1. Introduction

The well-known nonlinear Schrödinger (NLS) equation:

$$ \mathrm{i}{\mathrm{q}}_{\mathrm{t}}+{\mathrm{q}}_{\mathrm{x}\mathrm{x}}+2|\mathrm{q}{|}^2\mathrm{q}=0,$$(1)

can be regarded as the result of the coupled nonlinear Schrödinger (CNLS) equations:

$$ \mathrm{i}{\mathrm{q}}_{\mathrm{t}}=-{\mathrm{q}}_{\mathrm{x}\mathrm{x}}+2{\mathrm{q}}^2\mathrm{r},$$(2)

$$ \mathrm{i}{\mathrm{r}}_{\mathrm{t}}=-{\mathrm{r}}_{\mathrm{x}\mathrm{x}}-2{\mathrm{r}}^2\mathrm{q},$$(3)

under the reduction  $$ \mathrm{r}=-{\mathrm{q}}^{\ast }$$.Equation (2,3) belongs to the well-known AKNS hierarchy, where  $$ \mathrm{q}$$  and  $$ \mathrm{r}$$  are two slowly varying complex envelopes of the propagating waves, the subscripts  $$ x$$  and $$ t$$  denote the partial derivatives with respect to normalized distance and time respectively.

In 2013, Ablowitz and Mussilimani [1] introduced a "nonlocal" constraint on  $$ \mathrm{q}$$  and  $$ \mathrm{r}$$ , given by  $$ \mathrm{q}(\mathrm{x},\mathrm{t})={\mathrm{r}}^{\ast }(-\mathrm{x},-\mathrm{t})$$, which allows the derivation of the nonlocal nonlinear Schrödinger (NNLS) equation from Equation (2,3):

$$ \mathrm{i}{\mathrm{q}}_{\mathrm{t}}+{\mathrm{q}}_{\mathrm{x}\mathrm{x}}+2{\mathrm{q}}^2(\mathrm{x},\mathrm{t}){\mathrm{q}}^{\ast }(-\mathrm{x},\mathrm{t})=0.$$(4)

Equation (4) exhibits PT symmetry since the nonlinear term  $$ \mathrm{V}(\mathrm{x},\mathrm{t})=\mathrm{q}(\mathrm{x},\mathrm{t}){\mathrm{q}}^{\ast }(-\mathrm{x},\mathrm{t})$$  remains invariant under the PT transformation, i.e. $$ \mathrm{V}(\mathrm{x},\mathrm{t})={\mathrm{V}}^{\ast }(-\mathrm{x},\mathrm{t})$$. Compared to the NLS Equation (1), the key difference is that the nonlinear term  $$ 2\left|\mathrm{q}\right(\mathrm{x},\mathrm{t}\left){|}^2\mathrm{q}\right(\mathrm{x},\mathrm{t})$$  in Equation (1) is replaced by  $$ 2{\mathrm{q}}^2(\mathrm{x},\mathrm{t}){\mathrm{q}}^{\ast }(-\mathrm{x},\mathrm{t})$$  in the NNLS Equation (4), which reflects the nature of the anti-spatiotemporal nonlocal coupling between  $$ \mathrm{q}(\mathrm{x},\mathrm{t})$$  and  $$ {\mathrm{q}}^{\ast }(-\mathrm{x},\mathrm{t})$$ .

Ablowitz and Ladik [2] discovered the semi-discrete nonlinear Schrödinger (sd-NLS) equation:

$$ \mathrm{i}\frac{\mathrm{d}{\mathrm{Q}}_{\mathrm{n}}}{\mathrm{d}\mathrm{t}}+({\mathrm{Q}}_{\mathrm{n}+1}+{\mathrm{Q}}_{\mathrm{n}-1}-2{\mathrm{Q}}_{\mathrm{n}})+2\left|{\mathrm{Q}}_{\mathrm{n}}{|}^2\right({\mathrm{Q}}_{\mathrm{n}+1}+{\mathrm{Q}}_{\mathrm{n}-1})=0.$$(5)

In recent years, semi-discrete integrable systems have received increasing attention as mathematical models for various physical phenomena, including nonlinear optics, biology, ladder circuits, and lattice dynamics [3-6].

The studies above primarily focus on single-component continuous and discrete nonlinear Schrödinger equations. However, research on multi-component nonlinear Schrödinger (MNLS) equations has become a major topic of interest. MNLS equations are crucial dynamical systems in optics and mathematical physics, describing the simultaneous propagation of multiple nonlinear waves in a homogeneous medium. They find applications in plasma physics [7], quantum electronics [8], nonlinear optics [9], Bose-Einstein conde-nsates [10], and fluid dynamics [11].

Reference [12] presents the integrable multi-component form of the semi-discrete coupled nonlinear Schrödinger (sd-CNLS) equations:

$$ \mathrm{i}{\dot{{\mathrm{u}}_{\mathrm{n}}}}^{\left(\mathrm{j}\right)}+({\mathrm{u}}_{\mathrm{n}+1}^{\left(\mathrm{j}\right)}+{\mathrm{u}}_{\mathrm{n}-1}^{\left(\mathrm{j}\right)})(1-\sum _{\mathrm{k}=1}^{\mathrm{m}}{\mathrm{u}}_{\mathrm{n}}^{\left(\mathrm{k}\right)}{\mathrm{v}}_{\mathrm{n}}^{\left(\mathrm{k}\right)})-2{\mathrm{u}}_{\mathrm{n}}^{\left(\mathrm{j}\right)}=0,$$(6)

$$ \mathrm{i}{\dot{{\mathrm{v}}_{\mathrm{n}}}}^{\left(\mathrm{j}\right)}-({\mathrm{v}}_{\mathrm{n}+1}^{\left(\mathrm{j}\right)}+{\mathrm{v}}_{\mathrm{n}-1}^{\left(\mathrm{j}\right)})(1-\sum _{\mathrm{k}=1}^{\mathrm{m}}{\mathrm{v}}_{\mathrm{n}}^{\left(\mathrm{k}\right)}{\mathrm{u}}_{\mathrm{n}}^{\left(\mathrm{k}\right)})+2{\mathrm{v}}_{\mathrm{n}}^{\left(\mathrm{j}\right)}=0,\hspace{1em}\mathrm{j}=\mathrm{1,2},\dots ,\mathrm{m}.$$(7)

This study investigates the integrability of these equations, derives an infinite set of conservation laws, and constructs their  $$ \mathrm{N}$$ -soliton solutions via the inverse scattering method (ISM). However, to the best of our knowledge, no soliton solutions for this equation have been obtained using the Darboux transformation.

In this work, we consider the case $$ \mathrm{j}$$ and impose the reduction $$ {\mathrm{v}}_{\mathrm{n}}^{\left(\mathrm{j}\right)}=-{\mathrm{u}}_{\mathrm{n}}^{\left(\mathrm{j}\right)\ast }$$,which transforms Equation (6,7) into the following system:

$$ \mathrm{i}{\dot{\mathrm{u}}}^{\left(1\right)}+({\mathrm{u}}_{\mathrm{n}+1}^{\left(1\right)}+{\mathrm{u}}_{\mathrm{n}-1}^{\left(1\right)})(1+|{\mathrm{u}}_{\mathrm{n}}^{\left(1\right)}{|}^2+\left|{\mathrm{u}}_{\mathrm{n}}^{\left(2\right)}{|}^2\right)-2{\mathrm{u}}_{\mathrm{n}}^{\left(1\right)}=0,$$(8)

$$ \mathrm{i}{\dot{\mathrm{u}}}^{\left(2\right)}+({\mathrm{u}}_{\mathrm{n}+1}^{\left(2\right)}+{\mathrm{u}}_{\mathrm{n}-1}^{\left(2\right)})(1+|{\mathrm{u}}_{\mathrm{n}}^{\left(1\right)}{|}^2+\left|{\mathrm{u}}_{\mathrm{n}}^{\left(2\right)}{|}^2\right)-2{\mathrm{u}}_{\mathrm{n}}^{\left(2\right)}=0.$$(9)

We first construct the Darboux transformation for Equation (8,9) by employing a gauge transformation and the associated Lax pair. Subsequently, we derive the explicit solutions of Equation (8,9) and analyze their dynamical properties.

