This review article provides a concise summary of one- and two-dimensional models for the propagation of linear and nonlinear waves in fractional media. The basic models, which originate from Laskin’s fractional quantum mechanics and more experimentally relevant setups emulating fractional diffraction in optics, are based on the Riesz definition of fractional derivatives, which are characterized by the respective *Lévy indices.* Basic species of one-dimensional solitons, produced by the fractional models which include cubic or quadratic nonlinear terms, are outlined too. In particular, it is demonstrated that the variational approximation is relevant in many cases. A summary of the recently demonstrated experimental realization of the fractional group-velocity dispersion in fiber lasers is also presented.

The formal calculus based on the concept of fractional derivatives is a branch of formal mathematics that has been known in the course of ca. 200 years. Approximately 20 years ago, this concept had appeared in physics as an ingredient of fractional quantum mechanics, which is based on the Schrödinger equation with the fractional operator of kinetic energy, for the wave function of particles which, in their classical form, move by random *Lévy flights*. The fractionality of the Schrödinger equations of this type is characterized by the *Lévy index* (LI) $\alpha $, which takes values $0<\alpha \u22642$. It determines the form of the fractional kinetic-energy operator, as $ ( \u2212 \u2202 2 / \u2202 x 2 ) \alpha / 2$, the usual (nonfractional) quantum mechanics corresponding to $\alpha =2$. While fractional quantum mechanics remains far from experimental implementation, much interest has been drawn to the more recent proposal to emulate the fractional diffraction, modeled by the same equations, by means of the wave propagation in specially devised optical cavities. Many theoretical results have been reported for solitons, vortices, and other modes supported by the respective fractional Schrödinger equations, which include cubic or quadratic optical nonlinearities. Very recently, the first experimental implementation of the effective fractional group-velocity dispersion has been reported for the light propagation in a fiber-laser cavity. A great deal of the current interest in these theoretical and experimental studies suggest the relevance of providing a summary of the state of art in the area. Such a concise summary is offered by this article, which outlines basic models of the fractional wave propagation, and provides a brief overview of basic species of solitons produced by these models.

## I. INTRODUCTION: THE CONCEPT OF FRACTIONAL-ORDER DERIVATIVES

^{1}In Eq. (1), $ |k |$ implies that the same instability gain is produced by right- and left-traveling perturbations. In the framework of the phenomenological model,

^{2}one can introduce a model equation for a real order parameter $u ( x , t )$, which corresponds to the dispersion relation (1),

It is shown in detail below that, while there are different formal definitions of fractional differential operators, such as the one $ \u2212 \Delta $ in Eq. (4), the definitions which are relevant for the realization in physics are defined in terms of the action of the operators in the Fourier space, similar to the relation between Eqs. (1) and (2) in 1D or between Eqs. (3) and (4) in 2D.

^{3}and Joseph Liouville in 1832.

^{4}The development of these concepts has led to the abstract definition of what is known as the

*Caputo derivative*of non-integer order $\alpha $,

^{5,6}

*Riesz derivatives*,

^{7}see Eqs. (9) and (14) in Sec. II.

This paper aims to produce a concise overview of linear and nonlinear dynamics in physically relevant models of fractional media. Also included is a summary of the first experimental realization of the concept of the fractional derivatives, in the form of a fiber cavity, which emulates fractional group-velocity dispersion (GVD).^{8} The presentation is not comprehensive; in particular, while the main objective is to introduce models of linear and nonlinear fractional media that are relevant to physics, the summary of soliton solutions amounts to the presentation of a few examples, many other cases being only briefly mentioned. An earlier review of theoretical results for solitons in fractional models is provided in Ref. 9.

## II. THE ADVENT OF FRACTIONAL CALCULUS TO PHYSICS: FRACTIONAL QUANTUM MECHANICS

For the first time, fractional derivatives had appeared in the context of physically relevant models as the mathematical basis of *fractional quantum mechanics*. This theory was introduced by Laskin^{10,11} for nonrelativistic particles which move, at the classical level, by *Lévy flights* (random leaps).

