1. Introduction

Periodicity and localization are two fundamental but distinct characteristics of nonlinear waves. The periodicity is a commonality of all waves and dominates various wave phenomena [1,2], while the (self-induced) localization dominates the formation of nonlinear localized waves [3–5]. As a widespread kind of nonlinear localized wave, soliton has been attracting lots of attentions and has wide applications by virtue of its property of stable propagation [6–11]. Soliton can produce periodicity-dominated phenomena like interference [12–14] and diffraction [15,16], and it also has localization-dominated phenomena like self-bound propagation and elastic collision [17,18]. In the previous studies, the periodicity and localization of a soliton were always treated separately, and little attention has been paid to their combined effects. However, in nonlinear fibers with the fourth-order dispersion (FOD), the bright solitons with oscillating striped structures were theoretically discovered [19–23] and experimentally observed [24,25]. They exhibit both of noticeable periodicity and localization, and therefore can be considered as the results of competition between the two characteristics. As we know, the modified linear stability analysis (MLSA) method has been used to predict the quantitative dynamics of nonlinear waves, including their periodicity and localization [26,27]. Therefore, it provides the possibility to figure out the formation mechanism of striped solitons under the FOD and furthermore find more kinds of solitons in this system.

In this paper, we study the excitations of dark solitons in a nonlinear optical fiber with the second-order dispersion and FOD. We find that striped dark solitons (SDSs) and their bound states exist in this system, which exhibit more kinds of structures than the striped bright solitons (SBS) [22,23] and the multi-soliton bound states [20]. The single-soliton state of SDS contains two types, i.e., the ones with or without the total phase step, while the multi-dark-soliton bound states have different numbers of amplitude humps in the time-domain distribution. By the MLSA method, a SDS is considered as the result of the competition between periodicity and localization, and furthermore we quantitatively give its existence condition and oscillation frequency. Based on the derived oscillation frequency, we analyze the modulation instability (MI) of plane-wave background of SDS and give its stability condition. Besides, we apply the MLSA to the SBSs to provide a possible interpretation of their formation, and find that the pure-quartic soliton is a special kind of SBS when its periodicity and localization carry equal weight.

2. Model and the existence condition of striped solitons

In the slow-varying envelope approximation, the propagation of a linearly polarized light in a single-mode nonlinear fiber can be described by the following nonlinear Schrödinger equation with the second-order dispersion and FOD [8],

In the model  (2), some bright solitons and multi-soliton states have been studied [19–25,29,30], including their existence conditions, stability, and even analytical solutions. They have no plane waves as their background waves. Recently, dark solitons and multi-soliton states under the FOD have been preliminarily demonstrated [31,32], which exist on the plane-wave background and therefore are essentially different from the bright ones. In our work, the solitons we mainly concern about are the dark solitons. According to our analysis, the existence condition of solitons in the model (2) is shown in Fig. 1. The solitons with different structures correspond to different regions in the $\beta _2$-$\beta _4$ plane. The details are as follows.

In the above conditions, $\beta (>0)$ denotes the propagation constant of solitons. In other regions of the $\beta _2$-$\beta _4$ plane, no soliton exists. We define the quantity $\varphi$ by $\varphi =\tan ^{-1} ({\omega _s}/{\eta _s})$ in the range from $0$ to $\pi /2$ to describe the strength of a wave’s stripe characteristic. $\omega _s$ and $\eta _s$ are respectively the parameters determining the periodicity and localization of a wave, whose expressions can be seen in Eqs. (16) and (26) for SDS and SBS, respectively. With $\varphi$ increasing, the wave’s periodicity gets stronger and the wave’s localization gets weaker. The striped solitons with different shapes can be regarded as the results of the competition between periodicity and localization in different proportions. In particular, for a pure-quartic soliton [24], the two parameters $\omega _s$ and $\eta _s$ are equal, which indicates that its periodicity and localization carry equal weight. There is a counterintuitive result that the two parabolas for positive and negative $\beta _4$ have different widths. Besides of our findings, this result can be supported by the results from two other papers. Specifically, for dark solitons, reference can be made to the discussion beneath Eq. (4) in Ref. [31], and for bright solitons, reference can be made to Eq. (11) in Ref. [23].

