## Abstract

We study existence, bifurcation and stability of two-dimensional optical solitons in the framework of fractional nonlinear Schrödinger equation, characterized by its Lévy index, with self-focusing and self-defocusing saturable nonlinearities. We demonstrate that the fractional diffraction system with different Lévy indexes, combined with saturable nonlinearity, supports two-dimensional symmetric, antisymmetric and asymmetric solitons, where the asymmetric solitons emerge by way of symmetry breaking bifurcation. Different scenarios of bifurcations emerge with the change of stability: the branches of asymmetric solitons split off the branches of unstable symmetric solitons with the increase of soliton power and form a supercritical type bifurcation for self-focusing saturable nonlinearity; the branches of asymmetric solitons bifurcates from the branches of unstable antisymmetric solitons for self-defocusing saturable nonlinearity, featuring a convex shape of the bifurcation loops: an antisymmetric soliton loses its stability via a supercritical bifurcation, which is followed by a reverse bifurcation that restores the stability of the symmetric soliton. Furthermore, we found a scheme of restoration or destruction the symmetry of the antisymmetric solitons by controlling the fractional diffraction in the case of self-defocusing saturable nonlinearity.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

A well-known phenomenon is that a collimated light beam usually spreads during propagation in a linear optical medium due to diffraction. While in nonlinear optical media, the tendency for the optical beam to expand can be dynamically balanced by the opposing effect, which leads to the formation of spatial solitons [1,2]. These phenomena can be modeled by the linear and nonlinear Schrödinger equations (NLSEs), respectively, in the scalar and paraxial approximations (the slowly varying electric field amplitude), where the second-order derivative represents the diffraction effect for optical beams.

In recent years, much interest has been drawn to the propagation of optical beams governed by the fractional Schrödinger equation (FSE), where the fractional-order derivative implies the existence of unconventional diffraction effects. FSE was originally proposed by Hu and Kallianpur in a rigorous mathematical form [3], and by Laskin in quantum mechanics [4,5]. In the quantum theory, FSE was derived as a model in which Feynman path integrals over Brownian trajectories lead to the standard (non-fractional) Schrödinger equation, while path integrals over “skipping” Lévy trajectories leads to the space-fractional quantum mechanics [6,7].

The experimental realization of space-fractional quantum mechanics has been proposed by introducing a one-dimensional infinite-range tight-binding chain in a condensed-matter environment by Stickler [8], but it is challenging to a perform a direct identification. The analogy of formalisms between quantum mechanics and classical optics makes it possible to use optical waveguides as platform to investigate quantum concepts, which have been proposed in quantum mechanics but difficult to achieve in quantum mechanics, as in Ref. [9]. In fact, in the past ten years, research by direct analogy method, linear and nonlinear photonics have made important progress in at least two aspects: one is a PT-symmetric optical system and the other is topological photonics, see reviews [10–12].

In order to explore the propagation dynamics and eigenmodes of spatial optical beams, a more concrete idea was proposed based on optical cavities. Such a system can be governed by linear and nonlinear FSEs [13]. According to this approach, many striking properties have been revealed in the framework of FSE [14–27] . When the nonlinear effects have been added in the FSE [28,29] , recent theoretical works have predicted a variety of fractional solitons [30–59] . Two-dimensional symmetric and asymmetric solitons, as an important branch of soliton research, have been intensively investigated in conventional NLSEs [60–62], dissipative media modeled by complex Ginzburg-Landau equation [63], and parity-time-symmetric systems [64–66]. So far, such endeavors have focused mainly on conventional diffraction optics system. Moreover, although such a two-dimensional asymmetric soliton exists in a variety of systems, the restoration or destruction of the symmetry were controlled by nonlinearity [67]. Does the diffraction effect have an effect on the symmetry breaking of soliton? Thus, further explorations of physically relevant settings that allow maintaining such two-dimensional solitons and new regime of symmetry breaking bifurcations remains a relevant objective for further work.

