Rogue waves and solitons of the coherently-coupled nonlinear Schrödinger equations with the positive coherent coupling

Solitons, which are a kind of the localized waves, preserve their shapes during the propagation via the balance between the nonlinear and dispersive effects [1, 3–8]. Envelope solitons of the nonlinear Schrödinger (NLS)-type equations have been seen in nonlinear optics [1, 2], plasma physics [9] and Bose–Einstein condensation [10]. Besides, the Peregrine soliton has been used to describe the rogue wave (RW), a nonlinear wave of the NLS equation localized in both time and space [11]. The RWs have been observed experimentally in different physical systems [12], such as the optical fibers [13, 14], water-wave tank [15] and plasmas with negative ions [16]. Twist RW (TRW) pairs and soliton-like rational solutions have been found for the Sasa–Satsuma equation, which possess different properties from those for the NLS equation [17–19]. In the physical systems describing interacting wave components, such as the Bose–Einstein condensates and optical fibers, vector RWs have been introduced and studied [20–22]. Experimental demonstration of the optical dark RWs and second order RWs have also been provided [23, 24].

With the consideration of the simultaneous propagation of the optical pulses with different frequencies or polarizations in the birefringent fibers or multi-mode fibers, the coupled NLS equations have been investigated [25–34]. For example, the Manakov system has been claimed to the left- and right-polarized modes of the electromagnetic waves [34].

For the coupled NLS equations, there have been two categories of the coupling effects: incoherent coupling and coherent coupling [35–37]. It has been reported that the coherence happens when the nonlinear fiber is weakly birefringent [25, 35–38]. Simultaneous propagation of two optical pulses with different frequencies or polarizations in a weakly birefringent fiber has been governed by the following coherently-coupled NLS equations [33, 39, 40]:

$$\begin{eqnarray}&&{{iq}}_{1,z}+{q}_{1,{tt}}+2({| {q}_{1}| }^{2}+2{| {q}_{2}| }^{2}){q}_{1}-2{q}_{1}^{* }{q}_{2}^{2}=0,\end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray}&&{{iq}}_{2,z}+{q}_{2,{tt}}+2({| {q}_{2}| }^{2}+2{| {q}_{1}| }^{2}){q}_{2}-2{q}_{2}^{* }{q}_{1}^{2}=0,\end{eqnarray} \tag{ 1b }$$

where q₁ and q₂ are the slowly varying envelopes of two interacting optical modes, the variables z and t, respectively, correspond to the normalized distance and retarded time, the asterisk denotes the complex conjugate, the terms $| {q}_{1}{| }^{2}{q}_{1}$ and $| {q}_{2}{| }^{2}{q}_{2}$ represent the self-phase modulation (SPM), the terms $| {q}_{1}{| }^{2}{q}_{2}$ and $| {q}_{2}{| }^{2}{q}_{1}$ represent the XPM, and the terms ${q}_{1}^{* }{q}_{2}^{2}$ and ${q}_{2}^{* }{q}_{1}^{2}$ represent the coherent coupling terms governing the energy exchange between two axes of the fiber [39]. As the ratio between the coefficients of the XPM terms and the coherent coupling terms in equations (1) is −2, the coherent coupling has been called the negative coherent coupling [40]. Integrability properties and soliton solutions of equations (1) have been obtained [33].

Furthermore, researchers have concentrated their attention on the following coherently-coupled NLS equations [38, 39, 41]:

$$\begin{eqnarray}&&{{iq}}_{1,z}+{q}_{1,{tt}}+2({| {q}_{1}| }^{2}-2{| {q}_{2}| }^{2}){q}_{1}-2{q}_{1}^{* }{q}_{2}^{2}=0,\end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray}&&{{iq}}_{2,z}+{q}_{2,{tt}}+2(2{| {q}_{1}| }^{2}-{| {q}_{2}| }^{2}){q}_{2}+2{q}_{2}^{* }{q}_{1}^{2}=0,\end{eqnarray} \tag{ 2b }$$

obtained by means of the Ablowitz–Kaup–Newell–Segur (AKNS) technology [41]. Compared with that in equations (1), the ratio between the coefficients of the XPM and the coherent coupling terms in equations (2) is positive. Conserved quantities, Lax pair and Darboux transformation of equations (2) have been derived [41]. Vector solitons possessing single- or double-hump profiles of equations (2) as well as their interactions have been discussed via the Hirota method [39]. Bound-soliton, breather and RW solutions of equations (2) with the complex spectral parameters have been derived via the Darboux transformation (DT) [38].

However, the case that the spectral parameter is real has not been considered for the DT in [38], with respect to equations (2). In this paper, we will construct the binary DT (BDT) of equations (2) with the real or complex spectral parameters, and present the vector TRW pairs, solitons, periodic and soliton-like rational solutions of equations (2), which have not been obtained, to our knowledge. In section 2, we will give the formation of the one-fold BDT of equations (2). In section 3, vector analytic solutions of equations (2) from the non-zero seed solutions will be presented. Section 4 will be the discussion. Section 5 will be the conclusion.

Lax pair of equations (2) is [38, 41]

$$\begin{eqnarray}{\phi }_{t} & = & {\rm{M}}\phi =(\lambda {{\rm{M}}}_{0}+{{\rm{M}}}_{1})\phi ,\\ {\phi }_{z} & = & {\rm{N}}\phi =({\lambda }^{2}{{\rm{N}}}_{0}+\lambda {{\rm{N}}}_{1}+{{\rm{N}}}_{2})\phi ,\end{eqnarray} \tag{ 3 }$$

with

$$\begin{eqnarray*}{{\rm{M}}}_{0} & = & i\left(\begin{array}{cc}-{\rm{I}} & {\rm{O}}\\ {\rm{O}} & {\rm{I}}\end{array}\right),\quad {{\rm{M}}}_{1}=\left(\begin{array}{cc}{\rm{O}} & {\rm{Q}}\\ -{{\rm{Q}}}^{* } & {\rm{O}}\end{array}\right),\quad {\rm{Q}}=\left(\begin{array}{cc}{q}_{1} & {q}_{2}\\ -{q}_{2} & {q}_{1}\end{array}\right),\\ {{\rm{N}}}_{0} & = & 2i\left(\begin{array}{cc}-{\rm{I}} & {\rm{O}}\\ {\rm{O}} & {\rm{I}}\end{array}\right),\quad {{\rm{N}}}_{1}=2\left(\begin{array}{cc}{\rm{O}} & {\rm{Q}}\\ -{{\rm{Q}}}^{* } & {\rm{O}}\end{array}\right),\\ {{\rm{N}}}_{2} & = & i\left(\begin{array}{cc}{\mathrm{QQ}}^{* } & {{\rm{Q}}}_{t}\\ {{\rm{Q}}}_{t}^{* } & -{{\rm{Q}}}^{* }{\rm{Q}}\end{array}\right),\end{eqnarray*}$$

where $\phi ={({\zeta }_{1},{\zeta }_{2},{\zeta }_{3},{\zeta }_{4})}^{{\text{}}T}$ is a 4 × 1 vector, ζ_α's (α = 1, 2, 3, 4) are functions of z and $t,{{\rm{M}}}_{0},{{\rm{M}}}_{1},{{\rm{N}}}_{0}$, ${{\rm{N}}}_{1}$ and ${{\rm{N}}}_{2}$ are the 4 × 4 matrices, ${\rm{I}}$ is the 2 × 2 unit matrix, ${\rm{O}}$ is the 2 × 2 zero matrix, ${\rm{Q}}$ is a 2 × 2 matrix, λ is a complex spectral parameter, the superscript ${\text{}}T$ denotes the matrix transpose and '*' denotes the matrix complex conjugate. The compatibility condition ${{\rm{M}}}_{z}-{{\rm{N}}}_{t}+\mathrm{MN}-\mathrm{NM}=0$, which is known as the zero-curvature equation [42], reproduces equations (2).

