# Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results

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## Abstract

**:**

## 1. Introduction and the Basic Models

## 2. Basic Soliton Families in the One-Dimensional Fractional Medium

#### 2.1. Stationary States and Analysis of Their Stability

#### 2.2. The Quasi-Local Approximation for Modes Carried by a Rapidly Oscillating Continuous Wave

#### 2.3. The Scaling Relation and Variational Approximation (VA) for Soliton Families

#### 2.4. Numerical Findings

## 3. Further Results for Nonlinear Modes in One-Dimensional Fractional Waveguides

#### 3.1. Systems with Trapping Potentials

#### 3.2. Dissipative Solitons Produced by the Fractional Complex Ginzburg-Landau Equation (CGLE)

## 4. Vortex Modes in Two-Dimensional (2D) Fractional-Diffraction Settings

#### 4.1. Stationary Vortex Solitons

#### 4.2. Dynamics of Vortical Clusters

#### 4.3. Stabilization of 2D Solitons by the Trapping Potential

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Profiles of solitons with $\mu =-0.3$ (

**a**) and $-2.0$ (

**b**), predicted by the VA based on the Gaussian ansatz (23) and Euler–Lagrange equations (26) and their counterparts produced by the numerical solution of Equation (8) with Lévy index $\alpha =1.5$ and $g=1$. Reprinted with permission from Reference [39]. Copyright 2020 Elsevier.

**Figure 2.**Dependence of the soliton’s norm N on the propagation constant, $k\equiv -\mu $, for the Lévy index $\alpha =1.5$ and self-focusing coefficient $g=1$, as predicted by the VA based on the Gaussian ansatz (23) and Euler–Lagrange Equations (26). The corresponding dependence obtained from the numerical solution of Equation (8) is included too. Reprinted with permission from Reference [39]. Copyright 2020 Elsevier.

**Figure 3.**(

**a**) Dependence $N\left(\mu \right)$ and (

**b**) profiles of solitons labeled B1 and B2 in (

**a**) (corresponding to $\mu =-0.2$ and $-2$, respectively), as produced by numerical solution of Equation (8) in the free space ($V=0$), for a fixed value of the Lévy index, $\alpha =1.1$, and nonlinearity coefficient, $g=1$. (

**c**) Dependence $N\left(\alpha \right)$ and (

**d**) profiles of solitons labeled B3 and B4 in (

**c**) (corresponding to $\alpha =1.7$ and $1.1$, respectively) for a fixed propagation constant $-\mu =1.5$ and $g=1$. Blue and red segments in panels (

**a**,

**c**) denote subfamilies of stable and unstable solitons, respectively. The black dashed line in (

**a**) represents the analytical scaling relation (20). The value marked by the arrow is the $\mu $-independent one (27), predicted by the VA for the degenerate family of the quasi-Townes solitons. Its numerically found counterpart is given by Equation (30). The stability and evolution of the solitons labeled by B1–B4 are displayed in Figure 4. The plots are borrowed from Reference [47] (unpublished).

**Figure 4.**The perturbed evolution of the solitons corresponding to labels B1–B4 in Figure 3a,c. The respective values of the parameters are $\alpha =1.1$, $g=1$, $\mu =-0.2$ (

**a**); $\alpha =1.1$, $\mu =-2$ (

**b**); $\alpha =1.7$, $\mu =-1.5$ (

**c**); $\alpha =1.1$, $\mu =-1.5$ (

**d**). The evolution is plotted in the spatial domain $-10<x<+10$. Values ${\mathrm{t}}_{\mathrm{m}}$ indicate intervals of the scaled propagation distance in the respective panels, $0<z<{\mathrm{t}}_{\mathrm{m}}$. In panels (

**a**,

**c**), larger ${\mathrm{t}}_{\mathrm{m}}$ is taken to corroborate the full stability of the solitons. The plots are borrowed from Reference [47] (unpublished).