2. Darboux transformation

According to Ref [13], the auxiliary linear equations corresponding to the semi-discrete coupled local nonlinear Schrödinger Equation (8,9) are given by:

$$ {\mathrm{\Psi }}_{\mathrm{n}+1}={\mathrm{N}}_{\mathrm{n}}{\mathrm{\Psi }}_{\mathrm{n}},$$(10)

$$ \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{\mathrm{\Psi }}_{\mathrm{n}}={\mathrm{M}}_{\mathrm{n}}{\mathrm{\Psi }}_{\mathrm{n}}.$$(11)

The corresponding Lax pair is given as follows:

$$ {\mathrm{N}}_{\mathrm{n}}=\left(\begin{array}{cc}\mathrm{\lambda }\mathrm{I}& {\mathrm{Q}}_{\mathrm{n}}\\ {\mathrm{R}}_{\mathrm{n}}& {\mathrm{\lambda }}^{-1}\mathrm{I}\end{array}\right)$$(12)

$$ {\mathrm{M}}_{\mathrm{n}}=\mathrm{i}\left(\begin{array}{cc}{\mathrm{Q}}_{\mathrm{n}}{\mathrm{R}}_{\mathrm{n}-1}-\frac{1}{2}(\mathrm{\lambda }-\frac{1}{\mathrm{\lambda }}{)}^2\mathrm{I}& -\mathrm{\lambda }{\mathrm{Q}}_{\mathrm{n}}+\frac{1}{\mathrm{\lambda }}{\mathrm{Q}}_{\mathrm{n}-1}\\ \frac{1}{\mathrm{\lambda }}{\mathrm{R}}_{\mathrm{n}}-\mathrm{\lambda }{\mathrm{R}}_{\mathrm{n}-1}& -{\mathrm{V}}_{\mathrm{n}}{\mathrm{U}}_{\mathrm{n}-1}+\frac{1}{2}(\mathrm{\lambda }-\frac{1}{\mathrm{\lambda }}{)}^2\mathrm{I}\end{array}\right)$$(13)

We may choose

$$ {\mathrm{Q}}_{\mathrm{n}}=\left(\begin{array}{cc}{\mathrm{q}}_{\mathrm{n}}& {\mathrm{r}}_{\mathrm{n}}\\ {\mathrm{r}}_{\mathrm{n}}^{\ast }& {\mathrm{q}}_{\mathrm{n}}^{\ast }\end{array}\right),{\mathrm{R}}_{\mathrm{n}}=\left(\begin{array}{cc}-{\mathrm{q}}_{\mathrm{n}}^{\ast }& {\mathrm{r}}_{\mathrm{n}}\\ {\mathrm{r}}_{\mathrm{n}}^{\ast }& -{\mathrm{q}}_{\mathrm{n}}\end{array}\right),$$(14)

Thus, the Lax pair becomes:

$$ {\mathrm{N}}_{\mathrm{n}}=\left(\begin{array}{cccc}\mathrm{\lambda }& 0& {\mathrm{q}}_{\mathrm{n}}& {\mathrm{r}}_{\mathrm{n}}\\ 0& \mathrm{\lambda }& {\mathrm{r}}_{\mathrm{n}}^{\ast }& {\mathrm{q}}_{\mathrm{n}}^{\ast }\\ -{\mathrm{q}}_{\mathrm{n}}^{\ast }& {\mathrm{r}}_{\mathrm{n}}& {\mathrm{\lambda }}^{-1}& 0\\ {\mathrm{r}}_{\mathrm{n}}^{\ast }& -{\mathrm{q}}_{\mathrm{n}}& 0& {\mathrm{\lambda }}^{-1}\end{array}\right),$$(15)

$$ \begin{array}{c}{\mathrm{M}}_{\mathrm{n}}=\mathrm{i}\left(\begin{array}{cccc}-\frac{{\mathrm{\lambda }}^2+{\mathrm{\lambda }}^{-2}}{2}-{\mathrm{q}}_{\mathrm{n}}{\mathrm{q}}_{\mathrm{n}-1}^{\ast }+{\mathrm{r}}_{\mathrm{n}}{\mathrm{r}}_{\mathrm{n}-1}^{\ast }+1& -{\mathrm{q}}_{\mathrm{n}-1}{\mathrm{r}}_{\mathrm{n}}+{\mathrm{q}}_{\mathrm{n}}{\mathrm{r}}_{\mathrm{n}-1}& \frac{1}{\mathrm{\lambda }}{\mathrm{q}}_{\mathrm{n}-1}-\mathrm{\lambda }{\mathrm{q}}_{\mathrm{n}}& \frac{1}{\mathrm{\lambda }}{\mathrm{r}}_{\mathrm{n}-1}-\mathrm{\lambda }{\mathrm{r}}_{\mathrm{n}}\\ -{\mathrm{q}}_{\mathrm{n}-1}^{\ast }{\mathrm{r}}_{\mathrm{n}}^{\ast }+{\mathrm{q}}_{\mathrm{n}}^{\ast }{\mathrm{r}}_{\mathrm{n}-1}^{\ast }& -\frac{{\mathrm{\lambda }}^2+{\mathrm{\lambda }}^{-2}}{2}-{\mathrm{q}}_{\mathrm{n}-1}{\mathrm{q}}_{\mathrm{n}}^{\ast }+{\mathrm{r}}_{\mathrm{n}-1}{\mathrm{r}}_{\mathrm{n}}^{\ast }+1& -\mathrm{\lambda }{\mathrm{r}}_{\mathrm{n}}^{\ast }+\frac{1}{\mathrm{\lambda }}{\mathrm{r}}_{\mathrm{n}-1}^{\ast }& -\mathrm{\lambda }{\mathrm{q}}_{\mathrm{n}}^{\ast }+\frac{1}{\mathrm{\lambda }}{\mathrm{q}}_{\mathrm{n}-1}^{\ast }\\ -\frac{1}{\mathrm{\lambda }}{\mathrm{q}}_{\mathrm{n}}^{\ast }+\mathrm{\lambda }{\mathrm{q}}_{\mathrm{n}-1}^{\ast }& \frac{1}{\mathrm{\lambda }}{\mathrm{r}}_{\mathrm{n}}-\mathrm{\lambda }{\mathrm{r}}_{\mathrm{n}-1}& \frac{{\mathrm{\lambda }}^2+{\mathrm{\lambda }}^{-2}}{2}+{\mathrm{q}}_{\mathrm{n}-1}{\mathrm{q}}_{\mathrm{n}}^{\ast }-{\mathrm{r}}_{\mathrm{n}}{\mathrm{r}}_{\mathrm{n}-1}^{\ast }-1& {\mathrm{r}}_{\mathrm{n}-1}{\mathrm{q}}_{\mathrm{n}}^{\ast }-{\mathrm{r}}_{\mathrm{n}}{\mathrm{q}}_{\mathrm{n}-1}^{\ast }\\ \frac{1}{\mathrm{\lambda }}{\mathrm{r}}_{\mathrm{n}}^{\ast }-\mathrm{\lambda }{\mathrm{r}}_{\mathrm{n}-1}^{\ast }& -\frac{1}{\mathrm{\lambda }}{\mathrm{q}}_{\mathrm{n}}+\mathrm{\lambda }{\mathrm{q}}_{\mathrm{n}-1}& -{\mathrm{q}}_{\mathrm{n}-1}{\mathrm{r}}_{\mathrm{n}}^{\ast }+{\mathrm{q}}_{\mathrm{n}}{\mathrm{r}}_{\mathrm{n}-1}^{\ast }& \frac{{\mathrm{\lambda }}^2+{\mathrm{\lambda }}^{-2}}{2}+{\mathrm{q}}_{\mathrm{n}}{\mathrm{q}}_{\mathrm{n}-1}^{\ast }-{\mathrm{r}}_{\mathrm{n}-1}{\mathrm{r}}_{\mathrm{n}}^{\ast }-1\end{array}\right)\end{array}$$(16)