In this work, we aim to explore the existence, bifurcation, stability and dynamics of the two-dimensional solitons in fractional NLSE with saturable nonlinearities. This system supports two-dimensional symmetric, antisymmetric and asymmetric solitons. And, different scenarios of symmetry breaking bifurcations emerge with the change of stability. Furthermore, we find a new scheme of restoration or destruction the symmetry of two-dimensional optical solitons by controlling the fractional diffraction. The paper is organized as follows. The model and the methods of numerically found the two-dimensional symmetric, antisymmetric and asymmetric solitons, and stability analysis are introduced in Sec. 2., which is followed by the numerical results and discussions about the existence, bifurcation and stability, dynamics of two-dimensional solitons, as well as identification of stable and unstable solitons in Sec. 3. The paper is concluded by Sec. 4.

## 2. Model and methods

#### 2.1 Model and reduction

We consider the beam propagation along the $z$-axis in a nonlinear isotropic medium with the saturable nonlinear correction to the refractive index, which can be described by the fractional NLSE

Eq. (1) can be cast in a normalized form by means of additional rescaling, $\Psi \left ( \xi ,\eta ,\zeta \right ) =\left ( k_{0}\left \vert n_{2}\right \vert L_{d}\right ) ^{1/2}A\left ( x,y,z\right )$, where $L_{d}=2k_{0}n_{0}w_{0}^{\alpha }$ is the diffraction length, $\xi =x/w_{0}$, $\eta =y/w_{0}$, $\zeta =z/L_{d}$ are the normalized coordinates scaled to characteristic width $w_{0}$ of the input beam and the diffraction length $L_d$, respectively. The effective potential is $V\left ( \xi ,\eta \right ) =2k_{0}^{2}w_{0}^{\alpha }n_{0}\left [ n\left ( x,y\right ) -n_{0}\right ]$.

The normalized fractional NLSE is

Experimental setups which realize the beam propagation with fractional diffraction effect were proposed in Refs. [13] and [15]. The setting is designed as a Fabry-Perot resonator, with two convex lenses and two phase masks inserted into it. The first lens converts the input beam into the Fourier space, then a central phase mask, whose position defines the position of the system’s Fourier plane, performs the transformation of the beam in the dual space, as per Eq. (2), and, eventually, the second lens converts the output beam back from the dual domain into the real (coordinate) one. Thus, the fractional diffraction is effectively executed in the Fourier representation of the field amplitude. The saturable nonlinearity can be incorporated in the setup by inserting a piece of a saturable nonlinearity material in the cavity. To maintain the effectively local form of the nonlinearity, the material should be inserted between an edge mirror and the lens closest to it, where the beam’s propagation takes place in the spatial domain (rather than in the Fourier space). The saturable nonlinearity terms can be induced in a low pressure rubidium vapor contained within a glass cell, which was used for the creation of stable bright spatial solitons [72].

Next, we address stationary solutions to Eq. (3) with propagation constant $\beta$

where the function $\psi (\xi ,\eta )$ obeys the equationThe two-dimensional potential is assumed as the following form

Soliton solutions of Eq. (5) are characterized by the integral power (norm), which is a dynamical invariant of Eq. (3), defined as

The Hamiltonian of Eq. (3) is

#### 2.2 Method of solving solitary wave solutions

To obtain numerical soliton solutions of Eq. (5), we employed the Newton-conjugate-gradient method to the fractional NLSE. Accordingly, Eq. (5) is rewritten as

whereHere, the propagation constant $\beta$ is considered as a parameter with a fixed value, while solution $\psi (\xi ,\eta )$ is generated by means of Newton’s iterations

where $\psi _{n}$ is an approximate solution, and correction $\Delta \psi _{n}$ is computed from the linear Newton’s equation where $L_{1n}$ is the linearization operator $L_{1}$ corresponding to Eq. (10), evaluated with $\psi$ replaced by approximate solution $\psi _{n}$:In the framework of this scheme, Eq. (13) can be solved directly by dint of conjugate gradient iterations [74,75], which yields symmetric, antisymmetric, and asymmetric soliton solutions. A noteworthy fact is that the above numerical scheme is applicable to both of real function and complex function for the soliton solution $\psi (\xi ,\eta )$.

