© Wuhan University 2024

0 Introduction

Soliton is an analytic and localized wave-packet that scatters elastically^[1], whose stability is ensured by the conservation laws caused by integrability^[2]. Equations permitting soliton solutions are nonlinear and dispersive^[3], describing a wide range of nonlinear models ^[1], and they usually form hierarchies^[4]. Optical soliton theory attracts great attention due to its practical application in field of optical-fibre communication^[5,6]. The optical fibre permits the existence of dark and bright solitons, as well as vector solitons with two perpendicular polarizations^[7,8].

Letting $u\left(x,t\right)$ be the electric field of a short optical pulse, t the normalized propagation distance, and x the normalized time, then the modified nonlinear Schrödinger equation (MNLSE) is as follows:

$\mathrm{i}{\partial }_{t}u+{\partial }_x^{\mathrm{2}}u+\mathrm{i}{\partial }_x(|u{|}^{\mathrm{2}}u)+\mathrm{2}\gamma |u{|}^{\mathrm{2}}u=\mathrm{0}$(1)

which describes the propagation of a pulse in a long waveguide, and might be of use in the optical communication, in which $\gamma$ quantifies the nonlinearity of the refractive index. It was first proposed by Tzoar and Jain in 1981^[9] and Lashkin in 2007^[10]. Different from the nonlinear Schrödinger equation (NLSE), MNLSE contains two nonlinear correction terms, which are called the self-phase-modulation and self-steepen effect^[9], respectively, and proved to be essential to describe the distortion of a short pulse propagating in optical fibre^[11,12].

By a gauge-like transformation^[13] or a simple setting of the parameter $\gamma$ to zero, the MNLSE becomes the derivative Schrödinger equation (DNLSE):

$\mathrm{i}{\partial }_{t}v+{\partial }_x^{\mathrm{2}}v+\mathrm{i}{\partial }_x(|v{|}^{\mathrm{2}}v)=\mathrm{0}$(2)

where$v(x,t)$ stands for the normalized amplitude of the wave envelope of field^[10]. The MNLSE can be regarded as the DNLSE with perturbation. The DNLSE can be also used to describe the evolution of small-amplitude nonlinear Alfvén waves propagating quasi-parallelly with respect to the background of magnetic field^[14],whose zero-curvature condition leads to various equations such as a coupled Chen-Lee-Liu equation with given appropriate parameters^[2]. The connection between the DNLS and the coupled Chen-Lee-Liu equation makes it possible to transfer the solution of the former to that of the latter.

The case for nonvanishing boundary conditions (NVBC), apart from its correlation to physical systems in reality, also supports the existence of more varieties of solitons compared with the case for vanishing boundary conditions (VBC)^[13,15], but it is more complicated to tackle. For the DNLSE with NVBC, the inverse scattering transform (IST) encountered divergent integrals and the problem of sheet determination of multiple-value functions^[16], Bäcklund transformation obtains solution in terms of determinants^[17,18], and they both need modification to adapt for the NVBC case. The Hirota method can be applied to the NVBC case directly and give rise to explicit solution^[3]. Compared with the other methods, it is more straightforward and has fewer requirements.

High-order solitons describe the interaction among N solitons resulting from N-th multiple-poles instead of N simple-poles of the reflection coefficient in the IST method^[19],which leads to a particular chirp. The double-pole case of the NLS has been presented in the original work of Zakharov and Shabat^[20] but without much analysis, and then the N-poles soliton was studied in detail by dressing method^[21]. The high-order solitons are referred to as the simple pole solution (SPS) when the poles are the first order and as the double pole solution (DPS) when the poles are the second order^[22]. The DPS is also called a positon^[23,24]. Some equations prohibit the existence of high-order soliton since they only permit simple poles, such as Korteweg-de Vries equation^[19]. The Darboux transformation^{[19, 25-28]} and the Riemann-Hilbert (RH) method^[29,30] gives a Wronskian representation of the high-order solitons, while the Hirota method gives an explicit form utilizing a simple and quick limit of the SPS^[21,23,31]. The above cases mainly focus on the mKdV-sine Gordon equation under VBC. Ref. [19] investigated the high-order solitons of the DNLS via the Darboux method.

The N SPS of the MNLSE under VBC had been obtained via the IST method^[32], and the rogue wave solution under NVBC via the Hirota method ^[33]. The one SPS of the MNLSE was obtained under NVBC through the Hirota method in our previous paper^[33]. The DNLSE had also been extensively investigated. Its N SPS under the VBC and breather-type soliton solution under the NVBC had been studied using various method ^{[10,16,34-39]}. The high-order soliton of the DNLSE under VBC^[19], the high-order rational soliton under NVBC^[19], and the DPS under VBC^[27] had been obtained using the Darboux transformation.

In this paper, we derived the DPS of the MNLSE/DNLSE under standing-wave boundary condition in an explicit form via Hirota method. In Section 1, there is a brief introduction to Hirota method and the MNLSE is bilinearized. In Section 2, the 1-SPS and the 2-SPS of the MNLSE are obtained and that of the DNLSE are also obtained by setting $\gamma \to \mathrm{0}$. In Section 3, the DPS of the MNLSE is constructed as limit of two SPS with opposite signs and equal amplitudes. The DPS of the DNLSE is obtained with $\gamma \to \mathrm{0}$. In Section 4, some concluding remarks are made, and the influence of the perturbation of the DNLSE and the background oscillation strength is also discussed in detail.

