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Abstract
We prove that the dark solitons can be stable in the purely quintic nonlinear lattices, including the fundamental, tripole and five-pole solitons. These dark soliton families are generated on the periodic nonlinear backgrounds. The propagation constant affects the forms of these solitons, while the number of poles does not lead to the variation of the backgrounds. The dark solitons are stable only when the propagation constant is moderately large.
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1. Introduction
Solitons are formed by the balance of diffraction (or dispersion) and nonlinear effects [1–8]. Due to the collapse of solitons in the uniform nonlinear media [9,10], the inhomogeneous linear and nonlinear effects have been adopted to stabilize various types of soliton families [11–16]. For example, the periodic linear [17–24] and nonlinear potentials [25–29] are widely used to generate various types of solitons. The nonlinear lattices have been well understood now. They can be created by the photorefractive effect [30] in optics or by the Feshbach resonance [31,32] in Bose-Einstein condensates (BECs). Many types of bright solitons have been reported [33] in the cubic (Kerr) nonlinear lattices, such as the fundamental (single) and multipole solitons. Besides, the dark soliton families have been also demonstrated to exist in such periodic nonlinear structures [1]. It is well known that the uniform media can only generate the fundamental bright or dark solitons in nonlinear Schrödinger equation (NLSE). On the other hand, the nonlinear lattices can create both fundamental and multipole solitons, and the solitons can be located in the center of each period of the nonlinearity [1]. Note also that the soliton molecules, a type of multipole solitons (dipole solitons), have attracted more and more attentions in recent few years, and many relevant experiments have been reported [34–39].
In classic nonlinear optics, the light beam in the cubic (Kerr) nonlinearity has been widely investigated. Further, the experiments for optical solitons in quintic nonlinearity have been performed where the quintic nonlinearity is realized in colloidal waveguides which contain the suspensions of metallic nanoparticles [40]. After that, the quintic nonlinearity is attracting more and more attentions, and different types of solitons have been reported [41,42], such as bright solitons [43], multipole and vortex solitons [44]. In addition, singular solitons, a type of novel solitons, have also been predicted to be stabilized in the quintic nonlinearity [45]. These singular solitons feature sharp wave forms which are quite different from the common Gaussian-like profiles. Therefore, the studies of solitons in quintic nonlinearities are important and urgent. Currently, most of the reports on quintic nonlinearity focus on the bright solitons, and the dark solitons in such media require more attentions. Despite the solitons in cubic nonlinearity with a periodic linear potential have been reported recently [46], the dark solitons stabilized by the quintic nonlinear lattices (without any extra linear potential) are still missing. Here we propose that the quintic nonlinear lattices can be realized in colloidal waveguides under the suspensions of metallic nanoparticles [40] which also undergo the periodic photorefractive effect [30].
In this work, we focus on the dark solitons supported by purely quintic nonlinear lattices (without extra linear effect), and we find that the solitons can be found with the continuously varying value of propagation constant, while the solitons can only exist in the gap structure of propagation constant in Ref. [46]. In addition, we clearly display the backgrounds of these dark solitons in quintic nonlinearity in Fig. 1(a). The rest of this work is arranged as follows. In Sec. 2, we report our theoretical model and method. Then our numerical results on both the fundamental and multipole dark solitons are displayed in Sec. 3. Finally, we conclude the article in Sec. 4.
Fig. 1. Profiles of the backgrounds and dark soliton families with different values of propagation constant $b$ at $\mathrm {g}=1$: (a) backgrounds; (b) fundamental dark solitons; (c) tripole dark solitons; (d) five-pole dark solitons.
2. Theoretical model and method
In this work, the famous NLSE is used to describe the propagation of light beams in a quintic nonlinear lattice, written in a dimensionless form
Here $E$ represents the field amplitude, $z$ is the propagation distance, $\mathrm {g}>0$ stands for the quintic self-defocusing nonlinearity. Note that Eq. (1) can be also employed for the evolution of a BEC where $E$ and $z$ should be replaced by the wave function $\psi$ and time $t$ respectively.
