New types of exact quasi-soliton solutions in metamaterials

It is well known that nonlinear Schrödinger-like equations (NLSEs) are important physical models to describe many nonlinear phenomena and dynamic processes in the branches of physics such as nonlinear optics, Bose–Einstein condensates and plasma physics. The best known solutions for NLSEs are various types of bright and dark solitons, multipole solitons, soliton-like or quasi-solitons and so on [1–12]. Among these solutions, the quasi-soliton, which is a soliton-like nonlinear pulse with a stationary structure [5, 6], is very attractive due to its weak interactions, small peak powers and less rigorous formation conditions [5–9]. In optical fibers, some exact quasi-soliton solutions, such as bright and dark quasi-solitons, dark-like-bright solitons and double-kink solitons are presented under different combinations of dispersion and nonlinearity [6–12]. However, to the best of our knowledge, less attention has been paid to the study of quasi-solitons in MMs with dispersive permittivity and dispersive permeability.

During the past several years, artificial metamaterials (MMs) have attracted increasing interest due to their unique electromagnetic characteristics and promising applications [13–16]. The development of nonlinear MMs, especially left-handed MMs, has inspired extensive research on the propagation of pulses and the generation of solitons. So far, theoretical models for ultrashort pulse propagation in MMs have been established [17–20]. Some authors have investigated the modulation instability in MMs closely associated with the existence of solitons or solitary waves [21–23]. Bright and dark solitons, combined solitary waves and periodic waves have been analytically or numerically studied from different viewpoints [24–32]. In order to observe solitons in experiments, left-handed nonlinear transmission lines (NLTL), employed as nonlinear MMs, have been used to investigate the generation of solitons [33–36]. The trains of both bright and dark envelope solitons were observed in the left-handed NLTL MMs [33–35] and stable generation of soliton pulses was experimentally demonstrated in an active NLTL MM composed of a left-handed NLTL inserted into a ring resonator [35], in which the approach can be employed for the other types of active MMs. In addition, dark solitons in a practical left-handed NLTL MMs with series nonlinear capacitance are demonstrated by circuit analysis, which verified analytically that the left-handed NLTL could support dark solitons by tailoring the circuit parameters [36]. These theoretical and experimental studies show that it is practical and significant to search for new and possible solitons in MMs.

In this paper, we present three new types of exact quasi-soliton solutions by ansatz method based on a generalized NLSE describing the propagation of ultrashort pulses in MMs. The formation conditions, the existence regions and the features of the quasi-solitons are analytically investigated and the stabilities are numerically discussed. The significance of the results shown here is twofold. First, these three exact quasi-soliton solutions with an adjustable constant are new in MMs; Second, the obtained results indicate that there may exist abundant soliton solutions in MMs, which is expected to be confirmed in the future.

The propagation of ultrashort pulses in MMs is governed by a generalized NLSE with higher-order effects such as pseudo-quintic nonlinearity and self-steepening (SS) effect [17, 19]:

$$\begin{eqnarray}&&i\displaystyle \frac{\partial \psi }{\partial z}+\displaystyle \frac{{k}_{2}}{2}\displaystyle \frac{{\partial }^{2}\psi }{\partial {t}^{2}}+{p}_{3}{\left|\psi \right|}^{2}\psi -{p}_{5}{\left|\psi \right|}^{4}-i{s}_{1}\displaystyle \frac{\partial }{\partial t}({\left|\psi \right|}^{2}\psi )=0,\end{eqnarray} \tag{ 1 }$$