### A. Classical particles moving by Lévy flights

*Lévy flights*), rather than in the form of the usual Brownian random walk. In this regime, the mean distance $ |x |$ of the particle, which moves by 1D “flights” from the initial position, $x=0$, grows with time $t$ as

*Lévy index*(LI) $\alpha $ takes values

^{12}

*fractional Schrödinger equation*(FSE).

### B. The fractional linear Schrödinger equations for Lévy-flying particles and the fractional Riesz derivative

^{10,11}(see also Ref. 13). It is based on the fundamental formalism, which defines quantum mechanics by means of as Feynman’s path integration. This formalism represents the quantum dynamics of a particle as a result of the superposition of virtual motions along all randomly chosen trajectories (paths). The superposition is defined as the integral in the space of all paths, $\u223c\u222bexp\u2061 [ i S ( path ) ]d( path)$, where $S$ is the classical action corresponding to a particular path.

^{14}To apply this concept, Laskin considered the superposition of the paths which correspond not to the Brownian random walks but to the Lévy flights. In the 1D setting, the result is FSE for wave function $\Psi ( x , t )$. In the scaled form, it is written as

*Riesz derivative*,

^{7}which is defined as the juxtaposition of the direct and inverse Fourier transforms ( $x\u2192p\u2192x$) of the wave function,

Thus, the derivation of the physically relevant model, viz., FSE, leads to the relatively simple Riesz fractional derivative. As concerns “more sophisticated” varieties of fractional derivatives, defined on abstract mathematical grounds, such as the Caputo derivative (5), they do not emerge naturally in these physical contexts.

^{11}

*chemical potential*] are looked for in the usual form

^{15}

^{,}

^{15}

The fractional calculus has found another important application to physics in the form of fractional kinetics. This concept, which was elaborated by Zaslavsky *et al*.,^{16,17} addresses, in particular, the kinetic theory, based on the fractional Fokker–Planck equations, for dynamics which may be intermediate between integrable and purely chaotic. It predicts fundamental effects such as anomalous transport and superdiffusion and models real kinetic phenomena such as advection of particles in diverse physical settings. However, this topic is not considered in detail in the present article.

### C. A conjecture: Fractional Gross–Pitaevskii equations (FGPEs)

^{18}

^{,}

*fractional Gross–Pitaevskii equation*(FGPE), with the usual kinetic-energy operator in Eq. (18) replaced by its fractional counterpart defined as per Eq. (8) or (11), remains a challenging objective, a natural expectation is that the equation sought for will take the following form, in the scaled notation:

^{9}

*spin–orbit coupling*(SOC) between the components.

^{19}The respective system of coupled FGPEs in 2D is

^{20}

*Lagrangian*,

*Ansatz*for the approximate solutions is adopted as

^{21,22}). The norm of

*Ansatz*(25) is $N= ( \pi / 2 \beta ) ( A + 2 + A \u2212 2 / ( 2 \beta ) )$.

The *Ansatz* (25) represents the *semi-vortex* (SV) type, which implies that components $ u +$ and $ u \u2212$ carry vorticities $0$ and $1$, respectively^{23} [see an example shown by means of cross sections of the two components in Fig. 1(b)]. The SV structure is an exact feature of the 2D solitons shaped by SOC, which is not restricted by the applicability of VA.

*Ansatz*(25) in Lagrangian (24) produces the respective

*effective Lagrangian*,

*stable SVs*in the interval of LI values

*Ansatz*(25), may produce reasonable results even for the complex system including the fractional diffraction (kinetic-energy operator), SOC, and the nonlinear interactions.

An exact property of the system is that $ N c ( SV )(\alpha =1)=0$, i.e., no SVs exist at $\alpha \u22641$. In the case of $\alpha \u22121\u2192+0$, the SV exists with an infinitesimal amplitude, hence it can be found as a numerical solution of the linearized version of Eq. (22) with $\alpha =1$. Cross sections of components $ | u \xb1 |$ of the latter solution are displayed in Fig. 1(b). Actually, these plots adequately represent the shape of the SV solitons in the general case, when the nonlinearity is present.