Suppose that ${\phi }_{1}={({\varphi }_{1},{\varphi }_{2},{\varphi }_{3},{\varphi }_{4})}^{{\text{}}T}$ is a vector solution of Lax pair (3) corresponding to the complex spectral parameter λ₁, and φ_α's are the functions of z and t. We can infer that ${(-{\varphi }_{2},{\varphi }_{1},-{\varphi }_{4},{\varphi }_{3})}^{{\text{}}T}$ is a vector solution of Lax pair (3) corresponding to λ₁, and ${({\varphi }_{3}^{* },{\varphi }_{4}^{* },-{\varphi }_{1}^{* },-{\varphi }_{2}^{* })}^{{\text{}}T}$ is a vector solution corresponding to ${\lambda }_{1}^{* }$.

For simplicity, we use Φ₁, Φ₂, Φ₃ and Φ₄ to denote the 1 × 4 row vectors of the 4 × 4 martix Σ₁ as

$$\begin{eqnarray*}&&\left(\begin{array}{c}{{\rm{\Phi }}}_{1}\\ {{\rm{\Phi }}}_{2}\\ {{\rm{\Phi }}}_{3}\\ {{\rm{\Phi }}}_{4}\end{array}\right)={{\rm{\Sigma }}}_{1}=\left(\begin{array}{cccc}{\varphi }_{1} & -{\varphi }_{2} & {\varphi }_{3}^{* } & -{\varphi }_{4}^{* }\\ {\varphi }_{2} & {\varphi }_{1} & {\varphi }_{4}^{* } & {\varphi }_{3}^{* }\\ {\varphi }_{3} & -{\varphi }_{4} & -{\varphi }_{1}^{* } & {\varphi }_{2}^{* }\\ {\varphi }_{4} & {\varphi }_{3} & -{\varphi }_{2}^{* } & -{\varphi }_{1}^{* }\end{array}\right).\end{eqnarray*}$$

Via the dimensional reduction [43], we can build the one-fold BDT for equations (2) as

$$\begin{eqnarray}&&{q}_{1,[1]}={q}_{1}-2i\,{{\rm{\Phi }}}_{1}\,{{\rm{\Omega }}}^{-1}\,{{\rm{\Phi }}}_{4}^{\dagger },\end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray}&&{q}_{2,[1]}={q}_{2}-2i\,{{\rm{\Phi }}}_{1}\,{{\rm{\Omega }}}^{-1}\,{{\rm{\Phi }}}_{3}^{\dagger },\end{eqnarray} \tag{ 4b }$$

where the superscript '−1' denotes the matrix inverse, the superscript † denotes the conjugate transpose, q_1,[1] and q_2,[1] are the solutions of equations (2), the mark [1] represents the first iteration of q₁ or q₂, and Ω is the 4 × 4 matrix derived from the algebraic condition

$$\begin{eqnarray}&&{\rm{\Omega }}{{\rm{\Lambda }}}_{1}-{{\rm{\Lambda }}}_{1}^{\dagger }{\rm{\Omega }}={{\rm{\Sigma }}}_{1}^{\dagger }{{\rm{S}}}_{1}{{\rm{\Sigma }}}_{1},\end{eqnarray} \tag{ 5 }$$

where ${{\rm{S}}}_{1}=\left(\begin{array}{cccc}0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{array}\right),{{\rm{\Lambda }}}_{1}=\left(\begin{array}{cccc}{\lambda }_{1} & 0 & 0 & 0\\ 0 & {\lambda }_{1} & 0 & 0\\ 0 & 0 & {\lambda }_{1}^{* } & 0\\ 0 & 0 & 0 & {\lambda }_{1}^{* }\end{array}\right)$, and the details to determine Ω are described in the appendix.

To obtain the vector analytic solutions on the non-zero background of equations (2) via BDT (4), we use the following non-zero seed solutions of equations (2):

$$\begin{eqnarray}&&\left(\begin{array}{c}{q}_{1}\\ {q}_{2}\end{array}\right)={e}^{i\theta (t,z)}\left(\begin{array}{c}{{ic}}_{1}\\ {c}_{2}\end{array}\right),\end{eqnarray} \tag{ 6 }$$

where $\theta (t,z)={at}+2({c}_{1}^{2}-{c}_{2}^{2})z-{a}^{2}z,a$, c₁ and c₂ are the real constants. The vector solutions of Lax pair (3) corresponding to λ₁ can be expressed as

$$\begin{eqnarray}&&{\phi }_{1}=\left(\begin{array}{c}{\varphi }_{1}\\ {\varphi }_{2}\\ {\varphi }_{3}\\ {\varphi }_{4}\end{array}\right)=\left(\begin{array}{c}{k}_{1}\left({\mu }_{1}-{\lambda }_{1}-\tfrac{1}{2}a\right){e}^{{{iX}}_{1}+\tfrac{i\theta (z,t)}{2}}+{k}_{3}\left({\mu }_{2}-{\lambda }_{1}-\tfrac{1}{2}a\right){e}^{{{iX}}_{2}+\tfrac{i\theta (z,t)}{2}}\\ {k}_{2}\left({\mu }_{1}-{\lambda }_{1}-\tfrac{1}{2}a\right){e}^{{{iX}}_{1}+\tfrac{i\theta (z,t)}{2}}+{k}_{4}\left({\mu }_{2}-{\lambda }_{1}-\tfrac{1}{2}a\right){e}^{{{iX}}_{2}+\tfrac{i\theta (z,t)}{2}}\\ ({k}_{1}{c}_{1}+{{ik}}_{2}{c}_{2}){e}^{{{iX}}_{1}-\tfrac{i\theta (z,t)}{2}}+({k}_{3}{c}_{1}+{{ik}}_{4}{c}_{2}){e}^{{{iX}}_{2}-\tfrac{i\theta (z,t)}{2}}\\ (-{{ik}}_{1}{c}_{2}+{k}_{2}{c}_{1}){e}^{{{iX}}_{1}-\tfrac{i\theta (z,t)}{2}}+(-{{ik}}_{3}{c}_{2}+{k}_{4}{c}_{1}){e}^{{{iX}}_{2}-\tfrac{i\theta (z,t)}{2}}\end{array}\right),\end{eqnarray} \tag{ 7 }$$

where

$$\begin{eqnarray}&&{X}_{1}={\mu }_{1}t+(2{\lambda }_{1}-a){\mu }_{1}z,\end{eqnarray} \tag{ 8a }$$

$$\begin{eqnarray}&&{X}_{2}={\mu }_{2}t+(2{\lambda }_{1}-a){\mu }_{2}z,\end{eqnarray} \tag{ 8b }$$

k_α's (α = 1, 2, 3, 4) are the complex constants, and

$$\begin{eqnarray}{\mu }_{1} & = & -\sqrt{{\left({\lambda }_{1}+\displaystyle \frac{a}{2}\right)}^{2}+{c}_{1}^{2}-{c}_{2}^{2}},\\ {\mu }_{2} & = & \sqrt{{\left({\lambda }_{1}+\displaystyle \frac{a}{2}\right)}^{2}+{c}_{1}^{2}-{c}_{2}^{2}}.\end{eqnarray} \tag{ 9 }$$