#### 2.3 Linear stability analysis

To investigate the stability behaviors of these solitons, it is necessary to explore the perturbed solutions for a solitary wave solution $\psi \left ( \xi ,\eta \right )$ with the form

## 3. Results and discussion

#### 3.1 Two-dimensional symmetric, antisymmetric and asymmetric soliton solutions

We first consider the generic examples of soliton families. The two-dimensional symmetric, antisymmetric and asymmetric soliton solutions were produced using the Newton-conjugate-gradient method, applied to Eq. (5) with Lévy index $\alpha =1.5$, the self-focusing ($\sigma =+1$) and self-defocusing ($\sigma =-1$) saturable nonlinearities, and the strength of saturable nonlinearity $S=1$, while the potential $V(\xi ,\eta )$ is taken as per Eq. (6) with $V_{0}=4$, $\xi _{0}=1.5$, and $\chi _{_{0}}=1$.

Figure 1 displays typical symmetric, antisymmetric and asymmetric solutions of two-dimensional solitons with different values of norm for the self-focusing and self-defocusing saturable nonlinearities, respectively. Fig. 1(b) shows power curves $P(\beta )$ for stable and unstable symmetric soliton solutions, stable antisymmetric and stable asymmetric soliton solutions with the self-focusing saturable nonlinearity, where the stable asymmetric soliton solutions occurs when the integral power of symmetric solitons exceeds a critical value (which corresponds to the symmetry breaking bifurcation point), $P_{cr}\approx 1.25$, the respective propagation constant being $\beta _{cr}\approx 1.47$. The power curves of the symmetric and asymmetric soliton solutions form a pitchfork bifurcation (a supercritical bifurcation). Let us now consider the self-defocusing saturable nonlinearity. Fig. 1(c) shows power curves $P(\beta )$ for stable and unstable antisymmetric soliton solutions, stable symmetric and stable asymmetric soliton solutions of the self-defocusing saturable nonlinearity. Notice that the bifurcation originates from the branch of antisymmetric soliton solutions, contrary to the self-focusing saturable nonlinearity case, where it originated from the branch of symmetric soliton solutions. With increase of the integral power, stable antisymmetric soliton turns into an unstable one at the first critical point $P_{cr1}\approx 2.28$ with the propagation constant $\beta _{cr1}\approx 0.90$, where the first bifurcation (a supercritical bifurcation) occurs and a new branch of the power curve bifurcates from the base branch of antisymmetric solitons and crosses it at the intersection point $P_{c}\approx 15.66$ ($\beta _{c}\approx 0.543$), then merges into the base branch at the second critical point $P_{cr2}\approx 33.6$ with the corresponding propagation constant $\beta _{cr2}\approx 0.41$, and the second bifurcation (a reverse supercritical bifurcation) is produced. Beyond the second critical point, the unstable antisymmetric soliton turns back to a stable one, as shown in Fig. 1(c). The power curves of the antisymmetric solitons together with the asymmetric solitons form a bifurcation loops. It is worthy to note that, at the intersection point $P_{c}$, the antisymmetric and asymmetric solitons share the same propagation constant and the power. As examples, two-dimensional antisymmetric, symmetric, and asymmetric solitons with self-focusing saturable nonlinearity at $\beta =1.6$ are displayed in Fig. 1(d), (e), and (f), respectively. Fig. 1(g), (h), and (i) show the two-dimensional symmetric, antisymmetric, and asymmetric solitons with self-defocusing saturable nonlinearity at $\beta =0.543$, respectively.