Where  $$ {\mathrm{\Psi }}_{\mathrm{n}}$$  is a  $$ 4\times 4$$  matrix, and  $$ \mathrm{\lambda }$$  is the spectral parameter, which is independent of both  $$ \mathrm{n}$$  and  $$ \mathrm{t}$$ .

Substituting the Lax Equation (15,16) into the zero-curvature equation:

$$ \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{\mathrm{N}}_{\mathrm{n}}={\mathrm{M}}_{\mathrm{n}+1}{\mathrm{N}}_{\mathrm{n}}-{\mathrm{N}}_{\mathrm{n}}{\mathrm{M}}_{\mathrm{n}},$$(17)

we obtain the equations under investigation:

$$ \mathrm{i}{\mathrm{q}}_{\mathrm{n},\mathrm{t}}={\mathrm{q}}_{\mathrm{n}+1}-2{\mathrm{q}}_{\mathrm{n}}+{\mathrm{q}}_{\mathrm{n}-1}+\left({\left|{\mathrm{q}}_{\mathrm{n}}\right|}^2-{\left|{\mathrm{r}}_{\mathrm{n}}\right|}^2\right)\left({\mathrm{q}}_{\mathrm{n}+1}-{\mathrm{q}}_{\mathrm{n}-1}\right),$$(18)

$$ \mathrm{i}{\mathrm{r}}_{\mathrm{n},\mathrm{t}}={\mathrm{r}}_{\mathrm{n}+1}-2{\mathrm{r}}_{\mathrm{n}}+{\mathrm{r}}_{\mathrm{n}-1}+\left({\left|{\mathrm{q}}_{\mathrm{n}}\right|}^2-{\left|{\mathrm{r}}_{\mathrm{n}}\right|}^2\right)\left({\mathrm{r}}_{\mathrm{n}+1}+{\mathrm{r}}_{\mathrm{n}-1}\right).$$(19)

The Darboux transformation is an effective tool for constructing exact solutions of integrable nonlinear equations. To derive the Darboux transformation for Equation (18,19), we introduce a gauge transformation:

$$ {\mathrm{\Psi }}_{\mathrm{n}}\left[1\right]={\mathrm{T}}_{\mathrm{n}}{\mathrm{\Psi }}_{\mathrm{n}},$$(20)

where the transformation matrix  $$ {\mathrm{T}}_{\mathrm{n}}$$  is given by:

$$ {\mathrm{T}}_{\mathrm{n}}=\mathrm{\lambda }\mathrm{A}+{\mathrm{\lambda }}^{-1}\mathrm{B}+\mathrm{C}$$(21)

Where

$$$ \mathrm{A}=\left(\begin{array}{cccc}{\mathrm{a}}_{11,\mathrm{n}}& {\mathrm{a}}_{12,\mathrm{n}}& {\mathrm{a}}_{13,\mathrm{n}}& {\mathrm{a}}_{14,\mathrm{n}}\\ {\mathrm{a}}_{21,\mathrm{n}}& {\mathrm{a}}_{22,\mathrm{n}}& {\mathrm{a}}_{23,\mathrm{n}}& {\mathrm{a}}_{24,\mathrm{n}}\\ {\mathrm{a}}_{31,\mathrm{n}}& {\mathrm{a}}_{32,\mathrm{n}}& {\mathrm{a}}_{33,\mathrm{n}}& {\mathrm{a}}_{34,\mathrm{n}}\\ {\mathrm{a}}_{41,\mathrm{n}}& {\mathrm{a}}_{42,\mathrm{n}}& {\mathrm{a}}_{43,\mathrm{n}}& {\mathrm{a}}_{44,\mathrm{n}}\end{array}\right),\mathrm{B}=\left(\begin{array}{cccc}{\mathrm{b}}_{11,\mathrm{n}}& {\mathrm{b}}_{12,\mathrm{n}}& {\mathrm{b}}_{13,\mathrm{n}}& {\mathrm{b}}_{14,\mathrm{n}}\\ {\mathrm{b}}_{21,\mathrm{n}}& {\mathrm{b}}_{22,\mathrm{n}}& {\mathrm{b}}_{23,\mathrm{n}}& {\mathrm{b}}_{24,\mathrm{n}}\\ {\mathrm{b}}_{31,\mathrm{n}}& {\mathrm{b}}_{32,\mathrm{n}}& {\mathrm{b}}_{33,\mathrm{n}}& {\mathrm{b}}_{34,\mathrm{n}}\\ {\mathrm{b}}_{41,\mathrm{n}}& {\mathrm{b}}_{42,\mathrm{n}}& {\mathrm{b}}_{43,\mathrm{n}}& {\mathrm{b}}_{44,\mathrm{n}}\end{array}\right),\mathrm{C}=\left(\begin{array}{cccc}{\mathrm{c}}_{11,\mathrm{n}}& {\mathrm{c}}_{12,\mathrm{n}}& {\mathrm{c}}_{13,\mathrm{n}}& {\mathrm{c}}_{14,\mathrm{n}}\\ {\mathrm{c}}_{21,\mathrm{n}}& {\mathrm{c}}_{22,\mathrm{n}}& {\mathrm{c}}_{23,\mathrm{n}}& {\mathrm{c}}_{24,\mathrm{n}}\\ {\mathrm{c}}_{31,\mathrm{n}}& {\mathrm{c}}_{32,\mathrm{n}}& {\mathrm{c}}_{33,\mathrm{n}}& {\mathrm{c}}_{34,\mathrm{n}}\\ {\mathrm{c}}_{41,\mathrm{n}}& {\mathrm{c}}_{42,\mathrm{n}}& {\mathrm{c}}_{43,\mathrm{n}}& {\mathrm{c}}_{44,\mathrm{n}}\end{array}\right).$$$

By applying the gauge transformation, a spectral problem can be converted into another of the same type, transforming the spectral problem Equation (10,11) into:

$$ {\mathrm{\Psi }}_{\mathrm{n}+1}\left[1\right]={\mathrm{N}}_{\mathrm{n}}\left[1\right]{\mathrm{\Psi }}_{\mathrm{n}}\left[1\right],$$(22)

$$ \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{\mathrm{\Psi }}_{\mathrm{n}}\left[1\right]={\mathrm{M}}_{\mathrm{n}}\left[1\right]{\mathrm{\Psi }}_{\mathrm{n}}\left[1\right].$$(23)

