physically as reductions modeling spinor Bose-Einstein condensates with hyperfine spin $F=1$ and find applications in nonlinear optics. A uniformization variable is employed to map the underlying Riemann surface to the complex plane, enabling a complete characterization of the analyticity, symmetries, and asymptotic behaviors of the Jost functions and scattering data. Extending the established framework for simple and double poles, we show that $\mathrm{rank} P(x,t,z_n)=3$ requires a third-order zero of $\det a\left(z\right)$ at $z=z_n$, while $\mathrm{rank} P(x,t,z_n)=2$ necessitates a fourth-order zero—a nontrivial feature absent in lower-order cases. The discrete spectrum for both rank configurations is fully characterized, and the full singular behavior near a triple pole is derived, respecting the quartet symmetry $z_n$, $z_n^{*}$, $v{k}_0^2/z_n$, $v{k}_0^2/z_n^{*}$ imposed by the nonzero boundary conditions. Solving the resulting matrix Riemann-Hilbert problem with triple poles yields the potential reconstruction formula and, in the reflectionless case, explicit expressions for general N-triple-pole soliton solutions, with a detailed example for $N=1$ presented to illustrate the construction.

1. Introduction

Quantum mechanics, a cornerstone of modern physics, emerged from the inability of classical theory to describe microscopic systems. While quantum fluctuations are generally suppressed in the limit of a large number of particles, leading to classical behavior, genuine macroscopic quantum phenomena are rare. A prominent exception is the Bose–Einstein condensate (BEC). In recent decades, BEC systems have been widely explored from both theoretical and experimental perspectives [1,2,3,4,5,6,7,8].

In 1961, Gross and Pitaevskii independently formulated the mean-field theory for a dilute atomic gas BEC, resulting in the Gross–Pitaevskii (GP) equation

$$i\hslash {\displaystyle \frac{\partial }{\partial t}}\mathsf{\Psi }(\mathbf{r},t)=\left[-{\displaystyle \frac{{\hslash }^2{\nabla }^2}{2m}}+V_{\mathrm{ext}}\left(\mathbf{r}\right)+Q{\left|\mathsf{\Psi }\right|}^{ 2}\right]\mathsf{\Psi }(\mathbf{r},t),$$

(1)

where $\mathsf{\Psi }(\mathbf{r},t)$ is the condensate wave function, $V_{\mathrm{ext}}\left(\mathbf{r}\right)$ is the external trapping potential, $Q=4\pi N\hslash a_s/m$ is the coupling constant (with N the particle number, m the atomic mass, and $a_s$ the s-wave scattering length), and ${\left|\mathsf{\Psi }\right|}^{ 2}$ denotes the atomic density.

Under homogeneous or trap-free conditions, the GP equation reduces to the standard nonlinear Schrödinger (NLS) equation. Subsequent advances in optical trapping techniques have liberated the internal spin degrees of freedom, spurring extensive research on multicomponent BEC [9,10,11]. In the mean-field description that neglects collisions and thermal clouds, two interacting condensates can be modeled by a pair of coupled GP equations:

$$\begin{array}{cc}\hfill & i\hslash {\displaystyle \frac{\partial {\mathsf{\Psi }}_1}{\partial t}}=\left[-{\displaystyle \frac{{\hslash }^2}{2m_1}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}+V_1+Q_{11}{\left|{\mathsf{\Psi }}_1\right|}^2+Q_{12}{\left|{\mathsf{\Psi }}_2\right|}^2\right]{\mathsf{\Psi }}_1,\hfill \\ \hfill & i\hslash {\displaystyle \frac{\partial {\mathsf{\Psi }}_2}{\partial t}}=\left[-{\displaystyle \frac{{\hslash }^2}{2m_2}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}+V_2+Q_{21}{\left|{\mathsf{\Psi }}_1\right|}^2+Q_{22}{\left|{\mathsf{\Psi }}_2\right|}^2\right]{\mathsf{\Psi }}_2,\hfill \end{array}$$

(2)

where $Q_{ij}=2\pi {\hslash }^2{a}_{ij}/A{\mu }_{ij}$, with $a_{ij}$ being the three-dimensional scattering length, ${\mu }_{ij}={\displaystyle \frac{m_i{m}_j}{(m_i+m_j)}}$ the reduced mass, and A the cross-sectional area of the cigar-shaped BEC. Here, $m_i$ is the atomic mass in component i, $V_i$ the external trapping potential, and $N_i={\int }_{-\infty }^{\infty }{\left|{\mathsf{\Psi }}_i\right|}^{2}dx$ the number of atoms in component i.

In 1998, the group of W. Ketterle realized a Bose-Einstein condensate by confining ²³Na atomic gas with a single focused infrared laser beam as an optical dipole trap—a pioneering experimental achievement. The spin-exchange interactions among atoms give rise to rich spin textures and hyperfine structures within the condensate, leading to the distinct phenomenon known as a spinor BEC. The dynamics of a spinor BEC with three hyperfine states confined in an optical dipole trap is governed by a three-component coupled GP equation [12,13]:

$$\begin{array}{cc}\hfill & i\hslash {\displaystyle \frac{\partial {\varphi }_1}{\partial t}}=-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{{\partial }^2{\varphi }_1}{\partial x^2}}+\left({\overline{c}}_0+{\overline{c}}_2\right)\left[{\left|{\varphi }_1\right|}^2+{\left|{\varphi }_0\right|}^2\right]{\varphi }_1+\left({\overline{c}}_0-{\overline{c}}_2\right){\left|{\varphi }_{-1}\right|}^2{\varphi }_1+{\overline{c}}_2{\varphi }_{-1}^{*}{\varphi }_0^2,\hfill \\ \hfill & i\hslash {\displaystyle \frac{\partial {\varphi }_0}{\partial t}}=-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{{\partial }^2{\varphi }_0}{\partial x^2}}+\left({\overline{c}}_0+{\overline{c}}_2\right)\left[{\left|{\varphi }_1\right|}^2+{\left|{\varphi }_{-1}\right|}^2\right]{\varphi }_0+{\overline{c}}_0{\left|{\varphi }_0\right|}^2{\varphi }_0+2{\overline{c}}_2{\varphi }_0^{*}{\varphi }_1{\varphi }_{-1},\hfill \\ \hfill & i\hslash {\displaystyle \frac{\partial {\varphi }_{-1}}{\partial t}}=-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{{\partial }^2{\varphi }_{-1}}{\partial x^2}}+\left({\overline{c}}_0+{\overline{c}}_2\right)\left[{\left|{\varphi }_{-1}\right|}^2+{\left|{\varphi }_0\right|}^2\right]{\varphi }_{-1}+\left({\overline{c}}_0-{\overline{c}}_2\right){\left|{\varphi }_1\right|}^2{\varphi }_{-1}+{\overline{c}}_2{\varphi }_1^{*}{\varphi }_0^2,\hfill \end{array}$$

(3)

here ${\overline{c}}_0$ and ${\overline{c}}_2$ represent the mean-field and spin-exchange interaction strengths, respectively.

After nondimensionalization, this system reduces to a focusing-type matrix NLS equation. More generally, in the homogeneous case, an $F=1$ spinor BEC is modeled by the matrix NLS equation

$$i{Q}_t+Q_{xx}-2vQ{Q}^{†}Q=0_{2\times 2},v=\pm 1,$$

(4)

for appropriate interaction coefficients. Here, $Q(x,t)$ is a $2\times 2$ complex matrix-valued function, $Q^{†}$ denotes its Hermitian conjugate, and $0_{2\times 2}$ is the $2\times 2$ zero matrix [14,15]. In this work we consider the generalized matrix NLS equation

$$i{Q}_t+Q_{xx}-2QRQ=0_{2\times 2},R=\mathsf{\Sigma }{Q}^{†}\mathsf{\Omega },$$

(5)

where $\mathsf{\Omega }$ and $\mathsf{\Sigma }$ are $2\times 2$ Hermitian matrices whose specific choices determine the type of nonlinearity (focusing, defocusing, or mixed sign), as shown below. In the following, we refer to Equation (5) as the matrix NLS equation.

Under the gauge transformation [16]

$$Q(x,t)\to Q(x,t)e^{\mp 2i{k}_0^{2}t},$$

where $k_0>0$, Equation (5) becomes the equivalent form

$$i{Q}_t+Q_{xx}-2(QRQ-v{k}_0^2{I}_2)=0_{2\times 2},$$

(6)

where $I_2$ is the $2\times 2$ identity matrix. Depending on the sign parameter v, Equation (5) admits four distinct types of reductions, often categorized as follows:

• Case 1: (defocusing, $\mathsf{\Sigma }=\mathsf{\Omega }=\mathrm{diag}(1,1)$)

$$\begin{array}{cc}\hfill & i{q}_{1t}+q_{1xx}-2q_1\left({\left|q_1\right|}^2+2{\left|q_0\right|}^2\right)-2q_0^2{q}_{-1}^{*}=0,\hfill \\ \hfill & i{q}_{0t}+q_{0xx}-2q_0({\left|q_{-1}\right|}^2+{\left|q_0\right|}^2+{\left|q_1\right|}^2)-2q_{-1}{q}_0^{*}{q}_1=0,\hfill \\ \hfill & i{q}_{-1t}+q_{-1xx}-2q_{-1}({\left|q_{-1}\right|}^2+2{\left|q_0\right|}^2)-2q_0^2{q}_1^{*}=0.\hfill \end{array}$$

(7)

• Case 2: (focusing, $\mathsf{\Sigma }=-\mathsf{\Omega }=\mathrm{diag}(1,1)$)

$$\begin{array}{cc}\hfill & i{q}_{1t}+q_{1xx}+2q_1({\left|q_1\right|}^2+2{\left|q_0\right|}^2)+2q_0^2{q}_{-1}^{*}=0,\hfill \\ \hfill & i{q}_{0t}+q_{0xx}+2q_0({\left|q_{-1}\right|}^2+{\left|q_0\right|}^2+{\left|q_1\right|}^2)+2q_{-1}{q}_0^{*}{q}_1=0,\hfill \\ \hfill & i{q}_{-1t}+q_{-1xx}+2q_{-1}({\left|q_{-1}\right|}^2+2{\left|q_0\right|}^2)+2q_0^2{q}_1^{*}=0.\hfill \end{array}$$

(8)

• Case 3: (mixed sign(i), $\mathsf{\Sigma }=\mathsf{\Omega }=\mathrm{diag}(1,-1)$)

$$\begin{array}{cc}\hfill & i{q}_{1t}+q_{1xx}-2q_1\left({\left|q_1\right|}^2-2{\left|q_0\right|}^2\right)-2q_0^2{q}_{-1}^{*}=0,\hfill \\ \hfill & i{q}_{0t}+q_{0xx}-2q_0({\left|q_{-1}\right|}^2-{\left|q_0\right|}^2+{\left|q_1\right|}^2)+2q_{-1}{q}_0^{*}{q}_1=0,\hfill \\ \hfill & i{q}_{-1t}+q_{-1xx}-2q_{-1}({\left|q_{-1}\right|}^2-2{\left|q_0\right|}^2)-2q_0^2{q}_1^{*}=0.\hfill \end{array}$$

(9)

• Case 4: (mixed sign(ii), $\mathsf{\Sigma }=-\mathsf{\Omega }=\mathrm{diag}(1,-1)$)

$$\begin{array}{cc}\hfill & i{q}_{1t}+q_{1xx}+2q_1({\left|q_1\right|}^2-2{\left|q_0\right|}^2)+2q_0^2{q}_{-1}^{*}=0,\hfill \\ \hfill & i{q}_{0t}+q_{0xx}+2q_0({\left|q_{-1}\right|}^2-{\left|q_0\right|}^2+{\left|q_1\right|}^2)-2q_{-1}{q}_0^{*}{q}_1=0,\hfill \\ \hfill & i{q}_{-1t}+q_{-1xx}+2q_{-1}({\left|q_{-1}\right|}^2-2{\left|q_0\right|}^2)+2q_0^2{q}_1^{*}=0.\hfill \end{array}$$

(10)

Here, $v=1$ corresponds to Cases 1 and 3, while $v=-1$ corresponds to Cases 2 and 4.

Cases 1 and 2 (the defocusing and focusing reductions) have been studied extensively [17,18,19]. Complete soliton solutions for the mixed-sign vector NLS equation have also been reported under zero and nonzero boundary conditions [20,21,22]. However, the matrix case is fundamentally different. In the context of BEC, the signs of the coupling constants reflect those of the s-wave scattering lengths governing intraspecies and interspecies atomic interactions. Consequently, the nonlinear integrable equations in Cases 3 and 4 do not correspond to a physically valid model for three-component $F=1$ spinor BEC, unlike Cases 1 and 2. Instead, they find application in other physical settings, notably in nonlinear optics and in models of four-component fermionic condensates [23,24,25,26,27].

1.1. Motivation

The inverse scattering transform (IST) provides a powerful framework for solving integrable nonlinear evolution equations by linearizing them into a spectral problem. Recently, Ortiz and Prinari [28] systematically developed the IST and constructed simple-pole solutions for the matrix NLS Equation (5) under nonzero boundary conditions (NZBCs) at infinity. Subsequently, the IST for Equation (5) with both zero boundary conditions (ZBCs) and NZBCs, accounting for double zeros in the analytic scattering coefficients, was investigated in Ref. [29]. However, the case of triple poles—corresponding to higher-order spectral degeneracies and more intricate soliton dynamics—has remained largely unexplored. Constructing triple-pole solutions demands a more delicate analysis, including refined regularization and residue calculus within the associated Riemann-Hilbert (RH) problem. These solutions not only provide a stringent test of the completeness and robustness of the IST framework, but also reveal deeper algebraic structures inherent in the integrable hierarchy.

The extension from double to triple poles in the matrix setting introduces essential new features absent in scalar or vector counterparts. Most notably, there exists a forced correspondence between the order of zeros of $\det a\left(z\right)$ and the rank of the matrix $P(x,t,z_n)$—a feature already present at the double-pole level, but whose explicit realization at third order requires a systematic characterization. Specifically, as we show in this work:

• $\mathrm{rank}$ $P(x,t,z_n)=3$ requires a third-order zero of $\det a\left(z\right)$ at $z=z_n$;
• $\mathrm{rank}$ $P(x,t,z_n)=2$ requires a fourth-order zero of $\det a\left(z\right)$ at $z=z_n$.

These rank configurations give rise to distinct residue conditions and norming constant structures, which must be carefully derived and incorporated into the RH problem.

Furthermore, the physical relevance of NZBCs cannot be overstated. They describe systems with nontrivial backgrounds, such as multicomponent BEC in external potentials or light propagation in inhomogeneous media—settings where zero boundary conditions are no longer applicable. Thus, extending the IST to matrix NLS equations with triple poles under NZBCs is both mathematically necessary and physically meaningful.

The IST is ideally suited for triple-pole solitons in the matrix NLS equation under NZBCs. First, IST systematically characterizes the discrete spectrum through analytic scattering coefficients: the order of zeros of $\det a\left(z\right)$ directly dictates the pole structure, enabling rigorous treatment of higher-order poles via Laurent expansions. Second, the matrix nature is encoded in the scattering data—the rank of $P(x,t,z_n)$ emerges from linear dependence among Jost functions at the eigenvalue, revealing a correspondence between spectral degeneracy and eigenfunction structure inaccessible to purely algebraic methods. Third, NZBCs introduce a Riemann surface and continuous spectrum, seamlessly handled via a uniformization variable and the associated RH problem. This approach yields explicit soliton solutions in the reflectionless case, as well as spectral properties, trace formulae, and long-time asymptotics—offering a comprehensive understanding beyond mere solution construction.

