## Abstract

We investigate the existence and stability of in-phase three-pole and four-pole gap solitons in the fractional Schrödinger equation supported by one-dimensional parity-time-symmetric periodic potentials (optical lattices) with defocusing Kerr nonlinearity. These solitons exist in the first finite gap and are stable in the moderate power region. When the Lévy index decreases, the stable regions of these in-phase multipole gap solitons shrink. Below a Lévy index threshold, the effective multipole soliton widths decrease as the Lévy index increases. Above the threshold, these solitons become less localized as the Lévy index increases. The Lévy index cannot change the phase transition point of the PT-symmetric optical lattices. We also study transverse power flow in these multipole gap solitons.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The fractional Schrödinger equation (FSE) was proposed by Laskin to describe fractional quantum mechanics [1–3]. In 2015, the FSE was first introduced into optics by Longhi [4]. Based on the dynamics of laser light in aspherical optical resonators, an optical realization of the FSE was proposed. Later, the propagation dynamics of chirped Gaussian beams in the FSE with a harmonic potential were investigated [5], and zigzag and funnel-like propagation paths in one and two dimensions were discovered. Additionally, the propagations of beams in the FSE with one-dimensional (1D) and two-dimensional (2D) parity-time (PT)-symmetric periodic potentials (optical lattices) were studied [6]. Unique propagation properties were found in this study, including nondiffracting propagation and conical diffraction. For a PT-symmetric potential, the real and imaginary parts are even and odd functions, respectively [7,8].

In the nonlinear fractional Schrödinger equation (NFSE), the propagation dynamics of super-Gaussian beams have been investigated in one and two dimensions [9]. Gap solitons in the NFSE with a real optical lattice were first reported in [10]. Solitons can remain stable in finite gaps in focusing Kerr media. Multipole surface gap solitons in the NFSE with defocusing Kerr nonlinearity were also studied and these solitons are stable in the finite gap [11]. In a real 2D optical lattice, stable off-site and on-site vortex solitons in the NFSE were reported [12]. These solitons exist in the semi-infinite gap and are stable in the moderate power region. Additionally, as the Lévy index decreases, the stability region shrinks. Recently, stable four-pole, six-pole, and eight-pole in-phase solitons were also found in the NFSE with a real optical lattice [13]. Furthermore, the real nonlinear lattices can also support stable fundamental and multipole solitons in the NFSE [14].

In 2008, Musslimani *et al*. first investigated gap solitons in PT-symmetric optical lattices [15]. Subsequently, solitons in PT-symmetric potentials were studied widely [16–37]. PT-symmetric waveguides can support stable dark solitons [16,17]. Discrete and exact solitons were also found in PT-symmetric dual waveguides [18,19]. Fundamental solitons can be stable in PT-symmetric superlattices [20]. In PT-symmetric mixed linear-nonlinear optical lattices, the fundamental solitons remain stable in the low power region [21]. The fundamental and out-of-phase dipole solitons can also be stable in PT-symmetric optical lattices with nonlocal focusing nonlinearity [22,23]. In defocusing media, the PT-symmetric optical lattices can also support nonlocal fundamental solitons [24]. The stability of solitons in PT-symmetric optical lattices has also been analyzed carefully [25]. Stable vector solitons that is supported by the PT-symmetric optical lattices have been reported [26] and optical solitons in PT-symmetric lattices have been successfully observed in experiments [27].

Multipole (with three or more poles) solitons can be stable in PT-symmetric nonlinear optical lattices [28]. In PT-symmetric linear optical lattices, stable multipole gap solitons were reported in a competing cubic-quintic medium and in a defocusing Kerr medium [29,30]. PT-symmetric mixed linear-nonlinear optical lattices can also support stable multipole solitons [31]. In defocusing Kerr media, nonlocal in-phase multipole gap solitons with odd numbers of poles remain stable in PT-symmetric optical lattices [32]. In addition, 2D PT-symmetric optical lattices with defocusing Kerr nonlinearity can also support stable multipole solitons with even and odd numbers of poles [33]. Stable fundamental [34,35] and dipole [36] solitons were reported in the NFSE with 1D PT-symmetric aperiodic potentials or with optical lattices. We also discovered stable vector solitons in the NFSEs with 1D PT-symmetric optical lattices [37]. Moreover, dissipative surface solitons can also be stable in the NFSE [38]. Recently, stable nonlocal multipole solitons were first reported in the NFSE [39]. However, multipole gap solitons have not been investigated in the NFSE with PT-symmetric optical lattices to date.