 

Fig. 1. Existence condition of solitons under different second-order dispersion $\beta _2$ and FOD $\beta _4$. They contain the regions of SDS (striped dark soliton), DS (the traditional dark soliton), SBS (the striped bright soliton), and BS (the traditional bright soliton). The boundary of SDS region has the expression $\beta _4=3\beta _2^2/4\beta$, and the boundary of SBS region is $\beta _4=-3\beta _2^2/2\beta$, where $\beta$ is the propagation constant of solitons. The color scale describes the strength $\varphi$ of a soliton’s periodicity relative to its localization in the range from 0 to $\pi /2$. The black dashed line denotes the case of $\beta _2=0$, where a pure-quartic soliton can exist. In all cases, the propagation constant is set as $\beta =1$.

Download Full Size | PDF

In the next two sections, we will separately study SDS and SBS, including their existence condition, stripe characteristic, and formation mechanism.

3. Dark solitons and their bound states

3.1 Single-soliton state

At first, let us focus on the single-soliton states of SDSs and their counterparts without the total phase step. The term “the total phase step” is used to describe the phase difference of plane wave between the two sides of dark solitons, which is the key characteristic of dark solitons. It is known that the model (2) has the plane-wave solution,

 

Fig. 2. (a1-e1) Distribution of the dark solitons with the total phase step, when the second-order dispersion $\beta _2$ is (a1) $-1$, (b1) $-0.8$, (c1) $-0.3$, (d1) 0, and (e1) 1.5. Their wave functions $\psi$ and the amplitude $|\psi |$ are denoted by red dashed curves and blue solid curves. (a2-e2) Amplitude evolution of the corresponding dark solitons in (a1-e1). The horizontal lines are the spatial dividers between the solitons and the spontaneous oscillations induced by MI. (a3-e3) Except for showing the dark solitons without the total phase step, everything else is the same as depicted in (a1-e1). (a4-e4) Except for showing the dark solitons without the total phase step, everything else is the same as depicted in (a2-e2). Other parameters are set as $\beta _4=1$ and $\beta =1$.

Download Full Size | PDF

Later, we set their wave functions as the initial states to numerically integrate the model (2), by the split-step Fourier method [36]. The evolution results are shown in Fig. 2(a2) or (a4), for the dark solitons with or without the total phase step, respectively. The SDS with the total phase step stably propagates for seven unit distance. At its final evolution stage, some new waves can be seen. Because the plane-wave background is unstable under the perturbation of dark solitons, these waves are generally induced by the modulation instability (MI), which will be analyzed in detail below. The similar result is also obtained for the SDS without the total phase step, which propagates stably for six unit distance.

For some larger values of $\beta _2$, the results of distribution and evolution are shown in Figs. 2(b1-e1) and (b2-e2) for the dark solitons with the total phase step, and Figs. 2(b3-e3) and (b4-e4) for the dark solitons without the total phase step. With the increase of $\beta _2$, the SDS has less stripes but can propagate for a longer distance, especially when $\beta _2>0$. Note that the SDS without the total phase step can be regarded as two separated dark solitons when $\beta _2=1.5$, as shown in Fig. 2(e3). Here, we need to engage in some necessary discussion regarding the definition of stripe solitons. From Fig. 2, one can observe that as $\beta _2$ increases, the soliton’s striped features continuously weaken. It’s challenging to distinguish whether a soliton is with oscillatory tails or with obvious stripes, especially for the solitons with less obvious oscillatory features. Therefore, considering that oscillatory tails can be viewed as weak stripes, we temporarily refrain from distinguishing between these two types of solitons and refer to them collectively as "stripe solitons". In other words, as long as a soliton exhibits temporal periodicity, we refer to it as a stripe soliton in this paper.