1.2. Structure of This Paper

This paper is organized as follows. Section 2 reviews the direct scattering problem for Equation (6) under NZBCs, with emphasis on the characterization of the discrete spectrum for triple poles. We derive the associated residue conditions and norming constants for the two distinct cases: $\mathrm{rank}$ $P(x,t,z_n)=2$ and $\mathrm{rank}$ $P(x,t,z_n)=3$. Section 3 formulates the inverse problem as a matrix RH problem. Solving this RH problem in the reflectionless case yields the potential reconstruction formula and explicit expressions for general N-triple-pole soliton solutions. A detailed example for $N=1$ is presented in the Appendix A. Concluding remarks and directions for future research are given in Section 4.

2. Direct Scattering Problem

The direct scattering problem under NZBCs for systems with poles of different orders shares a common framework, differing principally in the characterization of the discrete spectrum. The direct scattering theory for the matrix NLS equation with NZBCs is well established for simple and double poles [28,29]; we therefore only briefly recall the essential elements before turning to the novel features associated with triple poles.

2.1. Lax Pair, Riemann Surface, and Uniformization Coordinate

Equation (6) admits the following Lax pair formulation:

$${\phi }_x=U\phi ,{\phi }_t=V\phi ,$$

(11)

where

$$U=-ik{\sigma }_4+\tilde{Q},V=-2i{k}^2{\sigma }_4+2k\tilde{Q}+i{\sigma }_4({\tilde{Q}}_x-{\tilde{Q}}^2+v{k}_0^2{I}_4),$$

(12)

and

$$\tilde{Q}=\left(\begin{array}{cc}{0}_{2\times 2}& Q\\ R& 0_{2\times 2}\end{array}\right),Q=\left(\begin{array}{cc}{q}_1(x,t)& q_0(x,t)\\ q_0(x,t)& q_{-1}(x,t)\end{array}\right),{\sigma }_4=\left(\begin{array}{cc}{I}_2& 0_{2\times 2}\\ 0_{2\times 2}& -I_2\end{array}\right),$$

(13)

with $R=\mathsf{\Sigma }{Q}^{†}\mathsf{\Omega }$ as in (5), and $I_4$ denotes the $4\times 4$ identity matrix. We consider Equation (6) under constant NZBCs:

$$Q(x,t)\to Q_{\pm },x\to \pm \infty .$$

(14)

The anticommutation relation $\tilde{Q}{\sigma }_4=-{\sigma }_4\tilde{Q}$ can be directly verified. Taking the limit $x\to \pm \infty$ and imposing the boundary condition (14) yields the asymptotic spectral problem

$${\phi }_x=U_{\pm }\phi ,U_{\pm }=-ik{\sigma }_4+\tilde{Q}.$$

(15)

From the boundary condition (14), it follows that

$${\tilde{Q}}_{\pm }^2=v{k}_0^2{I}_4,{\tilde{Q}}_{\pm }=\left(\begin{array}{cc}{0}_{2\times 2}& Q_{\pm }\\ R_{\pm }& 0_{2\times 2}\end{array}\right),$$

(16)

and consequently,

$$Q_{\pm }{R}_{\pm }=R_{\pm }{Q}_{\pm }=v{k}_0^2{I}_2.$$

(17)

The eigenvalues of $U_{\pm }$ and $V_{\pm }$ are $\pm i\lambda$ and $\pm 2ik\lambda$, respectively. To uniformize the two-sheeted Riemann surface, we introduce the variable

$$z=k+\lambda ,$$

(18)

where $\lambda$ is defined by ${\lambda }^2=k^2-v{k}_0^2$. The branch points of the Riemann surface are therefore at $k=\pm \sqrt{v}{k}_0^2$.

For Cases 1 and 3 ($v=1$), we parametrize

$$k-k_0=r_1{e}^{i{\theta }_1},k+k_0=r_2{e}^{i{\theta }_2},$$

(19)

which leads to the definition of $\lambda \left(k\right)$ on the two sheets:

$$\lambda \left(k\right)=\left\{\begin{array}{c}\sqrt{r_1{r}_2}{e}^{i({\theta }_1+{\theta }_2)/2},{\mathbb{C}}_I,\hfill \\ -\sqrt{r_1{r}_2}{e}^{i({\theta }_1+{\theta }_2)/2},{\mathbb{C}}_{II},\hfill \end{array}\right.$$

(20)

where $0<{\theta }_1<\pi$ and $-\pi <{\theta }_2<\pi$. ${\mathbb{C}}_I$ and ${\mathbb{C}}_{II}$ denoten the first and second sheets of the Riemann surface, respectively. Thus, one has

$$k={\displaystyle \frac{1}{2}}\left(z+{\displaystyle \frac{k_0^2}{z}}\right),\lambda ={\displaystyle \frac{1}{2}}\left(z-{\displaystyle \frac{k_0^2}{z}}\right).$$

(21)

Under Equation (21), the first sheet ${\mathbb{C}}_I$ is mapped conformally onto the upper half-plane $D^{+}=\{z\in \mathbb{C}:\mathrm{Im}z>0\}$, while the second sheet ${\mathbb{C}}_{II}$ is mapped onto the lower half-plane $D^{-}=\{z\in \mathbb{C}:\mathrm{Im}z<0\}$. A neighbourhood of $k=\infty$ on either sheet corresponds to a neighbourhood of either $z=0$ or $z=\infty$, depending on the sign of $\mathrm{Im}\lambda$.

For Cases 2 and 4 ($v=-1$), we employ the parametrization

$$k-i{k}_0=r_1{e}^{i{\theta }_1},k+i{k}_0=r_2{e}^{i{\theta }_2},$$

(22)

with $-{\displaystyle \frac{\pi }{2}}<{\theta }_1,{\theta }_2<{\displaystyle \frac{3\pi }{2}}$. The function $\lambda \left(k\right)$ is then given by

$$\lambda \left(k\right)=\left\{\begin{array}{c}\sqrt{r_1{r}_2}{e}^{i({\theta }_1+{\theta }_2)/2},{\mathbb{C}}_I,\hfill \\ -\sqrt{r_1{r}_2}{e}^{i({\theta }_1+{\theta }_2)/2},{\mathbb{C}}_{II},\hfill \end{array}\right.$$

(23)

and the relations inverse to Equation (18) become

$$k={\displaystyle \frac{1}{2}}\left(z-{\displaystyle \frac{k_0^2}{z}}\right),\lambda ={\displaystyle \frac{1}{2}}\left(z+{\displaystyle \frac{k_0^2}{z}}\right).$$

(24)

In this case, the branch cut on the Riemann surface is mapped onto the circle $C_0=\{z\in \mathbb{C}:|z|=k_0\}$. The first sheet ${\mathbb{C}}_I$ corresponds to the exterior of $C_0$ ($\left|z\right|>k_0$), while the second sheet ${\mathbb{C}}_{II}$ corresponds to its interior ($\left|z\right|<k_0$). The points at infinity map as $z\left({\infty }_I\right)=\infty$ and $z\left({\infty }_{II}\right)=0$. The regions $D^{+}$ and $D^{-}$ are now defined by the sign of $\left(\right|z{|}^2-k_0^2)\mathrm{Im}z$:

$$D^{+}={\{z\in \mathbb{C}:(\left|z\right|}^2-k_0^2)\mathrm{Im}z>0\},D^{-}={\{z\in \mathbb{C}:(\left|z\right|}^2-k_0^2)\mathrm{Im}z<0\}.$$

These correspond to $\mathrm{Im}\lambda >0$ and $\mathrm{Im}\lambda <0$, respectively.

2.2. Jost Functions and Analyticty

The asymptotic eigenvector matrices are given by

$$X_{\pm }=I_4-{\displaystyle \frac{i}{z}}{\sigma }_4{\tilde{Q}}_{\pm },$$

(25)

which, by construction, satisfy

$$U_{\pm }{X}_{\pm }=-i\lambda X_{\pm }{\sigma }_4.$$

(26)

From this definition, one finds $\det X_{\pm }\left(z\right)={\left({\displaystyle \frac{2\lambda }{\lambda +k}}\right)}^2={\left(\gamma \left(z\right)\right)}^2$, where $\gamma \left(z\right)=1-v{\displaystyle \frac{k_0^2}{z^2}}$. The inverse is given by

$$X_{\pm }^{-1}\left(z\right)={\displaystyle \frac{1}{\gamma \left(z\right)}}(I_4+{\displaystyle \frac{i}{z}}{\sigma }_4{\tilde{Q}}_{\pm }).$$

We note that $\gamma \left(z\right)$ (and hence $X_{\pm }\left(z\right)$) is singular at the branch points: $z=\pm k_0$ for $v=1$, and $z=\pm i{k}_0$ for $v=-1$; these points are excluded from the analysis.

The Jost solutions are defined as the following matrix-valued functions satisfying the asymptotic conditions:

$$\mathsf{\Phi }(x,t,z)=(\varphi (x,t,z),\overline{\varphi }(x,t,z)),\mathsf{\Psi }(x,t,z)=(\overline{\psi }(x,t,z),\psi (x,t,z)).$$

(27)

Here, $\varphi$, $\overline{\varphi }$ and $\psi$, $\overline{\psi }$ are each $4\times 2$ matrix functions. Their defining asymptotic behaviors (as $x\to \pm \infty$) are:

$$\varphi (x,t,z)\sim \left(\begin{array}{c}{I}_2\\ {\displaystyle \frac{i}{z}}{R}_{-}\end{array}\right)e^{-i\theta (x,t,z)},\overline{\varphi }(x,t,z)\sim \left(\begin{array}{c}-{\displaystyle \frac{i}{z}}{Q}_{-}\\ I_2\end{array}\right)e^{i\theta (x,t,z)},x\to -\infty ,$$

(28)

$$\mathit{\psi }(x,t,z)\sim \left(\begin{array}{c}{I}_2\\ {\displaystyle \frac{i}{z}}{R}_{+}\end{array}\right)e^{-i\theta (x,t,z)},\overline{\mathit{\psi }}(x,t,z)\sim \left(\begin{array}{c}-{\displaystyle \frac{i}{z}}{Q}_{+}\\ I_2\end{array}\right)e^{i\theta (x,t,z)},x\to +\infty ,$$

(29)

where $\theta (x,t,z)=\lambda (x+2kt)$. Consequently, the full matrix solutions $\mathsf{\Phi }$ and $\mathsf{\Psi }$ satisfy the asymptotic conditions

$$\mathsf{\Phi }(x,t,z)=X_{-}\left(z\right)e^{-i\theta (x,t,z){\sigma }_4}+o\left(1\right),x\to -\infty ,$$

(30)

$$\mathsf{\Psi }(x,t,z)=X_{+}\left(z\right)e^{-i\theta (x,t,z){\sigma }_4}+o\left(1\right),x\to +\infty .$$

(31)

It is standard to introduce the modified Jost functions, which remove the essential asymptotic oscillations:

$$\begin{array}{cc}\hfill & (M(x,t,z),\overline{M}(x,t,z))=\mathsf{\Phi }(x,t,z)e^{i\theta (x,t,z){\sigma }_4},\hfill \\ & (\overline{N}(x,t,z),N(x,t,z))=\mathsf{\Psi }(x,t,z)e^{i\theta (x,t,z){\sigma }_4}.\hfill \end{array}$$

(32)

From Equations (27)–(29) and (32), their asymptotic behaviors follow directly:

$$\begin{array}{cc}\hfill & (M(x,t,z),\overline{M}(x,t,z))\sim X_{-},x\to -\infty ,\hfill \\ \hfill & (\overline{N}(x,t,z),N(x,t,z))\sim X_{+},x\to +\infty .\hfill \end{array}$$

(33)

For $v=1$, the continuous spectrum in the original k-plane is ${\mathsf{\Sigma }}_k=\mathbb{R}\setminus [-k_0,k_0]$, which corresponds to ${\mathsf{\Sigma }}_z=\mathbb{R}$ in the uniformized z-plane. For $v=-1$, we have ${\mathsf{\Sigma }}_k=\mathbb{R}\cup i[-k_0,k_0]$ and ${\mathsf{\Sigma }}_z=\mathbb{R}\cup C_0$.

The Jost functions satisfy the following Volterra integral equations, derived from the equivalent Lax pair:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & (M(x,t,z),\overline{M}(x,t,z))=X_{-}+{\int }_{-\infty }^x{X}_{-}{e}^{i\lambda (y-x){\widehat{\sigma }}_4}[X_{-}^{-1}\Delta \tilde{Q}(y,t)(M(y,t,z),\overline{M}(y,t,z))]dy,\hfill \\ \hfill & (\overline{N}(x,t,z),N(x,t,z))=X_{+}+{\int }_{\infty }^x{X}_{+}{e}^{i\lambda (y-x){\widehat{\sigma }}_4}[X_{+}^{-1}\Delta \tilde{Q}(y,t)(\overline{N}(y,t,z),N(y,t,z))]dy,\hfill \end{array}\end{array}$$

(34)

where $\Delta Q_{\pm }=Q-Q_{\pm }$ for $x\to \pm \infty$, and ${\widehat{\sigma }}_4$ denotes the operator that acts on a $4\times 4$ matrix A as ${\widehat{\sigma }}_{4}A=[{\sigma }_4,A]={\sigma }_{4}A-A{\sigma }_4$. These integral equations imply the analyticity properties:

$$M(x,t,z),N(x,t,z)\in H\left(D^{+}\right),\overline{M}(x,t,z),\overline{N}(x,t,z)\in H\left(D^{-}\right).$$

Here, $H\left(D^{\pm }\right)$ denotes the set of analytic (holomorphic) functions on $D^{\pm }$.

2.3. Scattering Matrix and Reflection Coefficients

Since the Lax matrices are traceless, $tr\left(U\right)=tr\left(V\right)=0$, the determinants of the fundamental solutions are independent of x and t. One finds

$$\det \mathsf{\Phi }(x,t,z)=\det \mathsf{\Psi }(x,t,z)=\det X_{\pm }={\left(\gamma \left(z\right)\right)}^2.$$

(35)

Being fundamental solutions of the scattering problem, $\mathsf{\Phi }$ and $\mathsf{\Psi }$ are related by a constant (in x and t) scattering matrix $S\left(z\right)$:

$$\mathsf{\Phi }(x,t,z)=\mathsf{\Psi }(x,t,z)S\left(z\right),S\left(z\right)=\left(\begin{array}{cc}a\left(z\right)& \overline{b}\left(z\right)\\ b\left(z\right)& \overline{a}\left(z\right)\end{array}\right),z\in {\mathsf{\Sigma }}_z.$$

(36)

Here, $a\left(z\right),b\left(z\right),\overline{a}\left(z\right),\overline{b}\left(z\right)$ are $2\times 2$ matrix-valued functions, and the unimodularity condition $\det S\left(z\right)=1$ holds. Expanding Equation (36) column-wise yields the linear relations

$$\varphi =\psi b+\overline{\psi }a,\overline{\varphi }=\psi \overline{a}+\overline{\psi }\overline{b}.$$

(37)

Using these relations and Equation (35), one can express the determinants of the scattering coefficients as Wronskian ratios:

$$\det a\left(z\right)={\displaystyle \frac{Wr(\varphi ,\psi )}{Wr(\overline{\psi },\psi )}}={\displaystyle \frac{\det (\varphi ,\psi )}{{\left(\gamma \left(z\right)\right)}^2}},$$

(38)

$$\det \overline{a}\left(z\right)={\displaystyle \frac{Wr(\overline{\varphi },\overline{\psi })}{Wr(\overline{\psi },\psi )}}={\displaystyle \frac{\det (\overline{\psi },\overline{\varphi })}{{\left(\gamma \left(z\right)\right)}^2}}.$$

(39)

From the analyticity properties of the Jost functions and the Volterra integral Equation (34), it follows that $a\left(z\right)$ and $\det a\left(z\right)$ can be analytically extended to the region $D^{+}$, while $\overline{a}\left(z\right)$ and $\det \overline{a}\left(z\right)$ can be analytically extended to $D^{-}$.