In this article, we study the existence and the stability of both three-pole and four-pole gap solitons in the NFSE supported by 1D PT-symmetric optical lattices with defocusing Kerr nonlinearity. These multipole solitons exist in the first finite gap and cannot bifurcate from the lower edge of the first Bloch band. They are stable in the moderate power region. When the Lévy index decreases, the regions of stability of these multipole solitons become narrower. Below a Lévy index threshold, the effective widths of these multipole solitons can be reduced by increasing the Lévy index. Above this threshold, the increase of the Lévy index can broaden the effective widths of the multipole solitons. Furthermore, it is shown that the Lévy index cannot change the phase transition point of the PT-symmetric optical lattices and the transverse power flow in these multipole gap solitons is also examined.

## 2. Model

The normalized NFSE used to describe the light beam propagation in 1D PT-symmetric optical lattices with defocusing Kerr nonlinearity is given by [10–13,15]:

*U*,

*x*, and

*z*, respectively. ${( - \frac{{{\partial ^2}}}{{\partial {x^2}}})^{{\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right.} 2}}}$ and

*α*(1 <

*α≤*2) represent the fractional Laplacian and the Lévy index, respectively. In this article, we use the 1D PT-symmetric optical lattices with the form

*V*(

*x*) = 6[sin

^{2}(

*x*)+

*iW*

_{0}sin(2

*x*)], where

*W*

_{0}is the parameter that controls the relative amplitude of the imaginary part of the PT-symmetric optical lattices.

The stationary soliton solutions are assumed to have the form $U(x,z) = q(x){e^{i\mu z}}$, where *q*(*x*) is the complex function and *µ* is the real propagation constant. By substituting the above formula into Eq. (1), we can obtain the following equation:

To obtain the band structure, we consider the linear part of Eq. (2) as follows:

*µ*now is the propagation constant. According to the Bloch theorem [41], the eigenfunctions of Eq. (3) can be written as

*q*=

*f*(

_{k}*x*)e

*, where*

^{ikx}*f*(

_{k}*x*) is a periodic function that is given by

*f*(

_{k}*x*)=

*f*(

_{k}*x + T*). The period of the PT-symmetric periodic potential is also

*T*and

*k*($- \pi /T \le k \le \pi /T$) is the Bloch wave number. We then expand

*f*(

_{k}*x*) and

*V*(

*x*) into a series of plane waves:${f_k}(x) = \sum\nolimits_n {{C_n}} {e^{i{K_n}x}}$ and $V(x) = \sum\nolimits_m {{D_m}} {e^{i{K_m}x}}$. Here, ${K_n} = 2\pi n/T$ and ${D_m} = \int_T {V(x){e^{ - i{K_m}x}}} dx/T$. By inserting these series into Eq. (3), we can finally obtain the eigenvalue equation: Equation (4) is an eigenvalue problem in the form of a matrix. By solving Eq. (4) numerically, we can then obtain the band structure of the PT-symmetric optical lattices in the FSE. When

*W*

_{0}=0.1, the band structures for

*α*=1.6 and

*α*=1.3 are as shown in Figs. 1(a) and 1(d), respectively. The semi-infinite gaps for

*α*=1.6 and

*α*=1.3 are

*µ*≥4.1018 and

*µ*≥4.2625, respectively, while the first finite gaps when

*α*=1.6 and

*α*=1.3 are 1.1115≤

*µ*≤3.8046 and 1.4690≤

*µ*≤3.9156, respectively. We found that for different values of

*α*(1 <

*α≤*2), the phase transition point remains unchanged at $W_0^{th} = 0.5$. When $0 \le {W_0}\;<\;0.5$, the eigenvalue spectrum [

*µ*in Eq. (4)] is entirely real. Above the critical threshold, the bands become partially complex.