#### 3.2 Symmetry breaking bifurcations at different values of Lévy index and strength of saturable nonlinearity

To further explore how the values of Lévy index and strength of saturable nonlinearity effect the symmetry breaking bifurcations, we have numerically solved Eq. (5) with different values of Lévy index and strength of saturable nonlinearity and produced the power curves of soliton families.

The numerical results of the symmetry breaking bifurcations for the self-focusing saturable nonlinearity are summarized in Fig. 2, and the values of the critical power and the respective propagation constant at the symmetry breaking bifurcation points are collected in Table 1. First, the bifurcations built of the power curves of the symmetric and asymmetric soliton solutions are always the pitchfork bifurcations for the self-focusing saturable nonlinearity. It is independent of the values of the Lévy index and the strength of saturable nonlinearity (see Fig. 2 and Table 1). Secondly, the critical power of symmetric soliton at the bifurcation point decreases with the decrease of the values of the Lévy index. It is easy to explain, the shapes of two humps of the symmetric solitons become narrower and taller as the Lévy index decreases (see Visualization 1), hence the resultant symmetric solitons would be more easily broken. It is also found that the critical power of symmetric soliton at the bifurcation point increases with the increase of the strength of the self-focusing saturable nonlinearity. It resembles the standard NLSE ($\alpha =2$), where the increase of the strength of saturable nonlinearity is conducive to the stabilization of solitons.

We extend now to the case of the self-defocusing saturable nonlinearity. The symmetry breaking bifurcations are summarized in Fig. 3, and the values of the critical power and the respective propagation constants at the symmetry breaking bifurcation points are collected in Table 2. There are two cases for the strong saturable nonlinearity ($S=1$): the symmetry breaking bifurcation is completely suppressed for the Lévy index $\alpha =1.9$ (see Fig. 3(a)), and the pitchfork bifurcations turn into the bifurcation loops for the Lévy index $\alpha =1.5$ and $\alpha =1.1$ (see Figs. 3(b) and 3(c)), where an antisymmetric soliton loses its stability via a pitchfork bifurcation, which is followed by a reverse pitchfork bifurcation that restores the stability of the symmetric soliton with the increase of the power due to the saturable nonlinearity effect. The bifurcations built of the power curves of the antisymmetric and asymmetric soliton solutions are the reverse pitchfork bifurcations for the medium and low strength of saturable nonlinearities (see Figs. 3(d)–3(i)). It is also found that the critical power of antisymmetric soliton at the first bifurcation points decreases with the decrease of the values of the Lévy index (see Figs. 3(b)–3(i) and Table 2), where the shapes of two humps of the antisymmetric solitons become narrower and taller as the Lévy index decreases (see Visualization 2). Conversely, the critical power at the second bifurcation point increases with the decrease of the values of the Lévy index(see Figs. 3(b) and 3(c)). Similar to the self-focusing cases, the critical power always decreases with the decrease of the values of the Lévy index for the reverse pitchfork bifurcations in Figs. 3(d)–3(i). The increase of the strength of the self-defocusing saturable nonlinearity can also stabilize the antisymmetric solitons, so that the critical power of antisymmetric soliton at the bifurcation point increases, when the strength of saturable nonlinearity increases.

Getting back to the consideration of types of the symmetry breaking bifurcations, it can be cast in a more definite form by means of plots of asymmetry coefficient for the numerically found soliton solutions as a function of the propagation constant. The asymmetry coefficient is defined as

