## Abstract

We investigate the properties of double-hump solitons supported by the nonlinear Schrödinger equation featuring a combination of parity-time symmetry and fractional-order diffraction effect. Two classes of nonlinear states, i.e., out-of-phase and in-phase solitons are found. Each class contains two families of solitons originating from the same linear mode in both focusing and defocusing nonlinear Kerr media. The critical phase-transition point increases monotonously with increasing Lévy index. For strong gain and loss, out-of-phase solitons in focusing media are stable in a wide parameter window and are almost completely unstable in media with a defocusing nonlinearity. The stability of in-phase solitons is opposite to that of out-of-phase solitons. In-phase solitons in defocusing media are stable in their entire existence domains provided that the gain-loss strength is below a critical value. Meanwhile, the stability region shrinks with the decrease of Lévy index. We, thus, put forward the first example of spatial solitons in fractional dimensions with a parity-time symmetry.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In recent years, two promising extensions of standard quantum mechanics were paid special attentions. One is the parity-time (*𝒫𝒯*) symmetric (non-Hermitian) extension of quantum mechanics [1,2] and the other is the space-fractional quantum mechanics [3–5]. More recently, both of them were introduced into optics [6–8]. While *𝒫𝒯*-symmetric systems have now been intensively studied in both linear and nonlinear optical regimes [9], beam evolution in nonlinear fractional Schrödinger equation (NLFSE) is still poorly understood [10,11].

*𝒫𝒯*-symmetry requires that a Hamiltonian shares a common set of eigenfunction with the *𝒫𝒯* operator [1,2]. The parity operator *P̂* is defined through the operations *p̂* → − *p̂*, *x̂* → − *x̂*, and the time reversal operator *T̂* is defined as *p̂* → − *p̂*, *x̂* → *x̂*, and *î* → − *î*. According to the normalized Schrödinger equation, one obtains *T̂Ĥ* = *p̂*^{2}/2 + *V*^{*}(*x*), which results in the relation *ĤP̂T̂* =*p̂*^{2}/2 + *V*(*x*) and *P̂T̂Ĥ* = *p̂*^{2}/2 + *V*^{*} (−*x*). Therefore, when the external potential *V*(*x*) satisfies *V*(*x*) = *V*^{*}(−*x*), the system is *𝒫𝒯*-symmetric.

The non-Hermitian Hamiltonian displays entirely real spectra, provided that the gain-loss strength is below a phase-transition or symmetry-breaking point [1, 2]. Various families of optical solitons were explored theoretically and observed experimentally in *𝒫𝒯*-symmetric systems with different forms of potentials [12–22]. For a review of early works, see e.g., [9] and references therein. Nonlinear modes in partially *𝒫𝒯* symmetric potentials [23, 24] and non-parity-time-symmetric complex potentials [25] with all-real spectra were also reported.

On the other hand, beam propagation in fractional Schrödinger equation (FSE) is attracting a growing interest these days. When the Brownian trajectories in Feynman path integrals are replaced by Lévy flights, the behaviour of fractional field and fractional-spin particles are described by FSE [3–5]. The space-fractional quantum mechanics described by FSE is fundamentally important for understanding the phenomena involving in the fractional effects, including the fractional quantum Hall effect [26], the fractional Talbot effect [27], the fractional Josephson effect [28], and the fractional quantum oscillator [29].

In 2015, by considering the similarity between the FSE and the paraxial wave equation, Longhi introduced FSE into optics for the first time [8]. He proposed an optical scheme to emulate the fractional quantum harmonic oscillator and found dual Airy beams. This path-breaking work aroused a number of studies on beam dynamics in the framework of FSE. Zhang *et al.* reported the propagation properties of diffraction-free beams [30], chirped Gaussian beams [31] and *𝒫𝒯* symmetry [32] in the FSE with or without an external potential. “Accessible solitons” (linear modes) in a parabolic potential [33,34] and propagation management in a double-barrier potential [35] were also investigated in FSE. We should stress that, in nonlinear configurations, there are only two published papers in which propagation of super-Gaussian beams [10] and gap solitons [11] were addressed.

In this article, we predict the existence of double-hump solitons in the nonlinear Schrödinger equation featuring a fractional-order diffraction. Nonlinear states in focusing and defocusing media bifurcating from the same linear eigenmode of the corresponding linear system are revealed. The detailed linear stability analysis reveals that out-of-phase solitons can be stable in focusing media, and stable in-phase solitons can be supported by a defocusing Kerr nonlinearity.