1 Hirota Method and Bilinearization of MNLSE/DNLSE

We first introduce the bilinear derivative operator D^[3]

$\begin{array}{l}{\mathrm{D}}_t^m{\mathrm{D}}_x^{n}f(x,t)•g(x,t)={(\partial /\partial t-\partial /\partial t^{\text{'}})}^m\\ ·{(\partial /\partial x-\partial /\partial x^{\text{'}})}^{n}f(x,t)g(x^{\text{'}},t^{\text{'}}){|}_{t^{\text{'}}=t,x^{\text{'}}=x}\end{array}$(3)

or in a more convenient way, by an exponential identity:

$\begin{array}{l}\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{ϵ}{\mathrm{D}}_z\right)a\left(z\right)•b\left(z\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathrm{ϵ}{\partial }_y\right)a\left(z+y\right)b\left(z-y\right){|}_{y=\mathrm{0}}\\ =a\left(z+\mathrm{ϵ}\right)b\left(z-\mathrm{ϵ}\right)\end{array}$(4)

which resembles a translation operator. In above identity and (3), ($•$) between two functions means an ordered product, $\mathrm{ϵ}$ is a parameter and y is an auxiliary parameter. The bilinear derivative operator D has several important properties as follows:

${\mathrm{D}}_t^m{\mathrm{D}}_x^{n}f•g={\left(-\mathrm{1}\right)}^{m+n}{\mathrm{D}}_t^m{\mathrm{D}}_x^{n}g•f$(5)

${\mathrm{D}}_t^m{\mathrm{D}}_x^n{\mathrm{e}}^{{\eta }_{\mathrm{1}}}•{\mathrm{e}}^{{\eta }_{\mathrm{2}}}=({\omega }_{\mathrm{1}}-{\omega }_{\mathrm{2}}{)}^m(k_{\mathrm{1}}-k_{\mathrm{2}}{)}^n{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}}$(6)

where ${\eta }_j={\omega }_{j}t+k_{j}x+{\eta }_j^{\mathrm{0}},\left(j=\mathrm{1,2}\right)$.

Solutions of DNLSE and MNLSE have following typical form ^{[10,16,17,33,35,38,40]}

$u=g\overline{f}/f^{\mathrm{2}}=\left(g/f\right)\left(\overline{f}/f\right)$(7)

The Hirota method relies on some typical forms which are usually identical for equations in a same hierarchy^[3]. To bilinearize an equation is to transform the equation into the form that only contains the D operator acting upon the transformed variables without the ordinary/partial differential operators. u is performed Hirota transform upon the product of $g/f$ and $\overline{f}/f$, which are of similar structures. Assume $w=\alpha /\beta$, Then the bilinearized MNLSE can be obtained from the bilinearized partial derivatives of $w$, which are contained in the Taylor expansion of $\mathrm{e}\mathrm{x}\mathrm{p}\left(\delta \partial /\partial z\right)\left(\alpha /\beta \right)$. Since

$\begin{array}{l}\mathrm{e}\mathrm{x}\mathrm{p}\left(\delta \partial /\partial z\right)\left(\alpha /\beta \right)=\frac{\alpha \left(z+\delta \right)}{\beta \left(z+\delta \right)}\\ =\alpha \left(z+\delta \right)\beta \left(z-\delta \right)/\left[\beta \left(z+\delta \right)\beta \left(z-\delta \right)\right]\\ =\left(\mathrm{e}\mathrm{x}\mathrm{p}\left(\delta {\mathrm{D}}_z\right)\alpha •\beta \right){\left(\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\delta {\mathrm{D}}_z\right)\beta •\beta \right)}^{-\mathrm{1}}\end{array}$(8)

From the Taylor expansion of the final result of (8), some useful formulae for following bilinearized partial derivatives of w are obtained:

$\{\begin{array}{l}\left(\partial /\partial z\right)\left(\alpha /\beta \right)=\left({\mathrm{D}}_z\alpha •\beta \right)/{\beta }^{\mathrm{2}}\\ \left({\partial }^{\mathrm{2}}/\partial z^{\mathrm{2}}\right)\left(\alpha /\beta \right)=\left({\mathrm{D}}_z^{\mathrm{2}}\alpha •\beta \right)/{\beta }^{\mathrm{2}}-\left(\alpha /\beta \right)\left({\mathrm{D}}_z^{\mathrm{2}}\beta •\beta \right)/{\beta }^{\mathrm{2}}\end{array}$(9)

Insert (9) and (7) into the MNLSE (1), we have:

$\begin{array}{l}\mathrm{0}=\left(\mathrm{i}f\overline{f}/f^{\mathrm{4}}\right){\mathrm{D}}_{t}g•f+\left(\mathrm{i}fg/f^{\mathrm{4}}\right){\mathrm{D}}_t\overline{f}•f+\left(f\overline{f}/f^{\mathrm{4}}\right){\mathrm{D}}_x^{\mathrm{2}}g•f\\ -\left(\mathrm{2}g\overline{f}/f^{\mathrm{4}}\right){\mathrm{D}}_x^{\mathrm{2}}f•f+\left(\mathrm{2}/f^{\mathrm{4}}\right)\left({\mathrm{D}}_{x}g•f\right)\left({\mathrm{D}}_x\overline{f}•f\right)\\ +\left(gf/f^{\mathrm{4}}\right){\mathrm{D}}_x^{\mathrm{2}}\overline{f}•f+\left(\mathrm{2}\mathrm{i}g\overline{g}/f^{\mathrm{4}}\right){\mathrm{D}}_{x}g•f+\left(\mathrm{i}{g}^{\mathrm{2}}/f^{\mathrm{4}}\right){\mathrm{D}}_x\overline{g}•f\\ +\mathrm{2}\gamma g^{\mathrm{2}}\overline{g}/f^{\mathrm{3}}\end{array}$(10)