where $\psi \left(z,t\right)$ represents the complex envelope of the electric field, $t=cT/{\lambda }_{p}$ and $z=Z/{\lambda }_{p}$ are the respective normalized time and propagation distance, where ${\lambda }_{p}$ is the plasma wavelength. ${k}_{2}=\left(1/\beta n\right)\left[1/{V}_{g}^{2}-\alpha \gamma -\beta \left(\varepsilon \gamma \text{'}+\mu \alpha \text{'}\right)/4\pi \right]$ stands for the group-velocity dispersion (GVD). The value $\varpi =1.2646$ stands for the group-velocity dispersion (GVD), where $\beta =2\pi \varpi =2\pi \omega /{\omega }_{pe},$ ${V}_{g}=2n/(\varepsilon \gamma +\mu \alpha ),$ $\alpha =\partial [\varpi \varepsilon (\varpi )]/\partial \varpi ,$ $\alpha ^{\prime} ={\partial }^{2}[\varpi \varepsilon (\varpi )]/\partial {\varpi }^{2},$ $\gamma =\partial [\varpi \mu (\varpi )]/\partial \varpi$ and $\gamma ^{\prime} ={\partial }^{2}[\varpi \mu (\varpi )]/\partial {\varpi }^{2},$ respectively. In equation (1), ${p}_{3}=\beta \mu {\chi }^{(3)}/2n$, ${p}_{5}=\beta {\mu }^{2}{({\chi }^{(3)})}^{2}/8{n}^{3}$ and ${s}_{1}=({\chi }^{(3)}/2n)[\mu /{V}_{g}n-(\mu +\gamma )]$ denote the cubic, pseudo-quintic nonlinearity and the SS effect, respectively. It is worth noting that all linear and nonlinear terms in equation (1) are related to the dispersive permeability $\mu (\varpi ),$ which is regarded as a constant $\mu =1$ in conventional materials because of weak magnetization [4]. It has been demonstrated that the dispersive permeability $\mu (\varpi )$ plays an important role in ultrashort pulse propagation, leading to the difference between MMs and conventional materials [17–19]. Moreover, these coefficients of the terms in equation (1) may be engineered by designing the unit structure of MMs. This implies that there are more possibilities for the existence of solitary waves in MMs [17]. The dispersive dielectric permittivity $\varepsilon (\varpi )$ and magnetic permeability $\mu (\varpi )$ are described by the lossless Drude model $\varepsilon (\varpi )=1-{\varpi }^{-2}$ and $\mu (\varpi )=1-{\omega }_{p}^{2}{\varpi }^{-2}$ with ${\omega }_{p}={\omega }_{pm}/{\omega }_{pe},$ where ${\omega }_{pe}$ and ${\omega }_{pm}$ are the electric and magnetic plasma frequencies, respectively [17, 19]. For simplicity, here we neglect the losses since they may be reduced or compensated by introducing the gain and novel fabrication methods [37, 38]. Thus, for the typical value ${\omega }_{p}=0.8,$ the refraction index $n=\pm {[\varepsilon (\varpi )\mu (\varpi )]}^{1/2}$ is negative for $\varpi \lt 0.8$ and positive for $\varpi \gt 1.0.$ According to the expressions of above parameters, for self-focusing nonlinearity, ${p}_{3}\gt 0$ and ${p}_{5}\lt 0$ in the negative index region (NIR), ${p}_{3}\gt 0$ and ${p}_{5}\gt 0$ in the positive index region (PIR); while for self-defocusing nonlinearity, ${p}_{3}\lt 0$ and ${p}_{5}\lt 0$ in the NIR and ${p}_{3}\lt 0$ and ${p}_{5}\gt 0$ in the PIR. Moreover, all the model parameters in equation (1) are the function of the normalized frequency $\varpi ,$ whose specific dependence curves can be seen in figure 1 of reference [19]. Combined with the characteristics of these parameters, we will demonstrate that three new types of exact quasi-soliton solutions can exist in the anomalous dispersion regime of four regions of self-focusing and self-defocusing MMs. Hence for the convenience of our subsequent discussion, we define these four regions as Region I: in anomalous dispersion regime of self-focusing NIR; Region II: in anomalous dispersion regime of self-focusing PIR; Region III: in anomalous dispersion regime of self-defocusing NIR; Region IV: in anomalous dispersion regime of self-defocusing PIR. In general, equation (1) is not integrable, so we will seek new types of exact quasi-soliton solutions to equation (1) by ansatz method and investigate the formation conditions, the existence regions and the features of these solutions based on the parameter characteristics in MMs.