**Figure 5.**An example of a stable dipole-mode soliton produced by Equation (1) with Lévy index $\alpha =1$, parabolic trapping potential (32) with ${\mathsf{\Omega}}^{2}=1$, i.e., $V\left(x\right)=0.5{x}^{2}$, and nonlinearity strength $g=0.5$. The propagation constant of this state is $-\mu =1.16$. The stationary profiles of the soliton and its propagation, in terms of the local-power distribution, are displayed in the left and right panels, respectively. The solid line in the left panel represents the analytical fit provided by Equation (33) with $n=1$. Adapted with permission from Reference [34]. Copyright 2020 Elsevier. Panel (a) from the original figure is not included here, as it is not necessary.

**Figure 6.**(

**a**) Blue, red, and black lines show the norm (alias power, here denoted P) of symmetric, asymmetric, and antisymmetric solitons vs. their propagations constants (denoted $\beta \equiv -\mu $ here) in the fractional system based on Equation (1) with $g=1$ (the self-focusing nonlinearity) and the double-well potential (34). The solid and dotted segments of the blue line represent, respectively, stable and unstable subfamilies of symmetric solitons below and above the symmetry-breaking bifurcation. The branches of antisymmetric and asymmetric solitons are completely stable. Here and in Figure 7 and Figure 8, the parameters of the potential (34) are ${V}_{0}=1$, $w=1.4$, and ${x}_{0}=1.5$. Panels (

**b**–

**d**) display a pair of stable asymmetric solitons (mirror images of each other), an unstable symmetric soliton, and a stable anti-symmetric one, all taken at $\beta =1.75$ (these solitons correspond to dots b${}_{1,2}$, c, and d in panel (

**a**)). Reprinted with permission from Reference [42]. Copyright 2020 Elsevier.

**Figure 7.**(

**a**) Blue, red, and black lines show, respectively, the norm (power, here denoted P) of antisymmetric, asymmetric, and symmetric solitons vs. their propagations constants (denoted $\beta \equiv -\mu $ here) in the fractional system based on Equation (1) with the double-well potential (34) and $g=-1$ (the self-defocusing nonlinearity). Other parameters are the same as in Figure 6. The solid and dotted segments of the blue line represent, respectively, stable and unstable subfamilies of antisymmetric solitons below and above the symmetry-breaking bifurcation. The branches of antisymmetric and asymmetric solitons are completely stable. Panels (

**b**–

**d**) display a pair of stable asymmetric solitons (mirror images of each other), an unstable antisymmetric soliton, and a stable symmetric one, all taken at $\beta =1.12$ (these solitons correspond to dots b${}_{1,2}$, c, and d in panel (

**a**)). Reprinted with permission from Reference [42]. Copyright 2020 Elsevier.

**Figure 8.**The perturbed evolution of unstable symmetric (

**a**) and antisymmetric (

**b**) solitons, produced by simulations of Equation (1) with the double-well potential (34), Lévy index $\alpha =1.1$, and nonlinearity strength $g=+1$ (

**a**) and $-1$ (

**b**). The propagation constants of the unstable symmetric and antisymmetric solitons are $\mu =-1.62$ and $\mu =-1.12$, respectively. Reprinted with permission from Reference [42]. Copyright 2020 Elsevier.

**Figure 9.**Charts of different established states produced by simulations of the fractional CGLE with the cubic-quintic nonlinearity (35), plotted in planes of the most essential parameters: linear loss $\delta $ and Lévy index $\alpha $ ((

**a**), with fixed $\beta =0.1$), or the fractional-diffusion coefficient $\beta $ and $\alpha $ ((

**b**), with $\delta =0.1$). Other coefficients are $\epsilon =1.7$, $\mu =1$, and $\nu =0.115$. In the underdamped (low-loss) parameter area B, a uniform state extends, in direct simulations, to occupy the entire spatial domain. In areas C and D, stable dissipative solitons emerge (directly in D and via an initial unstable-evolution stage in C). The input decays to zero in the overdamped area E. Reprinted with permission from Reference [39]. Copyright 2020 Elsevier.