After algebraic manipulation, Eq.(10) becomes:

$\begin{array}{l}\mathrm{0}=f\overline{f}\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}\right)g•f+\mathrm{2}\gamma g^{\mathrm{2}}\overline{g}f-fg\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}\right)f•\overline{f}\\ +f^{-\mathrm{2}}\left\{{\mathrm{D}}_x{f}^{\mathrm{3}}•\left[g\left(\mathrm{2}{\mathrm{D}}_{x}f•\overline{f}-\mathrm{i}g\overline{g}\right)\right]\right\}\end{array}$(11)

By introducing an arbitrary function $\lambda$, the above equation is decomposed into following bilinear forms ^[3]:

$\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda \right)g•f=\mathrm{0}$(12)

$\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda \right)f•\overline{f}=\mathrm{2}\gamma g\overline{g}$(13)

${\mathrm{D}}_{x}f•\overline{f}=\left(\mathrm{i}/\mathrm{2}\right)g\overline{g}$(14)

From (13), we have

${\left|u\right|}^{\mathrm{2}}=g\overline{g}/f\overline{f}=\left(\mathrm{1}/\mathrm{2}\gamma \right)\left\{\left[\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}\right)f•\overline{f}\right]/f\overline{f}-\lambda \right\}$(15)

revealing that $\lambda$ depends on f, g, the sign of $\gamma$, and the boundary condition of $u\left(x,t\right)$.

2 1-SPS and 2-SPS of the MNLSE/DNLSE

Expand functions f and g in series of perturbation parameter $ϵ$ are

$f={\displaystyle \sum _{i=\mathrm{0}}^{\mathrm{\infty }}}{ϵ}^i{f}^{(i)},g={\displaystyle \sum _{i=\mathrm{0}}^{\mathrm{\infty }}}{ϵ}^i{g}^{(i)}$(16)

The j-th soliton would be in the form of

$f={\displaystyle \sum _{i=\mathrm{0}}^j}{ϵ}^i{f}^{(i)},g={\displaystyle \sum _{i=\mathrm{0}}^j}{ϵ}^i{g}^{(i)}$(17)

Inserting (16) into (12)-(14) leads to

$\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda \right)\left({\displaystyle \sum _{j=\mathrm{0}}^n}{g}^{(j)}•f^{(n-j)}\right)=\mathrm{0}$(18)

$\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)\left({\displaystyle \sum _{j=\mathrm{0}}^n}{f}^{(j)}•\overline{f^{(n-j)}}\right)=\mathrm{0}$(19)

${\mathrm{D}}_x\left({\displaystyle \sum _{j=\mathrm{0}}^n}{f}^{(j)}•\overline{f^{(n-j)}}\right)=\left(\mathrm{i}/\mathrm{2}\right)\left({\displaystyle \sum _{j=\mathrm{0}}^n}{g}^{(j)}•\overline{g^{(n-j)}}\right)$(20)

Set

$f^{(i)}=\mathrm{0},g^{(i)}=\mathrm{0},i>\mathrm{1}$(21)

so that only the terms related to the 1-SPS are remained in the series (16). Normally (21) could be satisfied by a deliberate choice of $f^{(\mathrm{0})}$ and $g^{(\mathrm{0})}$, which are hard to attain here due to extra nonlinearity in the MNLSE. Therefore, we set the higher perturbation of (16) to zero in the first place. Inserting (21) into (18)-(20), and after slight adjustment, we get

$\{\begin{array}{l}\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)f^{(\mathrm{0})}•\overline{f^{(\mathrm{0})}}=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)\left(f^{(\mathrm{0})}•\overline{f^{(\mathrm{1})}}+f^{(\mathrm{1})}•\overline{f^{(\mathrm{0})}}\right)=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)f^{(\mathrm{1})}•\overline{f^{(\mathrm{1})}}=\mathrm{0}\end{array}$(22)

$\{\begin{array}{l}\left(\mathrm{i}/\mathrm{2}\right)g^{(\mathrm{0})}\overline{g^{(\mathrm{0})}}={\mathrm{D}}_x{f}^{(\mathrm{0})}•\overline{f^{(\mathrm{0})}}\\ \left(\mathrm{i}/\mathrm{2}\right)\left(g^{(\mathrm{0})}\overline{g^{(\mathrm{1})}}+g^{(\mathrm{1})}\overline{g^{(\mathrm{0})}}\right)={\mathrm{D}}_x\left(f^{(\mathrm{0})}•\overline{f^{(\mathrm{1})}}+f^{(\mathrm{1})}•\overline{f^{(\mathrm{0})}}\right)\\ \left(\mathrm{i}/\mathrm{2}\right)g^{(\mathrm{1})}\overline{g^{(\mathrm{1})}}={\mathrm{D}}_x{f}^{(\mathrm{1})}•\overline{f^{(\mathrm{1})}}\end{array}$(23)