## III. THE EMULATION OF THE FRACTIONAL SCHRÖDINGER EQUATIONS (FSES) IN OPTICS

### A. Linear equations for the paraxial light propagation in optical systems emulating fractional diffraction

Realizations of the fractional quantum systems were proposed in solid-state settings, such as Levy crystals^{24} and exciton-polariton condensates in semiconductor microcavities.^{25} However, no experimental demonstration of the fractional quantum mechanics has been demonstrated thus far.

^{26}A scheme for the realization of this possibility was proposed by Longhi in 2015,

^{27}who considered the transverse light dynamics in optical cavities with the $4f$ (four-focal-lengths) structure. As shown in Fig. 2, the proposed setup incorporates two lenses and a phase mask, which is placed in the middle (Fourier) plane. The lenses perform the direct and inverse Fourier transforms of the light beam with respect to the single or two transverse coordinates, thus implementing Eq. (10) or (13) and the inverse form of these equations. The Fourier decomposition splits the beam into spectral components with transverse wavenumbers $ ( p , q )$, the distance of each component from the optical axis being, roughly speaking,

^{8}Thus, the light beam recombined by the right lens in Fig. 2 carries the phase structure corresponding to the action of the fractional diffraction.

In addition to that, the curved mirror at the left edge of the cavity in Fig. 2 introduces (prior to the action of the Fourier decomposition) a phase shift, which represents the action of potential $V ( x , y )$ in Eq. (11). The layer of the gain medium placed next to the mirror in Fig. 2 may be used to amplify particular modes in the transverse structure of the light beam.

The setup which is outlined above provides a single-step transformation of the optical beam. The continuous FSE (11) is then introduced as an approximation for many cycles of circulation of light in the cavity, assuming that each cycle introduces a small phase shift (29). The 1D version of the FSE is obtained as the obvious reduction of the 2D scheme.

### B. The emulation of the fractional diffraction in nonlinear optical systems: Fractional nonlinear Schrödinger equations (FNLSEs) in the spatial domain

#### 1. The cubic nonlinearity in 1D

*fractional solitons*, are characterized by their power [alias norm, cf. Eq. (23)],

*Vakhitov–Kolokolov*(VK) criterion, $dP/dk>0$

^{28,29}at $\alpha >1$, suggesting that the corresponding soliton family may be stable. The case of $\alpha =1$, which corresponds to $dP/dk=0$ in Eq. (36), implies the occurrence of the

*critical collapse*, which makes all solitons unstable, similar to the family of

*Townes solitons*produced by the 2D cubic nonlinear Schrödinger equation with the normal (non-fractional) diffraction ( $\alpha =2$).

^{29–31}For $\alpha <1$, relation (36) contradicts the VK criterion, yielding $dP/dk<0$, which implies strong instability of the solitons, driven by the

*supercritical collapse*.

^{29,30}

^{32,33}To this end, the Lagrangian of Eq. (32) is used [cf. Eq. (24)],

*Ansatz*for the solitons can be adopted as the usual Gaussian with width $W$ and amplitude $A$ [cf. Eq. (25)],

*Ansatz*(38) in Lagrangian (37) and integration yields the effective Lagrangian, in which the amplitude is eliminated in favor of the power by means of Eq. (39),

Typical examples of the fractional solitons with the parameters predicted by the Euler–Lagrange equations, $\u2202 L eff/\u2202P=\u2202 L eff/\u2202W=0$, and their comparison to numerical solutions obtained for the same values of $\alpha $ and $k$ are displayed in Fig. 3. Further, the characteristic of the soliton family in the form of the $P(k)$ dependence, as predicted by the VA and produced by the numerical solution, is plotted in Fig. 4. These results demonstrate that VA provides reasonable accuracy.