Combining Equations (10,11) and (20), we obtain:

$$ {\mathrm{N}}_{\mathrm{n}}\left[1\right]={\mathrm{T}}_{\mathrm{n}+1}{\mathrm{N}}_{\mathrm{n}}{\mathrm{T}}_{\mathrm{n}}^{-1},$$(24)

$$ {\mathrm{M}}_{\mathrm{n}}\left[1\right]=\left({\mathrm{T}}_{\mathrm{n},\mathrm{t}}+{\mathrm{T}}_{\mathrm{n}}{\mathrm{M}}_{\mathrm{n}}\right){\mathrm{T}}_{\mathrm{n}}^{-1}.$$(25)

Here,  $$ {\mathrm{N}}_{\mathrm{n}}\left[1\right]$$  and  $$ \mathrm{M}\mathrm{n}\left[1\right]$$  share the same structure as  $$ {\mathrm{N}}_{\mathrm{n}}$$  and  $$ {\mathrm{M}}_{\mathrm{n}}$$ . By comparing the coefficients of like powers of  $$ \mathrm{\lambda }$$  on both sides of Equation (24,25), the matrix  $$ {\mathrm{T}}_{\mathrm{n}}$$  can be simplified as follows:

$$ {\mathrm{T}}_{\mathrm{n}}=\left(\begin{array}{cccc}\mathrm{\lambda }+\frac{1}{\mathrm{\lambda }}{\mathrm{b}}_{11,\mathrm{n}}& \frac{1}{\mathrm{\lambda }}{\mathrm{b}}_{12,\mathrm{n}}& {\mathrm{c}}_{13,\mathrm{n}}& {\mathrm{c}}_{14,\mathrm{n}}\\ \frac{1}{\mathrm{\lambda }}{\mathrm{b}}_{12,\mathrm{n}}^{\ast }& \mathrm{\lambda }+\frac{1}{\mathrm{\lambda }}{\mathrm{b}}_{11,\mathrm{n}}^{\ast }& {\mathrm{c}}_{14,\mathrm{n}}^{\ast }& {\mathrm{c}}_{13,\mathrm{n}}^{\ast }\\ {\mathrm{c}}_{31,\mathrm{n}}& {\mathrm{c}}_{32,\mathrm{n}}& {\mathrm{a}}_{33,\mathrm{n}}\mathrm{\lambda }+\frac{1}{\mathrm{\lambda }}& {\mathrm{a}}_{34,\mathrm{n}}\mathrm{\lambda }\\ {\mathrm{c}}_{32,\mathrm{n}}^{\ast }& {\mathrm{c}}_{31,\mathrm{n}}^{\ast }& {\mathrm{a}}_{34,\mathrm{n}}^{\ast }\mathrm{\lambda }& {\mathrm{a}}_{33,\mathrm{n}}^{\ast }\mathrm{\lambda }+\frac{1}{\mathrm{\lambda }}\end{array}\right).$$(26)

By direct calculation, the following relation between the new and old potentials is obtained:

$$ {\mathrm{q}}_{\mathrm{n}}\left[1\right]={\mathrm{q}}_{\mathrm{n}}\hspace{0.17em}{\mathrm{a}}_{33,\mathrm{n}+1}^{\ast }-{\mathrm{r}}_{\mathrm{n}}\hspace{0.17em}{\mathrm{a}}_{34,\mathrm{n}+1}^{\ast }-{\mathrm{c}}_{31,\mathrm{n}+1}^{\ast },$$(27)

$$ {\mathrm{r}}_{\mathrm{n}}\left[1\right]={\mathrm{r}}_{\mathrm{n}}\hspace{0.17em}{\mathrm{a}}_{33,\mathrm{n}+1}-{\mathrm{q}}_{\mathrm{n}}\hspace{0.17em}{\mathrm{a}}_{34,\mathrm{n}+1}+{\mathrm{c}}_{32,\mathrm{n}+1}.$$(28)

Clearly,  $$ \det \mathrm{T}$$  is a quartic polynomial in  $$ \mathrm{\lambda }$$ , implying the existence of  $$ {\mathrm{\lambda }}_{\mathrm{j}}\mathrm{ }(\mathrm{j}=\mathrm{1,2},\mathrm{3,4})$$  such that  $$ \det \mathrm{T}=0$$ .

Moreover, when  $$ \mathrm{\lambda }={\mathrm{\lambda }}_1$$ , the vector

$$ {\mathrm{\phi }}_{\mathrm{n}}\left({\mathrm{\lambda }}_1\right)={\left({\mathrm{\phi }}_{\mathrm{n}}^1\left({\mathrm{\lambda }}_1\right),\mathrm{ }{\mathrm{\phi }}_{\mathrm{n}}^2\left({\mathrm{\lambda }}_1\right),\mathrm{ }{\mathrm{\phi }}_{\mathrm{n}}^3\left({\mathrm{\lambda }}_1\right),\mathrm{ }{\mathrm{\phi }}_{\mathrm{n}}^4\left({\mathrm{\lambda }}_1\right)\right)}^{\mathrm{\top }}$$

is a solution of Equation (10,11). Similarly, when  $$ \mathrm{\lambda }={\mathrm{\lambda }}_1^{\ast }$$ , the vector

$$ {\mathrm{\phi }}_{\mathrm{n}}\left({\mathrm{\lambda }}_1^{\ast }\right)={\left({{\mathrm{\phi }}_{\mathrm{n}}^2}^{\ast }\left({\mathrm{\lambda }}_1^{\ast }\right),\mathrm{ }{{\mathrm{\phi }}_{\mathrm{n}}^1}^{\ast }\left({\mathrm{\lambda }}_1^{\ast }\right),\mathrm{ }{{\mathrm{\phi }}_{\mathrm{n}}^4}^{\ast }\left({\mathrm{\lambda }}_1^{\ast }\right),\mathrm{ }{{\mathrm{\phi }}_{\mathrm{n}}^3}^{\ast }\left({\mathrm{\lambda }}_1^{\ast }\right)\right)}^{\mathrm{\top }}$$

also satisfies Equation (10,11). Therefore, for  $$ \mathrm{\lambda }={\mathrm{\lambda }}_{\mathrm{j}}\mathrm{ }(\mathrm{j}=\mathrm{1,2})$$, a fundamental set of solutions to Equation (10,11) is given by:

$$ {\mathrm{\phi }}_{\mathrm{n}}\left({\mathrm{\lambda }}_1\right)={\left({\mathrm{\phi }}_{\mathrm{n}}^1\left({\mathrm{\lambda }}_1\right),{\mathrm{\phi }}_{\mathrm{n}}^2\left({\mathrm{\lambda }}_1\right),{\mathrm{\phi }}_{\mathrm{n}}^3\left({\mathrm{\lambda }}_1\right),{\mathrm{\phi }}_{\mathrm{n}}^4\left({\mathrm{\lambda }}_1\right)\right)}^{\mathrm{\top }}$$ ,  $$ {\mathrm{\phi }}_{\mathrm{n}}\left({\mathrm{\lambda }}_1^{\ast }\right)={\left({{\mathrm{\phi }}_{\mathrm{n}}^2}^{\ast }\left({\mathrm{\lambda }}_1^{\ast }\right),{{\mathrm{\phi }}_{\mathrm{n}}^1}^{\ast }\left({\mathrm{\lambda }}_1^{\ast }\right),{{\mathrm{\phi }}_{\mathrm{n}}^4}^{\ast }\left({\mathrm{\lambda }}_1^{\ast }\right),{{\mathrm{\phi }}_{\mathrm{n}}^3}^{\ast }\left({\mathrm{\lambda }}_1^{\ast }\right)\right)}^{\mathrm{\top }}$$