#### 2. Theoretical model

The dimensionless nonlinear fractional Shrödinger equation for the scaled wave function *q*(*x*, *z*) is written in the normalized Cartesian coordinates *x* and *z* [11,36],

*∂*

^{2}/

*∂x*

^{2})

^{α/2}is the fractional Laplacian with

*α*being the Lévy index (1 <

*α*≤ 2). If

*α*= 2, Eq. (1) degenerates to the conventional NLSE.

*γ*= ∓1 represents the focusing and defocusing Kerr nonlinearity, respectively.

*V*(

*x*) =

*V*+

_{r}*iV*is an external complex potential whose real part

_{i}*V*stands for the linear refractive-index modulation and imaginary part

_{r}*V*denotes the gain and loss.

_{i}We consider a particular form of potential *V*(*x*) = *g*^{2}(*x*) + *g*(*x*) + *iχg′*(*x*) [21], where *g*(*x*) is an arbitrary real and even function and *χ* represents the strength of gain-loss component. For simplicity, we select *g*(*x*) = *A*[sech(*x* − *x*_{0}) + sech(*x* + *x*_{0})] which leads to *V _{r}*(

*x*) =

*A*

^{2}[sech(

*x*−

*x*

_{0}) + sech(

*x*+

*x*

_{0})]

^{2}+

*A*[sech(

*x*−

*x*

_{0}) + sech(

*x*+

*x*

_{0})] and

*V*(

_{i}*x*) = −

*Aχ*[sech(

*x*−

*x*

_{0}) tanh(

*x*−

*x*

_{0}) + sech(

*x*+

*x*

_{0}) tanh(

*x*+

*x*

_{0})]. Obviously, the condition

*V*(

*x*) =

*V*

^{*}(−

*x*) is satisfied and thus the system respects a

*𝒫𝒯*symmetry. Throughout the rest of this paper, the potential height

*A*is assumed as 1.5 and

*x*

_{0}is set as 1.8. The corresponding potential with

*χ*= 1 is plotted in Fig. 1(a). The refractive-index modulation is small compared with the unperturbed index and is of the order of nonlinear correction. The fractional-order diffraction effect may be implemented by the scheme proposed by Zhang

*et al.*[30]. Equation (1) conserves several quantities, including the power or energy flow: $U={\int}_{-\infty}^{\infty}{\left|q(x)\right|}^{2}dx$.

To explore the nonlinear states bifurcating from the linear modes, it is necessary to study the discrete eigen-spectra of the corresponding linear fractional system. The dispersion relation can be solved by considering solutions of the linear version of Eq. (1) in the form *q*(*x*, *z*) = *w*(*x*) exp(*ibz*), where *w*(*x*) is the eigenmode and *b* is the propagation constant or wave number, respectively.

Two typical examples of numerically calculated spectra are shown in Figs. 1(c, d) for systems with *α* = 1.3 and 1.7, respectively. In the *𝒫𝒯*-symmetric potential, one observes coalescence of two double degenerate real eigenvalues leading to the appearance of two double degenerate complex conjugate eigenvalues at *χ* = *χ*_{cr}. The *χ*_{cr} is the so-called phase-transition point or symmetry-breaking point of a *𝒫𝒯*-symmetric system. The spectrum is purely real below it and becomes complex above it. When Lévy index decreases from 2, the critical gain-loss strength *χ*_{cr} decreases monotonously [Fig. 1(b)]. The disappearance of *χ*_{cr} for *α* ≤ 1.12 means that there is no phase-transition point and the system now is *𝒫𝒯* unbreakable [37].

The stationary solutions of nonlinear modes of Eq. (1) can be searched by assuming *q*(*x*, *z*) = *w*(*x*) exp(*ibz*), where *w*(*x*) is complex and represents the profile of light field; *b* is a propagation constant. Substitution of the expression into Eq. (1) yields the following nonlinear equation:

## 3. Numerical results and discussions

For the sake of distinction, according to the relative phase relation between the humps of real parts, we term nonlinear modes with two out-of-phase real parts as out-of-phase solitons and solitons with two in-phase real parts as in-phase solitons.

In the system with *α* = 1.7, the critical phase-transition point *χ*_{cr} = 1.262 [Fig. 1(b)]. Thus, *χ* = 1 we choose is below the *χ*_{cr} and the linear spectrum of this potential is all-real. It contains seven positive discrete eigenvalues. The largest of them is 2.962, from which continuous families of *𝒫𝒯*-symmetric solitons can bifurcate out.