*quasi-Townes*solitons at $\alpha =1$, for which, as said above, the critical collapse takes place

^{33}

Numerical investigation of the stability of the fractional-soliton family demonstrates that some of them may be weakly unstable at values of LI which are relatively far from $\alpha =2$. The instability does not destroy the solitons, leading to the onset of small-amplitude intrinsic oscillations in them.^{34}

#### 2. Quadratic nonlinearity in 1D and 2D

In the case of quadratic (rather than cubic) self-focusing nonlinearity in the 1D space, the critical collapse takes place at $\alpha =1/2$, hence stable solitons may exist in the interval of $1/2<\alpha \u22642$,^{34} which is broader than its counterpart (27) in the case of the cubic self-focusing. In particular, the quadratic term may appear in Eq. (19) as $\u2212\epsilon |\psi |\psi $ with $\epsilon >0$, representing a correction to the 1D FGPE induced by quantum fluctuations around the mean-field state.^{35}

In the 2D space, the cubic self-focusing nonlinearity gives rise to the critical collapse, as mentioned above, for $\alpha =2$, and to the supercritical collapse at all values of $\alpha <2$, therefore, all the 2D fractional solitons are unstable 2D solitons, including ones with embedded vorticity, may be stabilized in the model including cubic self-focusing and quintic defocusing. As shown in Ref. 37, families of stable solitons produced by the 2D FNLSE with the cubic–quintic nonlinearity and $\alpha <2$ are qualitatively similar to their counterparts, which were studied in detail in the 2D equation with the same nonlinearity and normal (non-fractional) diffraction, corresponding to $\alpha =2$.^{38}

#### 3. Fast moving modes in the fractional medium: Reduction to the non-fractional equation

^{9}To this end, one can consider solutions built as a product of a slowly varying amplitude $\psi (x)$ and a rapidly oscillating continuous-waver (CW) carrier with large wavenumber $ P$,

*Ansatz*(45) in the nonlocally defined Riesz derivative in Eq. (9) leads to a quasi-local expression expanded in powers of small $1/ P$:

### C. Systems of fractional nonlinear Schrödinger equations with the cubic nonlinearity

#### 1. Fractional domain walls

*self-defocusing*cubic nonlinearity. The corresponding 1D system of coupled FNLSEs for wave amplitudes $\Psi $ and $\Phi $ is

^{26}Other positive values of $\beta $ are possible in photonic crystals.

^{26}

In addition to XPM, the two components may be coupled in Eq. (49) by linear terms with real coefficient $\lambda $. The linear coupling accounts for the twist of the waveguide in the case of the copropagation of linear polarizations or elliptic deformation of the waveguide in the case of circular polarizations.^{26}

^{40–42}which link different CW states at $x\u2192\xb1\u221e$, that are mirror images of each other,

*immiscibility*

^{43}of the two components

^{42,44}), substitution $ { U ( x ) , V ( x )}=(1/2) [ \lambda \u2212 k \u2213 W ( x ) ]$ reduces two Eq. (50) to a single one

#### 2. Couplers and spontaneous symmetry breaking

*self-focusing*acting in each equation, and linear coupling between the equations, introduces a dual-core optical waveguide

^{45,46}

*spontaneous symmetry breaking*(SSB), which destabilizes the obvious symmetric solitons, with $ U 1(x)= U 2(x)$, at a critical value of the soliton’s power, $P= P cr$, and replaces the unstable symmetric modes by asymmetric ones, with $ U 1(x)\u2260 U 2(x)$. The asymmetry of the solitons is naturally quantified by the normalized difference in powers of their components, i.e.,

The SSB phenomenology for solitons was studied in detail theoretically^{47–52} and was recently demonstrated experimentally in dual-core optical fibers^{53} with the second-order (non-fractional) GVD. A characteristic peculiarity of the SSB in this system is that the transition from the symmetric solitons to asymmetric ones is performed through a *subcritical bifurcation*^{54} (similar to a symmetry-breaking phase transition of the first kind), which means that a stable asymmetric soliton solution emerges at a value of $P$, which is slightly smaller than $ P cr$, i.e., in the *subcritical* range. In this case, the branch of the asymmetric soliton states, appearing at $P= P cr$, originally moves in the backward direction, being unstable (therefore, another name for this bifurcation is the *backward* one). Then, this branch turns forward, getting stable, at the turning point (see, e.g., the left panel of Fig. 7).