Based on the process in the appendix, Ω is expressed as

$$\begin{eqnarray}{\rm{\Omega }}=\left(\begin{array}{cccc}{f}_{1} & {f}_{2} & -{g}_{1}^{* }\,-\,{m}_{1}^{* } & -{g}_{2}^{* }\,-\,{m}_{2}^{* }\\ {f}_{2} & {-f}_{1} & -{g}_{2}^{* }\,-\,{m}_{2}^{* } & {g}_{1}^{* }\,+\,{m}_{1}^{* }\\ {g}_{1}+{m}_{1} & {g}_{2}+{m}_{2} & -{f}_{1} & {-f}_{2}\\ {g}_{2}+{m}_{2} & -{g}_{1}-{m}_{1} & -{f}_{2} & {f}_{1}\end{array}\right),\end{eqnarray} \tag{ 10 }$$

where m₁ and m₂ are the real constants, and f₁, f₂, g₁ and g₂ are expressed as follows. Two cases need to be considered:

3.1. λ₁ is complex with a non-zero imaginary part

Substituting expression (7) into expression (A.7), we have

$$\begin{eqnarray}{f}_{1} & = & 2({c}_{1}^{2}-{c}_{2}^{2})\left[\displaystyle \frac{{k}_{3}{k}_{4}^{* }+{k}_{4}{k}_{3}^{* }}{{\mu }_{2}-{\mu }_{2}^{* }+{\lambda }_{1}-{\lambda }_{1}^{* }}{e}^{i({X}_{2}-{X}_{2}^{* })}\right.\\ & & +\displaystyle \frac{{k}_{3}{k}_{2}^{* }+{k}_{4}{k}_{1}^{* }}{{\mu }_{2}-{\mu }_{1}^{* }+{\lambda }_{1}-{\lambda }_{1}^{* }}{e}^{i({X}_{2}-{X}_{1}^{* })}\\ & & +\displaystyle \frac{{k}_{2}{k}_{3}^{* }+{k}_{1}{k}_{4}^{* }}{{\mu }_{1}-{\mu }_{2}^{* }+{\lambda }_{1}-{\lambda }_{1}^{* }}{e}^{i({X}_{1}-{X}_{2}^{* })}\\ & & \left.+\displaystyle \frac{{k}_{1}{k}_{2}^{* }+{k}_{2}{k}_{1}^{* }}{{\mu }_{1}-{\mu }_{1}^{* }+{\lambda }_{1}-{\lambda }_{1}^{* }}{e}^{i({X}_{1}-{X}_{1}^{* })}\right],\end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray}{f}_{2} & = & 2({c}_{1}^{2}-{c}_{2}^{2})\left[\displaystyle \frac{{k}_{3}{k}_{3}^{* }-{k}_{4}{k}_{4}^{* }}{{\mu }_{2}-{\mu }_{2}^{* }+{\lambda }_{1}-{\lambda }_{1}^{* }}{e}^{i({X}_{2}-{X}_{2}^{* })}\right.\\ & & +\displaystyle \frac{{k}_{3}{k}_{1}^{* }-{k}_{4}{k}_{2}^{* }}{{\mu }_{2}-{\mu }_{1}^{* }+{\lambda }_{1}-{\lambda }_{1}^{* }}{e}^{i({X}_{2}-{X}_{1}^{* })}\\ & & +\displaystyle \frac{{k}_{1}{k}_{3}^{* }-{k}_{2}{k}_{4}^{* }}{{\mu }_{1}-{\mu }_{2}^{* }+{\lambda }_{1}-{\lambda }_{1}^{* }}{e}^{i({X}_{1}-{X}_{2}^{* })}\\ & & \left.+\displaystyle \frac{{k}_{1}{k}_{1}^{* }-{k}_{2}{k}_{2}^{* }}{{\mu }_{1}-{\mu }_{1}^{* }+{\lambda }_{1}-{\lambda }_{1}^{* }}{e}^{i({X}_{1}-{X}_{1}^{* })}\right].\end{eqnarray} \tag{ 11b }$$

Via limitation (A.5), we have

$$\begin{eqnarray}{g}_{1} & = & [2{c}_{1}{k}_{1}{k}_{2}-{{ic}}_{2}({k}_{1}^{2}-{k}_{2}^{2})]\alpha \\ & & +[2{c}_{1}{k}_{3}{k}_{4}-{{ic}}_{2}({k}_{3}^{2}-{k}_{4}^{2})]\beta \\ & & +[{{ic}}_{2}({k}_{2}{k}_{4}-{k}_{1}{k}_{3})+{c}_{1}({k}_{1}{k}_{4}+{k}_{2}{k}_{3})]\chi ,\end{eqnarray} \tag{ 12a }$$

$$\begin{eqnarray}{g}_{2} & = & [2{{ic}}_{2}{k}_{1}{k}_{2}+{c}_{1}({k}_{1}^{2}-{k}_{2}^{2})]\alpha \\ & & +[2{{ic}}_{2}{k}_{3}{k}_{4}+{c}_{1}({k}_{3}^{2}-{k}_{4}^{2})]\beta \\ & & +[{{ic}}_{2}({k}_{2}{k}_{3}+{k}_{1}{k}_{4})-{c}_{1}({k}_{2}{k}_{4}-{k}_{1}{k}_{3})]\chi ,\end{eqnarray} \tag{ 12b }$$

$$\begin{eqnarray}&&\alpha ={e}^{2{{iX}}_{1}}\left(-1+\displaystyle \frac{{\lambda }_{1}+\tfrac{a}{2}}{{\mu }_{1}}\right),\quad \beta ={e}^{2{{iX}}_{2}}\left(-1+\displaystyle \frac{{\lambda }_{1}+\tfrac{a}{2}}{{\mu }_{2}}\right),\end{eqnarray} \tag{ 12c }$$

$$\begin{eqnarray}&&\chi =-2+2i[(a+2{\lambda }_{1})t+(4{\lambda }_{1}^{2}-{a}^{2})z].\end{eqnarray} \tag{ 12d }$$