$\{\begin{array}{l}\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda \right)g^{(\mathrm{0})}•f^{(\mathrm{0})}=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda \right)\left(g^{(\mathrm{0})}•f^{(\mathrm{1})}+g^{(\mathrm{1})}•f^{(\mathrm{0})}\right)=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda \right)g^{(\mathrm{1})}•f^{(\mathrm{1})}=\mathrm{0}\end{array}$(24)

From our previous paper^[41], under the standing wave boundary condition:

$u\to \rho {\mathrm{e}}^{\mathrm{i}\left(b_{\mathrm{0}}-\mathrm{3}{k}_{\mathrm{0}}\right)x}{\mathrm{e}}^{\mathrm{i}{\varphi }_{\mathrm{0}}}$(25)

when $x\to \pm \mathrm{\infty }$ or $t\to \pm \mathrm{\infty }$, in which ${\varphi }_{\mathrm{0}}$ is a phase shift, solution of (22-24) is

$f^{(\mathrm{0})}={\mathrm{e}}^{\mathrm{i}{k}_{\mathrm{0}}x},f^{(\mathrm{1})}=f^{(\mathrm{0})}{\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{\eta },g^{(\mathrm{0})}=\rho {\mathrm{e}}^{\mathrm{i}{b}_{\mathrm{0}}x},g^{(\mathrm{1})}=g^{(\mathrm{0})}{\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{\eta }$(26)

where $\eta =\tau t+\kappa x+{\eta }^{(\mathrm{0})}$ and

$\{\begin{array}{l}{k}_{\mathrm{0}}=\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{2}}\\ b_{\mathrm{0}}=\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{2}}\pm \sqrt[]{\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{4}}+\mathrm{2}\gamma {\left|\rho \right|}^{\mathrm{2}}}\\ {\alpha }_{\mathrm{1}}^{(\mathrm{0})}={\xi }_{\mathrm{1}}^{(\mathrm{0})}+\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{c}\mathrm{o}{\mathrm{s}}^{\mathrm{2}}{\xi }_{\mathrm{1}}^{(\mathrm{0})}+\left(\mathrm{\Delta }-\mathrm{1}\right)\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}{\xi }_{\mathrm{1}}^{(\mathrm{0})}\right)\\ \mathrm{\Delta }=\mathrm{2}\left({\left|\rho \right|}^{\mathrm{2}}\mp \sqrt[]{{\left|\rho \right|}^{\mathrm{4}}+\mathrm{8}\gamma {\left|\rho \right|}^{\mathrm{2}}}+\mathrm{4}\gamma \right)/{\left|\rho \right|}^{\mathrm{2}}\\ \kappa =\left(\mathrm{1}/\mathrm{2}\right){\left|\rho \right|}^{\mathrm{2}}\left(\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_{\mathrm{1}}^{(\mathrm{0})}-\mathrm{c}\mathrm{o}\mathrm{s}{\xi }_{\mathrm{1}}^{(\mathrm{0})}\right)/\mathrm{s}\mathrm{i}\mathrm{n}{\xi }_{\mathrm{1}}^{(\mathrm{0})}\\ \tau =-\mathrm{4}\left(\gamma +k_{\mathrm{0}}\right)\kappa +\mathrm{c}\mathrm{o}\mathrm{t}{\xi }_{\mathrm{1}}^{(\mathrm{0})}{\kappa }^{\mathrm{2}}\end{array}$(27)

The unperturbed terms $f^{(\mathrm{0})},g^{(\mathrm{0})}$ correspond to the influence of the boundary condition, and the first-order perturbation terms $f^{(\mathrm{1})},g^{(\mathrm{1})}$ correspond to the one-soliton case. Hence the 1-SPS of the MNLSE is

$u=\rho {\mathrm{e}}^{\mathrm{i}\left(b_{\mathrm{0}}-\mathrm{3}{k}_{\mathrm{0}}\right)x}\left(\mathrm{1}+{\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{\eta }\right)\left(\mathrm{1}+{\mathrm{e}}^{-\mathrm{i}{\xi }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{\eta }\right)/{\left(\mathrm{1}+{\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{\eta }\right)}^{\mathrm{2}}$(28)

with $\eta =\tau t+\kappa x+{\eta }^{(\mathrm{0})}$ and (27). Here $\rho ,\gamma ,{\xi }_{\mathrm{1}}^{(\mathrm{0})}$ are viewed as variables on which the other parameters, $k_{\mathrm{0}},b_{\mathrm{0}},{\alpha }_{\mathrm{1}}^{(\mathrm{0})},\kappa ,\tau$, depend.