The stationary solutions $U$ can be solved by $E=U~{\rm exp}(ibz)$ in which $b$ denotes the real propagation constant, then the stationary equation can be written as
In the field of BEC, the propagation constant $b$ has meaning of chemical potential.
Next we discuss the linear stability analysis of these solitons. The field amplitude is perturbed by $E=[U(x)+p(x)\mathrm {exp}(\lambda z)+q^{\ast }(x)\mathrm {exp}(\lambda ^{\ast } z)]\mathrm {exp}(ibz).$ Then, after the substitution of the above expression into Eq. (1), the eigenvalue problem for the instability growth is given by
Here ${\rm Re}(\lambda )>0$ is the instability growth rate. In this work, the power integral for the dark solitons is defined as
where $U_{backg}$ stands for the background of the dark solitons.3. Numerical results
Now, we discuss our numerical results. Firstly, we display the background states, see Fig. 1(a). Here the blue and yellow lines portray the background with propagation constant $b=-1$ and $b=-12$, respectively. We define the depth of the background by $U_{\rm backgmax}-U_{\rm backgmin}$ where $U_{\rm backgmax}$ and $U_{\rm backgmin}$ are the maximal and minimal value of $U_{\rm backg}$. It is obvious that both amplitude and depth of the background increase with the increase of $|b|$. Then we present the waveforms of fundamental, tripole and five-pole dark solitons in the rest of Fig. 1. The profiles of fundamental dark solitons, whose number of poles (troughs) $N_p=1$, are reported in Fig. 1(b). Here the cupped parts (which are near the position of $x=0$) of fundamental dark solitons will shrink when $|b|$ increase. Furthermore, by comparing the results in Figs. 1(a,b), we can clearly find that the backgrounds of fundamental dark solitons are the same as the backgrounds in panel (a). This means that the change in structures (particularly the cupped parts) of fundamental dark solitons does not lead to the change of the backgrounds. It should be mentioned that the there is phase shift $\pi$ between neighboring intensity maxima of these multipole solitons in this work, that is, the sign of the both sides for the zero points of these dark solitons are different. For example, if the one in the left side of zero point is positive, then the other one in the right side must be negative.
To further explore how the structures of dark soliton families affect the backgrounds, we portray the waveforms of tripole dark solitons (number of poles $N_p=3$) with different values of $b$ in Fig. 1(c) where the cupped parts of tripole dark solitons would also shrink if $|b|$ increases. Here, the period of the poles for tripole dark solitons is $\Delta d=\pi$, and note that the period of the background is also $\pi$. The backgrounds of tripole dark solitons are also the same as the one shown in panel (a). The profiles of five-pole dark solitons ($N_p=5$) with different values of $b$ are displayed in Fig. 1(d) where the cupped parts of five-pole dark solitons would shrink when $|b|$ increases. Here, the period of the pole for five-pole dark solitons is $\Delta d=\pi$, same as the period of tripole dark solitons in panel (c). It should be stressed that the backgrounds of five-pole dark solitons are also the same as the one shown in panel (a), similar to the results of fundamental and tripole dark solitons. And it means that the structures of multipole dark solitons do not lead to any change of the backgrounds.
To further investigate the relationships between the backgrounds and dark soliton families, the amplitudes of the backgrounds $A_{backg}$ and fundamental dark solitons $A_{U}$ versus propagation constant $b$ for different values of $\mathrm {g}$ are presented in Figs. 2(a,b) respectively. Obviously, the curves of $A_{U}$ versus $b$ for both the cases with $\mathrm {g}=1$ and $\mathrm {g}=2$ are totally the same with the ones of $A_{backg}$ versus $b$. In addition, we have also studied the relationships for the amplitudes of tripole and five-pole dark solitons versus $b$, and have found that they are totally the same with the ones of backgrounds and fundamental solitons shown in Figs. 2(a,b) respectively. And it means that the structures of multipole dark solitons do not affect the waveforms of the backgrounds, and these dark soliton families are generated based on the backgrounds.