where the integral power of soliton solution in Eq. (8) is naturally split in contributions from the left and right regions along the $\xi$-axis withThe symmetry breaking bifurcation diagrams are displayed in Fig. 4 and Fig. 5 for the self-focusing and self-defocusing saturable nonlinearities, respectively. In Fig. 4, the symmetry breaking bifurcations are always supercritical. Fig. 5(a) shows that the symmetry of antisymmetric solitons haven’t been broken. Rather, the symmetry breaking bifurcations for the strong saturable nonlinearity with the Lévy index $\alpha =1.5$ and $\alpha =1.1$ are observed to be convex shapes of the bifurcation loops in Figs. 5(b) and 5(c), these closed loops are opened by a supercritical bifurcation (the first bifurcations) and closed by a reverse supercritical bifurcation (the second bifurcations). Fig. 5(a)–5(c) clearly show that the symmetry of the antisymmetric solitons can be restored or destructed by increasing or decreasing Lévy index. It indicates that the symmetry breaking can be controlled using the fractional diffraction. Note that, in NLSE with the conventional diffraction, the restoration or destruction of the soliton’s symmetry can only be realized by changing the strength of nonlinearity, see Ref. [67]. Furthermore, the reverse pitchfork bifurcations in Figs. 5(d)–5(i) are all the reverse supercritical bifurcation for the low and moderate strength of saturable nonlinearities. All the symmetry breaking bifurcation diagrams by the present system are of the supercritical bifurcations, featuring no bistability.

#### 3.3 Stability and dynamics

Conclusions concerning the stability and instability of the symmetric, antisymmetric and asymmetric soliton solutions for the self-focusing and self-defocusing saturable nonlinearities, respectively, were obtained by dint of the computation of eigenvalues of Eq. (16). As mentioned above, the linear problem of Eq. (16) includes a term of the fractional derivative, so it is convenient to solve it by the Fourier collocation method.

Antisymmetric soliton solutions, which are the fundamental states, are all stable within the range of the calculated propagation constants for the self-focusing saturable nonlinearity. The instability development of unstable symmetric soliton solutions were catalyzed by adding small initial perturbations to them. And, the largest linear instability growth rates of symmetric and asymmetric soliton solutions are displayed in Fig. 6. Obviously, the turning point of symmetric soliton solutions from stable state to unstable state is exactly the symmetry breaking bifurcation point. Furthermore, the range of symmetry breaking can be reduced by increasing the values both of the Lévy index and the strength of the saturable nonlinearity.

Typical examples of evolutions for the stable antisymmetric soliton solution, the unstable symmetric and the stable asymmetric soliton solutions are displayed in Fig. 7, Fig. 8 and Fig. 9, respectively. To corroborate predictions of the linear stability analysis, evolution of the stable antisymmetric soliton solution at $\beta =1.45$, near the bifurcation point ($\beta _{cr}=1.47$), has been examined in direct simulation of Eq. (3) under the action of random-noise initial perturbations with a relative amplitude of $5\%$. Eigenvalue spectrum of linear stability analysis is shown in Fig. 7(a), the initial ($\zeta =0$) and evolved ($\zeta =2000$) field intensities are displayed in Fig. 7(b) and 7(c), respectively. The evolution of the stable antisymmetric soliton solution is displayed by iso-intensity surface in 7(d). It is seen that, at least upto $\zeta =2000$, the antisymmetric soliton solution remains stable. Next, we address the unstable symmetric soliton solution at $\beta =1.45>$ $\beta _{cr}=1.47$. The corresponding eigenvalue spectrum of linear stability analysis is shown in Fig. 8(a). Field field intensities of different propagation lengths are displayed in Fig. 8(b)-Fig. 8(f), respectively. The evolution of the unstable symmetric soliton solution is displayed by iso-intensity surface in 8(g). The results show that evolution does not tend to convert the unstable symmetric soliton solution into an asymmetric one that would be spontaneously pinned to one hump of potential, instead, it develops periodic oscillations between the two humps of potential. The eigenvalue spectra of linear stability analysis and evolution of the stable asymmetric soliton solutions with the same propagation constant $\beta =1.45$ have been shown in Fig. 9. It is seen that the stable asymmetric soliton solution can robustly propagate at least upto $\zeta =2000$ under the action of random-noise initial perturbations with a relative amplitude of $5\%$.