To obtain the RW and TWR pair solutions, we take ${\lambda }_{1}\to -\tfrac{a}{2}\pm i\sqrt{{c}_{1}^{2}-{c}_{2}^{2}}$ and ${c}_{1}^{2}\gt {c}_{2}^{2}$. From expressions (9), we know that μ₂ tends to 0. In that case, g₁ and g₂ will be singular. To avoid that, we need k_α's to satisfy the following conditions:

$$\begin{eqnarray}&&2{c}_{1}{k}_{1}{k}_{2}-{{ic}}_{2}({k}_{1}^{2}-{k}_{2}^{2})=2{c}_{1}{k}_{3}{k}_{4}-{{ic}}_{2}({k}_{3}^{2}-{k}_{4}^{2}),\end{eqnarray} \tag{ 13a }$$

$$\begin{eqnarray}&&2{{ic}}_{2}{k}_{1}{k}_{2}+{c}_{1}({k}_{1}^{2}-{k}_{2}^{2})=2{{ic}}_{2}{k}_{3}{k}_{4}+{c}_{1}({k}_{3}^{2}-{k}_{4}^{2}),\end{eqnarray} \tag{ 13b }$$

and have

$$\begin{eqnarray}{g}_{1}^{{\prime} } & = & \mathop{\mathrm{lim}}\limits_{{\lambda }_{1}\to -\tfrac{a}{2}\pm i\sqrt{{c}_{1}^{2}-{c}_{2}^{2}}}{g}_{1}\\ & = & [2{c}_{1}{k}_{1}{k}_{2}-{{ic}}_{2}({k}_{1}^{2}-{k}_{2}^{2})+{c}_{1}({k}_{2}{k}_{3}+{k}_{1}{k}_{4})\\ & & +{{ic}}_{2}({k}_{2}{k}_{4}-{k}_{1}{k}_{3})]\chi ,\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}{g}_{2}^{{\prime} } & = & \mathop{\mathrm{lim}}\limits_{{\lambda }_{1}\to -\tfrac{a}{2}\pm i\sqrt{{c}_{1}^{2}-{c}_{2}^{2}}}{g}_{2}\\ & = & [2{{ic}}_{2}{k}_{1}{k}_{2}+{c}_{1}({k}_{1}^{2}-{k}_{2}^{2})+{{ic}}_{2}({k}_{2}{k}_{3}+{k}_{1}{k}_{4})\\ & & -{c}_{1}({k}_{2}{k}_{4}-{k}_{1}{k}_{3})]\chi .\end{eqnarray} \tag{ 15 }$$

Substituting ${g}_{1}^{{\prime} }$, ${g}_{2}^{{\prime} }$ and χ into BDT (4), we obtain the rational solutions corresponding to the RWs and TRW pairs.

3.2. λ₁ is real

We have either ${\mu }_{b}^{* }=-{\mu }_{b}$ (b = 1, 2) with ${\left({\lambda }_{1}+\tfrac{a}{2}\right)}^{2}+{c}_{1}^{2}\,\lt {c}_{2}^{2}$ or ${\mu }_{b}^{* }={\mu }_{b}$ with ${\left({\lambda }_{1}+\tfrac{a}{2}\right)}^{2}+{c}_{1}^{2}\gt {c}_{2}^{2}$. Expressions (11) can be written as

$$\begin{eqnarray}{f}_{1} & = & 2({c}_{1}^{2}-{c}_{2}^{2})\left[\displaystyle \frac{{k}_{3}{k}_{4}^{* }+{k}_{4}{k}_{3}^{* }}{{\mu }_{2}-{\mu }_{2}^{* }}{e}^{i({X}_{2}-{X}_{2}^{* })}\right.\\ & & +\displaystyle \frac{{k}_{3}{k}_{2}^{* }+{k}_{4}{k}_{1}^{* }}{{\mu }_{2}-{\mu }_{1}^{* }}{e}^{i({X}_{2}-{X}_{1}^{* })}+\displaystyle \frac{{k}_{2}{k}_{3}^{* }+{k}_{1}{k}_{4}^{* }}{{\mu }_{1}-{\mu }_{2}^{* }}{e}^{i({X}_{1}-{X}_{2}^{* })}\\ & & \left.+\displaystyle \frac{{k}_{1}{k}_{2}^{* }+{k}_{2}{k}_{1}^{* }}{{\mu }_{1}-{\mu }_{1}^{* }}{e}^{i({X}_{1}-{X}_{1}^{* })}\right],\end{eqnarray} \tag{ 16a }$$

$$\begin{eqnarray}{f}_{2} & = & 2({c}_{1}^{2}-{c}_{2}^{2})\left[\displaystyle \frac{{k}_{3}{k}_{3}^{* }-{k}_{4}{k}_{4}^{* }}{{\mu }_{2}-{\mu }_{2}^{* }}{e}^{i({X}_{2}-{X}_{2}^{* })}\right.\\ & & +\displaystyle \frac{{k}_{3}{k}_{1}^{* }-{k}_{4}{k}_{2}^{* }}{{\mu }_{2}-{\mu }_{1}^{* }}{e}^{i({X}_{2}-{X}_{1}^{* })}+\displaystyle \frac{{k}_{1}{k}_{3}^{* }-{k}_{2}{k}_{4}^{* }}{{\mu }_{1}-{\mu }_{2}^{* }}{e}^{i({X}_{1}-{X}_{2}^{* })}\\ & & \left.+\displaystyle \frac{{k}_{1}{k}_{1}^{* }-{k}_{2}{k}_{2}^{* }}{{\mu }_{1}-{\mu }_{1}^{* }}{e}^{i({X}_{1}-{X}_{1}^{* })}\right].\end{eqnarray} \tag{ 16b }$$

However, we should note that k_α's also have some restrictions to ensure that f₁ and f₂ are non-singular. When ${\mu }_{b}^{* }=-{\mu }_{b}$, i.e., ${\mu }_{1}^{* }={\mu }_{2}$ and ${\mu }_{2}^{* }={\mu }_{1}$, we have ${X}_{2}^{* }={X}_{1},{X}_{1}^{* }={X}_{2}=-{X}_{1}$ and k_α's in expressions (16) must satisfy that ${k}_{1}^{* }{k}_{3}-{k}_{2}^{* }{k}_{4}={k}_{2}^{* }{k}_{3}+{k}_{1}^{* }{k}_{4}=0$. When ${\mu }_{b}^{* }={\mu }_{b}$, we have ${X}_{1}^{* }={X}_{1},{X}_{2}^{* }={X}_{2}=-{X}_{1}$ and k_α's must satisfy that $| {k}_{1}{| }^{2}\,-| {k}_{2}{| }^{2}=| {k}_{3}{| }^{2}-| {k}_{4}{| }^{2}$ $=\,{k}_{1}{k}_{2}^{* }+{k}_{1}^{* }{k}_{2}$ $=\,{k}_{3}{k}_{4}^{* }+{k}_{3}^{* }{k}_{4}=0$.

Up to now, we have introduce the formalisms of the vector analytic solutions with some restrictions on k_α's in different cases. The solutions of equations (2) are also characterized by five real parameters a, c₁, c₂, m₁, m₂ and a complex spectral parameter λ₁. In the next section, we will discuss different types of the analytic solutions based on the formalisms.