Under the limit of $\gamma \to \mathrm{0}$, the solution of MNLSE degenerates into that of the DNLSE, and (28) results in the 1-SPS of the DNLSE:

$u=\rho \left(\mathrm{1}+{\mathrm{e}}^{\mathrm{i}\mathrm{3}{\xi }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{\eta }\right)\left(\mathrm{1}+{\mathrm{e}}^{-\mathrm{i}{\xi }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{\eta }\right)/{\left(\mathrm{1}+{\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{\eta }\right)}^{\mathrm{2}}$(29)

with $\eta =\tau t+\kappa x+{\eta }^{(\mathrm{0})}$ and

$\begin{array}{l}{k}_{\mathrm{0}}=\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{2}},b_{\mathrm{0}}=\left(\mathrm{3}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{2}},{\alpha }_{\mathrm{1}}^{(\mathrm{0})}=\mathrm{3}{\xi }_{\mathrm{1}}^{(\mathrm{0})};\\ \kappa =-{\left|\rho \right|}^{\mathrm{2}}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{2}{\xi }_{\mathrm{1}}^{(\mathrm{0})},\tau ={\left|\rho \right|}^{\mathrm{4}}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{2}{\xi }_{\mathrm{1}}^{(\mathrm{0})}\left(\mathrm{1}+\mathrm{2}\mathrm{c}\mathrm{o}{\mathrm{s}}^{\mathrm{2}}{\xi }_{\mathrm{1}}^{(\mathrm{0})}\right)\end{array}$(30)

As $\gamma \to \mathrm{0}$, the standing-wave NVBC of the MNLSE (25) becomes a constant NVBC of the DNLSE

$u\to \rho {\mathrm{e}}^{\mathrm{i}{\varphi }_{\mathrm{0}}}$(31)

when $x\to \pm \mathrm{\infty }$ or $t\to \pm \mathrm{\infty }$, where ${\varphi }_{\mathrm{0}}$ is a phase shift. Next, we derive the 2-SPS of the MNLSE/DNLSE. Set

$f^{(i)}=\mathrm{0},g^{(i)}=\mathrm{0},\left(i>\mathrm{2}\right)$(32)

so that only terms involving in the first and the second soliton are remained in the series (16). Similar to the SPS case, the unperturbed states $f^{(\mathrm{0})},g^{(\mathrm{0})}$ correspond to the influence of the boundary condition, the first-order perturbation $f^{(\mathrm{1})},g^{(\mathrm{1})}$ correspond to two solitons and the second-order perturbation $f^{(\mathrm{2})},g^{(\mathrm{2})}$ correspond to their interaction. Inserting (32) into (18-20), and after slight alteration, they become

$\{\begin{array}{l}\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda \right)g^{(\mathrm{0})}\cdot f^{(\mathrm{0})}=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda \right)\left(g^{(\mathrm{0})}\cdot f^{(\mathrm{1})}+g^{(\mathrm{1})}\cdot f^{(\mathrm{0})}\right)=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda \right)\left(g^{(\mathrm{0})}\cdot f^{(\mathrm{2})}+g^{(\mathrm{1})}\cdot f^{(\mathrm{1})}+g^{(\mathrm{2})}\cdot f^{(\mathrm{0})}\right)=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda \right)\left(g^{(\mathrm{1})}\cdot f^{(\mathrm{2})}+g^{(\mathrm{2})}\cdot f^{(\mathrm{1})}\right)=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda \right)g^{(\mathrm{2})}\cdot f^{(\mathrm{2})}=\mathrm{0}\end{array}$(33)

$\{\begin{array}{l}\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)f^{(\mathrm{0})}\cdot \overline{f^{(\mathrm{0})}}=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)\left(f^{(\mathrm{0})}\cdot \overline{f^{(\mathrm{1})}}+f^{(\mathrm{1})}\cdot \overline{f^{(\mathrm{0})}}\right)=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)\left(f^{(\mathrm{0})}\cdot \overline{f^{(\mathrm{2})}}+f^{(\mathrm{1})}\cdot \overline{f^{(\mathrm{1})}}+f^{(\mathrm{2})}\cdot \overline{f^{(\mathrm{0})}}\right)=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)\left(f^{(\mathrm{1})}\cdot \overline{f^{(\mathrm{2})}}+f^{(\mathrm{2})}\cdot \overline{f^{(\mathrm{1})}}\right)=\mathrm{0}\\ \left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)f^{(\mathrm{2})}\cdot \overline{f^{(\mathrm{2})}}=\mathrm{0}\end{array}$(34)

$\{\begin{array}{l}{\mathrm{D}}_x{f}^{(\mathrm{0})}\cdot \overline{f^{(\mathrm{0})}}=\left(\mathrm{i}/\mathrm{2}\right)g^{(\mathrm{0})}\overline{g^{(\mathrm{0})}}\\ {\mathrm{D}}_x\left(f^{(\mathrm{0})}\cdot \overline{f^{(\mathrm{1})}}+f^{(\mathrm{1})}\cdot \overline{f^{(\mathrm{0})}}\right)=\left(\mathrm{i}/\mathrm{2}\right)\left(g^{(\mathrm{0})}\overline{g^{(\mathrm{1})}}+g^{(\mathrm{1})}\overline{g^{(\mathrm{0})}}\right)\\ {\mathrm{D}}_x\left(f^{(\mathrm{0})}\cdot \overline{f^{(\mathrm{2})}}+f^{(\mathrm{1})}\cdot \overline{f^{(\mathrm{1})}}+f^{(\mathrm{2})}\cdot \overline{f^{(\mathrm{0})}}\right)\\ =\left(\mathrm{i}/\mathrm{2}\right)\left(g^{(\mathrm{0})}\overline{g^{(\mathrm{2})}}+g^{(\mathrm{1})}\overline{g^{(\mathrm{1})}}+g^{(\mathrm{2})}\overline{g^{(\mathrm{0})}}\right)\\ {\mathrm{D}}_x\left(f^{(\mathrm{1})}\cdot \overline{f^{(\mathrm{2})}}+f^{(\mathrm{2})}\cdot \overline{f^{(\mathrm{1})}}\right)=\left(\mathrm{i}/\mathrm{2}\right)\left(g^{(\mathrm{1})}\overline{g^{(\mathrm{2})}}+g^{(\mathrm{2})}\overline{g^{(\mathrm{1})}}\right)\\ {\mathrm{D}}_x{f}^{(\mathrm{2})}\cdot \overline{f^{(\mathrm{2})}}=\left(\mathrm{i}/\mathrm{2}\right)g^{(\mathrm{2})}\overline{g^{(\mathrm{2})}}\end{array}$(35)