**Figure 10.**The merger of initially separated stable dissipative solitons with zero phase difference between them, as produced by simulations of the fractional cubic-quintic CGLE (35), with parameters $\delta =0.3$, $\beta =0.1$, $\epsilon =1.7$, $\nu =0.115$, and $\mu =1$, and different values of the Lévy index: $\alpha =1.5$ (

**b**), $1.8$ (

**c**), and $2.0$ (

**d**). Adapted with permission from Reference [39]. Copyright 2020 Elsevier. Panel (a) from the original figure is not included here, as it is not necessary" at the end of the caption.

**Figure 11.**The propagation constant, $\beta \equiv -\mu $, vs. the norm (power) of the 2D solitons with vorticities $s=1,2$ and 3. The power is defined here as $P\equiv {2}^{2/\alpha}{N}_{2\mathrm{D}}$, as shown in Equation (40). The results are produced by the numerical solution of Equation (38) for fixed values of the Lévy index: $\alpha =1$ (

**a**), $1.3$ (

**b**), $1.5$ (

**c**), $1.7$ (

**d**), $1.9$ (

**e**), and 2 (

**f**). The latter case, corresponding to the usual two-dimensional NLSE, is included for comparison with the results produced by the fractional diffraction. Note that values of the power are shown on the logarithmic scale, which is different in different panels. Blue and red segments represent stable and unstable vortex states, respectively. Reprinted with permission from Reference [31]. Copyright 2020 Elsevier.

**Figure 12.**The threshold value of the power (norm), ${P}_{\mathrm{thr}}\equiv {2}^{2/\alpha}{N}_{\mathrm{thr}}^{\left(s\right)}\left(\alpha \right)$, below which Equation (38) cannot produce vortex solitons (see Equation (42)), vs. the Lévy index $\alpha $, for vorticities $s=1,2,$ and 3. In the limit of $\alpha =2$, which corresponds to the normal (non-fractional) diffraction, the threshold values ${N}_{\mathrm{thr}}^{\left(s\right)}(\alpha =2)$ coincide with the norms of the Townes solitons with the embedded vorticity, as shown in Equation (43). Reprinted with permission from Reference [31]. Copyright 2020 Elsevier.

**Figure 13.**Spontaneous splitting of unstable ring-shaped vortex solitons, with winding numbers s and propagation constants $\mu $, into sets of fragments, as produced by simulations of Equation (40) with $\alpha =1.5$: (

**a**) $\left(s,\mu =1,-0.03\right)$; (

**b**) $\left(2,-0.08\right)$; (

**c**) $\left(3,-0.095\right)$. The left and right panels display, respectively, the local power distribution in the input ($z=0$) and output taken at $z=500$. All panels show the domain of the $(x,y)$ plane of the size $\left(-150,+1.150\right)\times \left(-150,+1.150\right)$. Reprinted with permission from Reference [31]. Copyright 2020 Elsevier.

**Figure 14.**The evolution of necklace-shaped clusters initially built, as per Equations (44) and (45), of $M=5$ fundamental solitons and carrying overall vorticity $S=0$ (

**a**), $S=2$ (

**b**), and $S=1$ (

**c**,

**d**). The initial radii of the inputs and the angular velocity of their rotation, produced by the evolution are, respectively, $\left(R,\omega )=(11.25,0\right)$ (

**a**); $\left(10.35,0.0128\right)$ (

**b**); $\left(13,0.005\right)$ (

**c**); and $\left(10.35,0.0071\right)$ (

**d**). The results were produced by simulations of Equation (36) with Lévy index $\alpha =1$. In this figure, coordinates x, y and z are denoted, respectively, as $\xi $, $\eta $ and $\zeta $. The figure is borrowed from Reference [35].

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Malomed, B.A.
Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results. *Photonics* **2021**, *8*, 353.
https://doi.org/10.3390/photonics8090353

**AMA Style**

Malomed BA.
Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results. *Photonics*. 2021; 8(9):353.
https://doi.org/10.3390/photonics8090353

**Chicago/Turabian Style**

Malomed, Boris A.
2021. "Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results" *Photonics* 8, no. 9: 353.
https://doi.org/10.3390/photonics8090353