4.1. The RWs with complex spectral parameter

With k₁ = k₃ = 1 and k₂ = k₄ = 0 in conditions (13), we obtain the vector RW solutions

$$\begin{eqnarray}&&{q}_{1,[1]}={e}^{i\theta }\left[{{ic}}_{1}-2i(-2{ir},0,2{c}_{1},-2{{ic}}_{2}){{\rm{\Omega }}}^{-1}\left(\begin{array}{c}2{{ic}}_{2}\\ 2{c}_{1}\\ 0\\ 2{ir}\end{array}\right)\right],\end{eqnarray} \tag{ 17a }$$

$$\begin{eqnarray}&&{q}_{2,[1]}={e}^{i\theta }\left[{c}_{2}-2i(-2{ir},0,2{c}_{1},-2{{ic}}_{2}){{\rm{\Omega }}}^{-1}\left(\begin{array}{c}2{c}_{1}\\ -2{{ic}}_{2}\\ 2{ir}\\ 0\end{array}\right)\right],\end{eqnarray} \tag{ 17b }$$

where

$$\begin{eqnarray*}{\rm{\Omega }} & = & \left(\begin{array}{cccc}0 & -4{ir} & -2{{ic}}_{2}{\chi }^{* }-{m}_{1} & -2{c}_{1}{\chi }^{* }-{m}_{2}\\ -4{ir} & 0 & -2{c}_{1}{\chi }^{* }-{m}_{2} & 2{{ic}}_{2}{\chi }^{* }+{m}_{1}\\ -2{{ic}}_{2}\chi +{m}_{1} & 2{c}_{1}\chi +{m}_{2} & 0 & 4{ir}\\ 2{c}_{1}\chi +{m}_{2} & 2{{ic}}_{2}\chi -{m}_{1} & 4{ir} & 0\end{array}\right),\\ \chi & = & -2-8{{ir}}^{2}z+4r(2{az}-t),\quad r=\sqrt{{c}_{1}^{2}-{c}_{2}^{2}}.\end{eqnarray*}$$

When m₁ = m₂ = 0, solutions (17) can be reduced to

$$\begin{eqnarray}{q}_{1,[1]} & = & {{ic}}_{1}{e}^{i\theta }\left[1-\displaystyle \frac{4(4+{\chi }^{* }-\chi )}{4+\chi {\chi }^{* }}\right],\\ {q}_{2,[1]} & = & {c}_{2}{e}^{i\theta }\left[1-\displaystyle \frac{4(4+{\chi }^{* }-\chi )}{4+\chi {\chi }^{* }}\right],\end{eqnarray} \tag{ 18 }$$

which are the vector generalization of the Peregrine soliton, as shown in figure 1. In that case, q₁ is proportional to q₂. The amplitudes $| {q}_{b}|$'s (b = 1, 2) are peaked at $t=-\tfrac{1}{2r}$ with the maximum values as $3| {c}_{b}|$ at z = 0.

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Further inspection of Ω shows that we should restrict m₁ and m₂ to satisfy ${c}_{1}^{2}{m}_{1}^{2}+{c}_{2}^{2}{m}_{2}^{2}\lt 16{r}^{4}$ to guarantee $| {\rm{\Omega }}| \gt 0$, and promise that the RW solutions have no singular point.

Under the restriction ${c}_{1}^{2}{m}_{1}^{2}+{c}_{2}^{2}{m}_{2}^{2}\lt 16{r}^{4}$, we find that the centres of the RWs locate at $\left(-\tfrac{1}{2r}+\tfrac{{c}_{1}{m}_{2}}{8{r}^{3}},-\tfrac{{c}_{2}{m}_{1}}{16{r}^{4}}\right)$, and the amplitudes of the RWs can be expressed as

$$\begin{eqnarray}&&\left|{q}_{1,[1]}\left(-\displaystyle \frac{1}{2r}+\displaystyle \frac{{c}_{1}{m}_{2}}{8{r}^{3}},-\displaystyle \frac{{c}_{2}{m}_{1}}{16{r}^{4}}\right)\right|\\ &&\quad =\,3| {c}_{1}| \displaystyle \frac{\left|16{r}^{4}+\tfrac{1}{3}({c}_{1}^{2}{m}_{1}^{2}+{c}_{2}^{2}{m}_{2}^{2})-\tfrac{16{c}_{2}^{2}{m}_{2}}{3{c}_{1}}{r}^{2}\right|}{16{r}^{4}-({c}_{1}^{2}{m}_{1}^{2}+{c}_{2}^{2}{m}_{2}^{2})},\end{eqnarray} \tag{ 19a }$$

$$\begin{eqnarray}&&\left|{q}_{2,[1]}\left(-\displaystyle \frac{1}{2r}+\displaystyle \frac{{c}_{1}{m}_{2}}{8{r}^{3}},-\displaystyle \frac{{c}_{2}{m}_{1}}{16{r}^{4}}\right)\right|\\ &&\quad =\,3| {c}_{2}| \displaystyle \frac{\left|16{r}^{4}+\tfrac{1}{3}({c}_{1}^{2}{m}_{1}^{2}+{c}_{2}^{2}{m}_{2}^{2})-\tfrac{16{c}_{1}{m}_{2}}{3}{r}^{2}\right|}{16{r}^{4}-({c}_{1}^{2}{m}_{1}^{2}+{c}_{2}^{2}{m}_{2}^{2})}.\end{eqnarray} \tag{ 19b }$$

Hereby, with different m₁ and m₂, we obtain the eye-shaped RWs (analogous to the Peregrine soliton) in the q₁ component while the TRW pairs in the q₂ component.

Figures 2(a), (c) and (e) show the eye-shaped RWs, each of which has one hump and two valleys in the q₁ component, and the amplitudes of the RWs become higher with m₂ increasing. While in the q₂ component, the structures of the RWs are more complicated, as shown in figures 2(b), (d) and (f). With m₂ = 1, we illustrate a four-petaled RW which has two humps and two valleys in figure 2(b). With m₂ = 1.4, we illustrate a RW which has four valleys in figure 2(d). With m₂ = 1.6, we illustrate a RW which has one hump and four valleys in figure 2(f).

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In fact, figures 1(b), 2(b), (d) and (f) show that an eye-shaped RW splits into a pair of the RWs. Amplitudes of the RWs in the q₂ component first decrease then increase with the increasing m₂. While ${c}_{1}^{2}{m}_{1}^{2}+{c}_{2}^{2}{m}_{2}^{2}\lt 16{r}^{4}$, the RWs could not separate completely and be named as the TRW pairs [18].

With m₁ = 0.1, we illustrate an eye-shaped RWs in the q₁ component, while a TRW pair which have three valleys in the q₂ component, as shown in figure 3. Comparing figures 3(a)–(b) with figures 3(c)–(d), we can observe that the RWs in both of the two components extend in the z axis when m₁ = 0.4. When m₁ = −0.4 in figures 3(e) and (f), the structures of the eye-shaped RW in the q₁ component and TWR pair in the q₂ component are the same as those in figures 3(c) and (d), while the inclinations of the central peaks or valleys of the RWs are opposite. Thus we can conclude that, as $| {m}_{1}|$ decreases, the RWs will be reduced in the z dimension, and the sign of m₁ defines the inclination of the central peaks or valleys of the RWs.