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3.1. Bright quasi-soliton solution

Let us construct a bright quasi-soliton solution for equation (1) by assuming an ansatz:

$$\begin{eqnarray}&&\psi (z,t)=\displaystyle \frac{A\;\mathrm{sech}[\eta (t-\chi z)]}{\sqrt{1+B{\mathrm{sech}}^{2}[\eta (t-\chi z)]}}{e}^{i(\kappa z-{\rm{\Omega }}t)},\end{eqnarray} \tag{ 2 }$$

where $A$ is related to the amplitude of quasi-soliton; $\eta ,$ ${\rm{\Omega }},$ $\chi$ and $\kappa$ stand for the inverse pulse, the frequency shift, the inverse group velocity and the wave number, respectively; and $B$ is a real constant larger than minus one. For $B=0,$ the solution (2) can be simplified to a standard bright soliton $\psi (z,t)=A\;\mathrm{sech}[\eta (t-\chi z)]{e}^{i(\kappa z-{\rm{\Omega }}t)},$ which has been studied in reference [39]. For $B\ne 0,$ equation (2) is the form of a bright quasi-soliton. Substituting equation (2) into (1) and setting the coefficients of independent terms equal to zero, we can obtain five compatible equations. By solving these equations, we find when ${s}_{1}=0,$ equation (1) admits the quasi-soliton solution in the form of equation (2) with the following parameters:

$$\begin{eqnarray}&&A=\sqrt{\displaystyle \frac{3B(1+B){p}_{3}}{2(1\quad +\quad 2B){p}_{5}}},\end{eqnarray} \tag{ 3a }$$

$$\begin{eqnarray}&&\eta =A\sqrt{\displaystyle \frac{{p}_{3}}{(1\quad +\quad 2B){k}_{2}}},\end{eqnarray} \tag{ 3b }$$

$$\begin{eqnarray}&&\chi =-{\rm{\Omega }}{k}_{2},\end{eqnarray} \tag{ 3c }$$

$$\begin{eqnarray}&&\kappa =\displaystyle \frac{1}{2}({\eta }^{2}-{{\rm{\Omega }}}^{2}){k}_{2}.\end{eqnarray} \tag{ 3d }$$

Here the frequency shift ${\rm{\Omega }}$ is an arbitrary constant. From the equations (3a) and (3b), it is easy to see that the existence of the bright quasi-soliton (2) requires $B(1\quad +\quad 2B){p}_{3}{p}_{5}\gt 0$ and $(1\quad +\quad 2B){k}_{2}{p}_{3}\gt 0$ with the primary requirement $B\gt -1.$ Combined with the condition ${s}_{1}=0,$ we find the bright quasi-soliton (2) may exist in three regions of self-focusing and defocusing nonlinear MMs, as shown in table 1. Obviously, the constant $B$ plays an important role in the existence regions of the bright quasi-soliton (2). This implies that the bright quasi-soliton can be formed in such three regions if we choose suitable $B.$ Figures 1(a)–(c) present the distributions of the bright quasi-soliton (2) in the three existence regions, respectively. It is clear that in each existence region, the bright quasi-soliton (2) with different $B$ possesses different intensities and pulse widths that increase or decrease with the increasing of $B.$ We notice that the quasi-solitons with different $B$ in each region have the same velocities because of the soliton velocity $\chi =-{\rm{\Omega }}{k}_{2}$ independent of $B.$ In addition, the numerical confirmations of the analytical bright quasi-soliton (2) are depicted by circles, which well coincide with the analytical ones.

Table 1.  Existence regions of the bright quasi-soliton (2) in self-focusing and defocusing MMs.

Ranges of $B$	Ranges of ${k}_{2}$,${p}_{3}$ and ${p}_{5}$	Existence regions of bright quasi-soliton (2)
$-1\lt B\lt -1/2$	${k}_{2}\lt 0,$ ${p}_{3}\gt 0,$ ${p}_{5}\gt 0$	Region II
$-1/2\lt B\lt 0$	${k}_{2}\lt 0,$ ${p}_{3}\lt 0,$ ${p}_{5}\gt 0$	Region IV
$B\gt 0$	${k}_{2}\lt 0,$ ${p}_{3}\lt 0,$ ${p}_{5}\lt 0$	Region III