$$ {\mathrm{\psi }}_{\mathrm{n}}\left({\mathrm{\lambda }}_2\right)={\left({\mathrm{\psi }}_{\mathrm{n}}^1\left({\mathrm{\lambda }}_2\right),{\mathrm{\psi }}_{\mathrm{n}}^2\left({\mathrm{\lambda }}_2\right),{\mathrm{\psi }}_{\mathrm{n}}^3\left({\mathrm{\lambda }}_2\right),{\mathrm{\psi }}_{\mathrm{n}}^4\left({\mathrm{\lambda }}_2\right)\right)}^{\mathrm{\top }}$$ ,  $$ {\mathrm{\psi }}_{\mathrm{n}}\left({\mathrm{\lambda }}_2^{\ast }\right)={\left({{\mathrm{\psi }}_{\mathrm{n}}^2}^{\ast }\left({\mathrm{\lambda }}_2^{\ast }\right),{{\mathrm{\psi }}_{\mathrm{n}}^1}^{\ast }\left({\mathrm{\lambda }}_2^{\ast }\right),{{\mathrm{\psi }}_{\mathrm{n}}^4}^{\ast }\left({\mathrm{\lambda }}_2^{\ast }\right),{{\mathrm{\psi }}_{\mathrm{n}}^3}^{\ast }\left({\mathrm{\lambda }}_2^{\ast }\right)\right)}^{\mathrm{\top }}$$(29)

Thus, Equation (20) can be rewritten as:

$$ {\mathrm{\Psi }}_{\mathrm{n}}\left[1\right]=\left(\begin{array}{cccc}{\mathrm{T}}_{11}& {\mathrm{T}}_{12}& {\mathrm{T}}_{13}& {\mathrm{T}}_{14}\\ {\mathrm{T}}_{21}& {\mathrm{T}}_{22}& {\mathrm{T}}_{23}& {\mathrm{T}}_{24}\\ {\mathrm{T}}_{31}& {\mathrm{T}}_{32}& {\mathrm{T}}_{33}& {\mathrm{T}}_{34}\\ {\mathrm{T}}_{41}& {\mathrm{T}}_{42}& {\mathrm{T}}_{43}& {\mathrm{T}}_{44}\end{array}\right)\left(\begin{array}{cccc}{\mathrm{\phi }}_{\mathrm{n}}^1& {{\mathrm{\phi }}_{\mathrm{n}}^2}^{\ast }& {\mathrm{\psi }}_{\mathrm{n}}^1& {{\mathrm{\psi }}_{\mathrm{n}}^2}^{\ast }\\ {\mathrm{\phi }}_{\mathrm{n}}^2& {{\mathrm{\phi }}_{\mathrm{n}}^1}^{\ast }& {\mathrm{\psi }}_{\mathrm{n}}^2& {{\mathrm{\psi }}_{\mathrm{n}}^1}^{\ast }\\ {\mathrm{\phi }}_{\mathrm{n}}^3& {{\mathrm{\phi }}_{\mathrm{n}}^4}^{\ast }& {\mathrm{\psi }}_{\mathrm{n}}^3& {{\mathrm{\psi }}_{\mathrm{n}}^4}^{\ast }\\ {\mathrm{\phi }}_{\mathrm{n}}^4& {{\mathrm{\phi }}_{\mathrm{n}}^3}^{\ast }& {\mathrm{\psi }}_{\mathrm{n}}^4& {{\mathrm{\psi }}_{\mathrm{n}}^3}^{\ast }\end{array}\right).$$(30)

Therefore,when  $$ \mathrm{\lambda }={\mathrm{\lambda }}_{\mathrm{j}}(\mathrm{j}=\mathrm{1,2})$$  the vectors

$$$ {\mathrm{\phi }}_{\mathrm{n}}\left[1\right]\left({\mathrm{\lambda }}_1\right)=\mathrm{T}{\mathrm{\phi }}_{\mathrm{n}}\left({\mathrm{\lambda }}_1\right),\mathrm{ }{\mathrm{\phi }}_{\mathrm{n}}\left[1\right]\left({\mathrm{\lambda }}_1^{\ast }\right)=\mathrm{T}{\mathrm{\phi }}_{\mathrm{n}}\left({\mathrm{\lambda }}_1^{\ast }\right),$$$

$$$ {\mathrm{\psi }}_{\mathrm{n}}\left[1\right]\left({\mathrm{\lambda }}_2\right)=\mathrm{T}{\mathrm{\psi }}_{\mathrm{n}}\left({\mathrm{\lambda }}_2\right),\mathrm{ }{\mathrm{\psi }}_{\mathrm{n}}\left[1\right]\left({\mathrm{\lambda }}_2^{\ast }\right)=\mathrm{T}{\mathrm{\psi }}_{\mathrm{n}}\left({\mathrm{\lambda }}_2^{\ast }\right)$$$

are linearly dependent. There exist constants  $$ {\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)},{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)},{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}$$  , not all zero, such that:

$$ \begin{array}{cc}& {\mathrm{T}}_{11}{\mathrm{\phi }}_{\mathrm{n}}^1+{\mathrm{T}}_{12}{\mathrm{\phi }}_{\mathrm{n}}^2+{\mathrm{T}}_{13}{\mathrm{\phi }}_{\mathrm{n}}^3+{\mathrm{T}}_{14}{\mathrm{\phi }}_{\mathrm{n}}^4+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}({\mathrm{T}}_{11}{{\mathrm{\phi }}_{\mathrm{n}}^2}^{\ast }+{\mathrm{T}}_{12}{{\mathrm{\phi }}_{\mathrm{n}}^1}^{\ast }+{\mathrm{T}}_{13}{{\mathrm{\phi }}_{\mathrm{n}}^4}^{\ast }+{\mathrm{T}}_{14}{{\mathrm{\phi }}_{\mathrm{n}}^3}^{\ast })\\ & +{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}({\mathrm{T}}_{11}{\mathrm{\psi }}_{\mathrm{n}}^1+{\mathrm{T}}_{12}{\mathrm{\psi }}_{\mathrm{n}}^2+{\mathrm{T}}_{13}{\mathrm{\psi }}_{\mathrm{n}}^3+{\mathrm{T}}_{14}{\mathrm{\psi }}_{\mathrm{n}}^4)+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}({\mathrm{T}}_{11}{{\mathrm{\psi }}_{\mathrm{n}}^2}^{\ast }+{\mathrm{T}}_{12}{{\mathrm{\psi }}_{\mathrm{n}}^1}^{\ast }+{\mathrm{T}}_{13}{{\mathrm{\psi }}_{\mathrm{n}}^4}^{\ast }+{\mathrm{T}}_{14}{{\mathrm{\psi }}_{\mathrm{n}}^3}^{\ast })=0,\\ & {\mathrm{T}}_{21}{\mathrm{\phi }}_{\mathrm{n}}^1+{\mathrm{T}}_{22}{\mathrm{\phi }}_{\mathrm{n}}^2+{\mathrm{T}}_{23}{\mathrm{\phi }}_{\mathrm{n}}^3+{\mathrm{T}}_{24}{\mathrm{\phi }}_{\mathrm{n}}^4+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}({\mathrm{T}}_{21}{{\mathrm{\phi }}_{\mathrm{n}}^2}^{\ast }+{\mathrm{T}}_{22}{{\mathrm{\phi }}_{\mathrm{n}}^1}^{\ast }+{\mathrm{T}}_{23}{{\mathrm{\phi }}_{\mathrm{n}}^4}^{\ast }+{\mathrm{T}}_{24}{{\mathrm{\phi }}_{\mathrm{n}}^3}^{\ast })\\ & +{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}({\mathrm{T}}_{21}{\mathrm{\psi }}_{\mathrm{n}}^1+{\mathrm{T}}_{22}{\mathrm{\psi }}_{\mathrm{n}}^2+{\mathrm{T}}_{23}{\mathrm{\psi }}_{\mathrm{n}}^3+{\mathrm{T}}_{24}{\mathrm{\psi }}_{\mathrm{n}}^4)+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}({\mathrm{T}}_{21}{{\mathrm{\psi }}_{\mathrm{n}}^2}^{\ast }+{\mathrm{T}}_{22}{{\mathrm{\psi }}_{\mathrm{n}}^1}^{\ast }+{\mathrm{T}}_{23}{{\mathrm{\psi }}_{\mathrm{n}}^4}^{\ast }+{\mathrm{T}}_{24}{{\mathrm{\psi }}_{\mathrm{n}}^3}^{\ast })=0,\\ & {\mathrm{T}}_{31}{\mathrm{\phi }}_{\mathrm{n}}^1+{\mathrm{T}}_{32}{\mathrm{\phi }}_{\mathrm{n}}^2+{\mathrm{T}}_{33}{\mathrm{\phi }}_{\mathrm{n}}^3+{\mathrm{T}}_{34}{\mathrm{\phi }}_{\mathrm{n}}^4+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}({\mathrm{T}}_{31}{{\mathrm{\phi }}_{\mathrm{n}}^2}^{\ast }+{\mathrm{T}}_{32}{{\mathrm{\phi }}_{\mathrm{n}}^1}^{\ast }+{\mathrm{T}}_{33}{{\mathrm{\phi }}_{\mathrm{n}}^4}^{\ast }+{\mathrm{T}}_{34}{{\mathrm{\phi }}_{\mathrm{n}}^3}^{\ast })\\ & +{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}({\mathrm{T}}_{31}{\mathrm{\psi }}_{\mathrm{n}}^1+{\mathrm{T}}_{32}{\mathrm{\psi }}_{\mathrm{n}}^2+{\mathrm{T}}_{33}{\mathrm{\psi }}_{\mathrm{n}}^3+{\mathrm{T}}_{34}{\mathrm{\psi }}_{\mathrm{n}}^3)+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}({\mathrm{T}}_{31}{{\mathrm{\psi }}_{\mathrm{n}}^2}^{\ast }+{\mathrm{T}}_{32}{{\mathrm{\psi }}_{\mathrm{n}}^1}^{\ast }+{\mathrm{T}}_{33}{{\mathrm{\psi }}_{\mathrm{n}}^4}^{\ast }+{\mathrm{T}}_{34}{{\mathrm{\psi }}_{\mathrm{n}}^3}^{\ast })=0,\\ & \begin{array}{c}{\mathrm{T}}_{41}{\mathrm{\phi }}_{\mathrm{n}}^1+{\mathrm{T}}_{42}{\mathrm{\phi }}_{\mathrm{n}}^2+{\mathrm{T}}_{43}{\mathrm{\phi }}_{\mathrm{n}}^3+{\mathrm{T}}_{44}{\mathrm{\phi }}_{\mathrm{n}}^4+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}({\mathrm{T}}_{41}{{\mathrm{\phi }}_{\mathrm{n}}^2}^{\ast }+{\mathrm{T}}_{42}{{\mathrm{\phi }}_{\mathrm{n}}^1}^{\ast }+{\mathrm{T}}_{43}{{\mathrm{\phi }}_{\mathrm{n}}^4}^{\ast }+{\mathrm{T}}_{44}{{\mathrm{\phi }}_{\mathrm{n}}^3}^{\ast })\\ +{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}({\mathrm{T}}_{41}{\mathrm{\psi }}_{\mathrm{n}}^1+{\mathrm{T}}_{42}{\mathrm{\psi }}_{\mathrm{n}}^2+{\mathrm{T}}_{43}{\mathrm{\psi }}_{\mathrm{n}}^3+{\mathrm{T}}_{44}{\mathrm{\psi }}_{\mathrm{n}}^3)+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}({\mathrm{T}}_{41}{{\mathrm{\psi }}_{\mathrm{n}}^2}^{\ast }+{\mathrm{T}}_{42}{{\mathrm{\psi }}_{\mathrm{n}}^1}^{\ast }+{\mathrm{T}}_{43}{{\mathrm{\psi }}_{\mathrm{n}}^4}^{\ast }+{\mathrm{T}}_{44}{{\mathrm{\psi }}_{\mathrm{n}}^3}^{\ast })=0\end{array}\end{array}$$(31)

Rearranging these equations, we obtain:

$$ \left(\begin{array}{cccc}{\mathrm{T}}_{11}& {\mathrm{T}}_{12}& {\mathrm{T}}_{13}& {\mathrm{T}}_{14}\\ {\mathrm{T}}_{21}& {\mathrm{T}}_{22}& {\mathrm{T}}_{23}& {\mathrm{T}}_{24}\\ {\mathrm{T}}_{31}& {\mathrm{T}}_{32}& {\mathrm{T}}_{33}& {\mathrm{T}}_{34}\\ {\mathrm{T}}_{41}& {\mathrm{T}}_{42}& {\mathrm{T}}_{43}& {\mathrm{T}}_{44}\end{array}\right)\left(\begin{array}{c}1\\ {\mathrm{\beta }}_{\mathrm{j}}^{\left(1\right)}\\ {\mathrm{\beta }}_{\mathrm{j}}^{\left(2\right)}\\ {\mathrm{\beta }}_{\mathrm{j}}^{\left(3\right)}\end{array}\right)=0$$(32)

where

$$ \begin{array}{cc}{\mathrm{\beta }}_{\mathrm{j}}^{\left(1\right)}& =\frac{{\mathrm{\phi }}_{\mathrm{n}}^2+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}{{\mathrm{\phi }}_{\mathrm{n}}^1}^{\ast }+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}{\mathrm{\psi }}_{\mathrm{n}}^2+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}{{\mathrm{\psi }}_{\mathrm{n}}^1}^{\ast }}{{\mathrm{\phi }}_{\mathrm{n}}^1+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}{{\mathrm{\phi }}_{\mathrm{n}}^2}^{\ast }+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}{\mathrm{\psi }}_{\mathrm{n}}^1+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}{{\mathrm{\psi }}_{\mathrm{n}}^2}^{\ast }},\\ {\mathrm{\beta }}_{\mathrm{j}}^{\left(2\right)}& =\frac{{\mathrm{\phi }}_{\mathrm{n}}^3+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}{{\mathrm{\phi }}_{\mathrm{n}}^4}^{\ast }+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}{\mathrm{\psi }}_{\mathrm{n}}^3+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}{{\mathrm{\psi }}_{\mathrm{n}}^4}^{\ast }}{{\mathrm{\phi }}_{\mathrm{n}}^1+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}{{\mathrm{\phi }}_{\mathrm{n}}^2}^{\ast }+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}{\mathrm{\psi }}_{\mathrm{n}}^1+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}{{\mathrm{\psi }}_{\mathrm{n}}^2}^{\ast }},\\ {\mathrm{\beta }}_{\mathrm{j}}^{\left(3\right)}& =\frac{{\mathrm{\phi }}_{\mathrm{n}}^4+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}{{\mathrm{\phi }}_{\mathrm{n}}^3}^{\ast }+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}{\mathrm{\psi }}_{\mathrm{n}}^4+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}{{\mathrm{\psi }}_{\mathrm{n}}^3}^{\ast }}{{\mathrm{\phi }}_{\mathrm{n}}^1+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}{{\mathrm{\phi }}_{\mathrm{n}}^2}^{\ast }+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}{\mathrm{\psi }}_{\mathrm{n}}^1+{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}{{\mathrm{\psi }}_{\mathrm{n}}^2}^{\ast }}.\end{array}$$(33)