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Types of the vector RWs presented in this section are shown in table 1.

Table 1.  Types of the vector RWs.

q1 component	q2 component
An eye-shaped RW	An eye-shaped RW
An eye-shaped RW	A four-petaled RW
An eye-shaped RW	A TRW with four valleys
An eye-shaped RW	A TRW with one hump and four valleys
An eye-shaped RW	A TRW with three valleys

4.2. Solutions with real spectral parameter

If ${\left({\lambda }_{1}+\tfrac{a}{2}\right)}^{2}+{c}_{1}^{2}\lt {c}_{2}^{2}$, we have known that k_α's must satisfy that ${k}_{1}^{* }{k}_{3}-{k}_{2}^{* }{k}_{4}={k}_{2}^{* }{k}_{3}+{k}_{1}^{* }{k}_{4}=0$. We can set that k₁ = 1, k₂ = k₃ = k₄ = 0 or k₂ = k₄ = ik₁ = ik₃ = i for simplicity, and the solutions will correspond to the soliton solutions or the so-called 'degenerate soliton' solutions on the non-zero background, respectively. The degenerate soliton can be regarded as the nonlinear superposition of a soliton upon itself [44].

When k₁ = 1, k₂ = k₃ = k₄ = 0, we have

$$\begin{eqnarray}{q}_{1,[1]} & = & {{ic}}_{1}{e}^{i\theta }\left[1-\displaystyle \frac{{B}_{1}}{2}-\displaystyle \frac{({c}_{1}-{c}_{2}){B}_{1}}{4{c}_{1}}\tanh ({{iX}}_{1}+\vartheta )\right.\\ & & \left.-\displaystyle \frac{({c}_{1}+{c}_{2}){B}_{1}}{4{c}_{1}}\tanh ({{iX}}_{1}+{\vartheta }^{* })\right],\end{eqnarray} \tag{ 20a }$$

$$\begin{eqnarray}{q}_{2,[1]} & = & {c}_{2}{e}^{i\theta }\left[1-\displaystyle \frac{{B}_{2}^{* }}{2}-\displaystyle \frac{({c}_{2}-{c}_{1}){B}_{2}^{* }}{4{c}_{2}}\tanh ({{iX}}_{1}+\vartheta )\right.\\ & & \left.-\displaystyle \frac{({c}_{1}+{c}_{2}){B}_{2}^{* }}{4{c}_{2}}\tanh ({{iX}}_{1}+{\vartheta }^{* })\right],\end{eqnarray} \tag{ 20b }$$

where

$$\begin{eqnarray*}&&\vartheta =\displaystyle \frac{1}{2}\mathrm{In}\left[\displaystyle \frac{({c}_{2}-{c}_{1})\tfrac{2}{{B}_{1}}-({c}_{1}+{c}_{2})\tfrac{2}{{B}_{2}}}{{{im}}_{1}+{m}_{2}}\right],\quad {B}_{b}=\displaystyle \frac{2{\mu }_{b}}{{\lambda }_{1}+\tfrac{a}{2}+{\mu }_{b}}.\end{eqnarray*}$$

With ϑ = ϑ*,

$$\begin{eqnarray}&&{q}_{1,[1]}={{ic}}_{1}{e}^{i\theta }\left[1-\displaystyle \frac{{B}_{1}}{2}-\displaystyle \frac{{B}_{1}}{2}\tanh ({{iX}}_{1}+\vartheta )\right],\,\end{eqnarray} \tag{ 21a }$$

$$\begin{eqnarray}&&{q}_{2,[1]}={c}_{2}{e}^{i\theta }\left[1-\displaystyle \frac{{B}_{2}^{* }}{2}-\displaystyle \frac{{B}_{2}^{* }}{2}\tanh ({{iX}}_{1}+\vartheta )\right].\,\end{eqnarray} \tag{ 21b }$$

As ${B}_{2}^{* }={B}_{1},{q}_{1}$ is proportional to q₂. The vector soliton is grey typed [1], because the cavity depth of the soliton $| {q}_{b}|$ is $\tfrac{{c}_{b}\sqrt{{c}_{2}^{2}-{c}_{1}^{2}-{\left({\lambda }_{1}+\tfrac{a}{2}\right)}^{2}}}{\sqrt{{c}_{2}^{2}-{c}_{1}^{2}}}$, related to λ₁. Velocity of the grey soliton is v = −(2λ₁ − a), and the centre of the soliton locates on the line iX₁ + ϑ = 0.

With $\vartheta \ne {\vartheta }^{* }$, q₁ is not proportional to q₂ and the solitons we obtain in here will possess different profiles. Figure 4(a) shows a W-shaped soliton (in the q₁ component), while figure 4(b) shows a grey soliton (in the q₂ component). Figure 4(c) shows an anti-dark soliton (in the q₁ component) which has the form of a bright soliton on a non-zero background, while figure 4(d) shows an M-shaped soliton (in the q₂ component).

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When k₂ = k₄ = ik₁ = ik₃ = i, from expressions (12) we can see that q₁ and q₂ become the combination of the exponential and polynomial functions of z and t, and they form two coincident solitons with the same spectral parameter upon a first-order RW. In figure 5(a), the wave profile consists of two lines of the anti-dark soliton upon a RW which has one hump and two valleys, while in figure 5(b), the wave profile consists of two lines of the dark soliton upon one dark RW which has two valleys.

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If ${\left({\lambda }_{1}+\tfrac{a}{2}\right)}^{2}+{c}_{1}^{2}\gt {c}_{2}^{2}$, we have known that X₁ and X₂ in expressions (12) and (16) are the real functions and k_α's must satisfy that $| {k}_{1}{| }^{2}-| {k}_{2}{| }^{2}=| {k}_{3}{| }^{2}-| {k}_{4}{| }^{2}$ $={k}_{1}{k}_{2}^{* }+{k}_{1}^{* }{k}_{2}\,=$ ${k}_{3}{k}_{4}^{* }+{k}_{3}^{* }{k}_{4}=0$. When we choose k₁ = 1, k₂ = −i, k₃ = k₄ = 0, the solutions of equations (2) are the periodic solutions, as shown in figure 6. Compared with the elliptic periodic functions, we should notice that those periodic solutions obtained here are consist of the real trigonometric functions only, and 2μ₂ is the frequency of the solutions.

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When we choose ${k}_{1}=-{{ik}}_{2}=i,{k}_{4}={{ik}}_{3}=4{{ie}}^{i\tfrac{\pi }{6}}$, the solutions is the so-called 'degenerate solutions', which can be regarded as the nonlinear superposition of a periodic solution upon itself [45], as shown in figure 7. The wave profiles consist of a line of periodic peaks upon a periodic background.