Reflection coefficients are defined by

$$\rho \left(z\right)=b\left(z\right)a^{-1}\left(z\right),\overline{\rho }\left(z\right)=\overline{b}\left(z\right){\overline{a}}^{-1}\left(z\right),z\in {\mathsf{\Sigma }}_0,$$

(40)

where ${\mathsf{\Sigma }}_0={\mathsf{\Sigma }}_z\setminus \left\{\mathrm{branch}\mathrm{points}\right\}$. Using Equation (37) and the definitions of the modified Jost functions (32), one derives the following relations:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & M(x,t,z)a^{-1}\left(z\right)=\overline{N}(x,t,z)+e^{2i\theta (x,t,z)}N(x,t,z)\rho \left(z\right),\hfill \\ & \overline{M}(x,t,z){\overline{a}}^{-1}\left(z\right)=N(x,t,z)+e^{-2i\theta (x,t,z)}\overline{N}(x,t,z)\overline{\rho }\left(z\right).\hfill \end{array}\end{array}$$

(41)

Consequently, $M(x,t,z)a^{-1}\left(z\right)$ is meromorphic in the upper half-plane $D^{+}$, and $\overline{M}(x,t,z){\overline{a}}^{-1}\left(z\right)$ is meromorphic in the lower half-plane $D^{-}$, with poles occurring at the zeros of $\det a\left(z\right)$ and $\det \overline{a}\left(z\right)$, respectively.

2.4. Symmetries

Due to the specific matrix structure of the potential and the imposed boundary conditions, the Jost functions and scattering data inherit three fundamental symmetries. We present their implications in turn.

• First symmetry (Hermitian conjugation): $(k,\lambda )\to (k^{*},{\lambda }^{*})$

The Jost functions satisfy the involution property

$$\mathsf{\Phi }{(x,t,z)}^{-1}={\displaystyle \frac{1}{\gamma \left(z\right)}}{J}_v{\mathsf{\Phi }}^{†}(x,t,z^{*})J_v,\mathsf{\Psi }{(x,t,z)}^{-1}={\displaystyle \frac{1}{\gamma \left(z\right)}}{J}_v{\mathsf{\Psi }}^{†}(x,t,z^{*})J_v,$$

(42)

where the constant matrix $J_v$ is defined by

$$J_v=\left(\begin{array}{cc}{\mathsf{\Omega }}^{-1}& 2\times 2\\ 2\times 2& -\mathsf{\Sigma }\end{array}\right).$$

(43)

Consequently, the scattering matrix and reflection coefficients obey

$$S^{-1}\left(z\right)=J_v{S}^{†}\left(z^{*}\right)J_v,\overline{\rho }\left(z\right)={\mathsf{\Omega }}^{-1}{\rho }^{†}\left(z^{*}\right){\mathsf{\Sigma }}^{-1}.$$

(44)

If we denote the blocks of the inverse scattering matrix as

$$S^{-1}\left(z\right)=\left(\begin{array}{cc}\overline{c}\left(z\right)& d\left(z\right)\\ \overline{d}\left(z\right)& c\left(z\right)\end{array}\right),$$

(45)

then expanding Equation (44) yields the following relations between the block entries:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \overline{c}\left(z\right)={\mathsf{\Omega }}^{-1}{a}^{†}\left(z^{*}\right){\mathsf{\Omega }}^{-1},c\left(z\right)=\mathsf{\Sigma }{\overline{a}}^{†}\left(z^{*}\right)\mathsf{\Sigma },\hfill \\ \hfill & d\left(z\right)=-{\mathsf{\Omega }}^{-1}{b}^{†}\left(z^{*}\right)\mathsf{\Sigma },\overline{d}\left(z\right)=-\mathsf{\Sigma }{\overline{b}}^{†}\left(z^{*}\right){\mathsf{\Omega }}^{-1}.\hfill \end{array}\end{array}$$

(46)

Furthermore, the determinants satisfy the conjugation relation

$$\det \overline{a}\left(z\right)=\det a^{†}\left(z^{*}\right)={\left(\det a\left(z^{*}\right)\right)}^{*},z\in D^{-}.$$

(47)

• Second symmetry: $(k,\lambda )\to (k,-\lambda )$

For $z\in {\mathsf{\Sigma }}_z$, the Jost functions transform as

$$\mathsf{\Phi }(x,t,z)=-{\displaystyle \frac{i}{z}}\mathsf{\Phi }(x,t,v{k}_0^2/z){\sigma }_4{\tilde{Q}}_{-},\mathsf{\Psi }(x,t,z)=-{\displaystyle \frac{i}{z}}\mathsf{\Psi }(x,t,v{k}_0^2/z){\sigma }_4{\tilde{Q}}_{+},$$

(48)

leading to the scattering matrix symmetry

$$S(v{k}_0^2/z)={\sigma }_4{\tilde{Q}}_{+}S\left(z\right){\tilde{Q}}_{-}^{-1}{\sigma }_4.$$

(49)

The reflection coefficients consequently satisfy

$$\rho (v{k}_0^2/z)=-R_{+}\overline{\rho }\left(z\right)Q_{+}^{-1},\overline{\rho }(v{k}_0^2/z)=-Q_{+}\rho \left(z\right)R_{+}^{-1}.$$

(50)

• Third symmetry (transposition): $Q\to Q^T$

For $z\in {\mathsf{\Sigma }}_z$, the Jost functions obey the symplectic-like identity

$${\mathsf{\Phi }}^T(x,t,z){\widehat{\sigma }}_2\mathsf{\Phi }(x,t,z)={\mathsf{\Psi }}^T(x,t,z){\widehat{\sigma }}_2\mathsf{\Psi }(x,t,z)=\gamma \left(z\right){\widehat{\sigma }}_2,$$

(51)

where

$${\widehat{\sigma }}_2=\left(\begin{array}{cc}{0}_{2\times 2}& -i{I}_2\\ i{I}_2& 0_{2\times 2}\end{array}\right).$$

This implies a corresponding symmetry for the scattering matrix and reflection coefficients:

$$S^{-1}\left(z\right)={\widehat{\sigma }}_2{S}^T\left(z\right){\widehat{\sigma }}_2,$$

(52)

and

$${\rho }^T\left(z\right)=\rho \left(z\right),{\overline{\rho }}^T\left(z\right)=\overline{\rho }\left(z\right),\overline{\rho }\left(z\right)=\mathsf{\Sigma }{\rho }^{*}\left(z^{*}\right)\mathsf{\Omega }.$$

(53)

2.5. Asymptotics

The asymptotic expansions of the modified Jost functions as $z\to \infty$ are derived by substituting the WKB-type ansatz into the Lax pair equations. For $z\in D^{+}$ and $z\in D^{-}$, we obtain respectively:

$$M(x,t,z)=\left(\begin{array}{c}{I}_2+{\displaystyle \frac{i}{z}}{\int }_{-\infty }^x[Q(x^{\prime },t)R(x^{\prime },t)-v{k}_0^2{I}_2]d{x}^{\prime }+O(1/z^2)\\ \\ {\displaystyle \frac{i}{z}}R(x,t)+O(1/z^2)\end{array}\right),z\in D^{+},$$

(54)

$$\overline{M}(x,t,z)=\left(\begin{array}{c}-{\displaystyle \frac{i}{z}}Q(x,t)+O(1/z^2)\\ \\ I_2-{\displaystyle \frac{i}{z}}{\int }_{-\infty }^x[R(x^{\prime },t)Q(x^{\prime },t)-v{k}_0^2{I}_2]d{x}^{\prime }+O(1/z^2)\end{array}\right),z\in D^{-},$$

(55)

and

$$\overline{N}(x,t,z)=\left(\begin{array}{c}{I}_2+{\displaystyle \frac{i}{z}}{\int }_x^{\infty }[Q(x^{\prime },t)R(x^{\prime },t)-v{k}_0^2{I}_2]d{x}^{\prime }+O(1/z^2)\\ \\ {\displaystyle \frac{i}{z}}R(x,t)+O(1/z^2)\end{array}\right),z\in D^{-},$$

(56)

$$N(x,t,z)=\left(\begin{array}{c}-{\displaystyle \frac{i}{z}}Q(x,t)+O(1/z^2)\\ \\ I_2-{\displaystyle \frac{i}{z}}{\int }_x^{\infty }[R(x^{\prime },t)Q(x^{\prime },t)-v{k}_0^2{I}_2]d{x}^{\prime }+O(1/z^2)\end{array}\right),z\in D^{+}.$$

(57)

Similarly, the asymptotic behaviors as $z\to 0$ are found to be

$$M(x,t,z)=\left(\begin{array}{c}vQ{R}_{-}/k_0^2+O\left(z\right)\\ \\ i{R}_{-}/z+O\left(1\right)\end{array}\right),\overline{M}(x,t,z)=\left(\begin{array}{c}-i{Q}_{-}/z+O\left(1\right)\\ \\ vR{Q}_{-}/k_0^2+O\left(z\right)\end{array}\right),$$

(58)

$$\overline{N}(x,t,z)=\left(\begin{array}{c}vQ{R}_{+}/k_0^2+O\left(z\right)\\ \\ i{R}_{+}/z+O\left(1\right)\end{array}\right),N(x,t,z)=\left(\begin{array}{c}-i{Q}_{+}/z+O\left(1\right)\\ \\ vR{Q}_{+}/k_0^2+O\left(z\right)\end{array}\right).$$

(59)

Substituting the above asymptotics into the scattering relation (36) yields the large- and small-z behaviour of $S\left(z\right)$:

$$S\left(z\right)=I_4+O(1/z),z\to \infty ,$$

(60)

and

$$S\left(z\right)={\displaystyle \frac{v}{k_0^2}}\left(\begin{array}{cc}{Q}_{+}{R}_{-}& 0_{2\times 2}\\ 0_{2\times 2}& R_{+}{Q}_{-}\end{array}\right),z\to 0.$$

(61)

2.6. Discrete Spectrum and Residue Conditions

The discrete spectrum consists of the zeros of the analytic scattering coefficients. In this work, we focus on the case where $\det a\left(z\right)$ possesses a triple zero at $z=z_n$. From the symmetry relation (47), it follows that such zeros do not appear in isolation but rather in quartets:

$$\det a\left(z_n\right)=0\iff \det \overline{a}\left(z_n^{*}\right)=0\iff \det \overline{a}(v{k}_0^2/z_n)=0\iff \det a(v{k}_0^2/z_n^{*})=0.$$

(62)

Consequently, the discrete spectrum is defined as the set

$$\mathcal{Z}={\left\{z_n,z_n^{*},v{k}_0^2/z_n,v{k}_0^2/z_n^{*}\right\}}_{n=1}^N.$$

(63)

To avoid spectral singularities on the continuous spectrum ${\mathsf{\Sigma }}_z$, we assume $\det a\left(z\right)\ne 0$ and $\det \overline{a}\left(z\right)\ne 0$ for all $z\in {\mathsf{\Sigma }}_z$. To characterize the discrete eigenvalues, it is convenient to introduce the auxiliary $4\times 4$ matrices

$$\begin{array}{c}\hfill \begin{array}{c}\hfill P(x,t,z)=(\varphi (x,t,z),\psi (x,t,z)),\overline{P}(x,t,z)=(\overline{\psi }(x,t,z),\overline{\varphi }(x,t,z)).\end{array}\end{array}$$

(64)

As is proved in [30], for a discrete eigenvalue $z_n\in D^{+}\cap \mathcal{Z}$, the $\mathrm{rank}$ of $P(x,t,z_n)$ is directly linked to the $\mathrm{rank}$ of the scattering block $a\left(z\right)$. Specifically:

• $\mathrm{rank}$ $P(x,t,z_n)=2$ if and only if $\mathrm{rank}$ $a\left(z\right)=0$ (i.e., $a\left(z_n\right)=0_{2\times 2}$);
• $\mathrm{rank}$ $P(x,t,z_n)=3$ if and only if $\mathrm{rank}$ $a\left(z\right)=1$.

Analogous statements hold for the conjugate eigenvalue $z_n^{*}\in D^{-}\cap \mathcal{Z}$ and the matrix $\overline{P}$.

• Case of $\mathrm{rank}$ $P(x,t,z_n)=3$

Let $\alpha \left(z\right)$ and $\overline{\alpha }\left(z\right)$ denote the adjugate matrices of $a\left(z\right)$ and $\overline{a}\left(z\right)$, respectively. Since $z_n$ is a triple zero of $\det a\left(z\right)$, we have the linear dependence

$$\varphi (x,t,z_n)\alpha \left(z_n\right)=\psi (x,t,z_n)c_n,$$

(65)

where $c_n$ is a nonzero $2\times 2$ matrix. To capture the higher-order pole contributions, we differentiate Equation (65) with respect to the spectral parameter z (denoted by a prime) and obtain

$$\begin{array}{cc}\hfill & {\left(\varphi \left(z_n\right)\alpha \left(z_n\right)\right)}^{\prime }={\psi }^{\prime }\left(z_n\right)c_n+\psi \left(z_n\right)d_n,\hfill \\ \hfill & {\left(\varphi \left(z_n\right)\alpha \left(z_n\right)\right)}^{″}={\psi }^{″}\left(z_n\right)c_n+2{\psi }^{\prime }\left(z_n\right)d_n+\psi \left(z_n\right)e_n,\hfill \end{array}$$

(66)

where $d_n$, $e_n$ are additional $2\times 2$ matrices (higher-order scattering data). Using the relation $M=\mathsf{\Phi }{e}^{i\theta }$ and N from Equation (32), these relations become

$$\begin{array}{cc}\hfill & {\left(M(x,t,z_n)\alpha \left(z_n\right)\right)}^{\prime }=e^{2i\theta \left(z_n\right)}[N^{\prime }\left(z_n\right)c_n+N\left(z_n\right)(d_n+2i{\theta }^{\prime }\left(z_n\right)c_n)],\hfill \\ \hfill & {\left(M(x,t,z_n)\alpha \left(z_n\right)\right)}^{″}=e^{2i\theta \left(z_n\right)}[N\left(z_n\right)(e_n+4i{\theta }^{\prime }\left(z_n\right)d_n+2i{\theta }^{″}\left(z_n\right)c_n-4c_n{\left({\theta }^{\prime }\left(z_n\right)\right)}^2)\hfill \\ \hfill & +N^{″}\left(z_n\right)c_n+N^{\prime }\left(z_n\right)(2d_n+4i{\theta }^{\prime }\left(z_n\right)c_n)].\hfill \end{array}$$

(67)

To simplify notation, we write $M(x,t,z)$ as $M\left(z\right)$ in the remainder of this paper. To find the singular part of $M\left(z\right)a^{-1}\left(z\right)={\displaystyle \frac{M\left(z\right)\alpha \left(z\right)}{\det a\left(z\right)}}$ near $z=z_n$, we consider the function

$$g\left(z\right)={(z-z_n)}^3{\displaystyle \frac{M\left(z\right)\alpha \left(z\right)}{\det a\left(z\right)}},$$

(68)

where is analytic at $z_n$. Expanding both numerator and denominator in Taylor series, we have

$$\det a\left(z\right)={\displaystyle \frac{1}{3!}}{\left(\det a\right)}^{‴}\left(z_n\right){(z-z_n)}^3+{\displaystyle \frac{1}{4!}}{\left(\det a\right)}^{\left(4\right)}\left(z_n\right){(z-z_n)}^4+\cdots ,$$

(69)

and

$$M\left(z\right)\alpha \left(z\right)=M\alpha +{\left(M\alpha \right)}^{\prime }(z-z_n)+{\displaystyle \frac{1}{2!}}{\left(M\alpha \right)}^{″}{(z-z_n)}^2+{\displaystyle \frac{1}{3!}}{\left(M\alpha \right)}^{‴}{(z-z_n)}^3+\cdots ,$$