3.2 Analysis of existence condition and oscillation frequency

For the bright solitons in the model (2), the asymptotic analysis has been used to obtain their existence condition and oscillation characteristic [19,20,22,23]. However, considering that the dark solitons have plane-wave backgrounds, the asymptotic analysis cannot be applied directly to them. As shown in Fig. 2, SDSs have the background waves, and meanwhile they have both of periodicity and localization. For such a nonlinear wave, the MLSA method can quantitatively give its dynamical characteristics and guide its generation [26,27]. Thus, we will apply this method to analyze the existence condition and characteristics of fundamental SDSs. The foundation of this method, namely the traditional linear stability analysis, has been introduced in Refs. [3,8], and it plays a significant role in analyzing the MI in nonlinear systems. The MLSA method starts with the ansatz solution of a perturbed plane wave,

 

Fig. 3. (a) Phase of the propagating function $p_z(t,z)$ under different $\omega$ and $\eta$, when $\beta _2=-0.5$, $\beta _4=1$, and $\beta =1$. The black solid and dashed curves denote $V_{\rm en}=0$ and $V_{\rm st}=0$, respectively. The white point with the coordinate $(\omega _s, \eta _s)$ is an exceptional point of $\arg (p_z)$ function. The yellow curve with an arrow is the trajectory of this point with $\beta _2$ increasing. (b) Dependence of SDS’s oscillation frequency $\omega _s$ on $\beta _2$, when $\beta _4=1$. The red dots and the black curve are respectively the results from numerical evolution and the MLSA method. (c) Dependence of the gain value $G$ on $\omega$, when $\beta _2=-1$ (upper) or $\beta _2=-0.3$ (lower). The strong or weak MI is defined by whether $\omega _s$ locates in the MI band or not. (d) Distribution of MI (modulation instability) and MS (modulation stability) on the $\beta _2$-$\beta _4$ plane. The color scale describes the value of $g(\omega _s)$, i.e., Eq. (20).

Download Full Size | PDF

Therefore, to take the term $u^*$ into consideration, we assume the perturbation as the linear superposition of two conjugating components,

In the above expressions, $p_t$ and $p_z$ denote the partial derivative of $p(t,z)$ about $t$ and $z$, respectively. The function $p_z$ can be obtained by solving the eigenvalue of the left-hand matrix in Eq. (7):

There are two modes, where the eigenvalues $p_z$ are the opposite of each other. The mode with a positive eigenvalue is focused on here.

Now, our main aim is to find the condition that the shape of the perturbation remains unchanged in the propagating process, which is just the existence condition of SDS. We consider the function $p(t,z)$ has the following form at the initial distance,

By this approximation, the function $p_t(t,z)$ can be successfully transformed into a complex constant, and therefore offers great convenience for our analysis. Moreover, this approximation can be conveniently applied to the function $p_z$. Considering that the cases of $t\rightarrow \pm \infty$ have the same results, we take the case of $t\rightarrow -\infty$ as an example and can obtain

It describes the propagation characteristics of the perturbation. To be specific, the perturbation’s propagation constant and growing rate relative to the plane wave are respectively

Then, the velocity of its localized envelope and stripes are

From the insets of Fig. 2, we know that a SDS has a zero-velocity localized envelope and some zero-velocity stripes, which indicates $V_{\rm en}=0$ and $V_{\rm st}=0$. In Fig. 3(a), when $\beta _2=-0.5$, $\beta _4=1$, and $\beta =1$, we show the curves of $V_{\rm en}=0$ and $V_{\rm st}=0$ in the $\omega$-$\eta$ plane. The coordinate of their intersection point $(\omega _s,\eta _s)$ is just the oscillation frequency and steepness of SDS, and the expressions are

Thus, the parameters need to satisfy $\omega _s\neq 0$ and $\eta _s\neq 0$ for the existence of SDS, namely