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Similar to the process of obtaining the RWs in section 3.1, we carry out the same limit ${\lambda }_{1}\to -\tfrac{a}{2}\pm i\sqrt{{c}_{1}^{2}-{c}_{2}^{2}}$ with ${c}_{1}^{2}\lt {c}_{2}^{2}$, i.e., ${\mu }_{1}=-{\mu }_{2}\to 0$, and the periodic solutions become the rational solutions as long as k_α's satisfy the conditions

$$\begin{eqnarray}&&{k}_{1}{k}_{2}^{* }+{k}_{1}^{* }{k}_{2}={k}_{3}{k}_{4}^{* }+{k}_{3}^{* }{k}_{4},\,{k}_{3}{k}_{2}^{* }+{k}_{1}^{* }{k}_{4}={k}_{1}{k}_{4}^{* }+{k}_{3}^{* }{k}_{2},\end{eqnarray} \tag{ 22a }$$

$$\begin{eqnarray}&&{k}_{3}{k}_{1}^{* }-{k}_{2}^{* }{k}_{4}={k}_{1}{k}_{3}^{* }-{k}_{4}^{* }{k}_{2},\quad | {k}_{3}{| }^{2}-| {k}_{4}{| }^{2}=| {k}_{1}{| }^{2}-| {k}_{2}{| }^{2}\end{eqnarray} \tag{ 22b }$$

in expressions (16), and the longest period arises in that limit. For instance, we choose k₁ = k₃ = 1 and k₂ = k₄ = 0 to satisfy conditions (13) and (22), so that

$$\begin{eqnarray}{f}_{1}^{{\prime} } & = & \mathop{\mathrm{lim}}\limits_{{\lambda }_{1}\to -\tfrac{a}{2}\pm i\sqrt{{c}_{1}^{2}-{c}_{2}^{2}}}{f}_{1}\\ & = & 8i({c}_{1}^{2}-{c}_{2}^{2})({k}_{1}{k}_{2}^{* }+{k}_{1}^{* }{k}_{2})[t+(2{\lambda }_{1}-a)z],\end{eqnarray} \tag{ 23a }$$

$$\begin{eqnarray}{f}_{2}^{{\prime} } & = & \mathop{\mathrm{lim}}\limits_{{\lambda }_{1}\to -\tfrac{a}{2}\pm i\sqrt{{c}_{1}^{2}-{c}_{2}^{2}}}{f}_{2}\\ & = & 8i({c}_{1}^{2}-{c}_{2}^{2})(| {k}_{1}{| }^{2}-| {k}_{2}{| }^{2})[t+(2{\lambda }_{1}-a)z].\end{eqnarray} \tag{ 23b }$$

Substituting ${f}_{1}^{{\prime} }$, ${f}_{2}^{{\prime} }$ and ${g}_{1}^{{\prime} }$, ${g}_{2}^{{\prime} }$ into BDT (4), we obtain the rational solutions, as shown in figure 8. It can be seen that the rational solutions preserve their profiles during the evolution. We should note that, although figures 4 and 8 have the similar wave profiles, analytic forms of the solutions are different (the first of which is exponential while the latter one is rational). Velocity of the rational soliton is v = −(2λ₁ − a).

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In this paper, based on Lax pair (3), we have defined one-fold BDT (4) of equations (2) with condition (5), and constructed some vector analytic solutions of the coherently-coupled NLS equations with the positive coherent coupling terms, i.e., equations (3), which are related to the optical fiber communication.

Our main results from BDT (4) are proposed as follows:

• When the spectral parameter λ₁ is complex with non-zero imaginary part, we have obtained the RWs via solutions (17) with the parameters m₁ and m₂. Figure 1 has displayed the vector generalization of the Peregrine soliton with m₁ = m₂ = 0. With m₂ increasing, figures 2(a), (c) and (e) have displayed the eye-shaped RWs in the q₁ component which have one hump and two valleys. In the q₂ component, figure 2(b) has displayed a four-petaled RW which has two humps and two valleys, figure 2(d) has displayed a RW which has four valleys, and figure 2(f) has displayed a RW which has one hump and four valleys. In fact, figures 1(b), 2(b), (d) and (f) have shown that an eye-shaped RW splits in the q₂ component with m₂ increasing. Figure 3 has shown that the RWs in the two components are reduced in the z dimension with $| {m}_{1}|$ decreasing, and the inclinations of the RWs are dependent on the sign of m₁. Summarisation of the vector RWs obtained in this paper have been given in table 1.
• When λ₁ is real, we have obtained the soliton and soliton-like rational solutions on the non-zero background. Figure 4 has shown the soliton solutions via Solutions (20). Besides, via expressions (16), figure 5–7 have shown the degenerate soliton solutions, periodic solutions, and degenerate solutions, respectively. Figure 8 has shown the soliton-like rational solutions via expressions (23).

This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11272023 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.

In this section, we will determine the matrix Ω in BDT (4) under condition (5). Two 4 × 2 matrixes ϒ and ${{\rm{S}}}_{2}{{\rm{\Upsilon }}}^{* }$ are used to denote the first and last two columns of the 4 × 4 matrix Σ₁, respectively, so that Σ₁ can be written as

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{1}=({\rm{\Upsilon }},{{\rm{S}}}_{2}{{\rm{\Upsilon }}}^{* }),\end{eqnarray} \tag{ A.1 }$$

where ${\rm{\Upsilon }}=\left(\begin{array}{cc}{\varphi }_{1} & -{\varphi }_{2}\\ {\varphi }_{2} & {\varphi }_{1}\\ {\varphi }_{3} & -{\varphi }_{4}\\ {\varphi }_{4} & {\varphi }_{3}\end{array}\right),{{\rm{S}}}_{2}=\left(\begin{array}{cccc}0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\end{array}\right)$ is a 4 × 4 matrix. Ω and Λ₁ can be expressed as

$$\begin{eqnarray}{\rm{\Omega }}=\left(\begin{array}{cc}{{\rm{F}}}_{11} & {{\rm{M}}}_{11}\\ {{\rm{G}}}_{11} & {{\rm{N}}}_{11}\end{array}\right),\quad {{\rm{\Lambda }}}_{1}=\left(\begin{array}{cc}{\lambda }_{1}{\rm{I}} & {\rm{O}}\\ {\rm{O}} & {\lambda }_{1}^{* }{\rm{I}}\end{array}\right).\end{eqnarray} \tag{ A.2 }$$

Substituting expressions (A.2) into condition (5), we have

$$\begin{eqnarray}\left(\begin{array}{cc}({\lambda }_{1}-{\lambda }_{1}^{* }){{\rm{F}}}_{11} & ({\lambda }_{1}^{* }-{\lambda }_{1}^{* }){{\rm{M}}}_{11}\\ ({\lambda }_{1}-{\lambda }_{1}){{\rm{G}}}_{11} & ({\lambda }_{1}^{* }-{\lambda }_{1}){{\rm{N}}}_{11}\end{array}\right)=\left(\begin{array}{cc}{{\rm{\Upsilon }}}^{\dagger }{{\rm{S}}}_{1}{\rm{\Upsilon }} & {{\rm{\Upsilon }}}^{\dagger }{{\rm{S}}}_{1}{{\rm{S}}}_{2}{{\rm{\Upsilon }}}^{* }\\ -{{\rm{\Upsilon }}}^{T}{{\rm{S}}}_{2}{{\rm{S}}}_{1}{\rm{\Upsilon }} & {{\rm{\Upsilon }}}^{T}{{\rm{S}}}_{1}{{\rm{\Upsilon }}}^{* }\end{array}\right),\end{eqnarray} \tag{ A.3 }$$

where F₁₁, G₁₁, M₁₁, N₁₁ are the 2 × 2 matrices, and section I is the 2 × 2 unit matrix, O is the 2 × 2 zero matrix, and S₁S₂ = S₂S₁.