(70)

where $M\alpha \equiv M\left(z_n\right)\alpha \left(z_n\right)$. Substituting these expansions into $g\left(z\right)$ and simplifying gives the Taylor expansion of $g\left(z\right)$:

$$\begin{array}{cc}\hfill g\left(z\right)& ={\mathrm{P}}_{-3}\left[M\left(z\right)a^{-1}\left(z\right)\right]+{\mathrm{P}}_{-2}\left[M\left(z\right)a^{-1}\left(z\right)\right](z-z_n)+Res\left[M\left(z\right)a^{-1}\left(z\right)\right]{(z-z_n)}^2\hfill \\ & +O\left({(z-z_n)}^3\right).\hfill \end{array}$$

(71)

Here, $Res\left[M\left(z\right)a^{-1}\left(z\right)\right]$, $P_{-2}\left[M\left(z\right)a^{-1}\left(z\right)\right]$ and $P_{-3}\left[M\left(z\right)a^{-1}\left(z\right)\right]$ denote the coefficients of $(z-z_n)$, ${(z-z_n)}^2$ and ${(z-z_n)}^3$ in the Laurent expansion of $M\left(z\right)a^{-1}\left(z\right)$, respectively. The coefficients are obtained from the standard formulas for Taylor coefficients:

$$\begin{array}{cc}\hfill & \underset{z=z_n}{Res}\left[M\left(z\right)a^{-1}\left(z\right)\right]={\displaystyle \frac{1}{2!}}{g}^{″}\left(z_n\right)={\displaystyle \frac{1}{2!}}\underset{z\to z_n}{lim}{\displaystyle \frac{d^2}{d{z}^2}}\left[{(z-z_n)}^3{\displaystyle \frac{M\left(z\right)\alpha \left(z\right)}{\det a\left(z\right)}}\right],\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-2}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=\underset{z\to z_n}{lim}{\displaystyle \frac{d}{dz}}\left[{(z-z_n)}^3{\displaystyle \frac{M\left(z\right)\alpha \left(z\right)}{\det a\left(z\right)}}\right],\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-3}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=\underset{z\to z_n}{lim}{(z-z_n)}^3{\displaystyle \frac{M\left(z\right)\alpha \left(z\right)}{\det a\left(z\right)}}.\hfill \end{array}$$

(72)

Evaluating these limits yields

$$\begin{array}{cc}\hfill & \underset{z=z_n}{Res}\left[M\left(z\right)a^{-1}\left(z\right)\right]=[-{\displaystyle \frac{3{\left(M\left(z\right)\alpha \left(z\right)\right)}^{\prime }{\left(\det a\right)}^{\left(4\right)}\left(z\right)}{2{\left({\left(\det a\right)}^{‴}\left(z\right)\right)}^2}}+{\displaystyle \frac{3M\left(z\right)\alpha \left(z\right){\left({\left(\det a\right)}^{\left(4\right)}\left(z\right)\right)}^2}{8{\left({\left(\det a\right)}^{‴}\left(z\right)\right)}^3}}\hfill \\ \hfill & -{\displaystyle \frac{3M\left(z\right)\alpha \left(z\right){\left(\det a\right)}^{\left(5\right)}\left(z\right)}{10{\left({\left(\det a\right)}^{‴}\left(z\right)\right)}^2}}+{\displaystyle \frac{3{\left(M\left(z\right)\alpha \left(z\right)\right)}^{″}}{{\left(\det a\right)}^{‴}\left(z\right)}}{]}_{z=z_n},\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-2}}\left[M\left(z\right)a^{-1}\left(z\right)\right]={\left[-{\displaystyle \frac{3M\left(z\right)\alpha \left(z\right){\left(\det a\right)}^{\left(4\right)}\left(z\right)}{2{\left({\left(\det a\right)}^{‴}\left(z\right)\right)}^2}}+{\displaystyle \frac{6{\left(M\left(z\right)\alpha \left(z\right)\right)}^{\prime }}{{\left(\det a\right)}^{‴}\left(z\right)}}\right]}_{z=z_n},\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-3}}\left[M\left(z\right)a^{-1}\left(z\right)\right]={\left[{\displaystyle \frac{6M\left(z\right)\alpha \left(z\right)}{{\left(\det a\right)}^{‴}\left(z\right)}}\right]}_{z=z_n}.\hfill \end{array}$$

(73)

To express these conditions in a compact and useful form, we define the following constant matrices derived from the scattering data and the previously introduced matrices $c_n,d_n,e_n$

$$\begin{array}{cc}\hfill & A_n={\displaystyle \frac{6c_n}{{\left(\det a\right)}^{‴}\left(z_n\right)}},B_n=\left[2d_n-{\displaystyle \frac{c_n{\left(\det a\right)}^{\left(4\right)}\left(z_n\right)}{2{\left(\det a\right)}^{‴}\left(z_n\right)}}\right]{\displaystyle \frac{3}{{\left(\det a\right)}^{‴}\left(z_n\right)}},\hfill \\ \hfill & \begin{array}{c}\hfill C_n=\left[{\displaystyle \frac{{\left({\left(\det a\right)}^{\left(4\right)}\left(z_n\right)\right)}^2}{8{\left({\left(\det a\right)}^{‴}\left(z_n\right)\right)}^2}}-{\displaystyle \frac{d_n{c}_n^{-1}{\left(\det a\right)}^{\left(4\right)}\left(z_n\right)}{2{\left(\det a\right)}^{‴}\left(z_n\right)}}-{\displaystyle \frac{{\left(\det a\right)}^{\left(5\right)}\left(z_n\right)}{10{\left(\det a\right)}^{‴}\left(z_n\right)}}+e_n{c}_n^{-1}\right]{\displaystyle \frac{3}{{\left(\det a\right)}^{‴}\left(z_n\right)}}.\end{array}\hfill \end{array}$$

Substituting the above constant matrices and Equation (67) into Equation (73), we obtain

$$\begin{array}{cc}\hfill & \underset{z=z_n}{Res}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2i\theta \left(z_n\right)}[{\displaystyle \frac{1}{2}}{N}^{″}\left(z_n\right)A_n+N^{\prime }\left(z_n\right)\left(B_n+2i{\theta }^{\prime }{A}_n\right)\hfill \\ \hfill & +N\left(z_n\right)\left(C_n+2i{\theta }^{\prime }{B}_n+(i{\theta }^{″}-2{\left({\theta }^{\prime }\right)}^2)A_n\right)],\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-2}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2\mathrm{i}\theta \left(z_n\right)}\left[N^{\prime }\left(z_n\right)A_n+N\left(z_n\right)(B_n+2i{\theta }^{\prime }{A}_n)\right],\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-3}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2\mathrm{i}\theta \left(z_n\right)}N\left(z_n\right)A_n.\hfill \end{array}$$

Furthermore, we combine the above equations with the phase derivatives to define the norming constants:

$$\begin{array}{cc}\hfill & C_{n,1}=C_n+2i{\theta }^{\prime }{B}_n+(i{\theta }^{″}-2{\left({\theta }^{\prime }\right)}^2)A_n,C_{n,2}=B_n+2i{\theta }^{\prime }{A}_n,C_{n,3}=A_n.\hfill \end{array}$$

(74)

In terms of these constants, the singular behavior of $M\left(z\right)a^{-1}\left(z\right)$ takes its final, compact form:

$$\begin{array}{cc}\hfill & \underset{z=z_n}{Res}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2i\theta \left(z_n\right)}\left[{\displaystyle \frac{1}{2}}{N}^{″}\left(z_n\right)C_{n,3}+N^{\prime }\left(z_n\right)C_{n,2}+N\left(z_n\right)C_{n,1}\right],\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-2}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2\mathrm{i}\theta \left(z_n\right)}\left[N^{\prime }\left(z_n\right)C_{n,3}+N\left(z_n\right)C_{n,2}\right],\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-3}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2i\theta \left(z_n\right)}N\left(z_n\right)C_{n,3}.\hfill \end{array}$$

(75)

An entirely analogous derivation for the conjugate pole $z=z_n^{*}\in D^{-}\cap \mathcal{Z}$ provides the corresponding singular behavior for $\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)$. The results are structurally identical to (73) under the substitutions $M\to \overline{M}$, $\alpha \to \overline{\alpha }$, $\det a\left(z\right)\to \det \overline{a}\left(z\right)$, and evaluation at $z=z_n^{*}$

$$\begin{array}{cc}\hfill & \underset{z=z_n^{*}}{Res}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=[{\displaystyle \frac{3\overline{M}\left(z\right)\overline{\alpha }\left(z\right){\left({\left(\det \overline{a}\right)}^{\left(4\right)}\left(z\right)\right)}^2}{8{\left({\left(\det \overline{a}\right)}^{‴}\left(z\right)\right)}^3}}-{\displaystyle \frac{3{\left(\overline{M}\left(z\right)\overline{\alpha }\left(z\right)\right)}^{\prime }{\left(\det \overline{a}\right)}^{\left(4\right)}\left(z\right)}{2{\left({\left(\det \overline{a}\right)}^{‴}\left(z\right)\right)}^2}}\hfill \\ \hfill & -{\displaystyle \frac{3\overline{M}\left(z\right)\overline{\alpha }\left(z\right){\left(\det \overline{a}\right)}^{\left(5\right)}\left(z\right)}{10{\left({\left(\det \overline{a}\right)}^{‴}\left(z\right)\right)}^2}}+{\displaystyle \frac{3{\left(\overline{M}\left(z\right)\overline{\alpha }\left(z\right)\right)}^{″}}{{\left(\det \overline{a}\right)}^{‴}\left(z\right)}}{]}_{z=z_n^{*}},\hfill \\ \hfill & \underset{z=z_n^{*}}{{\mathrm{P}}_{-2}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]={\left[-{\displaystyle \frac{3\overline{M}\left(z\right)\overline{\alpha }\left(z\right){\left(\det \overline{a}\right)}^{\left(4\right)}\left(z\right)}{2{\left({\left(\det \overline{a}\right)}^{‴}\left(z\right)\right)}^2}}+{\displaystyle \frac{6{\left(\overline{M}\left(z\right)\overline{\alpha }\left(z\right)\right)}^{\prime }}{{\left(\det \overline{a}\right)}^{‴}\left(z\right)}}\right]}_{z=z_n^{*}},\hfill \\ \hfill & \underset{z=z_n^{*}}{{\mathrm{P}}_{-3}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]={\left[{\displaystyle \frac{6\overline{M}\left(z\right)\overline{\alpha }\left(z\right)}{{\left(\det \overline{a}\right)}^{‴}\left(z\right)}}\right]}_{z=z_n^{*}}.\hfill \end{array}$$

(76)

These lead to the definition of the norming constants ${\overline{C}}_{n,1}$, ${\overline{C}}_{n,2}$, ${\overline{C}}_{n,3}$, defined exactly as in (74) but with $c_n$, $d_n$, $e_n$ replaced by their conjugate counterparts ${\overline{c}}_n$, ${\overline{d}}_n$, ${\overline{e}}_n$. The derivatives of $\det a$ replaced by those of $\det \overline{a}$ evaluated at $z=z_n^{*}$, and ${\theta }^{\prime }\left(z_n\right)$, ${\theta }^{″}\left(z_n\right)$ replaced by ${\theta }^{\prime }\left(z_n^{*}\right)$, ${\theta }^{″}\left(z_n^{*}\right)$. The final compact form is then:

$$\begin{array}{cc}\hfill & \underset{z=z_n^{*}}{Res}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left(z_n^{*}\right)}\left[{\displaystyle \frac{1}{2}}{\overline{N}}^{″}\left(z_n^{*}\right){\overline{C}}_{n,3}+{\overline{N}}^{\prime }\left(z_n^{*}\right){\overline{C}}_{n,2}+\overline{N}\left(z_n^{*}\right){\overline{C}}_{n,1}\right],\hfill \\ \hfill & \underset{z=z_n^{*}}{{\mathrm{P}}_{-2}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left(z_n^{*}\right)}[{\overline{N}}^{\prime }\left(z_n^{*}\right){\overline{C}}_{n,3}+\overline{N}\left(z_n^{*}\right){\overline{C}}_{n,2}],\hfill \\ \hfill & \underset{z=z_n^{*}}{{\mathrm{P}}_{-3}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left(z_n^{*}\right)}\overline{N}\left(z_n^{*}\right){\overline{C}}_{n,3}.\hfill \end{array}$$

(77)

By symmetry (49), the points ${\widehat{z}}_n=v{k}_0^2/z_n^{*}$ and $\widehat{\overline{z}}=v{k}_0^2/z_n$ also belong to the discrete spectrum. At these points, there exist $2\times 2$ matrices ${\widehat{c}}_n$ and ${\widehat{\overline{c}}}_n$ (and corresponding higher-order constants ${\widehat{d}}_n$, ${\widehat{e}}_n$, ${\widehat{\overline{d}}}_n$, ${\widehat{\overline{e}}}_n$) that satisfy relations analogous to Equations (65) and (66). The singular behaviour of $M{a}^{-1}$ and $\overline{M}{\overline{a}}^{-1}$ at these points can be obtained by applying the same derivative formulas and symmetry transformations. And thus, we can obtain the corresponding residue conditions:

$$\begin{array}{cc}\hfill & \underset{z={\widehat{z}}_n}{Res}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2i\theta \left({\widehat{z}}_n\right)}\left[{\displaystyle \frac{1}{2}}{N}^{″}\left({\widehat{z}}_n\right){\widehat{C}}_{n,3}+N^{\prime }\left({\widehat{z}}_n\right){\widehat{C}}_{n,2}+N\left({\widehat{z}}_n\right){\widehat{C}}_{n,1}\right],\hfill \\ \hfill & \underset{z={\widehat{z}}_n}{{\mathrm{P}}_{-2}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2\mathrm{i}\theta \left({\widehat{z}}_n\right)}[N^{\prime }\left({\widehat{z}}_n\right){\widehat{C}}_{n,3}+N\left({\widehat{z}}_n\right){\widehat{C}}_{n,2}],\hfill \\ \hfill & \underset{z={\widehat{z}}_n}{{\mathrm{P}}_{-3}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2i\theta \left({\widehat{z}}_n\right)}N\left({\widehat{z}}_n\right){\widehat{C}}_{n,3},\hfill \end{array}$$

(78)

and

$$\begin{array}{cc}\hfill & \underset{z={\widehat{z}}_n^{*}}{Res}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left({\widehat{z}}_n^{*}\right)}\left[{\displaystyle \frac{1}{2}}{\overline{N}}^{″}\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,3}+{\overline{N}}^{\prime }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,2}+\overline{N}\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,1}\right],\hfill \\ \hfill & \underset{z={\widehat{z}}_n^{*}}{{\mathrm{P}}_{-2}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left({\widehat{z}}_n^{*}\right)}[{\overline{N}}^{\prime }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,3}+\overline{N}\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,2}],\hfill \\ \hfill & \underset{z={\widehat{z}}_n^{*}}{{\mathrm{P}}_{-3}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left({\widehat{z}}_n^{*}\right)}\overline{N}\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,3},\hfill \end{array}$$

(79)

where

$$\begin{array}{cc}\hfill & {\widehat{A}}_n={\displaystyle \frac{6{\widehat{c}}_n}{{\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)}},{\widehat{B}}_n=\left[2{\widehat{d}}_n-{\displaystyle \frac{{\widehat{c}}_n{\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)}{2{\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)}}\right]{\displaystyle \frac{3}{{\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)}},\hfill \\ \hfill & {\widehat{C}}_n=\left[{\displaystyle \frac{{\left({\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)\right)}^2}{8{\left({\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)\right)}^2}}-{\displaystyle \frac{{\widehat{d}}_n{\widehat{c}}_n^{-1}{\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)}{2{\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)}}-{\displaystyle \frac{{\left(\det a\right)}^{\left(5\right)}\left({\widehat{z}}_n\right)}{10{\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)}}+{\widehat{e}}_n{\widehat{c}}_n^{-1}\right]{\displaystyle \frac{3}{{\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)}},\hfill \\ \hfill & {\widehat{C}}_{n,1}={\widehat{C}}_n+2i{\theta }^{\prime }{\widehat{B}}_n+(i{\theta }^{″}-2{\left({\theta }^{\prime }\right)}^2){\widehat{A}}_n,{\widehat{C}}_{n,2}={\widehat{B}}_n+2i{\theta }^{\prime }{\widehat{A}}_n,{\widehat{C}}_{n,3}={\widehat{A}}_n,\hfill \end{array}$$

(80)

and the definition of norming constants ${\widehat{\overline{C}}}_{n,1}$, ${\widehat{\overline{C}}}_{n,2}$, ${\widehat{\overline{C}}}_{n,3}$, defined exactly as in Equation (80) but with ${\widehat{c}}_n$, ${\widehat{d}}_n$, ${\widehat{e}}_n$ replaced by their conjugate counterparts ${\widehat{\overline{c}}}_n$, ${\widehat{\overline{d}}}_n$, ${\widehat{\overline{e}}}_n$. The derivatives of $\det a$ replaced by those of $\det \overline{a}$ evaluated at $z={\widehat{z}}_n^{*}$, and ${\theta }^{\prime }\left({\widehat{z}}_n\right)$, ${\theta }^{″}\left({\widehat{z}}_n\right)$ replaced by ${\theta }^{\prime }\left({\widehat{z}}_n^{*}\right)$, ${\theta }^{″}\left({\widehat{z}}_n^{*}\right)$.