*Ansatz*, with width $W$, power $P$, and angle $\chi $ accounting for the norm distribution between the components is

*Ansatz*, the asymmetry parameter (58) is

*Ansatz*(61) yields

^{51,52}Note that the SSB point for the usual coupler is known in an exact form, which can be found without the use of VA,

^{47}

^{46}

The VA predictions following from Eqs. (65) and (67) for different values of LI $\alpha $ are plotted in Fig. 7, along with results produced by the numerical solution of Eq. (56). It is observed that the subcritical character of the SSB bifurcation becomes more pronounced with the decrease of $\alpha $. In the limit of $\alpha =1$, the dependence is available solely in the VA form, given by Eq. (68), while the numerical solutions were obtained only for $\alpha \u22651.2$.^{46} The VA-predicted $\Theta (P)$ dependence for $\alpha =1$, given by Eq. (68), takes the *extreme subcritical form* (defined as per Ref. 55, in which branch of the asymmetric solitons, going backward from the SSB point, never turns forward, always remaining unstable). The dependence (68) terminates at point $\Theta =1$, i.e., at $P=(6/\pi )ln\u20612\u22481.324$ (see the left panel in Fig. 7), as definition (58) does not admit values $ |\Theta |>1$.

In addition to that, Fig. 8 displays a set of numerically generated dependences $P(k)$ for different fixed values of LI. The SSB points are those separating stable and unstable segments of the symmetric-soliton branches, from which ones representing unstable asymmetric solitons stem. The stability boundaries, observed on the asymmetric branches for $\alpha =2.0,1.8,1.6,$ and $1.4$, correspond to the above-mentioned turning points, while the branches for $\alpha =1.2$ and $1.1$ do not reach those points; therefore, they seem completely unstable in Fig. 8.

The analysis of the SSB phenomenology was also developed for moving (tilted) two-component solitons (recall it is a nontrivial extension of the analysis because the fractional diffraction breaks the system’s Galilean invariance), demonstrating that the symmetry-breaking bifurcation keeps its subcritical character, and values of the power at the bifurcation point become lower than for the quiescent solitons.^{46} Collisions between moving solitons were studied too, with a conclusion that the collisions with small relative velocities lead to elastic rebound, which is followed by strongly inelastic symmetry-breaking interactions at intermediate values of the velocities, and, eventually, by restoration of the elasticity in collisions between fast solitons.^{46}

The analysis of SSB in solitons was also developed for the single-component FNLSE (30) with a symmetric double-well potential $V(x)$.^{40} On the contrary to what is shown here, in that case, the SSB bifurcation is of the *supercritical* (*forward*) type (i.e., it represents a symmetry-breaking phase transition of the second type). Furthermore, the same FNLSE with the self-defocusing nonlinearity [ $g<0$ in Eq. (30)] does not break the symmetry of the ground state trapped by the double-well potential, but the increase of $ |g |$ leads to spontaneous breaking of the *antisymmetry* of the first excited state trapped by the same potential.

## IV. THE FRACTIONAL GROUP-VELOCITY DISPERSION (GVD) IN FIBER CAVITIES: THE FRACTIONAL SCHRÖDINGER EQUATION IN THE TEMPORAL DOMAIN, AND ITS EXPERIMENTAL REALIZATION

Thus far, no experimental realization of the effective fractional diffraction for light beams in the spatial domain has been reported. Another option for the implementation of the concept of the fractional propagation in optics is to resort to the transmission of light pulses in the *temporal domain*, i.e., in fiber-based setups, with an effective fractional GVD. This option has been realized recently,^{8} thus providing the first experimental implementation of a fractional medium in any physical setting.