Fig. 2. Amplitudes of background $A_{\rm backg}$ (a) and fundamental dark solitons $A_{\rm U}$ (b) versus $b$ for different values of $\mathrm {g}$. The power integral $\Delta P$ for fundamental (c) and tripole (d) dark solitons versus $b$ for different values of $\mathrm {g}$. The solid and dashed lines in panel (c,d) represent the stable and unstable domains respectively. The propagation dynamics of the dark solitons marked by S1-S4 are portrayed in Figs. 3(a, g, c, i) respectively.
Next we explore the stability of these nonlinear waves in Eq. (2), and we have found that all the backgrounds in such model are stable, at least up to $b=-24$. On the other hand, these dark soliton families are stable only when $|b|$ are moderate large, including the fundamental, tripole, and five-pole solitons. Thus, we conclude that such periodic nonlinear waves maybe the fundamental modes in such model, and the dark soliton families are the excited modes. The power integral $\Delta P$ versus propagation constant $b$ for tripole and five-pole dark solitons are displayed in Figs. 2(c,d) respectively. Here in panels (c,d), the solid and dashed lines represent the stable and unstable areas for these soliton families. It is interesting that the parameter $\mathrm {g}$ do not affect the stability domains for these soliton families. For example, the stable domains of fundamental dark solitons for both cases $\mathrm {g}=1$ and $\mathrm {g}=2$ are about $-11.7<b<-3.5$, and the ones of tripole dark solitons for both cases $\mathrm {g}=1$ and $\mathrm {g}=2$ are about $-9.6<b<-5.7$. The propagation dynamics of the dark solitons labelled by S1-S4 is shown in Figs. 3(a, g, c, i), respectively.
Fig. 3. Perturbed propagations of stable dark soliton families at $b=-8$: (a) fundamental dark soliton; (b) dipole dark soliton; (c) tripole dark soliton; (d) quadrupole dark soliton; (e) five-pole dark soliton; (f) six-pole dark soliton. Perturbed propagations of unstable dark soliton families at $b=-1$: (g) fundamental dark soliton; (h) dipole dark soliton; (i) tripole dark soliton; (j) quadrupole dark soliton; (k) five-pole dark soliton; (l) six-pole dark soliton. $\mathrm {g}=1$ for all panels. $x\in [-10\pi, 10\pi ]$.
Now we focus on the propagation of the dark soliton families. The perturbed evolution of various types of these dark soliton families is reported in Fig. 3. It should be mentioned that the top and bottom rows of Fig. 3 display the stable and unstable propagations respectively. Note also that the first column (from left to right side) to the last column of Fig. 3 show the propagations of dark solitons with $N_p=1$ to $N_p=6$, respectively. For example, the first column of Fig. 3 [panels (a,g)] shows the propagations of fundamental dark solitons ($N_p=1$), and the third column [panels (c,i)] presents the ones of tripole dark solitons ($N_p=3$). One can clearly see that the stable dark soliton families keep their waveforms and amplitude during the long-distance propagations, as depicted in the top row of Fig. 3. On the other hand, the waveforms of unstable dark soliton families diverge from inside to outside during the propagations.