The symmetry $z\to v{k}_0^2/z$ induces relations between the norming constants at the symmetric points. After a careful computation (detailed in Appendix B), we obtain the following expressions for the norming constants:

$$\begin{array}{cc}\hfill & {\widehat{C}}_{n,1}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,1}{R}_{+}-{\displaystyle \frac{2}{{\left(z_n^{*}\right)}^3}}{R}_{+}({\overline{C}}_{n,2}-i{\theta }^{\prime }\left(z_n^{*}\right){\overline{C}}_{n,3})R_{+}+{\displaystyle \frac{3}{{\left(z_n^{*}\right)}^4}}{R}_{+}{\overline{C}}_{n,3}{R}_{+},\hfill \\ \hfill & {\widehat{C}}_{n,2}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,2}{R}_{+}-{\displaystyle \frac{3}{{\left(z_n^{*}\right)}^3}}{R}_{+}{\overline{C}}_{n,3}{R}_{+},{\widehat{C}}_{n,3}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,3}{R}_{+}.\hfill \end{array}$$

(81)

II.

Case of $\mathrm{rank}$ $P(x,t,z_n)=2$

This case corresponds to $\mathrm{rank}$ $a\left(z_n\right)=0$, i.e., $a\left(z_n\right)=0_{2\times 2}$ (and similarly $\overline{a}\left(z_n^{*}\right)=0_{2\times 2}$). Consequently, the Jost functions are linearly dependent at the eigenvalue:

$$\varphi (x,t,z_n)=\psi (x,t,z_n)b_n,\overline{\varphi }(x,t,z_n^{*})=\overline{\psi }(x,t,z_n^{*}){\overline{b}}_n,$$

(82)

where $b_n$ and ${\overline{b}}_n$ are nonsingular $2\times 2$ matrices.

A crucial subtlety must be addressed. If $\det a\left(z\right)$ possessed only a third-order zero at $z_n$, the meromorphic matrix function $M{a}^{-1}$ would exhibit at most a double pole. To obtain a genuine triple pole—the central object of this paper, a stronger condition is indispensable: $\det a\left(z\right)$ must have a fourth-order zero at $z=z_n$. We therefore introduce the function

$$f\left(z\right)={(z-z_n)}^4{\displaystyle \frac{M\left(z\right)\alpha \left(z\right)}{\det a\left(z\right)}},$$

(83)

where is analytic at $z_n$. Expanding both numerator and denominator in Taylor series gives

$$\det a\left(z\right)={\displaystyle \frac{1}{4!}}{\left(\det a\right)}^{\left(4\right)}\left(z_n\right){(z-z_n)}^4+{\displaystyle \frac{1}{5!}}{\left(\det a\right)}^{\left(5\right)}\left(z_n\right){(z-z_n)}^5+\cdots ,$$

(84)

and

$$\begin{array}{cc}\hfill & M\left(z\right)\alpha \left(z\right)=M\alpha +{\left(M\alpha \right)}^{\prime }(z-z_n)+{\displaystyle \frac{1}{2!}}{\left(M\alpha \right)}^{″}{(z-z_n)}^2+{\displaystyle \frac{1}{3!}}{\left(M\alpha \right)}^{‴}{(z-z_n)}^3\hfill \\ & +{\displaystyle \frac{1}{4!}}{\left(M\alpha \right)}^{\left(4\right)}{(z-z_n)}^4+\cdots .\hfill \end{array}$$

(85)

Substituting these expansions into Equation (83) and simplifying yields the Taylor expansion of $f\left(z\right)$:

$$\begin{array}{cc}\hfill f\left(z\right)& ={\mathrm{P}}_{-4}\left[M\left(z\right)a^{-1}\left(z\right)\right]+{\mathrm{P}}_{-3}\left[M\left(z\right)a^{-1}\left(z\right)\right](z-z_n)+{\mathrm{P}}_{-2}\left[M\left(z\right)a^{-1}\left(z\right)\right]{(z-z_n)}^2\hfill \\ & +Res\left[M\left(z\right)a^{-1}\left(z\right)\right]{(z-z_n)}^3+O\left({(z-z_n)}^4\right).\hfill \end{array}$$

(86)

Here, $Res\left[M\left(z\right)a^{-1}\left(z\right)\right]$, ${\mathrm{P}}_{-2}\left[M\left(z\right)a^{-1}\left(z\right)\right]$, ${\mathrm{P}}_{-3}\left[M\left(z\right)a^{-1}\left(z\right)\right]$ and ${\mathrm{P}}_{-4}\left[M\left(z\right)a^{-1}\left(z\right)\right]$ denote the coefficients of $(z-z_n)$, ${(z-z_n)}^2$, ${(z-z_n)}^3$ and ${(z-z_n)}^4$ in the Laurent expansion of $M\left(z\right)a^{-1}\left(z\right)$, respectively. The coefficients are obtained from the standard formulas for Taylor coefficients:

$$\begin{array}{cc}\hfill & \underset{z=z_n}{Res}\left[M\left(z\right)a^{-1}\left(z\right)\right]={\displaystyle \frac{1}{3!}}\underset{z\to z_n}{lim}{\displaystyle \frac{d^3}{d{z}^3}}\left[{(z-z_n)}^4{\displaystyle \frac{M\left(z\right)\alpha \left(z\right)}{\det a\left(z\right)}}\right],\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-2}}\left[M\left(z\right)a^{-1}\left(z\right)\right]={\displaystyle \frac{1}{2!}}\underset{z\to z_n}{lim}{\displaystyle \frac{d^2}{d{z}^2}}\left[{(z-z_n)}^4{\displaystyle \frac{M\left(z\right)\alpha \left(z\right)}{\det a\left(z\right)}}\right],\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-3}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=\underset{z\to z_n}{lim}{\displaystyle \frac{d}{dz}}\left[{(z-z_n)}^4{\displaystyle \frac{M\left(z\right)\alpha \left(z\right)}{\det a\left(z\right)}}\right],\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-4}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=\underset{z\to z_n}{lim}{(z-z_n)}^4{\displaystyle \frac{M\left(z\right)\alpha \left(z\right)}{\det a\left(z\right)}}.\hfill \end{array}$$

(87)

After a straightforward (though lengthy) computation, the singular part of $M{a}^{-1}$ at $z=z_n$ can again be cast into the compact form (74), but the norming constants $C_{n,1}$, $C_{n,2}$ and $C_{n,3}$ are now defined by more elaborate expressions that involve $b_n$, $d_n$, $e_n$ and the derivatives of $\det a\left(z\right)$ up to sixth order. Explicitly,

$$\begin{array}{cc}\hfill & \underset{z=z_n}{Res}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2i\theta \left(z_n\right)}\left[{\displaystyle \frac{1}{2}}{N}^{″}\left(z_n\right)C_{n,3}+N^{\prime }\left(z_n\right)C_{n,2}+N\left(z_n\right)C_{n,1}\right],\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-2}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2i\theta \left(z_n\right)}[N^{\prime }\left(z_n\right)C_{n,3}+N\left(z_n\right)C_{n,2}],\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-3}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2i\theta \left(z_n\right)}N\left(z_n\right)C_{n,3},\hfill \\ \hfill & \underset{z=z_n}{{\mathrm{P}}_{-4}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=0_{4\times 2}.\hfill \end{array}$$

(88)

The auxiliary constants are constructed from the matrices $b_n$, $d_n$, $e_n$ and the derivatives of $\det a\left(z\right)$

$$\begin{array}{cc}\hfill & A_n={\displaystyle \frac{24b_n{\alpha }^{\prime }\left(z_n\right)}{{\left(\det a\right)}^{\left(4\right)}\left(z_n\right)}},B_n=d_n{b}_n^{-1}-{\displaystyle \frac{{\left(\det a\right)}^{\left(5\right)}\left(z_n\right)}{5{\left(\det a\right)}^{\left(4\right)}\left(z_n\right)}},C_n={\displaystyle \frac{12b_n{\alpha }^{″}\left(z_n\right)}{{\left(\det a\right)}^{\left(4\right)}\left(z_n\right)}},\hfill \\ & D_n={\displaystyle \frac{1}{2}}{e}_n{b}_n^{-1}-{\displaystyle \frac{{\left(\det a\right)}^{\left(6\right)}\left(z_n\right)}{30{\left(\det a\right)}^{\left(4\right)}\left(z_n\right)}},E_n={\displaystyle \frac{4b_n{\alpha }^{‴}\left(z_n\right)}{{\left(\det a\right)}^{\left(4\right)}\left(z_n\right)}}.\hfill \end{array}$$

(89)

The norming constants for the triple-pole (fourth-order zero) case are then defined by

$$\begin{array}{cc}\hfill C_{n,1}& =A_n\left[\left(2i{\theta }^{\prime }\left(z_n\right)-{\displaystyle \frac{{\left(\det a\right)}^{\left(5\right)}\left(z_n\right)}{5{\left(\det a\right)}^{\left(4\right)}\left(z_n\right)}}\right)B_n+i{\theta }^{″}\left(z_n\right)-2{\left({\theta }^{\prime }\left(z_n\right)\right)}^2+D_n\right]\hfill \\ & +C_n(B_n+2i{\theta }^{\prime }\left(z_n\right))+E_n,\hfill \\ \hfill C_{n,2}& =A_n\left(B_n+2i{\theta }^{\prime }\left(z_n\right)\right)+C_n,C_{n,3}=A_n.\hfill \end{array}$$

(90)

The corresponding quantities for the conjugate pole $z_n^{*}$ are obtained by the usual substitutions $(M,\alpha ,\det a,z_n)⟶(\overline{M},\overline{\alpha },\det \overline{a},z_n^{*})$ and $\theta ⟶-\theta$. They take the form

$$\begin{array}{cc}\hfill & \underset{z=z_n^{*}}{Res}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left(z_n^{*}\right)}\left[{\displaystyle \frac{1}{2}}{\overline{N}}^{″}\left(z_n^{*}\right){\overline{C}}_{n,3}+{\overline{N}}^{\prime }\left(z_n^{*}\right){\overline{C}}_{n,2}+\overline{N}\left(z_n^{*}\right){\overline{C}}_{n,1}\right],\hfill \\ \hfill & \underset{z=z_n^{*}}{{\mathrm{P}}_{-2}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left(z_n^{*}\right)}[{\overline{N}}^{\prime }\left(z_n^{*}\right){\overline{C}}_{n,3}+\overline{N}\left(z_n^{*}\right){\overline{C}}_{n,2}],\hfill \\ \hfill & \underset{z=z_n^{*}}{{\mathrm{P}}_{-3}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left(z_n^{*}\right)}\overline{N}\left(z_n^{*}\right){\overline{C}}_{n,3},\hfill \\ \hfill & \underset{z=z_n^{*}}{{\mathrm{P}}_{-4}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=0_{4\times 2},\hfill \end{array}$$

(91)

where ${\overline{C}}_{n,1}$, ${\overline{C}}_{n,2}$, ${\overline{C}}_{n,3}$ are defined analogously to Equation (77) with all quantities replaced by their conjugate counterparts.

Similarly, for ${\widehat{z}}_n=v{k}_0^2/z_n^{*}$ and $\widehat{\overline{z}}=v{k}_0^2/z_n$, we have

$$\begin{array}{cc}\hfill & \underset{z={\widehat{z}}_n}{Res}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2i\theta \left({\widehat{z}}_n\right)}\left[{\displaystyle \frac{1}{2}}{N}^{″}\left({\widehat{z}}_n\right){\widehat{C}}_{n,3}+N^{\prime }\left({\widehat{z}}_n\right){\widehat{C}}_{n,2}+N\left({\widehat{z}}_n\right){\widehat{C}}_{n,1}\right],\hfill \\ \hfill & \underset{z={\widehat{z}}_n}{{\mathrm{P}}_{-2}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2i\theta \left({\widehat{z}}_n\right)}[N^{\prime }\left({\widehat{z}}_n\right){\widehat{C}}_{n,3}+N\left({\widehat{z}}_n\right){\widehat{C}}_{n,2}],\hfill \\ \hfill & \underset{z={\widehat{z}}_n}{{\mathrm{P}}_{-3}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=e^{2i\theta \left({\widehat{z}}_n\right)}N\left({\widehat{z}}_n\right){\widehat{C}}_{n,3},\hfill \\ \hfill & \underset{z={\widehat{z}}_n}{{\mathrm{P}}_{-4}}\left[M\left(z\right)a^{-1}\left(z\right)\right]=0_{4\times 2},\hfill \end{array}$$

(92)

and

$$\begin{array}{cc}\hfill & \underset{z={\widehat{z}}_n^{*}}{Res}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left({\widehat{z}}_n^{*}\right)}\left[{\displaystyle \frac{1}{2}}{\overline{N}}^{″}\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,3}+{\overline{N}}^{\prime }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,2}+\overline{N}\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,1}\right],\hfill \\ \hfill & \underset{z={\widehat{z}}_n^{*}}{{\mathrm{P}}_{-2}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left({\widehat{z}}_n^{*}\right)}[{\overline{N}}^{\prime }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,3}+\overline{N}\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,2}],\hfill \\ \hfill & \underset{z={\widehat{z}}_n^{*}}{{\mathrm{P}}_{-3}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=e^{-2i\theta \left({\widehat{z}}_n^{*}\right)}\overline{N}\left({\widehat{z}}_n^{*}\right){\widehat{\overline{C}}}_{n,3},\hfill \\ \hfill & \underset{z={\widehat{z}}_n^{*}}{{\mathrm{P}}_{-4}}\left[\overline{M}\left(z\right){\overline{a}}^{-1}\left(z\right)\right]=0_{4\times 2},\hfill \end{array}$$

(93)

where

$$\begin{array}{cc}\hfill & {\widehat{A}}_n={\displaystyle \frac{24{\widehat{b}}_n{\alpha }^{\prime }\left({\widehat{z}}_n\right)}{{\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)}},{\widehat{B}}_n={\widehat{d}}_n{\widehat{b}}_n^{-1}-{\displaystyle \frac{{\left(\det a\right)}^{\left(5\right)}\left({\widehat{z}}_n\right)}{5{\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)}},{\widehat{C}}_n={\displaystyle \frac{12{\widehat{b}}_n{\alpha }^{″}\left({\widehat{z}}_n\right)}{{\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)}},\hfill \\ & {\widehat{D}}_n={\displaystyle \frac{1}{2}}{\widehat{e}}_n{\widehat{b}}_n^{-1}-{\displaystyle \frac{{\left(\det a\right)}^{\left(6\right)}\left({\widehat{z}}_n\right)}{30{\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)}},{\widehat{E}}_n={\displaystyle \frac{4{\widehat{b}}_n{\alpha }^{‴}\left({\widehat{z}}_n\right)}{{\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)}},\hfill \end{array}$$

