Nonlinear waves in optical and matter-wave media: A ... · linear modes in parity-time-symmetric systems, rogue waves in scalar, vectorial, and multidimensional nonlinear systems,

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NONLINEAR WAVES IN OPTICAL AND MATTER-WAVE MEDIA:A TOPICAL SURVEY OF RECENT THEORETICAL AND EXPERIMENTAL

RESULTS

BORIS A. MALOMED1,2,∗, DUMITRU MIHALACHE3

1Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering,Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel

2Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, IsraelCorresponding author∗: [email protected]

3Department of Theoretical Physics,Horia Hulubei National Institute for Physics and Nuclear Engineering,

Reactorului 30, RO-077125, P.O.B. MG-6, Bucharest-Magurele, Romania

Received February 19, 2019

Abstract. We present a survey of some recent theoretical and experimental stud-ies on nonlinear waves that form and propagate in a broad class of optical and matter-wave media. The article is structured as a resource paper that outlines a large seriesof results for spatiotemporal optical solitons, ultrashort (few-cycle) optical pulses, non-linear modes in parity-time-symmetric systems, rogue waves in scalar, vectorial, andmultidimensional nonlinear systems, and matter-wave solitons and vortices.

Key words: nonlinear waves, spatiotemporal optical solitons, few-cycle opticalpulses, parity-time-symmetric nonlinear waves, rogue waves,matter-wave solitons.

1. INTRODUCTION

The formation, propagation, and stability of nonlinear waves in optical andmatter-wave media is a research field of broad interest in the context of both fun-damental and applied studies, as can be seen from books and review articles on thetopic [1–17] and a series of influential works published in the course of the past threedecades – see, in particular, Refs. [18–37].

In this topical-resource paper we focus on some recent theoretical and expe-rimental studies dedicated to nonlinear waves in a variety of physical settings in-volving optical and matter-wave media. The paper is structured in six Sections, thefirst one being an introduction, and the last one presenting a conclusion. Other Sec-tions are dedicated to nonlinear waves in dispersive and/or diffractive optical media(Sec. 2), nonlinear localized structures in parity-time (PT )-symmetric systems (Sec.3), rogue waves (RWs) in various settings (Sec. 4), and localized matter-wave modes(Sec. 5). The chapters are, largely, mutually independent and can be used as resourcereferences in the corresponding research fields.

Romanian Journal of Physics 64, 106 (2019)

Article no. 106 Boris A. Malomed, Dumitru Mihalache 2

Among the plethora of nonlinear-wave patterns, two- and three-dimensional(2D and 3D) ones stand out due to their fundamentally important role in many physi-cal settings; see a recent viewpoint [9] on multidimensional solitons and their legacyin contemporary atomic, molecular, and optical physics and the review articles [11]and [16] on well-established results and novel findings in the area of multidimen-sional solitons in nonlinear optics and atomic Bose-Einstein condensates (BECs).

The phenomena of the formation of multidimensional solitons in optical mediaand their robust propagation over distances that are much longer that the underlyingdiffraction/dispersion lengths, were, in particular, investigated in detail in connectionto the fascinating possibility of using spatiotemporal optical solitons (alias “lightbullets” [18]) as data bits in all-optical processing devices [4, 20]. Other potentialapplications of temporal, spatial, and spatiotemporal solitons were elaborated in thecontext of the use of dissipative solitons in mode-locked lasers [38], supercontinuumgeneration in photonic crystal fibers [39, 40] and the use of ultrashort (few-cycle)solitons for the same purpose [41, 42], and the possibility of using temporal cavitysolitons as data carriers in all-optical buffers [43]. A very important application thathas drawn a great deal of current interest is the use of dissipative temporal solitonscirculating in microresonators for the generation of frequency combs [44, 45]; for areview on dissipative Kerr solitons in optical microresonators, see a recent paper byKippenberg et al. [46]. Applications of 3D matter-wave solitons are also envisaged– in particular, in highly precise interferometry [47–49].

The paper is organized as follows. In Sec. 2 we overview the recent research ac-tivity in the area of multidimensional solitons in optical media, namely: i) theoreticaland experimental studies of linear and nonlinear light bullets; ii) theoretical studiesof 2D and 3D solitons forming in carbon nanotubes, and iii) recent theoretical andexperimental activities in the area of ultrashort (few-cycle) optical pulses. Section3 overviews recent results on nonlinear localized modes in PT -symmetric opticalsystems. In Sec. 4 we briefly review recent work on RWs in scalar, vectorial, andmultidimensional physical settings. Multidimensional matter-wave localized states,including recently predicted and experimentally created “quantum droplets”, are ad-dressed in Sec. 5. Section 6 concludes the article.

2. NONLINEAR WAVES IN DISPERSIVE AND/OR DIFFRACTIVE OPTICAL MEDIA

It is well known that in nonlinear dispersive and/or diffractive optical mediathe multidimensional solitons (2D and 3D ones) are self-trapped as a result of thebalance between linear and nonlinear effects. The combined effects of diffractionand group-velocity dispersion tend to broaden the optical wave packet in both spatialand temporal dimensions. The nonlinear effects tend to induce self-focusing or defo-cusing of the optical waveform. In particular, temporal solitons form and propagate

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3 Nonlinear waves in optical and matter-wave media: A topical survey Article no. 106

stably over long distances in monomode optical fibers. The fiber solitons are one-dimensional (1D) waveforms and can be stable if certain conditions are fulfilled, see,for example, the book by Kivshar and Agrawal [1]. However, the multidimensionaloptical solitons (2D and 3D ones) can be stable only in specially selected physicalsettings. The self-focusing cubic (Kerr-type) effect leads to the unavoidable collapseof the optical wave, see a review paper by Berge [50] on the wave collapse withspecific applications to optics and plasmas, and a book by Fibich [51].