It should be emphasized that the separation distance of all the multipole dark solitons reported in Figs. 1–3 is $\Delta d=\pi$. Here one may ask if the multipole dark solitons with different separation distances can be stabilized in such model? From our further study, we find that the multipole dark solitons can be stabilized with $\Delta d=\pi$, $\Delta d=2\pi$, and $\Delta d=3\pi$. Next we start to discuss the results of multipole dark solitons with $\Delta d=2\pi$ and $\Delta d=3\pi$ which are reported in Figs. 4 and 5. Portrayed in Figs. 4(a,b) are the profiles of tripole and five-pole dark solitons respectively. Note also that the blue and yellow lines in Figs. 4(a,b) are the waveforms of multipole dark solitons with $b=-1$ and $b=-12$ respectively, and these parameters are the same used in Fig. 1. It is obvious that the cupped parts of these multipole dark solitons shrink when $|b|$ increases, similar to the counterparts in Fig. 1. It should be mentioned that the obvious differences between Figs. 4(a,b) and Figs. 1(c,d) are the backgrounds, that is, some raised parts in the interval of each pole in these dark soliton families would appear in the former, while it would not happen in the latter. It can be also understood that the multipole dark solitons in Figs. 4(a,b) are also formed on the backgrounds in Fig. 1(a), here the separation distance of cupped parts is $\Delta d=2\pi$ now, not the $\Delta d=\pi$ in Figs. 1(c,d). The similarity can be also found in the results of tripole and five-pole dark solitons with $\Delta d=3\pi$ which are displayed in Figs. 4(c,d) respectively. According to Figs. 4(c,d), the cupped parts of multipole dark solitons with $\Delta d=3\pi$ would also shrink when $|b|$ increases. Furthermore, the backgrounds of such dark solitons are different from the ones of $\Delta d=\pi$ and $\Delta d=2\pi$, that is, two gaussian like pulses would raise in the interval of each pole in these dark soliton families. This can be also understood that the multipole dark solitons in Figs. 4(c,d) are formed on the backgrounds in Fig. 1(a), and the separation distance of cupped parts is $\Delta d=3\pi$ here.
Fig. 4. Profiles of multipole dark solitons with different values of $b$ and different separation distances $\Delta d$ between each pole at $\mathrm {g}=1$: (a) for tripole dark solitons with $\Delta d=2\pi$; (b) for five-pole dark solitons with $\Delta d=2\pi$; (c) for tripole dark solitons with $\Delta d=3\pi$; (d) for five-pole dark solitons with $\Delta d=3\pi$.
Fig. 5. Perturbed propagations of stable dark soliton families at $b=-8$: (a) tripole dark soliton with $\Delta d=2\pi$; (b) five-pole dark soliton with $\Delta d=2\pi$; (c) five-pole dark soliton with $\Delta d=3\pi$. Perturbed propagations of unstable dark soliton families at $b=-1$: (d) tripole dark soliton with $\Delta d=2\pi$; (e) five-pole dark soliton with $\Delta d=2\pi$; (f) five-pole dark soliton with $\Delta d=3\pi$. $\mathrm {g}=1$ for all panels. $x\in [-10\pi, 10\pi ]$
Finally we discuss the propagations of the multipole dark solitons with $\Delta d=2\pi$ and $\Delta d=3\pi$. Such propagations are reported in Fig. 5 where the top and bottom rows present the stable and unstable propagations respectively. Note also that the left, middle, and right columns portray the results for the tripole solitons with $\Delta d=2\pi$, five-pole solitons with $\Delta d=2\pi$ and five-pole solitons with $\Delta d=3\pi$, respectively. From Fig. 5, the stable multipole dark solitons always keep their waveforms and amplitude during the propagations, while distortion happen in the waveforms of unstable dark solitons from inside to outside during the propagations.
4. Conclusion
To conclude, we have surveyed the dark soliton existence in the purely quintic nonlinear lattices (without any extra linear potential) which can be realized in the context of nonlinear optics and BEC. In such model, a type of periodic nonlinear waves are robust and the dark soliton families are formed based on these backgrounds. In addition, we also find that the number $N_p$ of poles of these multipole dark solitons can be arbitrary, such as $N_p=1,2,3,\ldots$. Despite the cupped parts of these dark soliton families would shrink if the propagation constant $|b|$ increases, the structures of dark soliton families would not lead to the variations of the backgrounds. These backgrounds are completely stable, while the dark solitons are stable only in some range of propagation constant. Note also that the stable domains of multipole solitons are narrower than the ones of fundamental solitons. Finally, we have demonstrated that the separation distances of each pole of multipole dark solitons can be double or triple to the period of the backgrounds (also the same as the nonlinear lattices).
Funding
National Natural Science Foundation of China (12265015, 61827815, 62205224); Postdoctoral Research Start-up Funds of Zhejiang Normal University (ZC304021962).
Acknowledgments
We thank Prof. Dmitry V. Skryabin for improving the writing of this manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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