The former is independent of $\beta _2$, and the latter is independent of $\beta$. Thus, the point $(\omega _s,\eta _s)$ always locates on a circle when $\beta _4$ and $\beta$ are fixed, and it moves towards the positive axis of $\eta$ with $\beta _2$ increasing, as shown in Fig. 3(a). More interestingly, this point is just an exceptional point of the eigenvalue $p_z$, which is related to the non-Hermiticity of the matrix in Eq. (7). By calculating the contour integral of its argument, one can obtain that its winding number is $\pi$ [37]. In eigenvalue problems, an exceptional point is a singularity where the two eigenvalues coalesce and each of them undergoes the transition from a real value to a complex. With the development of non-Hermitian physics, the exceptional point plays an important role in various systems, like optics, photonics, and Bose–Einstein condensates [38,39]. However, for the relationship between SDS and exceptional point, a deeper understanding needs further investigations. Next, to verify the effectiveness of our prediction, the dependence of $\omega _s$ on $\beta _2$ is studied. When $\beta _4=1$ and $\beta =1$, we numerically measure the oscillation frequency of SDSs under different values of $\beta _2$ and show them by the red dots in Fig. 3(b). One can see that $\omega _s$ gradually decreases into 0 with $\beta _2$ increasing. To obtain them, we measure the time interval between the first and second peaks on the right side of SDS as its time period $T_s$, and then the oscillation frequency can be calculated by $\omega _s=2\pi /T_s$, as shown in the upper row of Fig. 2(a). On the other hand, we also show the analytic prediction from the MLSA method by the black curve. There are good agreements between the two results.

3.3 Analysis of propagation stability

Another problem worthy of consideration is the propagation stability of SDSs. From the insets of Fig. 2, one can see that some spontaneous oscillations emerge in the propagating process of SDSs. It is a widespread phenomenon in the nonlinear systems with MI, and therefore prompts us to analyze the MI in the model (2). As we know, different types of perturbations can produce different phenomena under the influence of MI. Generally speaking, a purely periodic perturbation can generate Akhmediev breathers [5,40]; a purely localized one can generate a rogue wave [41] or Kuznetsov-Ma breather [42], and induce the spontaneous oscillations in a triangular region [43,44]. (The word "purely" in front of "perturbation" is used to distinguish them with a periodic-localized perturbation.) A SDS at the initial distance can be seen as a periodic-localized perturbation on a plane wave, so we need to separately consider the influence of MI on its periodic and localized components. For its periodic component, we know that the traditional growing rate $g$ can be obtained by the modified growing rate $G$ when $\eta =0$, namely

If the SDS’s oscillation frequency $\omega _s$ locates in the MI band, its periodicity will intensively impact the stability of plane wave, whose corresponding MI distribution is shown as the upper plot in Fig. 3(c). Thus, we calculate the growing rate when $\omega =\omega _s$ by

3.4 Multi-soliton bound state

In this subsection, we focus on the multi-dark-soliton bound states. In Ref. [20], the bound states of several bright solitons under the FOD have been found, and their existence condition and stability were analyzed. Here, we will find the bound states of dark solitons by the biconjugate gradient method [33]. When $\beta _2=0$, $\beta _4=1$, and $\beta =1$, the distributions of wave functions of the multi-dark-soliton bound states are shown as the upper row in Fig. 4. There are many kinds of bound states, which have different number of amplitude humps, such as (a) two small humps, (b) six small humps, (c) two big humps, (d) six big humps, and (e) two small and two big humps. The bound states include but are not limited to these five kinds, and their counterparts without the total phase step also exist but are not shown in this paper. Next, we study their numerical evolution to test their propagation stability, which are shown as the lower row in Fig. 4. These bound states can always propagate for a long distance, and no MI phenomenon emerges.

 

Fig. 4. The multi-dark-soliton bound states. The upper row is the distribution of their wave functions $\psi$ (red dashed curves) and amplitude $|\psi |$ (blue solid curves), when $\beta _2=0$, $\beta _4=1$, and $\beta =1$. The lower row is their corresponding amplitude evolution.