To test the stability of the multipole gap solitons in the NFSE, the perturbations *g*(*x*) and *t*(*x*) are added to the solitons as follows [25]:

*δ*is the perturbation growth rate. By substituting Eq. (5) into Eq. (1) and then linearizing it, the following coupled eigenvalue equations can then be obtained:

*δ*exist with Re(

*δ*) > 0, the multipole gap solitons are linearly unstable; otherwise, the solitons are linearly stable.

## 3. Numerical results

First, we take *W*_{0}=0.1 and *α*=1.6. A continuous family of in-phase three-pole solitons then exists in the first finite gap. Their existence domain is 1.12≤*µ*≤3.67 and this family of three-pole gap solitons cannot bifurcate from the lower edge of the first Bloch band. Figure 2(a) shows the soliton power versus the propagation constant (*µ*). The in-phase three-pole solitons are stable in the moderate power region, where 1.68≤*µ*≤3.02. We use *µ*=1.8 as the stable case for discussion. The profile of this soliton is depicted in Fig. 2(b), where the blue and red lines represent the real and imaginary parts of the soliton, respectively. We also introduce the parameter $S = (i/2)(qq_x^\ast{-} {q^\ast }{q_x})$, which is associated with the transverse power flow density [15]. Figure 2(c) shows that *S* is not negative everywhere, so the direction of the transverse power flows from gain toward loss regions varies across the lattices. By solving Eq. (6) numerically, we then obtain the linear-stability spectrum of the in-phase three-pole soliton, as shown in Fig. 2(d). This spectrum indicates that the soliton can propagate stably, because all the eigenvalues are on the imaginary axis [Re(*δ*) = 0]. The stable propagation of the perturbed in-phase three-pole soliton [where random noise with 5% of the soliton amplitude is added to the simulation of Eq. (1)] is depicted in Fig. 2(e). In the high power region where 1.12≤*µ*≤1.67 and in the low power region where 3.03≤*µ*≤3.67, the in-phase three-pole solitons are unstable. The linear-stability spectra of the three-pole solitons for *µ*=1.15 and *µ*=3.5 [as shown in Figs. 2(f) and 2(h), respectively] clearly demonstrate that these two solitons cannot be stable, because there are values of Re(*δ*) > 0. Figures 2(g) and 2(i) show the corresponding unstable propagations of the two perturbed in-phase three-pole solitons.

Reduction of the Lévy index (*α*) to 1.3, a continuous family of in-phase three-pole solitons can also be found in the first finite gap. These solitons exist in the 1.47≤*µ*≤3.74 region and are stable in the moderate power region of 2.04≤*µ*≤3.08. Both the existence and stability regions are narrower when compared with those when *α*=1.6. We select *µ*=2.5 as the stable case for presentation here. The profile of the in-phase three-pole soliton is shown in Fig. 3(b). Figures 3(c) and 3(d) both indicate that the soliton is stable. In the high power region where 1.47≤*µ*≤2.03 and in the low power region where 3.09≤*µ*≤3.74, these in-phase three-pole solitons are unstable. We use *µ*=1.8 as an unstable case in the high power region. The unstable propagation of the perturbed soliton in this case is shown in Fig. 3(g). We also select *µ*=3.6 as the unstable case for the low power region. The linear-stability spectrum of the in-phase three-pole soliton and the unstable propagation of the perturbed soliton are illustrated in Figs. 3(h) and 3(i), respectively.