*reduced time*,

^{26}$\tau =t\u2212z/ V gr$, where $ V gr$ is the group velocity of the carrier wave. In this equation, the fractional GVD is determined by LI $\alpha $ [cf. Eq. (8)], $D$ is the corresponding coefficient, and the usual derivatives represent the second- and higher-order GVD, for $k=2$ and $k\u22653$, respectively, which act in the setup along with the fractional GVD.

The fiber-cavity setup, which was used for the experimental emulation of the fractional GVD is displayed in Fig. 9. It incorporates two holograms, with the left one shaping the input pulse. The second hologram, installed at the central position, was designed as the element similar to the phase mask in the setup presented above in Fig. 2 (it is displayed in the rotated form, to make its internal structure visible). It imposes the differential phase shift onto spectral components of the light signal, simulating the action of the fractional GVD as per Eq. (71) [cf. Eq. (29)].

^{8}Typical values of the dispersion coefficients in Eq. (71) are

^{26}although the case of the normal GVD, corresponding to $ \beta 2>0$, was considered too. The results were summarized as functions of two control parameters, namely, LI $\alpha $ and effective GVD length $ L GVD$ [which corresponds to $z$ in Eq. (71)]. The plane of these parameters is schematically defined in the inset to Fig. 9, where $\alpha $ varies from $0$ to $2$, while positive and negative values of $ L GVD$ correspond to the anomalous and normal second-order GVD.

The dynamics displayed in Fig. 10 is reversible, therefore, examples of the splitting, such as ones exhibited in panels Q1 and Q3, imply the possibility of *fusion* of colliding pulses into a single one, if the propagation distance $z$ varies in the opposite direction.

For the comparison’s sake, panels B2 in Fig. 10 demonstrate that, unlike the setup including the fractional GVD, the usual one, with $ L GVD=0$ and $\alpha =2$ [see area B2 in the inset in Fig. 9(b)], very quickly leads to the full destruction of the input pulse (note that largest propagation distance in panels B2 is $0.02$ km). Thus, the fractional GVD makes the outcome of the light propagation in the fiber cavity is much more nontrivial.

A remarkable fact is close proximity between the experimental findings and the corresponding theoretical results, which are presented, respectively, in rows (b) and (a) of Fig. 10. In addition to that, Fig. 11 reports a set of theoretical results for the same dispersion parameters as in Eq. (72), but with another value of the LI, $\alpha =1$. In particular, Fig. 11(a) demonstrates that, under the action of fractional-only GVD ( $ L GVD=0$), the evolution of the input is generally similar to that in the case when the usual second-order GVD is present too, but the fractional term is the dominant one (case Q1 in Fig. 10): the input pulse quickly splits in a pair of secondary ones. Such a setting with the purely fractional GVD was not realized experimentally, as the usual GVD cannot be eliminated in the real setup. Further, if strong anomalous or normal second-order GVD is included, corresponding to $ L GVD=+10$ and $\u221210$ in Figs. 11(b) and 11(c), respectively, the outcomes are also akin to what is shown for similar cases in panels Q2 and Q4 of Fig. 10: the fragmentation of the input into a non-expanding multi-jet pattern in the former case, or violent fission of the input in two loosely bound pulses in the latter case.

The experiments reported in Ref. 8 were also performed for the input pulse with the phase including a cubic term, $\u223c \tau 3$. It is well known that, in the framework of the usual linear Schrödinger equation with the second-order GVD, this term initiated the propagation of a self-accelerating self-bending Airy wave.^{56,57} In the interval of LI values $1\u2264\alpha \u22641.80$, the results reported in Ref. 8 (not shown here in detail) demonstrate splitting of the input into a self-bending quasi-Airy wave and an additional one propagating along a straight trajectory.