(94)

and

$$\begin{array}{cc}\hfill & {\widehat{C}}_{n,1}={\widehat{A}}_n\left[(2i{\theta }^{\prime }\left({\widehat{z}}_n\right)-{\displaystyle \frac{{\left(\det a\right)}^{\left(5\right)}\left({\widehat{z}}_n\right)}{5{\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)}}){\widehat{B}}_n+i{\theta }^{″}\left({\widehat{z}}_n\right)-2{\left({\theta }^{\prime }\left({\widehat{z}}_n\right)\right)}^2+{\widehat{D}}_n\right]\hfill \\ & +{\widehat{C}}_n({\widehat{B}}_n+2i{\theta }^{\prime }\left({\widehat{z}}_n\right))+{\widehat{E}}_n,\hfill \\ & {\widehat{C}}_{n,2}={\widehat{A}}_n({\widehat{B}}_n+2i{\theta }^{\prime }\left({\widehat{z}}_n\right))+{\widehat{C}}_n,{\widehat{C}}_{n,3}={\widehat{A}}_n.\hfill \end{array}$$

(95)

The norming constants ${\widehat{\overline{C}}}_{n,1}$, ${\widehat{\overline{C}}}_{n,2}$, ${\widehat{\overline{C}}}_{n,3}$ are defined analogously to the case $\mathrm{rank}$ $P(x,t,z_n)=3$. By calculation, we obtain the symmetry of the norming constant.

$$\begin{array}{cc}\hfill & {\widehat{C}}_{n,1}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,1}{R}_{+}-{\displaystyle \frac{2}{{\left(z_n^{*}\right)}^3}}{R}_{+}({\overline{C}}_{n,2}-i{\theta }^{\prime }\left(z_n^{*}\right){\overline{C}}_{n,3})R_{+}+{\displaystyle \frac{3}{{\left(z_n^{*}\right)}^4}}{R}_{+}{\overline{C}}_{n,3}{R}_{+},\hfill \\ \hfill & {\widehat{C}}_{n,2}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,2}{R}_{+}-{\displaystyle \frac{3}{{\left(z_n^{*}\right)}^3}}{R}_{+}{\overline{C}}_{n,3}{R}_{+},{\widehat{C}}_{n,3}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,3}{R}_{+}.\hfill \end{array}$$

(96)

The derivation is provided in Appendix B. We find the symmetries of the norming constants coincide with those in Equation (81).

3. Inverse Scattering Problem

3.1. Riemann-Hilbert Problem

The inverse problem is formulated as a matrix RH problem for sectionally meromorphic functions. We introduce the following sectionally meromorphic matrices:

$${\mu }^{+}(x,t,z)=(M{a}^{-1},N),{\mu }^{-}(x,t,z)=(\overline{N},\overline{M}{\overline{a}}^{-1}),$$

(97)

where ${\mu }^{\pm }(x,t,z)$ are analytic in $D^{\pm }$, respectively, except at the discrete eigenvalues where they possess poles. From the scattering relation (36), these matrices satisfy the following jump condition across the contour ${\mathsf{\Sigma }}_z$:

$${\mu }^{-}(x,t,z)={\mu }^{+}(x,t,z)(I_4-G(x,t,z)),$$

(98)

with the jump matrix $G(x,t,z)$ given by

$$G(x,t,z)=\left(\begin{array}{cc}{0}_{2\times 2}& -e^{-2i\theta (x,t,z)}\overline{\rho }\left(z\right)\\ e^{2i\theta (x,t,z)}\rho \left(z\right)& \rho \left(z\right)\overline{\rho }\left(z\right)\end{array}\right).$$

(99)

The asymptotic behaviour of ${\mu }^{\pm }\left(z\right)$ follows directly from the expansions of the modified Jost functions and scattering coefficients obtained in Section 2.5. As $z\to \infty$ (in the respective half-plane),

$${\mu }^{\pm }=I_4+o(1/z),$$

(100)

while as $z\to 0$,

$${\mu }^{\pm }=-{\displaystyle \frac{i}{z}}{\sigma }_4{\tilde{Q}}_{+}+o\left(1\right).$$

(101)

To obtain an explicit integral representation for the solution, we regularize the RH problem by subtracting the pole contributions. Define ${\xi }_j=z_j$ for $j=1,2,\dots ,N$, and ${\xi }_j={\widehat{z}}_{j-N}$ for $j=N+1,N+2,\dots ,2N$, encompassing all discrete eigenvalues in $D^{+}$. The contribution from the poles of $M{a}^{-1}$ is explicitly subtracted from ${\mu }^{+}$. Applying the Cauchy projection operator to the regularized RH problem then yields the following integral equation for $\overline{N}\left(z\right)$:

$$\begin{array}{cc}\hfill \overline{N}\left(z\right)& =\left({\displaystyle \genfrac{}{}{0pt}{}{I_2}{\frac{i}{z}{R}_{+}}}\right)+\sum _{j=1}^{2N}{\displaystyle \frac{\underset{z=z_j}{\mathrm{Res}}\left(M{a}^{-1}\right)}{z-{\xi }_j}}+\sum _{j=1}^{2N}{\displaystyle \frac{\underset{z=z_j}{{\mathrm{P}}_{-2}}\left(M{a}^{-1}\right)}{{\left(z-{\xi }_j\right)}^2}}+\sum _{j=1}^{2N}{\displaystyle \frac{\underset{z=z_j}{{\mathrm{P}}_{-3}}\left(M{a}^{-1}\right)}{{\left(z-{\xi }_j\right)}^3}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathsf{\Sigma }}_z}{\displaystyle \frac{e^{2i\theta \left(\xi \right)}N\left(\xi \right)\rho \left(\xi \right)}{\xi -(z-i0)}}d\xi ,\hfill \end{array}$$

(102)

where the residues and pole coefficients are given by Equations (75) and (88). Similarly, an integral equation for $N\left(z\right)$ is obtained by considering the poles of $\overline{M}{\overline{a}}^{-1}$ in $D^{-}$:

$$\begin{array}{cc}\hfill N\left(z\right)=& \left(\begin{array}{c}-{\displaystyle \frac{i}{z}}{Q}_{+}\\ I_2\end{array}\right)+\sum _{j=1}^{2N}{\displaystyle \frac{\underset{z=z_j^{*}}{\mathrm{Res}}\left(\overline{M}{\overline{a}}^{-1}\right)}{z-{\xi }_j^{*}}}+\sum _{j=1}^{2N}{\displaystyle \frac{\underset{z=z_j^{*}}{{\mathrm{P}}_{-2}}\left(\overline{M}{\overline{a}}^{-1}\right)}{{\left(z-{\xi }_j^{*}\right)}^2}}+\sum _{j=1}^{2N}{\displaystyle \frac{\underset{z=z_j^{*}}{{\mathrm{P}}_{-3}}\left(\overline{M}{\overline{a}}^{-1}\right)}{{\left(z-{\xi }_j^{*}\right)}^3}}\hfill \\ \hfill & -{\displaystyle \frac{1}{2\pi i}}{\int }_{{\mathsf{\Sigma }}_z}{\displaystyle \frac{e^{-2i\theta \left(\xi \right)}\overline{N}\left(\xi \right)\overline{\rho }\left(\xi \right)}{\xi -(z+i0)}}d\xi .\hfill \end{array}$$

(103)

Here, ${\xi }_j^{*}=z_j^{*}$ for $j=1,2,\dots ,N$, and ${\xi }_j^{*}={\widehat{z}}_{j-N}^{*}$ for $j=N+1,N+2,\dots ,2N$. The corresponding constants $C_{j,i}$, ${\widehat{C}}_{j,i}$, ${\overline{C}}_{j,i}$, ${\widehat{\overline{C}}}_{j,i}$ ($i=1,2,3$) are defined in Section 2.6.

The potential $Q(x,t)$ is reconstructed from the large-z asymptotics of the solution to the RH problem. Comparing the $O\left(z^{-1}\right)$ term in the expansion of (102) or (103) with the asymptotic Formula (57) yields:

$$\begin{array}{cc}\hfill Q(x,t)& =Q_{+}+i\sum _{j=1}^{2N}{e}^{-2i\theta \left({\xi }_j^{*}\right)}\left[{\displaystyle \frac{1}{2}}{\overline{N}}_{j,2}^{up}\left({\xi }_j^{*}\right){\overline{C}}_{j,3}+{\overline{N}}_{j,1}^{up}\left({\xi }_j^{*}\right){\overline{C}}_{j,2}+{\overline{N}}_{j,0}^{up}\left({\xi }_j^{*}\right){\overline{C}}_{j,1}\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{\pi }}{\int }_{{\mathsf{\Sigma }}_z}{e}^{-2i\theta (x,t,\xi )}{\overline{N}}^{up}(x,t,\xi )\overline{\rho }\left(\xi \right)d\xi ,\hfill \end{array}$$

(104)

$$\begin{array}{cc}\hfill R(x,t)& =R_{+}-i\sum _{j=1}^{2N}{e}^{2i\theta \left({\xi }_j\right)}\left[{\displaystyle \frac{1}{2}}{N}_{j,2}^{dn}\left({\xi }_j\right)C_{j,3}+N_{j,1}^{dn}\left({\xi }_j\right)C_{j,2}+N_{j,0}^{dn}\left({\xi }_j\right)C_{j,1}\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{\pi }}{\int }_{{\mathsf{\Sigma }}_z}{e}^{2i\theta (x,t,\xi )}{N}^{dn}(x,t,\xi )\rho \left(\xi \right)d\xi .\hfill \end{array}$$

(105)

In Equations (102) and (103), we have used the block decomposition $N(x,t)={(N^{up},N^{dn})}^T$ and $\overline{N}(x,t)={({\overline{N}}^{up},{\overline{N}}^{dn})}^T$, where the superscripts up and dn denote the upper and lower $2\times 2$ blocks, respectively. The coefficients $N_{j,k}$ and ${\overline{N}}_{j,k}$ are defined as the derivatives of these blocks at the discrete eigenvalues:

$$\begin{array}{cc}\hfill & N_{n,0}(x,t)=N(x,t,z_n),{\overline{N}}_{n,0}(x,t)=\overline{N}(x,t,z_n^{*}),\hfill \\ \hfill & N_{n,1}(x,t)=(\partial N/\partial z)(x,t,z_n),{\overline{N}}_{n,1}(x,t)=(\partial \overline{N}/\partial z)(x,t,z_n^{*}),\hfill \\ \hfill & N_{n,2}(x,t)=({\partial }^{2}N/\partial z^2)(x,t,z_n),{\overline{N}}_{n,2}(x,t)=({\partial }^2\overline{N}/\partial z^2)(x,t,z_n^{*}).\hfill \end{array}$$

The symmetry relations for the potential, $R=\mathsf{\Sigma }{Q}^{†}\mathsf{\Omega }$, together with the asymptotic behaviors $N^{dn}(x,t,z)\sim I_2$ and ${\overline{N}}^{up}(x,t,z)\sim I_2$ as $x\to \infty$, impose constraints on the norming constants. By comparing the structures of the reconstruction Formulas (102) and (103), we deduce the following symmetry:

$${\overline{C}}_{j,n}={\mathsf{\Omega }}^{-1}{C}_{j,n}^{†}{\mathsf{\Sigma }}^{-1},n=1,2,3,j=1,\cdots ,2N.$$

(106)

If, in addition, the potential satisfies the transposition reduction $Q(x,t)=Q^T(x,t)$ (equivalently $R(x,t)=R^T(x,t)$), the norming constants inherit the symmetry:

$${\overline{C}}_{j,n}={\overline{C}}_{j,n}^T,C_{j,n}=C_{j,n}^T,n=1,2,3,j=1,\cdots ,2N.$$

(107)

3.2. Reflectionless Case and N-Triple-Pole Soliton Solutions

In the reflectionless case, where the reflection coefficients vanish ($\rho \left(z\right)=\overline{\rho }\left(z\right)=0_{2\times 2}$), the RH problem simplifies considerably. The integral terms in Equations (102) and (103) vanish, reducing the problem to a closed algebraic system for the eigenfunctions. Focusing on the upper blocks ($up$-components), we obtain the following system:

$$\begin{array}{cc}\hfill {\overline{N}}^{up}\left(z\right)=& I_2+\sum _{j=1}^{2N}{\displaystyle \frac{e^{2i\theta \left({\xi }_j\right)}\left[\frac{1}{2}{N}_{j,2}^{up}(x,t)C_{j,3}+N_{j,1}^{up}(x,t)C_{j,2}+N_{j,0}^{up}(x,t)C_{j,1}\right]}{z-{\xi }_j}}\hfill \\ \hfill +& \sum _{j=1}^{2N}{\displaystyle \frac{e^{2i\theta \left({\xi }_j\right)}(N_{j,1}^{up}(x,t)C_{j,3}+N_{j,0}^{up}(x,t)C_{j,2})}{{\left(z-{\xi }_j\right)}^2}}+\sum _{j=1}^{2N}{\displaystyle \frac{e^{2i\theta \left({\xi }_j\right)}{N}_{j,0}^{up}(x,t)C_{j,3}}{{\left(z-{\xi }_j\right)}^3}},\hfill \end{array}$$

(108)

and

$$\begin{array}{cc}\hfill N^{up}\left(z\right)=& -{\displaystyle \frac{i}{z}}{Q}_{+}+\sum _{j=1}^{2N}{\displaystyle \frac{e^{-2i\theta \left({\xi }_j^{*}\right)}\left[\frac{1}{2}{\overline{N}}_{j,2}^{up}(x,t){\overline{C}}_{j,3}+{\overline{N}}_{j,1}^{up}(x,t){\overline{C}}_{j,2}+{\overline{N}}_{j,0}^{up}(x,t){\overline{C}}_{j,1}\right]}{z-{\xi }_j^{*}}}\hfill \\ \hfill +& \sum _{j=1}^{2N}{\displaystyle \frac{e^{-2i\theta \left({\xi }_j^{*}\right)}({\overline{N}}_{j,1}^{up}(x,t){\overline{C}}_{j,3}+{\overline{N}}_{j,0}^{up}(x,t){\overline{C}}_{j,2})}{{\left(z-{\xi }_j^{*}\right)}^2}}+\sum _{j=1}^{2N}{\displaystyle \frac{e^{-2i\theta \left({\xi }_j^{*}\right)}{\overline{N}}_{j,0}^{up}(x,t){\overline{C}}_{j,3}}{{\left(z-{\xi }_j^{*}\right)}^3}}.\hfill \end{array}$$

(109)

Here, for brevity, we have written $N_{j,k}^{up}=N_{j,k}^{up}(x,t)$ and similarly for ${\overline{N}}_{j,k}^{up}$.

Theorem 1.

(General N-triple-pole soliton solution). In the reflectionless case, the general N-triple-pole soliton solution of the matrix NLS Equation (6) under nonzero boundary conditions is reconstructed via

$$Q(x,t)=Q_{+}+i\sum _{n=1}^{2N}{e}^{-2i\theta \left({\xi }_n^{*}\right)}\left[{\displaystyle \frac{1}{2}}{\overline{N}}_{n,2}^{up}\left({\xi }_n^{*}\right){\overline{C}}_{n,3}+{\overline{N}}_{n,1}^{up}\left({\xi }_n^{*}\right){\overline{C}}_{n,2}+{\overline{N}}_{n,0}^{up}\left({\xi }_n^{*}\right){\overline{C}}_{n,1}\right].$$

(110)

The functions ${\overline{N}}_{j,2}^{up}(x,t)$, ${\overline{N}}_{j,1}^{up}(x,t)$ and ${\overline{N}}_{j,0}^{up}(x,t)$ are determined by solving the linear algebraic system formed by evaluating Equation (108) at the points $z={\xi }_k^{*}$ for $k=1,\dots ,2N$.