By substituting Equation (26) into Equation (32), the following linear algebraic system is obtained:

$$ \left(\begin{array}{cccc}{\mathrm{\lambda }}_{\mathrm{j}}+\frac{1}{{\mathrm{\lambda }}_{\mathrm{j}}}{\mathrm{b}}_{11,\mathrm{n}}& \frac{1}{{\mathrm{\lambda }}_{\mathrm{j}}}{\mathrm{b}}_{12,\mathrm{n}}& {\mathrm{c}}_{13,\mathrm{n}}& {\mathrm{c}}_{14,\mathrm{n}}\\ \frac{1}{{\mathrm{\lambda }}_{\mathrm{j}}}{\mathrm{b}}_{12,\mathrm{n}}^{\ast }& {\mathrm{\lambda }}_{\mathrm{j}}+\frac{1}{{\mathrm{\lambda }}_{\mathrm{j}}}{\mathrm{b}}_{11,\mathrm{n}}^{\ast }& {\mathrm{c}}_{14,\mathrm{n}}^{\ast }& {\mathrm{c}}_{13,\mathrm{n}}^{\ast }\\ {\mathrm{c}}_{31,\mathrm{n}}& {\mathrm{c}}_{32,\mathrm{n}}& {\mathrm{a}}_{33,\mathrm{n}}{\mathrm{\lambda }}_{\mathrm{j}}+\frac{1}{{\mathrm{\lambda }}_{\mathrm{j}}}& {\mathrm{a}}_{34,\mathrm{n}}{\mathrm{\lambda }}_{\mathrm{j}}\\ {\mathrm{c}}_{32,\mathrm{n}}^{\ast }& {\mathrm{c}}_{31,\mathrm{n}}^{\ast }& \hspace{0.17em}{\mathrm{a}}_{34,\mathrm{n}}^{\ast }{\mathrm{\lambda }}_{\mathrm{j}}& {\mathrm{a}}_{33,\mathrm{n}}^{\ast }{\mathrm{\lambda }}_{\mathrm{j}}+\frac{1}{{\mathrm{\lambda }}_{\mathrm{j}}}\end{array}\right)\left(\begin{array}{c}1\\ {\mathrm{\beta }}_{\mathrm{j}}^{\left(1\right)}\\ {\mathrm{\beta }}_{\mathrm{j}}^{\left(2\right)}\\ {\mathrm{\beta }}_{\mathrm{j}}^{\left(3\right)}\end{array}\right)=0.$$(34)

By appropriately selecting  $$ {\mathrm{\lambda }}_{\mathrm{j}},{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)},{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)},{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}(\mathrm{j}=\mathrm{1,2},\mathrm{3,4})$$, the determinant of the coefficient matrix in the linear algebraic system (33) can be ensured to be nonzero. Under this condition, the variables  $$ {\mathrm{a}}_{33,\mathrm{n}},\mathrm{ }{\mathrm{a}}_{34,\mathrm{n}},\mathrm{ }{\mathrm{b}}_{11,\mathrm{n}},\mathrm{ }{\mathrm{b}}_{12,\mathrm{n}},\mathrm{ }{\mathrm{c}}_{13,\mathrm{n}},\mathrm{ }{\mathrm{c}}_{14,\mathrm{n}},\mathrm{ }{\mathrm{c}}_{31,\mathrm{n}},\mathrm{ }{\mathrm{c}}_{32,\mathrm{n}}$$  can be uniquely determined by (34).

3. Explicit exact solutions

This section focuses on obtaining exact solutions of Equation (18,19) through the application of the Darboux transformation (27,28).

Initially, we select the seed solution  $$ {\mathrm{q}}_{\mathrm{n}}=0,{\mathrm{r}}_{\mathrm{n}}=0.$$  In this case, the matrices  $$ {\mathrm{N}}_{\mathrm{n}}$$  and  $$ {\mathrm{M}}_{\mathrm{n}}$$  are given by:

$$ {\mathrm{N}}_{\mathrm{n}}=\left(\begin{array}{cccc}\mathrm{\lambda }& 0& 0& 0\\ 0& {\mathrm{\lambda }}^{\ast }& 0& 0\\ 0& 0& {\mathrm{\lambda }}^{-1}& 0\\ 0& 0& 0& {{\mathrm{\lambda }}^{\ast }}^{-1}\end{array}\right),$$(35)

$$ {\mathrm{M}}_{\mathrm{n}}=\mathrm{i}\left(\begin{array}{cccc}-\frac{{\mathrm{\lambda }}^2+{\mathrm{\lambda }}^{-2}}{2}+1& 0& 0& 0\\ 0& -\frac{{{\mathrm{\lambda }}^{\ast }}^2+{{\mathrm{\lambda }}^{\ast }}^{-2}}{2}+1& 0& 0\\ 0& 0& \frac{{\mathrm{\lambda }}^2+{\mathrm{\lambda }}^{-2}}{2}-1& 0\\ 0& 0& 0& \frac{{{\mathrm{\lambda }}^{\ast }}^2+{{\mathrm{\lambda }}^{\ast }}^{-2}}{2}-1\end{array}\right).$$(36)

Thus, the spatial spectral problem  $$ {\mathrm{\Psi }}_{\mathrm{n}+1}={\mathrm{N}}_{\mathrm{n}}{\mathrm{\Psi }}_{\mathrm{n}}$$  and the temporal spectral problem  $$ \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{\mathrm{\Psi }}_{\mathrm{n}}={\mathrm{M}}_{\mathrm{n}}{\mathrm{\Psi }}_{\mathrm{n}}$$  admit the following four fundamental solutions:

$$ \begin{array}{cccc}& {\mathrm{\phi }}_{\mathrm{n}}\left(\mathrm{\lambda }\right)=\left(\begin{array}{c}{\mathrm{\lambda }}^{\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(-\mathrm{ }\frac{{\mathrm{\lambda }}^2+{\mathrm{\lambda }}^{-2}}{2}\mathrm{ }+1)\mathrm{t}}\\ 0\\ 0\\ 0\end{array}\right),& & {\mathrm{\psi }}_{\mathrm{n}}\left(\mathrm{\lambda }\right)=\left(\begin{array}{c}0\\ {{\mathrm{\lambda }}^{\ast }}^{\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(-\mathrm{ }\frac{{{\mathrm{\lambda }}^{\ast }}^2+{{\mathrm{\lambda }}^{\ast }}^{-2}}{2}\mathrm{ }+1)\mathrm{t}}\\ 0\\ 0\end{array}\right)\\ & & & \\ & {\mathrm{x}}_{\mathrm{n}}\left(\mathrm{\lambda }\right)=\left(\begin{array}{c}0\\ 0\\ {\mathrm{\lambda }}^{-\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(\frac{{\mathrm{\lambda }}^2+{\mathrm{\lambda }}^{-2}}{2}-1)\mathrm{t}}\\ 0\end{array}\right),& & {\mathrm{y}}_{\mathrm{n}}\left(\mathrm{\lambda }\right)=\left(\begin{array}{c}0\\ 0\\ 0\\ {{\mathrm{\lambda }}^{\ast }}^{-\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(\frac{{{\mathrm{\lambda }}^{\ast }}^2+{{\mathrm{\lambda }}^{\ast }}^{-2}}{2}-1)\mathrm{t}}\end{array}\right).\end{array}$$(37)