Finally, we consider the case of the self-defocusing saturable nonlinearity for the stability and dynamics of the soliton solutions. Symmetric soliton solutions (the fundamental states for the self-defocusing saturable nonlinearity) are all stable within the range of the calculated propagation constants. The details of the largest linear instability growth rates of antisymmetric and asymmetric soliton solutions are displayed in Fig. 10. Apparently, the symmetry breaking can be suppressed by increasing the values both of the Lévy index and the strength of the saturable nonlinearity. Especially, both the antisymmetric and asymmetric soliton solutions are all stable in Fig. 10(a) for Lévy index $\alpha =1.9$ and the strength of the saturable nonlinearity $S=1$.

Similarly, typical examples of evolutions for the stable symmetric soliton solution, the unstable antisymmetric and the stable asymmetric soliton solutions are displayed in Fig. 11, Fig. 12 and Fig. 13, respectively. In Fig. 11, the stable symmetric soliton solution with $\beta =0.92$, near the first bifurcation point ($\beta _{cr1}=0.90$), has been examined in direct simulation of Eq. (3) under the action of random-noise initial perturbations with a relative amplitude of $5\%$. The results of both the eigenvalue spectrum of linear stability analysis and the numerical evolution confirm that this symmetric soliton solution can robustly propagation. We have also considered the unstable antisymmetric and stable asymmetric soliton solutions with the same propagation constant $\beta =0.55$, ($\beta _{cr1}=0.90>\beta =0.55>$ $\beta _{cr2}=0.41$), the results are summarized in Fig. 12 and Fig. 13. Without the addition of initial perturbations, the unstable antisymmetric soliton solution develops oscillations in Fig. 12. The stable asymmetric soliton solution can robustly propagate at least upto $\zeta =2000$ under the action of random-noise initial perturbations with a relative amplitude of $5\%$ in Fig. 13.

## 4. Conclusion

We have numerically investigated the existence, bifurcation and stability of two-dimensional optical solitons in the framework of fractional nonlinear Schrödinger equation, which includes the fractional diffraction, characterized by Lévy index, and the self-focusing and self-defocusing saturable nonlinearities. The two-dimensional symmetric, antisymmetric and asymmetric solitons have been numerically found, where the branches of the asymmetric solitons were bifurcated from the base branch by way of symmetry breaking, featuring no bistability. The supercritical type bifurcations have been found for the self-focusing saturable nonlinearity. For the case of self-defocusing saturable nonlinearity, the bifurcation forms a convex shape of loop with $S=1$ at $\alpha =1.5$ and $\alpha =1.1$, respectively. At the high strength of self-defocusing saturable nonlinearities, the symmetry of the antisymmetric solitons can be restored or destructed by controlling the fractional diffraction. At the low and medium strength of self-defocusing saturable nonlinearities, the supercritical bifurcations have the shape of the reverse pitchforks. Their stability and instability regions, for different values of the Lévy index $\alpha$ and the strength of saturable nonlinearity, have been exactly predicted by linear stability analysis and confirmed by direct simulation. The results indicate that the increase of the values of Lévy index and the strength of saturable nonlinearity contribute to increase at different levels the stabilization of soliton in such a system.

These findings suggest other interesting scenarios for the existence, bifurcation and stability of the optical solitons generalizing the present one in fractional dimensions. In particular, it is relevant to extend the analysis for the symmetry breaking of two-dimensional solitons in a PT-symmetric system. Bridging the two areas could uncover the reciprocity between fractional diffraction effect and non-Hermiticity in complex systems.

## Funding

National Natural Science Foundation of China (11805141, 12075210); Applied Basic Research Program of Shanxi Province (201901D211424); Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (STIP) (2019L0782); “1331 Project” Key Innovative Research Team of Taiyuan Normal University (I0190364); Natural Science Foundation of Zhejiang Province (LR20A050001).

## Disclosures

The authors declare no conflicts of interest.

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