3.2. Dark quasi-soliton solution

Similarly, we take an ansatz of dark quasi-soliton for equation (1) as follows:

$$\begin{eqnarray}&&\psi (z,t)=\displaystyle \frac{A\;\mathrm{tanh}[\eta (t-\chi z)]}{\sqrt{1+B{\mathrm{sech}}^{2}[\eta (t-\chi z)]}}{e}^{i(\kappa z-{\rm{\Omega }}t)},\end{eqnarray} \tag{ 4 }$$

here $A,$ $\eta ,$ ${\rm{\Omega }},$ $\chi$ and $\kappa$ stand for the same soliton parameters as in the solution (2), respectively. The value of $B$ is still a real constant larger than minus one. For the special case $B=0,$ the ansatz (4) is reduced to a standard dark soliton $\psi (z,t)=A\;\mathrm{tanh}[\eta (t-\chi z)]{e}^{i(\kappa z-{\rm{\Omega }}t)},$ which has been studied in reference [19]; equation (4) with $B\ne 0$ is the form of a dark quasi-soliton. Substituting equation (4) into (1) and solving the obtained six compatible equations, we find that when ${s}_{1}=0,$ equation (1) admits the dark quasi-soliton solution in the form of equation (4) with the corresponding parameters as follows:

$$\begin{eqnarray}&&A=\sqrt{\displaystyle \frac{3B{p}_{3}}{2(2B-1){p}_{5}}},\end{eqnarray} \tag{ 5a }$$

$$\begin{eqnarray}&&\eta =A\sqrt{\displaystyle \frac{(B+1){p}_{3}}{(2B-1){k}_{2}}},\end{eqnarray} \tag{ 5b }$$

$$\begin{eqnarray}&&\chi =-{\rm{\Omega }}{k}_{2},\end{eqnarray} \tag{ 5c }$$

$$\begin{eqnarray}&&\kappa ={A}^{2}{p}_{3}-{A}^{4}{p}_{5}-\displaystyle \frac{1}{2}{{\rm{\Omega }}}^{2}{k}_{2}.\end{eqnarray} \tag{ 5d }$$

From the expressions of equations (5a) and (5b), it is easy to know that $B(2B-1){p}_{3}{p}_{5}\gt 0$ and $(2B-1){k}_{2}{p}_{3}\gt 0$ must be satisfied for the existence of the dark quasi-soliton (4). Combined the characteristics of model parameters with these conditions, it is found that the dark quasi-soliton (4) may also exist in three regions of self-focusing and defocusing MMs, as illustrated in table 2. Similar to the bright quasi-soliton (2), $B$ is an important parameter for the formation of the dark quasi-soliton (4), and has an impact on the intensity and pulse width of the quasi-soliton, while it has nothing to do with its velocity and wave number, which may be seen from equations (5a) and (5b). This means that the dark quasi-solitons with different $B$ in each existence region can propagate at the same velocities and different pulse widths when their intensities are normalized by ${|\psi (z,t)|}^{2}/{A}^{2},$ as shown in figure 2. Furthermore, it is easy to infer that the dark quasi-solitons in different existence regions have different pulse widths and velocities because of different normalized frequencies corresponding to different parameters of dispersion and nonlinearity. Also, the numerical distributions in each existence region are in agreement with the analytical results, which can be seen in the corresponding figures with circles.

Table 2.  Existence regions of the dark quasi-soliton (4) in self-focusing and defocusing MMs.

Ranges of $B$	Ranges of ${k}_{2}$,${p}_{3}$ and ${p}_{5}$	Existence regions of dark quasi-soliton (4)
$-1\lt B\lt 0$	${k}_{2}\lt 0,$ ${p}_{3}\gt 0,$ ${p}_{5}\gt 0$	Region II
$0\lt B\lt 1/2$	${k}_{2}\lt 0,$ ${p}_{3}\gt 0,$ ${p}_{5}\lt 0$	Region I
$B\gt 1/2$	${k}_{2}\lt 0,$ ${p}_{3}\lt 0,$ ${p}_{5}\lt 0$	Region III

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3.3. Bright-grey quasi-soliton solution

Motivated by above bright and dark quasi-soliton solutions, we take another ansatz for equation (1) as follows:

$$\begin{eqnarray}&&\psi (z,t)=\displaystyle \frac{A}{\sqrt{1+B{\mathrm{tanh}}^{2}[\eta (t-\chi z)]}}{e}^{i(\kappa z-{\rm{\Omega }}t)},\end{eqnarray} \tag{ 6 }$$