Typical examples of out-of-phase solitons in defocusing media are shown in Figs. 2(a, c). The humps of real parts coincide with the real humps of potentials. While the real parts are odd symmetric about *x* = 0, the imaginary parts are even symmetric. When *b* is small, the amplitude of imaginary parts is comparable with that of real parts [Fig. 2(a)]. At fixed *χ*, the peaks of solitons grow with the decrease of propagation constant. They decrease with the growth of *χ* for fixed *b* [Fig. 2(c)].

Under a focusing nonlinearity (*γ* = 1), the peaks increases rapidly with the growth of *b* and solitons become more localized [Fig. 2(b)]. Meanwhile, in contrast to the cases in defocusing media, imaginary parts are small compared with real parts. The profiles of solitons at *b* = 3.6, *χ* = 1 in systems with different Lévy indices are shown in Fig. 2(d). One immediately finds that the variation of *α* does not influence the distribution of solitons obviously. However, as we will show later, this slight difference plays a key role on the stability of solitons.

The dependence of soliton power on the propagation constant is shown in Fig. 3(a). Similar to the solitons in conventional nonlinear Shrödinger equation, power is a monotonously decreasing function of *b* in defocusing media and a monotonously increasing function in media with a focusing nonlinearity. Out-of-phase solitons in both focusing and defocusing media originate from the same linear mode. While solitons in focusing media can exist for an arbitrary large *b*, there is a lower cutoff of propagation constant *b*_{co} in defocusing media. Soliton solutions cannot be found if *b* < *b*_{co}.

To shed more lights on the properties of solitons, we numerically calculate the ratio between the amplitude of the imaginary parts and the real parts of out-of-phase solitons and the results are illustrated in the top plot of Fig. 3(b). The *A*_{ratio} decreases with the growth of *b*. The smooth transition of *A*_{ratio} at *b* = 2.962 indicates again that solitons in focusing and defocusing media bifurcates from the same linear mode. The decreasing effective width of solitons (defined as ${W}_{\text{eff}}=\sqrt{{\int}_{-\infty}^{\infty}{\left|w\right|}^{2}{x}^{2}dx/U}$) implies that the localization degree of solitons become stronger with the growth of *b* and approaches to a fixed value as *b* → ∞.

To study the stability of solitons in the system described by NLFSE, we consider the eigenvalues of small perturbations on the stationary solutions in the form: *q*(*x*, *z*) = {*w*(*x*) + [*u*(*x*) − *v*(*x*)] exp(*λz*) + [*u*(*x*) + *v*(*x*)]^{*} exp(*λ*^{*}*z*)} exp(*ibz*), where *w* is the stationary solution solved from Eq. (2); *u*, *v* ≪ 1 are the infinitesimal perturbations, and superscript * denotes the complex conjugation. Substituting the perturbed solution into Eq. (1) and linearizing it around *w* yield a coupled linear equations of an eigenvalue problem [38]:

*λ*equal zero.

Out-of-phase solitons are stable when *b* ∈ [2.94, 6.05] [Fig. 3(a)]. Note that there exists a very narrow stable region ([2.94, 2.96]) where solitons are supported by the defocusing nonlinearity. When the power of solitons in focusing media exceeds a critical value, an oscillatory instability appears. The maximum growth rate shown in Fig. 3(c) also manifests that solitons can be stable in a relatively wide parameter window, even when *χ* = 1 is close to the phase transition point *χ*_{cr} = 1.26. Decreasing *χ* will result in the expansion of stability region of solitons in both focusing and defocusing nonlinear media. In the system with Lévy index *α* = 1.3, *χ* = 1 is beyond the corresponding symmetry-breaking point *χ*_{cr} = 0.976. Though the stability domain is compressed in the region [2.98 3.38], we still find stable solitons in focusing media [Fig. 3(d)]. It indicates that stable solitons are possible when the spectra of the corresponding linear system are completely complex. This phenomenon may be explained by the nonlinearly induced *𝒫𝒯* transition reported in [39].

Now, we consider another class of double-hump solitons whose real parts are in-phase. Contrary to the out-of-phase solitons, the humps of the real parts of in-phase solitons joint together and the imaginary parts exhibit an odd symmetry [Fig. 4]. Under defocusing nonlinearity, with the decrease of *b*, the amplitude of imaginary parts grows faster than that of real parts [Fig. 4(a)]. At the same *b* value, the humps of solitons with a small *χ* are higher than those of solitons with a large *χ* [Fig. 4(c)]. In focusing media, soliton becomes more localized. The profiles of solitons in systems with different Lévy indices do not vary obviously [Fig. 4(d)], just similar to out-of-phase solitons. We stress again the slight difference is crucial on the stability properties of solitons, since it can lead to the shift of phase transition point which significantly changes the stability.