Download Full Size | PDF

We stress that the “dispersion waves” in Figs. 2(a-c) are the manifestation of the MI of plane-wave backgrounds, which is induced by the unstable frequency sideband from the periodicity of dark solitons. As shown in Fig. 3(a), the MI can be strong, weak, or absent in different cases, and therefore the dark solitons can propagate for a short or long distance under the influence of MI. Considering that they could be observed in the optical fibers with different length, our calculations about dark solitons can provide the theoretical guidance for their observations. Particularly, when $\beta _4\geq 0$, the dark solitons and their bound states are stable and propagate for a long distance, as shown in Figs. 2(d,e) and Fig. 4. In addition, the traditional linear stability analysis can also be used to obtain the linear Eq. (5) and the classical dispersion law (19). However, they can give us only the MI condition of a perturbed plane wave, but cannot give anything about striped solitons. This is the reason why we use the MLSA method to analyze the SDSs.

4. Additional discussion on bright solitons

As we know, the bright solitons with stripe structures in the model (2) have been studied in Refs. [22,23,25], including their existence condition and oscillation frequency. Here, we will study them by the MLSA method to explore their similarities and differences with dark solitons, while also providing a more illustrative demonstration of the formation mechanism of stripe solitons.

Firstly, by the biconjugate gradient method, we show the distribution of wave functions of SBS in Fig. 5(a) when $\beta _2=0$, $\beta _4=-1$, where the propagation constant is $\beta =1$. A bright soliton with oscillating tails can be seen, which is just the so-called pure-quartic soliton [24]. Then, when we change $\beta _2$ to $0.75$, a bright soliton with many stripes appears in Fig. 5(b). Unlike the SDS, the SBS has on background of plane wave. Thus, our calculation of MLSA directly starts with the perturbation,

After taking the limitations of $t\rightarrow -\infty$, the propagation function becomes

It indicates that the propagation constant of this wave is

It is worth noting that the formation mechanism of stripes in SDS and SBS is different. For SDS, the stripes originate from the interaction between a perturbation and a zero-velocity plane wave; for SBS, the stripes are caused by the interaction between two components with opposite frequencies. Thus, the velocity of stripes in SBS is not equal to $V_{\rm st}=K/\omega$. Nevertheless, the formation of localized envelopes is not related to the background wave, so $V_{\rm en}$ can still describe the velocity of localized envelopes in SBS. Its expression is

 

Fig. 5. (a) Distribution of the SBS’s wave function $\psi$ (red dashed curves) and its amplitude $|\psi |$ (blue solid curves), when $\beta _2=0$, $\beta _4=-1$ and $\beta =1$. (b) Same as (a) except for $\beta _2=0.75$. (c) In the $\omega$-$\eta$ plane, the characteristic parameters $(\omega _s, \eta _s)$ is the intersection point between the curves of $V_{\rm en}=0$ (black solid curve) and different $\beta$ value. The blue solid, purple dashed, and red solid curves denote $\beta =1$, $3/8$, and $1/8$, respectively. The parameters are $\beta _2=0.5$ and $\beta _4=-1$. (d) Dependence of SBS’s oscillation frequency $\omega _s$ on $\beta _2$, when $\beta _4=1$ and $\beta =1$. The red dots and the black solid curve are respectively the results of numerical solutions and the MLSA method [see Eq. (26)].

Download Full Size | PDF

For a certain propagation constant $\beta$, by solving $V_{\rm en}=0$ and Eq. (24), we can obtain that the oscillation frequency and steepness of SBS,

As shown in Fig. 5(c), the point $(\omega _s, \eta _s)$ is the intersection of the black curve ($V_{\rm en}=0$) and the blue curve ($\beta =1$), when $\beta _2=0.5$ and $\beta _4=-1$. We also depict the curves of $\beta =3/8$ and $\beta =1/8$: the former is the critical case where the intersection exists just right, and the latter has no intersection with the curve of $V_{\rm en}=0$. It indicates that the SBS cannot exist when $\beta <3/8$, and we also obtain the same result in our numerical simulations. Then, considering that $\omega _s\neq 0$ and $\eta _s\neq 0$, we can obtain that the existence condition of SBS is