We choose the unperturbed state of f and g as

$f^{(\mathrm{0})}={\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{0}}}={\mathrm{e}}^{\mathrm{i}{k}_{\mathrm{0}}x};g^{(\mathrm{0})}=\rho {\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{0}}}=\rho {\mathrm{e}}^{\mathrm{i}{b}_{\mathrm{0}}x}$(36)

which is similar to the case of SPS of the MNLSE, then from the first equations of Eqs.(33)-(35), which are the same as the 1-SPS case, the values of $k_{\mathrm{0}},b_{\mathrm{0}},\lambda$ can be gotten as follows:

$\begin{array}{l}{k}_{\mathrm{0}}=\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{2}};b_{\mathrm{0}}=\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{2}}\pm \sqrt[]{\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{4}}+\mathrm{2}\gamma {\left|\rho \right|}^{\mathrm{2}}}\\ \lambda =-\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{2}}-\mathrm{2}\gamma {\left|\rho \right|}^{\mathrm{2}}\end{array}$(37)

$f^{(\mathrm{1})},g^{(\mathrm{1})}$ should consist of terms involving in two solitons, and we write them as:

$\{\begin{array}{l}{f}^{(\mathrm{1})}=a_{\mathrm{1}}{f}^{(\mathrm{0})}{\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}}+a_{\mathrm{2}}{f}^{(\mathrm{0})}{\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{2}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}}\equiv a_{\mathrm{1}}{f}_{\mathrm{1}}^{(\mathrm{1})}+a_{\mathrm{2}}{f}_{\mathrm{2}}^{(\mathrm{1})}\\ g^{(\mathrm{1})}=b_{\mathrm{1}}{g}^{(\mathrm{0})}{\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}}+b_{\mathrm{2}}{g}^{(\mathrm{0})}{\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{2}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}}\equiv b_{\mathrm{1}}{g}_{\mathrm{1}}^{(\mathrm{1})}+b_{\mathrm{2}}{g}_{\mathrm{2}}^{(\mathrm{1})}\end{array}$(38)

If $f_j^{(m)},g_j^{(m)}\left(m,j=\mathrm{1,2}\right)$ in (38) satisfy the same equations as that in 1-SPS case, i.e., (22)-(24), then the second equation of (33) yields

$\begin{array}{l}\mathrm{0}=\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)\left(\overline{a_{\mathrm{1}}}{f}^{(\mathrm{0})}\cdot \overline{{f_{\mathrm{1}}}^{(\mathrm{1})}}+a_{\mathrm{1}}{f_{\mathrm{1}}}^{(\mathrm{1})}\cdot \overline{f^{(\mathrm{0})}}\right)\\ +\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)\left(\overline{a_{\mathrm{2}}}{f}^{(\mathrm{0})}\cdot \overline{{f_{\mathrm{2}}}^{(\mathrm{1})}}+a_{\mathrm{2}}{f_{\mathrm{2}}}^{(\mathrm{1})}\cdot \overline{f^{(\mathrm{0})}}\right)\end{array}$(39)

By simple comparison with the second equation of (22), Eq.(39) is satisfied when

$\overline{a_{\mathrm{1}}}=a_{\mathrm{1}},\overline{a_{\mathrm{2}}}=a_{\mathrm{2}}$(40)

which means $a_{\mathrm{1}},a_{\mathrm{2}}$ are real. Similarly, from the second equation of (34), we have

$a_{\mathrm{1}}=b_{\mathrm{1}},a_{\mathrm{2}}=b_{\mathrm{2}}$(41)

Eqs. (40) and (41) render the second equation of (35) satisfied automatically.

We can express $f^{(\mathrm{2})},g^{(\mathrm{2})}$ involving in the interaction between two solitons as

$\{\begin{array}{l}{f}^{(\mathrm{2})}=f^{(\mathrm{0})}{a}_{\mathrm{1}}{a}_{\mathrm{2}}A{\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{1}}^{(\mathrm{0})}+\mathrm{i}{\xi }_{\mathrm{2}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}}\\ g^{(\mathrm{2})}=g^{(\mathrm{0})}{b}_{\mathrm{1}}{b}_{\mathrm{2}}B{\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{1}}^{(\mathrm{0})}+\mathrm{i}{\alpha }_{\mathrm{2}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}}\end{array}$(42)

where ${\eta }_j={\tau }_{j}t+{\kappa }_{j}x+{\eta }_j^{(\mathrm{0})}$,$j=\mathrm{1},\mathrm{2}$, A, B correspond to the strength of interaction. Henceforth the fifth equation of (33) is satisfied automatically:

$\begin{array}{l}\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)f^{(\mathrm{2})}•\overline{f^{(\mathrm{2})}}\\ ={\left|a_{\mathrm{1}}{a}_{\mathrm{2}}A\right|}^{\mathrm{2}}\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)f^{(\mathrm{0})}•\overline{f^{(\mathrm{0})}}=\mathrm{0}\end{array}$(43)

Similarly, the fifth equation of (34) is satisfied automatically and the fifth equation of (35) is satisfied when

$A^{\mathrm{2}}={\left|B\right|}^{\mathrm{2}}$(44)

The fourth equation of (33) yields

$\begin{array}{l}\mathrm{0}=\left(\mathrm{i}{\mathrm{D}}_t+{\mathrm{D}}_x^{\mathrm{2}}-\lambda +\mathrm{4}\mathrm{i}\gamma {\mathrm{D}}_x\right)\left(f^{(\mathrm{1})}\cdot \overline{f^{(\mathrm{2})}}+f^{(\mathrm{2})}\cdot \overline{f^{(\mathrm{1})}}\right)\\ =\left(\mathrm{i}{\tau }_{\mathrm{2}}+\mathrm{4}\mathrm{i}{k}_{\mathrm{0}}{\kappa }_{\mathrm{2}}+{\kappa }_{\mathrm{2}}^{\mathrm{2}}+\mathrm{4}\mathrm{i}\gamma {\kappa }_{\mathrm{2}}\right){\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{2}}^{(\mathrm{0})}}\left(A-\overline{A}\right){\mathrm{e}}^{{\eta }_{\mathrm{1}}}\\ +\left(\mathrm{i}{\tau }_{\mathrm{1}}+\mathrm{4}\mathrm{i}{k}_{\mathrm{0}}{\kappa }_{\mathrm{1}}+{\kappa }_{\mathrm{1}}^{\mathrm{2}}+\mathrm{4}\mathrm{i}\gamma {\kappa }_{\mathrm{1}}\right){\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{1}}^{(\mathrm{0})}}\left(A-\overline{A}\right){\mathrm{e}}^{{\eta }_{\mathrm{2}}}\end{array}$(45)

Since ${\mathrm{e}}^{{\eta }_{\mathrm{1}}},{\mathrm{e}}^{{\eta }_{\mathrm{2}}}$ are independent, then

$A=\overline{A}$(46)

that is, A is real. Similarly, from the fourth equation of (34), we have

$A=B$(47)

then the fourth equation of (35) is satisfied automatically. Now we derive the interaction strength of the two solitons, A. The remaining equations, the third equation of (33)-(35), should contain the same information and parameter A derived from each of them should be identical. From the third equation of (33), we have

$A=\mathrm{1}+\mathrm{4}{\kappa }_{\mathrm{1}}{\kappa }_{\mathrm{2}}\mathrm{s}\mathrm{i}\mathrm{n}{\xi }_{\mathrm{1}}^{(\mathrm{0})}\mathrm{s}\mathrm{i}\mathrm{n}{\xi }_{\mathrm{2}}^{(\mathrm{0})}(\mathrm{2}{\kappa }_{\mathrm{1}}{\kappa }_{\mathrm{2}}\mathrm{c}\mathrm{o}\mathrm{s}\left({\xi }_{\mathrm{1}}^{(\mathrm{0})}+{\xi }_{\mathrm{2}}^{(\mathrm{0})}\right)$

${-{\kappa }_{\mathrm{2}}^{\mathrm{2}}\mathrm{s}\mathrm{i}\mathrm{n}{\xi }_{\mathrm{1}}^{(\mathrm{0})}/\mathrm{s}\mathrm{i}\mathrm{n}{\xi }_{\mathrm{2}}^{(\mathrm{0})}-{\kappa }_{\mathrm{1}}^{\mathrm{2}}\mathrm{s}\mathrm{i}\mathrm{n}{\xi }_{\mathrm{2}}^{(\mathrm{0})}/\mathrm{s}\mathrm{i}\mathrm{n}{\xi }_{\mathrm{1}}^{(\mathrm{0})})}^{-\mathrm{1}}$(48)

The second equation of (34) and (35) yield the same result of A. Thus we get the 2-SPS of the MNLSE under NVBC as follows

$u_{\mathrm{2}}=g_{\mathrm{2}}{\overline{f}}_{\mathrm{2}}/f_{\mathrm{2}}^{\mathrm{2}}$(49)

here,

$\begin{array}{l}{f}_{\mathrm{2}}={\mathrm{e}}^{\mathrm{i}{k}_{\mathrm{0}}x}\left(\mathrm{1}+a_{\mathrm{1}}{\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}}+a_{\mathrm{2}}{\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{2}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}}+a_{\mathrm{1}}{a}_{\mathrm{2}}A{\mathrm{e}}^{\mathrm{i}{\xi }_{\mathrm{1}}^{(\mathrm{0})}+\mathrm{i}{\xi }_{\mathrm{2}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}}\right)\\ g_{\mathrm{2}}=\rho {\mathrm{e}}^{\mathrm{i}{b}_{\mathrm{0}}x}\left(\mathrm{1}+a_{\mathrm{1}}{\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{1}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}}+a_{\mathrm{2}}{\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{2}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{2}}}+a_{\mathrm{1}}{a}_{\mathrm{2}}A{\mathrm{e}}^{\mathrm{i}{\alpha }_{\mathrm{1}}^{(\mathrm{0})}+\mathrm{i}{\alpha }_{\mathrm{2}}^{(\mathrm{0})}}{\mathrm{e}}^{{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{2}}}\right)\end{array}$(50)