However, as $({\lambda }_{1}^{* }-{\lambda }_{1}^{* }){{\rm{M}}}_{11}=({\lambda }_{1}-{\lambda }_{1}){{\rm{G}}}_{11}=0$ and ${{\rm{\Upsilon }}}^{\dagger }{{\rm{S}}}_{1}{{\rm{S}}}_{2}{{\rm{\Upsilon }}}^{* }=-{{\rm{\Upsilon }}}^{T}{{\rm{S}}}_{2}{{\rm{S}}}_{1}{\rm{\Upsilon }}=0$, we cannot calculate G₁₁ and M₁₁. The limit technique will help us to handle this problem [46]. We suppose that

$$\begin{eqnarray}{\rm{\Omega }} & = & \left(\begin{array}{cc}\displaystyle \frac{{{\rm{\Upsilon }}}^{\dagger }{{\rm{S}}}_{1}{\rm{\Upsilon }}}{{\lambda }_{1}-{\lambda }_{1}^{* }} & -\mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{1}}\displaystyle \frac{{{\rm{\Psi }}}^{\dagger }{{\rm{S}}}_{1}{{\rm{S}}}_{2}{{\rm{\Upsilon }}}^{* }}{{\lambda }^{* }-{\lambda }_{1}^{* }}-{{\rm{\Xi }}}_{1}^{* }\\ \mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{1}}\displaystyle \frac{{{\rm{\Psi }}}^{T}{{\rm{S}}}_{1}{{\rm{S}}}_{2}{\rm{\Upsilon }}}{\lambda -{\lambda }_{1}}+{{\rm{\Xi }}}_{1} & \displaystyle \frac{{{\rm{\Upsilon }}}^{T}{{\rm{S}}}_{1}{{\rm{\Upsilon }}}^{* }}{{\lambda }_{1}^{* }-{\lambda }_{1}}\end{array}\right)\\ & = & \left(\begin{array}{cc}{{\rm{F}}}_{11} & -{{\rm{G}}}_{11}^{* }-{{\rm{\Xi }}}_{1}^{* }\\ {{\rm{G}}}_{11}+{{\rm{\Xi }}}_{1} & {{\rm{F}}}_{11}^{* }\end{array}\right),\end{eqnarray} \tag{ A.4 }$$

where ${\rm{\Psi }}={\rm{\Upsilon }}{| }_{{\lambda }_{1}=\lambda }$ is a 4 × 2 matrix, and Ξ₁ is a 2 × 2 complex constant matrix.

G₁₁ can be expressed as

$$\begin{eqnarray}{{\rm{G}}}_{11}=\mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{1}}\displaystyle \frac{{{\rm{\Psi }}}^{T}{{\rm{S}}}_{1}{{\rm{S}}}_{2}{\rm{\Upsilon }}}{\lambda -{\lambda }_{1}}=\left(\begin{array}{cc}{g}_{1} & {g}_{2}\\ {g}_{2} & -{g}_{1}\end{array}\right),\end{eqnarray} \tag{ A.5 }$$

where g₁ and g₂ are the complex functions of z and t.

As Ξ₁ is introduced via the limitation in expression (A.5), we set Ξ₁ similar to G₁₁ as

$$\begin{eqnarray}{{\rm{\Xi }}}_{1}=\left(\begin{array}{cc}{m}_{1} & {m}_{2}\\ {m}_{2} & -{m}_{1}\end{array}\right),\end{eqnarray} \tag{ A.6 }$$

where m₁ and m₂ are the complex constants.

F₁₁ can be expressed as

$$\begin{eqnarray}{{\rm{F}}}_{11}=\displaystyle \frac{{{\rm{\Upsilon }}}^{\dagger }{{\rm{S}}}_{1}{\rm{\Upsilon }}}{{\lambda }_{1}-{\lambda }_{1}^{* }}=\left(\begin{array}{cc}{f}_{1} & {f}_{2}\\ {f}_{2} & -{f}_{1}\end{array}\right),\end{eqnarray} \tag{ A.7 }$$

where ${f}_{1}=\tfrac{1}{{\lambda }_{1}-{\lambda }_{1}^{* }}({\varphi }_{1}{\varphi }_{2}^{* }+{\varphi }_{1}^{* }{\varphi }_{2}+{\varphi }_{3}{\varphi }_{4}^{* }+{\varphi }_{3}^{* }{\varphi }_{4})$, ${f}_{2}\,=\tfrac{1}{{\lambda }_{1}-{\lambda }_{1}^{* }}(| {\varphi }_{1}{| }^{2}-| {\varphi }_{2}{| }^{2}+| {\varphi }_{3}{| }^{2}-| {\varphi }_{4}{| }^{2})$, and ${f}_{1}^{* }\,=\,-{f}_{1},{f}_{2}^{* }\,=-{f}_{2}$. Thus, ${{\rm{F}}}_{11}^{* }$ can be expressed as

$$\begin{eqnarray}{{\rm{F}}}_{11}^{* }=\left(\begin{array}{cc}-{f}_{1} & -{f}_{2}\\ -{f}_{2} & {f}_{1}\end{array}\right).\end{eqnarray} \tag{ A.8 }$$

If the spectral parameter ${\lambda }_{1}^{* }={\lambda }_{1}$, we have $({\lambda }_{1}-{\lambda }_{1}^{* }){{\rm{F}}}_{11}\,=({\lambda }_{1}^{* }-{\lambda }_{1}){{\rm{F}}}_{11}^{* }=0$. To keep the non-singularity of Ω, we require ϒ to satisfy ${{\rm{\Upsilon }}}^{\dagger }{{\rm{S}}}_{1}{\rm{\Upsilon }}=0$. In that case, we can suppose that

$$\begin{eqnarray}{\rm{\Omega }}=\mathop{\mathrm{lim}}\limits_{\lambda \to {\lambda }_{1}}\displaystyle \frac{1}{\lambda -{\lambda }_{1}}\left(\begin{array}{cc}-{{\rm{\Psi }}}^{\dagger }{{\rm{S}}}_{1}{\rm{\Upsilon }} & -{{\rm{\Psi }}}^{\dagger }{{\rm{S}}}_{1}{{\rm{S}}}_{2}{{\rm{\Upsilon }}}^{* }-{{\rm{\Xi }}}_{1}^{* }\\ {{\rm{\Psi }}}^{T}{{\rm{S}}}_{1}{{\rm{S}}}_{2}{\rm{\Upsilon }}+{{\rm{\Xi }}}_{1} & -{{\rm{\Psi }}}^{T}{{\rm{S}}}_{1}{{\rm{\Upsilon }}}^{* }\end{array}\right).\end{eqnarray} \tag{ A.9 }$$

For simplicity, m₁ and m₂ are regarded as the real constants.