where $A,$ $\eta ,$ ${\rm{\Omega }},$ $\chi$ and $\kappa$ still represent the same soliton parameters as in equations (2) and (4). The free constant $B$ is also required larger than minus one. For $B=0$ in this case, equation (6) is reduced to a plane wave. Different from the bright solution (2) and the dark solution (4), equation (6) may represent a bright quasi-soliton with an intensity ${|\psi |}^{2}={A}^{2}/\{1+B{\mathrm{tanh}}^{2}[\eta (t-\chi z)]\}$ for $B\gt 0$ or a grey quasi-soliton with an intensity ${|\psi |}^{2}={A}^{2}/\{1-|B|{\mathrm{tanh}}^{2}[\eta (t-\chi z)]\}$ for $-1\lt B\lt 0.$ That is to say, the quasi-soliton (6) is a bright soliton with a platform or a grey soliton with a nonzero dip for the case $B\ne 0.$ Following a similar solving process, we find that when ${s}_{1}=0,$ equation (1) admits the quasi-soliton solution (6) with the parameters:

$$\begin{eqnarray}&&A=\sqrt{\displaystyle \frac{3(1+B){p}_{3}}{2(3+B){p}_{5}}},\end{eqnarray} \tag{ 7a }$$

$$\begin{eqnarray}&&\eta =A\sqrt{-\displaystyle \frac{B{p}_{3}}{(3+B)(1+B){k}_{2}}},\end{eqnarray} \tag{ 7b }$$

$$\begin{eqnarray}&&\chi =-{\rm{\Omega }}{k}_{2},\end{eqnarray} \tag{ 7c }$$

$$\begin{eqnarray}&&\kappa =-\displaystyle \frac{{k}_{2}}{2}(2{\eta }^{2}+{{\rm{\Omega }}}^{2}+\displaystyle \frac{3{\eta }^{2}}{B}).\end{eqnarray} \tag{ 7d }$$

Similarly, the existence of the quasi-soliton (6) requires that ${p}_{3}{p}_{5}\gt 0$ and $B{k}_{2}{p}_{5}\lt 0$ with $B\gt -1$ simultaneously. Considering the condition ${s}_{1}=0,$ it is easy to find that the quasi-soliton (6) may exist in two regions and exhibit a bright or grey quasi-soliton form depending on the dispersion or nonlinearity, as well as the values of $B,$ which are shown in table 3. The distributions of the quasi-soliton (6) in each region are presented in figure 3, respectively. Obviously, the bright ones have a platform in figure 3(a) and the grey ones have a nonzero dip in figure 3(b). Also, the quasi-solitons with different $B$ possess different platforms (backgrounds) and pulse widths for the bright (grey) form when we adopt normalized treatment ${|\psi (z,t)|}^{2}/{A}^{2}.$ These features can be seen clearly from figure 3. We also perform numerical experiments for the quasi-soliton (6) in each region, as correspondingly shown with circles in figures 3(a) and (b), which are in good agreement with the analytical prediction.

Table 3.  Existence regions of the quasi-soliton (6) in self-focusing and defocusing MMs.

Ranges of $B$	Ranges of ${k}_{2}$,${p}_{3}$ and ${p}_{5}$	Existence regions of quasi-soliton (6)	Intensity profiles
$-1\lt B\lt 0$	${k}_{2}\lt 0,$ ${p}_{3}\lt 0,$ ${p}_{5}\lt 0$	Region III	Grey quasi-soliton
$B\gt 0$	${k}_{2}\lt 0,$ ${p}_{3}\gt 0,$ ${p}_{5}\gt 0$	Region II	Bright quasi-soliton

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It should be mentioned that all three types of quasi-soliton solutions are obtained under the condition ${s}_{1}=0$ corresponding to $\varpi =0.6325$ in NIR and $\varpi =1.2649$ in PIR for both self-focusing and self-defocusing MMs, respectively. However, the normalized frequency of incident pulses may fluctuate in real communication, resulting in nonzero SS parameter. On the other hand, solitons may be disturbed in the transmission by initial perturbation such as white noise, pulse width, etc. Therefore, it is necessary to investigate the stability of these quasi-solitons under the finite initial perturbation and nonzero SS parameter resulting from frequency fluctuation. Here we take the third type of quasi-soliton (6) as an example to investigate the stability by numerical simulation.