Our derived existence condition (27) and oscillation frequency (Eq. (26)) of SBSs agree well with the results given by Ref. [23]. It indicates that for SBSs, the MLSA method are equivalent to the direct asymptotic analysis, due to the absence of plane-wave background. Nevertheless, besides of the above results, the MLSA method can also provide the interpretation for the formation mechanism of SBSs. It is known that the wave function (21) has two components with opposite frequencies. When both of their localized envelopes have the zero velocity, i.e., $V_{\rm en}=0$, a SBS will be generated. It means that the formation of a SBS is the result of synchronous propagation of the two components. To confirm this guess, for a wave function with frequency $\omega$ and steepness $\eta$, we show the dependence of $V_{\rm en}$ on $\beta _4$ in Fig. 6(a), when $\omega =2.1678$, $\eta =0.4466$, and $\beta _2=0.75$. When $\beta _4=-1$, we have $V_{\rm en}=0$. The initial state has the form of

 

Fig. 6. (a) Dependence of $V_{\rm en}$ on $\beta _4$ when $\omega =2.1678$, $\eta =0.4466$, and $\beta _2=0.75$. The insets are the amplitude evolution of the initial state (28) when $\beta _4=-2$, $-1$, and $-0.5$. (b) Dependence of $V_{\rm en}$ on $\omega$ when $\eta =0.4466$, $\beta _2=0.75$, and $\beta _4=-1$. The insets are the amplitude evolution of the initial state (28) when $\omega =1$, $2.1678$, and $3$.

Download Full Size | PDF

Similarly, we also show the dependence of $V_{\rm en}$ on $\omega$ in Fig. 6(b), when $\eta =0.4466$, $\beta _2=0.75$, and $\beta _4=-1$. The evolution plots in the cases of $\omega =1$, $2.1678$, and $3$ are shown as the insets in Fig. 6. Only in the case of $\omega =2.1678$, the SBS can be generated. In the cases of $\omega \neq 2.1678$, the two components propagate towards opposite directions.

Finally, we summarize the characteristics of different nonlinear waves in Table 1. They contains the traditional solitons, the striped solitons, the pure-quartic solitons, and periodic waves. We recall that the quantity $\varphi$ is defined by $\varphi =\tan ^{-1} ({\omega _s}/{\eta _s})$ to describe the strength of a wave’s periodicity relative to its localization, and $\omega _s$ and $\eta _s$ are the parameters determining the periodicity and localization of a wave, respectively. A traditional soliton has the localization but no the periodicity (i.e., $\omega _s=0$ and $\eta _s>0$), while a periodic wave has the periodicity but no the localization (i.e., $\omega _s>0$ and $\eta _s=0$). A striped soliton has both of the periodicity and localization, and therefore it corresponds to $\omega _s>0$ and $\eta _s>0$. The pure-quartic soliton is a particular case of the striped solitons when the periodicity and localization carry equal weight, i.e., $\omega _s=\eta _s$, so its proportion of periodicity and localization is $\omega _s/\eta _s=1$ and the characteristic quantity is $\varphi =\pi /4$.

Table 1. Characteristics of several nonlinear waves

View Table

5. Conclusion

In summary, the excitations of dark solitons in a nonlinear fiber with the second-order dispersion and FOD are investigated. In this system, we find the SDSs, their counterparts without the total phase step, and some multi-dark-soliton bound states. With the help of the MLSA method, we quantitatively analyze the periodicity and localization of the SDSs under different coefficients of second-order dispersion and FOD. Their oscillation frequency is successfully predicted and the corresponding MI is analyzed, which can make the propagation of SDSs unstable. For the SBSs, though their major characteristics have been studied in the previous works [22,23], we provide a new possible perspective to understand their formation mechanism. Our method and results can provide the theoretical guidance for the generations and manipulations of striped solitons in nonlinear fibers, and furthermore bring more possibility for their applications in soliton communications and optical measurements. Considering that stripe solitons have been also found in spinor Bose-Einstein condensates [45–48], we expect that our method can be applied in these systems and help with the observation of matter-wave striped solitons. We also note that the competition between periodicity and localization of a soliton is similar to the well-known wave-particle duality to some extent. Understanding the similarity between them is a challenging but might be important task, which is worthy of future’s study.