A detailed example for the fundamental case $N=1$ is presented in the Appendix A.

4. Discussion

In this paper, we have systematically developed the inverse scattering transform for the matrix NLS equation under NZBCs, focusing on the case where the discrete spectrum contains triple poles. Two distinct scenarios arise from the rank of the matrix $P(x,t,z_n)$: the case $\mathrm{rank}$ $P(x,t,z_n)=3$ requires a third-order zero of $\det a\left(z\right)$ at $z=z_n$, while $\mathrm{rank}$ $P(x,t,z_n)=2$ necessitates a fourth-order zero. For each case, we derived the corresponding residue conditions and established the symmetries of the associated norming constants, thereby completing the discrete spectral analysis for triple poles. Solving the resulting matrix RH problem in the reflectionless case yielded explicit closed-form expressions for the general N-triple-pole soliton solutions.

The extension from double to triple poles in the matrix setting is by no means a mere parameter adjustment. The matrix structure introduces a forced correspondence between the order of zeros of $\det a\left(z\right)$ and the rank of the eigenfunction matrix $P(x,t,z_n)$—a feature already present at the double-pole level, but whose explicit realization at third order requires:

• handling third-order poles in the matrix Riemann–Hilbert problem under two distinct reduction classes;
• deriving Laurent coefficients up to order three for both admissible rank cases ($\mathrm{rank}$ $P=2,3$);
• constructing norming constants $C_{n,1},C_{n,2},C_{n,3}$ that absorb all higher-order scattering data;
• respecting the quartet symmetry $z_n,z_n^{*},v{k}_0^2/z_n,v{k}_0^2/z_n^{*}$ imposed by the boundary conditions.

Building on these results, several directions merit further investigation. First, a natural extension is to establish a unified framework for constructing soliton solutions with arbitrary-order poles under nonzero boundary conditions. Such an extension would complete the hierarchy of higher-order poles within the IST formalism and reveal deeper algebraic structures inherent in integrable systems. Second, the long-time asymptotics of solutions in the presence of higher-order discrete spectra constitutes a challenging yet important problem. The RH formalism established herein provides an ideal starting point for this endeavor, following the seminal work of Deift-Zhou and subsequent developments [31,32,33,34].

Beyond these mathematical generalizations, the matrix NLS equation under NZBCs is not merely a mathematical abstraction—it is the mean-field model for $F=1$ spinor BEC in the homogeneous trap-free regime, and also arises in the description of four-wave mixing processes in nonlinear optics. Equation (6) can be reduced to two-component or even scalar NLS systems via appropriate parameter choices, retaining significant physical content—from optical spatiotemporal solitons to effective descriptions of bilayer graphene [35]. These connections provide fertile ground for further interdisciplinary exploration. In particular, Cases 3 and 4—though not directly applicable to $F=1$ condensates—also warrant attention. They find important applications in other physical contexts, such as multicolor optical solitons and models of superconducting phenomena [35]. Elucidating these connections and investigating the role of triple-pole structures in such systems would constitute a valuable extension of the present work.

Appendix B.1. Relations for the Rank-3 Case

We begin by introducing norming constants at the symmetric point $z\to v{k}_0^2/z$. In analogy with Equations (65) and (66), we define ${\widehat{c}}_n$, ${\widehat{d}}_n$, ${\widehat{e}}_n$ via

$$\varphi \left({\widehat{z}}_n\right)\alpha \left({\widehat{z}}_n\right)=\psi \left({\widehat{z}}_n\right){\widehat{c}}_n,\overline{\varphi }\left({\widehat{z}}_n^{*}\right)\overline{\alpha }\left({\widehat{z}}_n^{*}\right)=\overline{\psi }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{c}}}_n,$$

(A5)

together with the higher-order relations

$$\begin{array}{cc}\hfill {\left(\varphi \left({\widehat{z}}_n\right)\alpha \left({\widehat{z}}_n\right)\right)}^{\prime }& ={\psi }^{\prime }\left({\widehat{z}}_n\right){\widehat{c}}_n+\psi \left({\widehat{z}}_n\right){\widehat{d}}_n,\hfill \\ \hfill {\left(\varphi \left({\widehat{z}}_n\right)\alpha \left({\widehat{z}}_n\right)\right)}^{″}& ={\psi }^{″}\left({\widehat{z}}_n\right){\widehat{c}}_n+2{\psi }^{\prime }\left({\widehat{z}}_n\right){\widehat{d}}_n+\psi \left({\widehat{z}}_n\right){\widehat{e}}_n,\hfill \end{array}$$

(A6)

and analogous formulas for the conjugates.

Using the symmetry (48) and the relations between the adjugate matrices obtained from Equation (49), we have

$$\begin{array}{cc}\hfill & \alpha \left(z_n\right)={\displaystyle \frac{v}{k_0^2}}\mathrm{cof}\left(R_{-}\right)\overline{\alpha }\left({\widehat{z}}_n^{*}\right)\mathrm{cof}\left(Q_{+}\right),\overline{\alpha }\left(z_n^{*}\right)={\displaystyle \frac{v}{k_0^2}}\mathrm{cof}\left(Q_{-}\right)\alpha \left({\widehat{z}}_n\right)\mathrm{cof}\left(R_{+}\right),\hfill \\ \hfill & \alpha \left({\widehat{z}}_n\right)={\displaystyle \frac{v}{k_0^2}}\mathrm{cof}\left(R_{-}\right)\overline{\alpha }\left(z_n^{*}\right)\mathrm{cof}\left(Q_{+}\right),\overline{\alpha }\left({\widehat{z}}_n^{*}\right)={\displaystyle \frac{v}{k_0^2}}\mathrm{cof}\left(Q_{-}\right)\alpha \left({\widehat{z}}_n\right)\mathrm{cof}\left(R_{+}\right),\hfill \end{array}$$

(A7)

where $\mathrm{cof}\left(Q_{\pm }\right)$ and $\mathrm{cof}\left(R_{\pm }\right)$ denote the adjugate matrices of $Q_{\pm }$ and $R_{\pm }$, respectively. Now we express $\varphi \left(z_n\right)\alpha \left(z_n\right)$, in two different ways. On one hand, using Equations (48) and (A5),

$$\varphi \left(z_n\right)\alpha \left(z_n\right)={\displaystyle \frac{i}{z_n}}\overline{\varphi }\left({\widehat{z}}_n^{*}\right)R_{-}\alpha \left(z_n\right)={\displaystyle \frac{iv}{z_n{k}_0^2}}\overline{\varphi }\left({\widehat{z}}_n^{*}\right)R_{-}\mathrm{cof}\left(R_{-}\right)\overline{\alpha }\left({\widehat{z}}_n^{*}\right)\mathrm{cof}\left(Q_{+}\right).$$

(A8)

Applying Equation (A7) to replace $\alpha \left(z\right)$ and noting that $R_{-}\mathrm{cof}\left(R_{-}\right)=\det R_{-}{I}_2$, we obtain

$$\varphi \left(z_n\right)\alpha \left(z_n\right)={\displaystyle \frac{iv\det R_{-}}{k_0^2{z}_n}}\overline{\psi }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{c}}}_n\mathrm{cof}\left(Q_{+}\right).$$

(A9)

On the other hand, from Equations (48) and (65),

$$\varphi \left(z_n\right)\alpha \left(z_n\right)=\psi \left(z_n\right)c_n=-{\displaystyle \frac{i}{z_n}}\overline{\psi }\left({\widehat{z}}_n^{*}\right)Q_{+}{c}_n.$$

(A10)

Comparing Equations (A9) and (A10) yields

$${\widehat{\overline{c}}}_n=-{\displaystyle \frac{v{k}_0^2}{\det Q_{+}\det R_{-}}}{Q}_{+}{c}_n{Q}_{+}.$$

(A11)

Similarly, by Equations (48), (A5) and (A7), we can obtain

$${\widehat{c}}_n=-{\displaystyle \frac{v{k}_0^2}{\det Q_{-}\det R_{+}}}{R}_{+}{\overline{c}}_n{R}_{+}.$$

(A12)

Following the same procedure and using the differentiated relations (A6) together with (66), we derive

$$\begin{array}{cc}\hfill & {\widehat{\overline{d}}}_n=-{\displaystyle \frac{v{k}_0^2}{\det Q_{+}\det R_{-}}}{Q}_{+}{d}_n{Q}_{+},{\widehat{d}}_n=-{\displaystyle \frac{v{k}_0^2}{\det Q_{-}\det R_{+}}}{R}_{+}{\overline{d}}_n{R}_{+},\hfill \\ \hfill & {\widehat{\overline{e}}}_n=-{\displaystyle \frac{v{k}_0^2}{\det Q_{+}\det R_{-}}}{Q}_{+}{e}_n{Q}_{+},{\widehat{e}}_n=-{\displaystyle \frac{v{k}_0^2}{\det Q_{-}\det R_{+}}}{R}_{+}{\overline{e}}_n{R}_{+}.\hfill \end{array}$$

(A13)

Next, we need the relations between the determinants of the scattering matrices and their derivatives. By Equations (49) and (A7), we have

$$\det a\left({\widehat{z}}_n\right)={\displaystyle \frac{\det Q_{+}\det R_{-}}{k_0^4}}\det \overline{a}\left(z_n^{*}\right),det\overline{a}\left({\widehat{z}}_n^{*}\right)={\displaystyle \frac{det{Q}_{-}det{R}_{+}}{k_0^4}}deta\left(z_n\right).$$

(A14)

Differentiating these identities repeatedly with respect to z and evaluating at the relevant points yield the following connections:

$$\begin{array}{cc}\hfill & {\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)=-{\displaystyle \frac{v{\left(z_n^{*}\right)}^6}{k_0^{10}}}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{‴}\left(z_n^{*}\right),\hfill \\ \hfill & {\left(\det \overline{a}\right)}^{‴}\left({\widehat{z}}_n^{*}\right)=-{\displaystyle \frac{v{\left(z_n\right)}^6}{k_0^{10}}}\det Q_{-}\det R_{+}{\left(\det a\right)}^{‴}\left(z_n\right),\hfill \end{array}$$

(A15)

$$\begin{array}{cc}\hfill & {\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)={\displaystyle \frac{12{\left(z_n^{*}\right)}^7}{k_0^{12}}}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{‴}\left(z_n^{*}\right)+{\displaystyle \frac{{\left(z_n^{*}\right)}^8}{k_0^{12}}}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{\left(4\right)}\left(z_n^{*}\right),\hfill \\ \hfill & {\left(\det \overline{a}\right)}^{\left(4\right)}\left({\widehat{z}}_n^{*}\right)={\displaystyle \frac{12{\left(z_n\right)}^7}{k_0^{12}}}\det Q_{-}\det R_{+}{\left(\det a\right)}^{‴}\left(z_n\right)+{\displaystyle \frac{{\left(z_n\right)}^8}{k_0^{12}}}\det Q_{-}\det R_{+}{\left(\det a\right)}^{\left(4\right)}\left(z_n\right),\hfill \end{array}$$

(A16)

and

$$\begin{array}{cc}\hfill & {\left(\det a\right)}^{\left(5\right)}\left({\widehat{z}}_n\right)=[-{\displaystyle \frac{v{z}^{10}}{k_0^{14}}}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{\left(5\right)}\left(z\right)-{\displaystyle \frac{20v{z}^9}{k_0^{14}}}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{\left(4\right)}\left(z\right)\hfill \\ \hfill & -{\displaystyle \frac{120v{z}^8}{k_0^{14}}}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{‴}\left(z\right){]}_{z=z_n^{*}},\hfill \\ \hfill & {\left(\det \overline{a}\right)}^{\left(5\right)}\left({\widehat{z}}_n^{*}\right)=[-{\displaystyle \frac{v{z}^{10}}{k_0^{14}}}\det Q_{-}\det R_{+}{\left(\det a\right)}^{\left(5\right)}\left(z\right)-{\displaystyle \frac{20v{z}^9}{k_0^{14}}}\det Q_{-}\det R_{+}{\left(\det a\right)}^{\left(4\right)}\left(z\right)\hfill \\ \hfill & -{\displaystyle \frac{120v{z}^8}{k_0^{14}}}\det Q_{-}\det R_{+}{\left(\det a\right)}^{‴}\left(z\right){]}_{z=z_n}.\hfill \end{array}$$

(A17)

By Equation (80), we finally obtain the following symmetry relations for ${\widehat{A}}_n$ and ${\widehat{\overline{A}}}_n$:

$$\begin{array}{cc}\hfill & {\widehat{A}}_n={\displaystyle \frac{-6v{k}_0^2{R}_{+}{\overline{c}}_n{R}_{+}{k}_0^{10}}{-v{\left(z_n^{*}\right)}^6\det Q_{-}\det R_{+}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{‴}\left(z_n^{*}\right)}}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{A}}_n{R}_{+},\hfill \\ \hfill & {\widehat{\overline{A}}}_n={\displaystyle \frac{-6v{k}_0^2{Q}_{+}{c}_n{Q}_{+}{k}_0^{10}}{-v{\left(z_n\right)}^6\det Q_{-}\det R_{+}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{‴}\left(z_n\right)}}={\displaystyle \frac{1}{{\left(z_n\right)}^2}}{Q}_{+}{A}_n{Q}_{+},\hfill \end{array}$$

(A18)

noting that $\det Q_{\pm }\det R_{\pm }=k_0^4$. It follows similarly that

$$\begin{array}{cc}\hfill & {\widehat{B}}_n={\displaystyle \frac{6k_0^{10}}{v{z}^6\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{‴}\left(z\right)}}[{\displaystyle \frac{R_{+}{\overline{c}}_n{R}_{+}[12z^7{\left(\det \overline{a}\right)}^{‴}\left(z\right)+z^8{\left(\det \overline{a}\right)}^{\left(4\right)}\left(z\right)]}{4z^6\det Q_{-}\det R_{+}{\left(\det \overline{a}\right)}^{‴}\left(z\right)}}\hfill \\ & +{\displaystyle \frac{v{k}_0^2{R}_{+}{\overline{d}}_n{R}_{+}}{\det Q_{-}\det R_{+}}}{]}_{z=z_n^{*}}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{B}}_n{R}_{+}-{\displaystyle \frac{3}{{\left(z_n^{*}\right)}^3}}{R}_{+}{\overline{A}}_n{R}_{+},\hfill \\ \hfill & {\widehat{\overline{B}}}_n={\displaystyle \frac{6k_0^{10}}{v{z}^6\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{‴}\left(z\right)}}[{\displaystyle \frac{Q_{+}{c}_n{Q}_{+}[12z^7{\left(\det \overline{a}\right)}^{‴}\left(z\right)+z^8{\left(\det \overline{a}\right)}^{\left(4\right)}\left(z\right)]}{4z^6\det Q_{-}\det R_{+}{\left(\det \overline{a}\right)}^{‴}\left(z\right)}}\hfill \\ & +{\displaystyle \frac{v{k}_0^2{Q}_{+}{d}_n{Q}_{+}}{\det Q_{-}\det R_{+}}}{]}_{z=z_n}={\displaystyle \frac{1}{{\left(z_n\right)}^2}}{Q}_{+}{B}_n{Q}_{+}-{\displaystyle \frac{3}{{\left(z_n\right)}^3}}{Q}_{+}{A}_n{Q}_{+},\hfill \end{array}$$