4.1. The stability under nonzero SS parameter

As mentioned above, the constant $B$ is related to the background intensity of the quasi-soliton (6). Here we consider two cases: i) the stability under the same SS parameters and different constant $B,$ as shown in figures 4(a) and (b); ii) the stability under the same constant $B$ and different SS parameters, as shown in figures 4(c) and (d). By comparing the curves in figure 4(a), we find that under the same SS parameters, the bright quasi-soliton with smaller $B$ can stably propagate keeping its shape unchanged after propagating 40 ${\lambda }_{p}.$ As $B$ is increasing, the bright quasi-soliton gradually broadens and then splits into two pulses. For the grey quasi-soliton, similar evolutions can be seen from figure 4(b), namely, with the increase of magnitude of $B,$ it gradually evolves into two dips. In other words, the smaller the magnitude of $B,$ the stabler its propagation. This means that when the SS parameter is nonzero, we can achieve stable bright and grey quasi-solitons by choosing suitable $B.$ So we take $B=0.03$ for the bright form and $B=-0.03$ for the grey form to discuss the stability under case ii); these are shown in figures 4(c) and (d), respectively. It can be seen clearly that the intensities of the bright ones are changed slightly under nonzero SS parameters corresponding to the frequency fluctuations $\varpi =1.2652$ and $\varpi =1.2646,$ and the intensities of the grey ones are affected more than those of the bright ones even if the incident frequency perturbations are small, i.e. $\varpi =0.6327$ and $\varpi =0{\rm{.6323}}.$ This means that the grey quasi-soliton is more sensitive to the frequency fluctuation. So we can conclude that the quasi-soliton (6) may propagate stably by choosing suitable constant $B$ under nonzero SS parameters originating from the frequency fluctuation of the incident pulse.

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4.2. The stability under pulse width fluctuation

We further investigate the stability of the quasi-soliton (6) under pulse width fluctuation. According to the results presented in figures 4(a) and (b), we still take $B=0.03$ for bright form and $B=-0.03$ for grey form. Figure 5 displays the numerical evolutions of the quasi-soliton (6) by adding a pulse width fluctuation of $5\%.$ It can be seen from figures 5(a) and (b) that the features of the bright and grey quasi-solitons remain nearly unchanged after propagating 40 ${\lambda }_{p}$ except for slight changes in the intensities and pulse widths. This indicates that both the bright and the grey quasi-solitons can maintain their shapes unchanged under finite fluctuation of pulse width.

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In conclusion, we have considered a generalized NLSE with dispersive permittivity and permeability describing the propagation of ultrashort pulses in MMs, and presented three new types of exact quasi-soliton solutions with a constant B by ansatz method. Based on the Drude model, we have investigated the existence regions and the characteristics of these solutions. It is found that by choosing suitable range of B, these bright and dark (grey) quasi-solitons can exist in wider regions in MMs than those in conventional materials. Also, as the constant B is also associated with the amplitude and the pulse width for each type of quasi-soliton, we may control the intensity and pulse width of the soliton by adjusting B. Such features imply that it is possible to form a quasi-soliton for an incident pulse with variable pulse width and intensity in MMs. As an example, we have numerically investigated the stability of the quasi-soliton (6) and found that the quasi-soliton (6) in both bright form and grey form can stably propagate under slight perturbations of nonzero SS parameter and the finite fluctuation of pulse width by choosing suitable B. The obtained results are important to explore much richer solitary waves in MMs. Finally, two things should be pointed out. i) Although the results we presented here are limited to theoretical analysis, we hope they may provide a theoretical reference for experimental verification in the future in view of the characteristics of artificial MMs, composed of various different unit structures, with tunable linear and nonlinear parameters. Moreover, the experiments and analytical approximation of practical NLTL MMs [33–36, 40] confirm that bright and dark envelope solitons can be observed and formed in the experiments. These results mean that MMs can be utilized to create suitable experimental circumstances for generating solitons [41]. ii) The results presented here are obtained without including the Raman effect. If the Raman effect is considered, there may not exist such exact solutions as presented above, and thus perturbation theory needs to be applied for the dynamics of bright and dark solitons [4, 42]. The characteristics of the above three types of bright and dark quasi-solitons under the perturbation of the Raman effect will be discussed in a separate paper.

This work is supported by the National Natural Science Foundation of China (NSFC) (grants 61178013, 61271160), and Selected Project of Overseas Science and Technology Activities of Shanxi Province (grant 201301).