All the experimental results demonstrating the implementation of the effective fractional GVD, summarized above and reported in detail in Ref. 8, have been obtained in the linear regime. The interplay of the strongly fractional GVD, with LI values $\alpha =1$ and $\alpha =0.2$, and the fiber’s self-focusing nonlinearity is the subject of a very recent experimental work.^{58} In particular, it demonstrates spectral bifurcations of ultrashort optical pulses in the fiber-laser cavity, in the form of spontaneous transitions between the pulses with single- and multi-lobe structures.

## V. CONCLUSION

The aim of this paper is to present a concise summary of dynamical models of 1D and 2D media with fractional diffraction or dispersion, both linear and nonlinear ones. Two well-substantiated physical models of this type are fractional quantum mechanics for particles which, at the classical level, move by random Lévy flights,^{10,11,13} and the emulation of the effective fractional diffraction in optical cavities.^{27} These models include the Riesz fractional derivatives^{7} or the respective 2D fractional Laplacian, which are defined as per Eqs. (9) and (14). The Riesz derivative with LI (Lévy index) $\alpha $ amounts to the multiplication of the Fourier transform of the underlying wave field, $ \Psi ^(p)$, by $ |p | \alpha $. Equations (9) and (14) demonstrate that the Riesz derivatives and their 2D counterparts are represented, in the explicit form, not by differential operators, but rather by nonlocal integral ones.

The paper presents basic types of solitons produced by the nonlinear fractional models in a brief form, without an objective to produce a systematic review of the solitons in fractional systems. The absolute majority of theoretical results for the fractional solitons have been obtained by means of numerical methods.^{59–80} Nevertheless, the present review includes some quasi-analytical results based on the VA (variational approximation). Actually, the comparison with numerical findings demonstrates that the VA works surprisingly well too in these relatively complex nonlinear nonlocal models.

In addition to the theoretical results, the article includes a summary of the recently reported first experimental realization of the fractional optics. It is based on the fiber-cavity setup, in which effective fractional GVD (group-velocity dispersion) is implemented by means of a specially designed phase plate (computer-generated hologram).^{8}

There are essential topics dealing with the dynamics of fractional media which are not considered in this paper, which is not designed as a comprehensive review. One of such topics is the dynamics of fractional discrete systems.^{59–61} Others, which have attracted much interest recently (and may be a relevant subject for a more extensive review), are dissipative solitons and self-trapped vortices in models based on fractional complex Ginzburg–Landau equations,^{32,62–64} as well as solitons in fractional parity-time-symmetric systems.^{65–67}

There are many possibilities for further theoretical and, especially, experimental developments in this area. In particular, on the theoretical side, a challenging objective (which is mentioned in this paper) is to derive equations of the FGPE (fractional Gross–Pitaevskii equation) type for a BEC (Bose–Einstein condensate) of quantum particles, which are individually governed by the fractional Schrödinger equation. On the experimental side, obvious objectives are to realize the fractional diffraction in spatial-domain optics and to create the predicted fractional solitons in such settings.

## DEDICATION

The paper is devoted to the celebration of the 80th birthday of David K. Campbell.

## ACKNOWLEDGMENTS

I am grateful to the Editors of this special issue of Chaos, celebrating the 80th birthday of David K. Campbell, for the invitation to submit a contribution. I would like to thank my colleagues, with whom I had a chance to collaborate on topics addressed in this brief review: Y. Cai, G. Dong, Y. He, E. Karimi, S. Kumar, S. Liu, H. Long, X. Lu, P. Li, J. Li, T. Mayteevarunyoo, D. Mihalache, Y. Qiu, H. Sakaguchi, D. Seletskiy, D. Strunin, Q. Wang, J. Zeng, L. Zeng, L. Zhang, Y. Zhang, Q. Zhu, Y. Zhu, and X. Zhu. My work on this topic was supported, in part, by the Israel Science Foundation through grant No. 1695/22.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Boris A. Malomed:** Conceptualization (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Validation (lead); Writing – original draft (lead); Writing – review & editing (lead).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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