(A19)

and

$$\begin{array}{cc}\hfill & {\widehat{C}}_n=\left[{\displaystyle \frac{{\left({\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)\right)}^2}{16{\left({\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)\right)}^2}}-{\displaystyle \frac{{\widehat{d}}_n{\widehat{c}}_n^{-1}{\left(\det a\right)}^{\left(4\right)}\left({\widehat{z}}_n\right)}{4{\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)}}-{\displaystyle \frac{{\left(\det a\right)}^{\left(5\right)}\left({\widehat{z}}_n\right)}{20{\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)}}+{\displaystyle \frac{{\widehat{e}}_n{\widehat{c}}_n^{-1}}{2}}\right]{\displaystyle \frac{6}{{\left(\det a\right)}^{‴}\left({\widehat{z}}_n\right)}}\hfill \\ \hfill & ={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_n{R}_{+}-{\displaystyle \frac{2}{{\left(z_n^{*}\right)}^3}}{R}_{+}{\overline{B}}_n{R}_{+}+{\displaystyle \frac{3}{{\left(z_n^{*}\right)}^4}}{R}_{+}{\overline{A}}_n{R}_{+},\hfill \\ \hfill & {\widehat{\overline{C}}}_n=\left[{\displaystyle \frac{{\left({\left(\det \overline{a}\right)}^{\left(4\right)}\left({\widehat{z}}_n^{*}\right)\right)}^2}{16{\left({\left(\det \overline{a}\right)}^{‴}\left({\widehat{z}}_n^{*}\right)\right)}^2}}-{\displaystyle \frac{{\widehat{\overline{d}}}_n{\widehat{\overline{c}}}_n^{-1}{\left(\det \overline{a}\right)}^{\left(4\right)}\left({\widehat{z}}_n^{*}\right)}{4{\left(\det \overline{a}\right)}^{‴}\left({\widehat{z}}_n^{*}\right)}}-{\displaystyle \frac{{\left(\det \overline{a}\right)}^{\left(5\right)}\left({\widehat{z}}_n^{*}\right)}{20{\left(\det \overline{a}\right)}^{‴}\left({\widehat{z}}_n^{*}\right)}}+{\displaystyle \frac{{\widehat{\overline{e}}}_n{\widehat{\overline{c}}}_n^{-1}}{2}}\right]{\displaystyle \frac{6}{{\left(\det \overline{a}\right)}^{‴}\left({\widehat{z}}_n^{*}\right)}}\hfill \\ \hfill & ={\displaystyle \frac{1}{{\left(z_n\right)}^2}}{Q}_{+}{C}_n{Q}_{+}-{\displaystyle \frac{2}{{\left(z_n\right)}^3}}{Q}_{+}{B}_n{Q}_{+}+{\displaystyle \frac{3}{{\left(z_n\right)}^4}}{Q}_{+}{A}_n{Q}_{+}.\hfill \end{array}$$

(A20)

Substituting the above equations into Equation (80), one has

$$\begin{array}{cc}\hfill & {\widehat{C}}_{n,1}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,1}{R}_{+}-{\displaystyle \frac{2}{{\left(z_n^{*}\right)}^3}}{R}_{+}({\overline{C}}_{n,2}-i{\theta }^{\prime }\left(z_n^{*}\right){\overline{C}}_{n,3})R_{+}+{\displaystyle \frac{3}{{\left(z_n^{*}\right)}^4}}{R}_{+}{\overline{C}}_{n,3}{R}_{+},\hfill \\ \hfill & {\widehat{C}}_{n,2}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,2}{R}_{+}-{\displaystyle \frac{3}{{\left(z_n^{*}\right)}^3}}{R}_{+}{\overline{C}}_{n,3}{R}_{+},{\widehat{C}}_{n,3}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,3}{R}_{+}.\hfill \end{array}$$

(A21)

Appendix B.2. Relations for the Rank-2 Case

We now perform an analogous analysis for the case $\mathrm{rank}$ $P=2$. At the symmetric points we introduce ${\widehat{c}}_n$, ${\widehat{d}}_n$, ${\widehat{e}}_n$ via

$$\varphi (x,t,{\widehat{z}}_n)=\psi (x,t,{\widehat{z}}_n){\widehat{b}}_n,\overline{\varphi }(x,t,{\widehat{z}}_n^{*})=\overline{\psi }(x,t,{\widehat{z}}_n^{*}){\widehat{\overline{b}}}_n,$$

(A22)

and the corresponding higher-order relations

$$\begin{array}{cc}& {\varphi }^{\prime }\left({\widehat{z}}_n\right)={\psi }^{\prime }\left({\widehat{z}}_n\right){\widehat{b}}_n+\psi \left({\widehat{z}}_n\right){\widehat{d}}_n,\hfill \\ & {\varphi }^{″}\left({\widehat{z}}_n\right)={\psi }^{″}\left({\widehat{z}}_n\right){\widehat{b}}_n+2{\psi }^{\prime }\left({\widehat{z}}_n\right){\widehat{d}}_n+\psi \left({\widehat{z}}_n\right){\widehat{e}}_n.\hfill \end{array}$$

(A23)

Using the symmetry (48) together with (A22), we can relate $\varphi \left(z_n\right)$ and $\psi \left(z_n\right)$ to quantities at the symmetric points:

$$\varphi (x,t,z_n)={\displaystyle \frac{i}{z_n}}\overline{\varphi }(x,t,{\widehat{z}}_n^{*})R_{-}={\displaystyle \frac{i}{z_n}}\overline{\psi }(x,t,{\widehat{z}}_n^{*}){\widehat{\overline{b}}}_n{R}_{-}.$$

(A24)

On the other hand, from Equation (82),

$$\varphi (x,t,z_n)=\psi (x,t,z_n)b_n=-{\displaystyle \frac{i}{z_n}}\overline{\psi }(x,t,{\widehat{z}}_n^{*})Q_{+}{b}_n.$$

(A25)

By comparing these two results, we obtain

$${\widehat{\overline{b}}}_n=-{\displaystyle \frac{v}{k_0^2}}{Q}_{+}{b}_n{Q}_{-}.$$

(A26)

Similarly, by Equations (48), (82), and (A22), we have

$${\widehat{b}}_n=-{\displaystyle \frac{v}{k_0^2}}{R}_{+}{\overline{b}}_n{R}_{-}.$$

(A27)

From Equation (82), it follows that

$$\begin{array}{cc}\hfill {\varphi }^{\prime }\left(z_n\right)& =-{\displaystyle \frac{i}{{\left(z_n\right)}^2}}\overline{\psi }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{b}}}_n{R}_{-}+{\displaystyle \frac{i}{z_n}}\left(-{\displaystyle \frac{v{k}_0^2}{{\left(z_n\right)}^2}}{\overline{\psi }}^{\prime }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{b}}}_n+\overline{\psi }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{d}}}_n\right)R_{-}\hfill \\ \hfill & ={\displaystyle \frac{i}{{\left(z_n\right)}^2}}\overline{\psi }\left({\widehat{z}}_n^{*}\right)Q_{+}{b}_n-{\displaystyle \frac{i}{z_n}}{\overline{\psi }}^{\prime }\left({\widehat{z}}_n^{*}\right){\displaystyle \frac{v{k}_0^2}{{\left(z_n\right)}^2}}{Q}_{+}{b}_n-{\displaystyle \frac{i}{z_n}}\overline{\psi }\left({\widehat{z}}_n^{*}\right)Q_{+}{d}_n,\hfill \end{array}$$

(A28)

and

$$\begin{array}{cc}\hfill & {\varphi }^{″}\left(z_n\right)={\displaystyle \frac{2i}{{\left(z_n\right)}^3}}\overline{\psi }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{b}}}_n{R}_{-}-{\displaystyle \frac{2i}{{\left(z_n\right)}^2}}\left(-{\displaystyle \frac{v{k}_0^2}{{\left(z_n\right)}^2}}{\overline{\psi }}^{\prime }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{b}}}_n+\overline{\psi }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{d}}}_n\right)R_{-}\hfill \\ \hfill & +{\displaystyle \frac{i}{z_n}}(2{\overline{\psi }}^{\prime }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{d}}}_n+{\overline{\psi }}^{″}\left({\widehat{z}}_n^{*}\right){\widehat{\overline{b}}}_n+\overline{\psi }\left({\widehat{z}}_n^{*}\right){\widehat{\overline{e}}}_n)R_{-}\hfill \\ \hfill & ={\displaystyle \frac{2i}{{\left(z_n\right)}^2}}\overline{\psi }\left({\widehat{z}}_n^{*}\right)Q_{+}{d}_n-{\displaystyle \frac{2i}{z_n}}{\overline{\psi }}^{\prime }\left({\widehat{z}}_n^{*}\right){\displaystyle \frac{v{k}_0^2}{{\left(z_n\right)}^2}}{Q}_{+}{d}_n-{\displaystyle \frac{i}{z_n}}\overline{\psi }\left({\widehat{z}}_n^{*}\right)Q_{+}{e}_n\hfill \\ \hfill & +\left(-{\displaystyle \frac{2i}{{\left(z_n\right)}^3}}\overline{\psi }\left({\widehat{z}}_n^{*}\right)+{\displaystyle \frac{2i}{{\left(z_n\right)}^2}}{\overline{\psi }}^{\prime }\left({\widehat{z}}_n^{*}\right)-{\displaystyle \frac{i}{z_n}}{\overline{\psi }}^{″}\left({\widehat{z}}_n^{*}\right)\right)Q_{+}{b}_n.\hfill \end{array}$$

(A29)

From the above equations, we obtain that

$$\begin{array}{cc}\hfill & {\widehat{\overline{d}}}_n=-{\displaystyle \frac{v}{k_0^2}}{Q}_{+}{d}_n{Q}_{-},{\widehat{d}}_n=-{\displaystyle \frac{v}{k_0^2}}{R}_{+}{\overline{d}}_n{R}_{-},\hfill \\ \hfill & {\widehat{\overline{e}}}_n=-{\displaystyle \frac{v}{k_0^2}}{Q}_{+}{e}_n{Q}_{-},{\widehat{e}}_n=-{\displaystyle \frac{v}{k_0^2}}{R}_{+}{\overline{e}}_n{R}_{-}.\hfill \end{array}$$

(A30)

By Equation (A17), we derive

$$\begin{array}{cc}\hfill & {\left(\det a\right)}^{\left(6\right)}\left({\widehat{z}}_n\right)=[{\displaystyle \frac{1300z^9}{k_0^{16}}}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{‴}\left(z\right)+{\displaystyle \frac{300z^{10}}{k_0^{16}}}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{\left(4\right)}\left(z\right)\hfill \\ \hfill & +{\displaystyle \frac{30z^{11}}{k_0^{16}}}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{\left(5\right)}\left(z\right)+{\displaystyle \frac{z^{12}}{k_0^{16}}}\det Q_{+}\det R_{-}{\left(\det \overline{a}\right)}^{\left(6\right)}\left(z\right){]}_{z=z_n^{*}},\hfill \\ \hfill & {\left(\det \overline{a}\right)}^{\left(5\right)}\left({\widehat{z}}_n^{*}\right)=[{\displaystyle \frac{1300z^9}{k_0^{16}}}\det Q_{-}\det R_{+}{\left(\det a\right)}^{‴}\left(z\right)+{\displaystyle \frac{300z^{10}}{k_0^{16}}}\det Q_{-}\det R_{+}{\left(\det a\right)}^{\left(4\right)}\left(z\right)\hfill \\ \hfill & +{\displaystyle \frac{30z^{11}}{k_0^{16}}}\det Q_{-}\det R_{+}{\left(\det a\right)}^{\left(5\right)}\left(z\right)+{\displaystyle \frac{z^{12}}{k_0^{16}}}\det Q_{-}\det R_{+}{\left(\det a\right)}^{\left(6\right)}\left(z\right){]}_{z=z_n}.\hfill \end{array}$$

(A31)

Differentiating Equation (A7) with respect to z gives

$$\begin{array}{cc}\hfill & {\alpha }^{\prime }\left({\widehat{z}}_n\right)=-{\displaystyle \frac{{\left(z_n^{*}\right)}^2}{k_0^4}}\mathrm{cof}\left(R_{-}\right){\overline{\alpha }}^{\prime }\left(z_n^{*}\right)\mathrm{cof}\left(Q_{+}\right),\hfill \\ \hfill & {\alpha }^{″}\left({\widehat{z}}_n\right)={\displaystyle \frac{2{\left(z_n^{*}\right)}^3}{v{k}_0^6}}\mathrm{cof}\left(R_{-}\right){\overline{\alpha }}^{\prime }\left(z_n^{*}\right)\mathrm{cof}\left(Q_{+}\right)+{\displaystyle \frac{{\left(z_n^{*}\right)}^4}{v{k}_0^6}}\mathrm{cof}\left(R_{-}\right){\overline{\alpha }}^{″}\left(z_n^{*}\right)\mathrm{cof}\left(Q_{+}\right),\hfill \\ \hfill & {\alpha }^{‴}\left({\widehat{z}}_n\right)=-{\displaystyle \frac{6{\left(z_n^{*}\right)}^4}{k_0^8}}\mathrm{cof}\left(R_{-}\right){\overline{\alpha }}^{\prime }\left(z_n^{*}\right)\mathrm{cof}\left(Q_{+}\right)-{\displaystyle \frac{6{\left(z_n^{*}\right)}^5}{k_0^8}}\mathrm{cof}\left(R_{-}\right){\overline{\alpha }}^{″}\left(z_n^{*}\right)\mathrm{cof}\left(Q_{+}\right)\hfill \\ \hfill & -{\displaystyle \frac{{\left(z_n^{*}\right)}^6}{k_0^8}}\mathrm{cof}\left(R_{-}\right){\overline{\alpha }}^{‴}\left(z_n^{*}\right)\mathrm{cof}\left(Q_{+}\right).\hfill \end{array}$$

(A32)

Using the definitions in Equation (94) and the above derivations, we conclude that

$$\begin{array}{cc}\hfill & {\widehat{A}}_n={\displaystyle \frac{R_{+}{\overline{A}}_n{R}_{+}}{{\left(z_n^{*}\right)}^2}},{\widehat{B}}_n={\overline{B}}_n-{\displaystyle \frac{4}{z_n^{*}}},{\widehat{C}}_n={\displaystyle \frac{R_{+}{\overline{C}}_n{R}_{+}}{{\left(z_n^{*}\right)}^2}}+{\displaystyle \frac{R_{+}{\overline{A}}_n{R}_{+}}{{\left(z_n^{*}\right)}^3}},\hfill \\ \hfill & {\widehat{D}}_n={\overline{D}}_n-{\displaystyle \frac{10}{{\left(z_n^{*}\right)}^2}}-{\displaystyle \frac{{\left(\det \overline{a}\right)}^{\left(5\right)}\left(z_n^{*}\right)}{z_n^{*}{\left(\det \overline{a}\right)}^{\left(4\right)}\left(z_n^{*}\right)}},{\widehat{E}}_n={\displaystyle \frac{R_{+}{\overline{E}}_n{R}_{+}}{{\left(z_n^{*}\right)}^2}}+{\displaystyle \frac{2R_{+}{\overline{C}}_n{R}_{+}}{{\left(z_n^{*}\right)}^3}}+{\displaystyle \frac{R_{+}{\overline{A}}_n{R}_{+}}{{\left(z_n^{*}\right)}^4}}.\hfill \end{array}$$

(A33)

Substituting the above expressions into Equation (95) yields the symmetry of the norming constant:

$$\begin{array}{cc}\hfill & {\widehat{C}}_{n,1}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,1}{R}_{+}-{\displaystyle \frac{2}{{\left(z_n^{*}\right)}^3}}{R}_{+}({\overline{C}}_{n,2}-i{\theta }^{\prime }\left(z_n^{*}\right){\overline{C}}_{n,3})R_{+}+{\displaystyle \frac{3}{{\left(z_n^{*}\right)}^4}}{R}_{+}{\overline{C}}_{n,3}{R}_{+},\hfill \\ \hfill & {\widehat{C}}_{n,2}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,2}{R}_{+}-{\displaystyle \frac{3}{{\left(z_n^{*}\right)}^3}}{R}_{+}{\overline{C}}_{n,3}{R}_{+},{\widehat{C}}_{n,3}={\displaystyle \frac{1}{{\left(z_n^{*}\right)}^2}}{R}_{+}{\overline{C}}_{n,3}{R}_{+}.\hfill \end{array}$$

(A34)