When *W*_{0}=0.1 and *α*=1.6, a continuous family of in-phase four-pole solitons can also exist in the first finite gap. These solitons exist in the region where 1.12≤*µ*≤3.66. This family of in-phase four-pole solitons also cannot bifurcate from the lower edge of the first Bloch band. It is different from the in-phase four-pole solitons in 2D PT-symmetric optical lattices with defocusing Kerr nonlinearity, which can bifurcate from the lower edge of the first Bloch band [33]. In the moderate power region where 1.69≤*µ*≤2.88, these in-phase four-pole solitons are stable. When compared with the three-pole solitons, the stability region is also narrower. In the case where *µ*=1.8, the profile of the in-phase four-pole soliton is shown in Fig. 4(b). This soliton is stable, as demonstrated in Figs. 4(d) and 4(e). We also select *µ*=1.15 and *µ*=3.6 as the unstable cases in the high and low power regions, respectively. Figures 4(g) and 4(i) show the corresponding unstable propagations of the two perturbed in-phase four-pole solitons, respectively.

When *α* is reduced to 1.3, a continuous family of in-phase four-pole solitons can also be found in the first finite gap. Their existence and stability regions are given by 1.47≤*µ*≤3.73 and 2.06≤*µ*≤2.92, respectively. These two regions are also narrower when compared with the family of in-phase four-pole solitons with *α*=1.6. As a stable case, Fig. 5(b) shows the profile of the in-phase four-pole soliton when *µ*=2.5. The corresponding stable propagation of the perturbed soliton is illustrated in Fig. 5(d). In the high power, the solitons are unstable. Figure 5(g) shows the unstable propagation of the perturbed in-phase four-pole soliton when *µ*=1.8. As the unstable case in the low power region, Fig. 5(i) shows the unstable propagation of the perturbed soliton when *µ*=3.6.

Figures 6(a) and 6(b) show the stability regions (gray domains) of the three-pole and four-pole solitons with different values of the Lévy index. The stability domains shrink with the decease of the Lévy index. Additionally, the families of multipole solitons with outer two lobes (out-of-phase with the inner poles) [30] can also be found in this model, but they are unstable. Moreover, fundamental and in-phase dipole solitons can also exist in the first finite gap. The fundamental solitons can bifurcate from the lower edge of the first Bloch band but the in-phase dipole solitons cannot. They are stable in the low and moderate power regions, respectively. With the increase of the Lévy index, the stability regions of the fundamental and in-phase dipole solitons also widen.

The integral form factor is defined as $\chi \textrm{ = }\int_{ - \infty }^\infty {{{|q |}^4}dx/{{(\int_{ - \infty }^\infty {{{|q |}^2}dx} )}^2}}$ and is inversely proportional to the effective soliton width [10,43]. When *W*_{0}=0.1 and *µ*=1.8, the form factors of the in-phase three-pole and four-pole solitons are plotted versus the Lévy index as shown in Figs. 6(c) and 6(d), respectively. There is a threshold value for *α.* Below this threshold, the effective widths of the in-phase multipole solitons become narrower with increasing *α*. Above the threshold, the increase of the Lévy index can broaden the effective multipole soliton widths. These results have not been reported previously.

## 4. Conclusion

In conclusion, we have investigated the existence and the stability of in-phase three-pole and four-pole solitons in the NFSE supported by 1D PT-symmetric optical lattices with defocusing Kerr nonlinearity. These solitons exist in the first finite gap and remain stable in the moderate power region. The decrease of the Lévy index is found to narrow the stability regions of these multipole solitons. There is a Lévy index threshold and the increase of the Lévy index can narrow the effective widths of these multipole solitons below the threshold. However, above the threshold, the increase of the Lévy index will broaden the effective soliton widths. Furthermore, the Lévy index cannot change the phase transition of the PT-symmetric optical lattices. The transverse power flow of these multipole solitons has also been examined.

## Funding

National Natural Science Foundation of China (11774068, 61675001); Natural Science Foundation of Guangdong Province (2017A030311025); Department of Education of Guangdong Province (2014KZDXM059, 2018KZDXM044).

## Disclosures

The authors declare no conflicts of interest.

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