We addressed there is a lower cutoff of propagation constant for out-of-phase solitons in defocusing media. In-phase solitons also have lower cutoffs below which no nonlinear modes can be found. While the lower cutoff increases with the gain-loss strength, the upper cutoff (corresponding to the largest discrete eigenvalue of the linear system) decreases slowly with the growth of *χ* [Fig. 5(b)]. In other words, under a defocusing nonlinearity, the existence domain of solitons (in the region between the *b*_{low} and *b*_{up} curves) shrinks with *χ*.

The stability property of in-phase solitons is opposite to that of out-of-phase solitons. More concretely, the stability domain of in-phase solitons in a system with *χ* = 1, *α* = 1.7 is *b* ∈ [1.876, 3.092] [Fig. 5(c)]. This region can be divided into two parts: one is *b* ∈ [1.876, 3.056) and another is *b* ∈ [3.056, 3.092]. The former belongs to the defocusing case and the latter narrow region is under the focusing nonlinearity. There also exists a very narrow stability region near *b*_{co}. We conclude that in-phase solitons are stable in a wide parameter window in defocusing media, contrary to out-of-phase solitons.

If one decreases the gain-loss strength *χ*, the stability region will expand quickly. For the system with *α* = 1.7 and a defocusing nonlinearity, in-phase solitons are stable in their entire existence domain provided that *χ* < 0.85. A representative example of the stability analysis result of solitons in a potential with *χ* = 0.6 is displayed in Fig. 5(c). Though the growth rate of solitons under the focusing nonlinearity resembles that of solitons in a potential with *χ* = 1.0, the instability of solitons in defocusing media disappears completely when *χ* = 0.6. The detailed spectra of the linearization operator of Eqs. (3) on solitons marked in Fig. 5(c) are shown in Fig. 5(d).

To verify the predictions of linear-stability analysis results, we exhaustively simulate the evolution of solitons by the split-step Fourier method. The direct numerical simulation results are in excellent agreements with the predictions of the linear stability analysis. Several representative examples are displayed in Fig. 6. The unstable out-of-phase soliton at *b* = 2.8 in defocusing media exchanges the energy between its two humps [Fig. 6(a)]. Out-of-phase solitons in focusing media with low or moderate power can propagate stably for an arbitrary distance [Fig. 6(b)]. Beyond the phase-transition point, stable solitons in focusing media are also possible [Fig. 6(c)]. The peaks of unstable in-phase solitons in a focusing medium grow abruptly at certain *z* [Fig. 6(d)]. At *χ* = 0.6, all in-phase solitons are stable in their whole existence domain, provided the nonlinearity is defocusing [Fig. 6(f)].

Finally, we should note that an interesting phenomenon occurring in *𝒫𝒯*−symmetric optical potentials in the form of *V*(*x*) = *g*^{2}(*x*)+*g*(*x*)+*iχg′*(*x*) with *g*(*x*) = *A* exp[−(*x* −*x*_{0})^{2}]+*A* exp[−(*x* + *x*_{0})^{2}] [21] cannot be found in the framework of NLFSE. In other words, no asymmetric solitons can be supported by the potential in the present paper even when *α* = 2.

## 4. Conclusions

In conclusion, we studied the properties of double-hump solitons supported by the nonlinear fractional Schrödinger equation with a *𝒫𝒯*−symmetric potential. Nonlinear states originating from the same linear modes in both focusing and defocusing Kerr media were revealed. The stability properties of out-of-phase solitons are contrary to those of in-phase solitons. At relatively large gain-loss strength, while stable out-of-phase solitons can be found in focusing media, stable in-phase solitons can be found in defocusing media. With the decrease of gain-loss strength, the stability domains of both classes of solitons expand. The variation of Lévy index changes the phase transition point of the system and thus alters the stability of solitons. We also found that, although the stability region is narrow, stable solitons beyond the symmetry-breaking point are possible in the present system. Our results and analysis methods and techniques can be easily generalized into the research on the dynamics of optical solitons in periodically *𝒫𝒯*−symmetric fractional system or matter-wave solitons in the fractional Gross-Pitaevskii equation with an external potential.

## Funding

National Natural Science Foundation of China (NSFC) (11374268, 11704339).

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