here A is given by (48) and

$\begin{array}{l}{k}_{\mathrm{0}}=\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{2}};b_{\mathrm{0}}=\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{2}}\pm \sqrt[]{\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{4}}+\mathrm{2}\gamma {\left|\rho \right|}^{\mathrm{2}}}\\ {\alpha }_j^{(\mathrm{0})}={\xi }_j^{(\mathrm{0})}+\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{c}\mathrm{o}{\mathrm{s}}^{\mathrm{2}}{\xi }_j^{(\mathrm{0})}+\left(\mathrm{\Delta }-\mathrm{1}\right)\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}{\xi }_j^{(\mathrm{0})}\right)\\ \mathrm{\Delta }=\mathrm{2}\left({\left|\rho \right|}^{\mathrm{2}}\mp \sqrt[]{{\left|\rho \right|}^{\mathrm{4}}+\mathrm{8}\gamma {\left|\rho \right|}^{\mathrm{2}}}+\mathrm{4}\gamma \right)/{\left|\rho \right|}^{\mathrm{2}}\\ {\kappa }_j=\left(\mathrm{1}/\mathrm{2}\right){\left|\rho \right|}^{\mathrm{2}}\left(\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_j^{(\mathrm{0})}-\mathrm{c}\mathrm{o}\mathrm{s}{\xi }_j^{(\mathrm{0})}\right)/\mathrm{s}\mathrm{i}\mathrm{n}{\xi }_j^{(\mathrm{0})}\\ {\tau }_j=-\mathrm{4}\left(\gamma +k_{\mathrm{0}}\right){\kappa }_j+\mathrm{c}\mathrm{o}\mathrm{t}{\xi }_j^{(\mathrm{0})}{\kappa }_j^{\mathrm{2}},j=\mathrm{1},\mathrm{2}\end{array}$

The parameters $\rho ,\gamma ,{\xi }_j^{(\mathrm{0})},a_j$ can be adjusted to fit the experiment result, and the other constants can be derived.

Given appropriate parameters, the space-time evolution of $\left|u\right|$, the modulus of the electric field of the short light pulse, is depicted in Fig. 1. It shows the elastic scattering of the two solitons in the 2-SPS solution, which split into two individual solitons as $t\to \pm \mathrm{\infty }$^[16]. When $\mathrm{0}<{\xi }_j^{(\mathrm{0})}<\mathrm{\pi }/\mathrm{2},\left(j=\mathrm{1,2}\right)$, and $a_{\mathrm{1}}<\mathrm{0},a_{\mathrm{2}}<\mathrm{0}$, a bright-bright soliton is derived; but when $a_{\mathrm{1}}<\mathrm{0},a_{\mathrm{2}}>\mathrm{0}$ or $a_{\mathrm{1}}>\mathrm{0},a_{\mathrm{2}}<\mathrm{0}$, a bright-dark soliton is derived; and when $a_{\mathrm{1}}>\mathrm{0},a_{\mathrm{2}}>\mathrm{0}$, a dark-dark soliton is derived as in Fig. 1.

Fig. 1 Elastic scattering of the two solitons in the 2-SPS solution (a) The bright-bright soliton, ${\mathit{a}}_{\mathrm{1}}=-\mathrm{1},{\mathit{a}}_{\mathrm{2}}=-\mathrm{1}$; (b) The bright-dark soliton, ${\mathit{a}}_{\mathrm{1}}=-\mathrm{1},{\mathit{a}}_{\mathrm{2}}=\mathrm{1}$; (c) The dark-dark soliton, ${\mathit{a}}_{\mathrm{1}}=\mathrm{1},{\mathit{a}}_{\mathrm{2}}=\mathrm{1}$;

When $\gamma \to \mathrm{0}$, the MNLSE degenerates into the DNLSE, the solution remains the same form as (49) and (50), only with

$\begin{array}{l}A=\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}\left({\xi }_{\mathrm{1}}^{(\mathrm{0})}-{\xi }_{\mathrm{2}}^{(\mathrm{0})}\right)/\mathrm{s}\mathrm{i}{\mathrm{n}}^{\mathrm{2}}\left({\xi }_{\mathrm{1}}^{(\mathrm{0})}+{\xi }_{\mathrm{2}}^{(\mathrm{0})}\right);k_{\mathrm{0}}=\left(\mathrm{1}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{2}}\\ b_{\mathrm{0}}=\left(\mathrm{3}/\mathrm{4}\right){\left|\rho \right|}^{\mathrm{2}};{\alpha }_j^{(\mathrm{0})}=\mathrm{3}{\xi }_j^{(\mathrm{0})}\\ {\kappa }_j=-{\left|\rho \right|}^{\mathrm{2}}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{2}{\xi }_j^{(\mathrm{0})};{\tau }_j={\left|\rho \right|}^{\mathrm{4}}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{2}{\xi }_j^{(\mathrm{0})}\left(\mathrm{1}+\mathrm{2}\mathrm{c}\mathrm{o}{\mathrm{s}}^{\mathrm{2}}{\xi }_j^{(\mathrm{0})}\right)\end{array}$(51)