Substituting these fundamental solutions into Equation (33), we obtain:

$$ \begin{array}{cc}& {\mathrm{\beta }}_{\mathrm{j}}^{\left(1\right)}=\frac{{\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}{{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^{\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(-\mathrm{ }\frac{{{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^2+{{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^{-2}}{2}\mathrm{ }+1)\mathrm{t}}}{{\mathrm{\lambda }}_{\mathrm{j}}^{\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(-\mathrm{ }\frac{{\mathrm{\lambda }}_{\mathrm{j}}^2+{\mathrm{\lambda }}_{\mathrm{j}}^{-2}}{2}\mathrm{ }+\mathrm{ }1)\mathrm{t}}}={\mathrm{\alpha }}_{\mathrm{j}}^{\left(1\right)}{{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^{\mathrm{n}}{\mathrm{\lambda }}_{\mathrm{j}}^{-\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(-\mathrm{ }\frac{{{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^2+{{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^{-2}}{2}\mathrm{ }+\mathrm{ }\frac{{\mathrm{\lambda }}_{\mathrm{j}}^2+{\mathrm{\lambda }}_{\mathrm{j}}^{-2}}{2})\mathrm{t}},\\ & {\mathrm{\beta }}_{\mathrm{j}}^{\left(2\right)}=\frac{{\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}{\mathrm{\lambda }}_{\mathrm{j}}^{-\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(\frac{{\mathrm{\lambda }}_{\mathrm{j}}^2+{\mathrm{\lambda }}_{\mathrm{j}}^{-2}}{2}\mathrm{ }-1)\mathrm{t}}}{{\mathrm{\lambda }}_{\mathrm{j}}^{\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(-\mathrm{ }\frac{{\mathrm{\lambda }}_{\mathrm{j}}^2+{\mathrm{\lambda }}_{\mathrm{j}}^{-2}}{2}\mathrm{ }+1)\mathrm{t}}}={\mathrm{\alpha }}_{\mathrm{j}}^{\left(2\right)}{\mathrm{\lambda }}_{\mathrm{j}}^{-2\mathrm{n}}{\mathrm{e}}^{\mathrm{i}({\mathrm{\lambda }}_{\mathrm{j}}^2+{\mathrm{\lambda }}_{\mathrm{j}}^{-2}-2)\mathrm{t}},\\ & {\mathrm{\beta }}_{\mathrm{j}}^{\left(3\right)}=\frac{{\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}{{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^{-\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(\frac{{{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^2+{{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^{-2}}{2}\mathrm{ }-1)\mathrm{t}}}{{\mathrm{\lambda }}_{\mathrm{j}}^{\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(-\mathrm{ }\frac{{\mathrm{\lambda }}_{\mathrm{j}}^2+{\mathrm{\lambda }}_{\mathrm{j}}^{-2}}{2}\mathrm{ }+1)\mathrm{t}}}={\mathrm{\alpha }}_{\mathrm{j}}^{\left(3\right)}\left({{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^{-\mathrm{n}}{\mathrm{\lambda }}_{\mathrm{j}}^{-\mathrm{n}}\right){\mathrm{e}}^{\mathrm{i}(\frac{{{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^2+{{\mathrm{\lambda }}_{\mathrm{j}}^{\ast }}^{-2}}{2}\mathrm{ }+\mathrm{ }\frac{{\mathrm{\lambda }}_{\mathrm{j}}^2+{\mathrm{\lambda }}_{\mathrm{j}}^{-2}}{2}\mathrm{ }-2)\mathrm{t}}.\end{array}$$(38)

Thus, the system of Equation (34) transforms into:

$$ \{\begin{array}{c}\begin{array}{c}({\mathrm{\lambda }}_{\mathrm{j}}+{\mathrm{b}}_{11,\mathrm{n}}{\mathrm{\lambda }}_{\mathrm{j}}^{-1})+{\mathrm{b}}_{12,\mathrm{n}}{\mathrm{\lambda }}_{\mathrm{j}}^{-1}{\mathrm{\beta }}_{\mathrm{j}}^{\left(1\right)}+{\mathrm{c}}_{13,\mathrm{n}}{\mathrm{\beta }}_{\mathrm{j}}^{\left(2\right)}+{\mathrm{c}}_{14,\mathrm{n}}{\mathrm{\beta }}_{\mathrm{j}}^{\left(3\right)}=0\\ {\mathrm{b}}_{12,\mathrm{n}}^{\ast }{\mathrm{\lambda }}_{\mathrm{j}}^{-1}+({\mathrm{\lambda }}_{\mathrm{j}}+{\mathrm{b}}_{11,\mathrm{n}}^{\ast }{\mathrm{\lambda }}_{\mathrm{j}}^{-1}){\mathrm{\beta }}_{\mathrm{j}}^{\left(1\right)}+{\mathrm{c}}_{14,\mathrm{n}}^{\ast }{\mathrm{\beta }}_{\mathrm{j}}^{\left(2\right)}+{\mathrm{c}}_{13,\mathrm{n}}^{\ast }{\mathrm{\beta }}_{\mathrm{j}}^{\left(3\right)}=0\end{array}\\ \begin{array}{c}{\mathrm{c}}_{31,\mathrm{n}}+{\mathrm{c}}_{32,\mathrm{n}}{\mathrm{\beta }}_{\mathrm{j}}^{\left(1\right)}+({\mathrm{a}}_{33,\mathrm{n}}{\mathrm{\lambda }}_{\mathrm{j}}+{\mathrm{\lambda }}_{\mathrm{j}}^{-1}){\mathrm{\beta }}_{\mathrm{j}}^{\left(2\right)}+{\mathrm{a}}_{34,\mathrm{n}}{\mathrm{\lambda }}_{\mathrm{j}}{\mathrm{\beta }}_{\mathrm{j}}^{\left(3\right)}=0\\ {\mathrm{c}}_{32,\mathrm{n}}^{\ast }+{\mathrm{c}}_{31,\mathrm{n}}^{\ast }{\mathrm{\beta }}_{\mathrm{j}}^{\left(1\right)}+{\mathrm{a}}_{34,\mathrm{n}}^{\ast }{\mathrm{\lambda }}_{\mathrm{j}}{\mathrm{\beta }}_{\mathrm{j}}^{\left(2\right)}+({\mathrm{a}}_{33,\mathrm{n}}^{\ast }{\mathrm{\lambda }}_{\mathrm{j}}+{\mathrm{\lambda }}_{\mathrm{j}}^{-1}){\mathrm{\beta }}_{\mathrm{j}}^{\left(3\right)}=0\end{array}\end{array}$$(39)

Rewriting the third and fourth equations, we obtain:

$$ {\mathrm{c}}_{31,\mathrm{n}}+{\mathrm{\beta }}_{\mathrm{j}}^{\left(1\right)}{\mathrm{c}}_{32,\mathrm{n}}+{\mathrm{\lambda }}_{\mathrm{j}}{\mathrm{\beta }}_{\mathrm{j}}^{\left(2\right)}{\mathrm{a}}_{33,\mathrm{n}}+{\mathrm{\lambda }}_{\mathrm{j}}{\mathrm{\beta }}_{\mathrm{j}}^{\left(3\right)}{\mathrm{a}}_{34,\mathrm{n}}=-{\mathrm{\lambda }}_{\mathrm{j}}^{-1}{\mathrm{\beta }}_{\mathrm{j}}^{\left(2\right)}.$$(40)