While fundamental (zero-vorticity) solitons are destabilized by the wave col-lapse in 2D and 2D settings, bright solitons with embedded vorticity are destroyed byinstability that breaks the axial symmetry of these topological modes, leading to theirsplitting into several fragments (typically, these are fundamental solitons) [16, 17].The stabilization of fundamental and vortex solitons in 2D and 3D settings is a pro-foundly important issue from both the theoretical and experimental point of view, aswell as for potential applications. To stabilize zero-vorticity solitons, non-Kerr non-linearities are widely used, of quadratic or saturable types, in which the collapse isarrested in 2D and 3D geometries. However, vortex solitons cannot be stabilized withthe help of these nonlinearities [52–55]. Another method uses effective potentialsinduced by photonic lattices, which readily stabilize both 2D and 3D fundamentaland vortex solitons [16, 17]. In media with competing optical nonlinearities (e.g.,self-focusing cubic terms in combination with self-defocusing quintic ones), vortexsolitons may be stable in some parts of their existence domains; we mention here theexperimental results demonstrating robust trapping of fundamental [56] and vortex(2+1)D solitons [57] in such media. In particular, transient stabilization of the vortexsoliton in the latter case was supported by nonlinear losses. In this connection, itis relevant to mention a new technique for the stabilization of weakly localized vor-tex beams (strictly speaking, ones with an infinite total energy) in Kerr media withself-focusing nonlinearity by means of higher-order dissipative terms representingmultiphoton absorption [58]. Kartashov et al. [59] have recently investigated theexistence and stability domains for rotating vortex clusters in optical media with in-homogeneous defocusing cubic nonlinearity growing toward the periphery. A seriesof stable vortex clusters nested in a common localized envelope have been shown toexist in this setting.

Zeng et al. [60] have recently introduced 1D and 2D models of optical andmatter-wave media with self-repulsive cubic nonlinearity, whose local strength issubject to spatial modulation. Flat-top solitons of various types, including fundamen-tal ones, 1D multipoles, and 2D vortices have been produced analytically and numer-ically in that model, and their stability domains have been identified by means of thelinear-stability analysis and direct numerical simulations. The problem of sponta-neous vortex nucleation in nonlocal nonlinear media has been recently addressed byBiloshytskyi et al. [61]. The formation and stability of vortex rings and vortex lines,

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emerging in the course of propagation of self-trapped wave beams in nonlocal non-linear media, have been investigated numerically. These types of vortex complexescan be seen as building blocks for spontaneous or engineered creation of complextopological patterns representing structured light, see Ref. [61] for further details.Liang [62] has studied the revolving and spinning of optical patterns by two coaxialspiraling elliptic beams in nonlocal nonlinear optical media, both analytically, using avariational method, and numerically. The self-induced periodic interfering behaviorof dual Airy beams in strongly nonlocal media has been studied by Zang et al. [63].The analytical and numerical results show that these dual Airy beams in stronglynonlocal media exhibit a periodic focusing and defocusing behavior, and form in-terference fringes between the corresponding focusing and defocusing positions, seeRef. [63] for details.

A plethora of management techniques can also be used to stabilize multidimen-sional fundamental solitons; see article [9] for a discussion of key physical mecha-nisms that may be used for producing robust multidimensional optical solitons.

2.1. LINEAR AND NONLINEAR LIGHT BULLETS: RECENT THEORETICAL ANDEXPERIMENTAL STUDIES

In what follows we briefly overview some recent theoretical and experimentalstudies of light-bullet formation and stability in both linear and nonlinear physicalsettings. Recently, a series of theoretical works dealing with spatiotemporal loca-lized modes in the free space have been published. In particular, Peng et al. [64]have considered controllable Airy-Airy vortex light bullets (2D vortex solitons of theAiry-Airy type in a system with quadratic nonlinearity were investigated too [65]).Generation, control, and manipulation of self-accelerating Airy-Laguerre-Gaussianlight bullets in strongly nonlocal 2D nonlinear media have been investigated, bothanalytically and numerically, by Wu et al. [66]. Salamin [67] has provided approxi-mate solutions for ultrashort tightly-focused radially polarized laser pulses in plasma,in the form of Bessel-Bessel light bullets.

Self-accelerating Airy-Ince-Gaussian and Airy-helical-Ince-Gaussian spatio-temporal wave packets in strongly nonlocal nonlinear media have been studied ana-lytically and numerically by Peng et al. [68]. Zhong et al. [69] have investigated3D Airy light bullets in self-defocusing nonlinear media. The 3D localized Airy-Cartesian and Airy-helical-Cartesian wavepackets in free space have been analy-tically investigated in a recent work by Huang and Deng [70]. Salamin [71] hasrecently reported analytic expressions for wave fields of ultrashort tightly-focusedBessel pulses with an arbitrary order (i.e., Bessel light bullets) in plasma. Self-trapped pulsed beams with finite power in nonlinear Kerr media excited by time-diffracting, spatiotemporal beams have been studied by Porras [72]. We also referhere to recent theoretical and experimental studies of spatiotemporal wave packets,

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which are pulsed optical beams that propagate diffraction- and dispersion-free, asa consequence of tight correlations between their spatial and temporal spectra, seeRefs. [73, 74] for details. In the framework of a graded-index model, the coexistenceof collapse and stable spatiotemporal solitons in multimode optical fibers has beeninvestigated by Shtyrina et al. [75], using a variational approach. The existence ofstable dipole-mode spatiotemporal solitons has been also predicted in Ref. [75].

Finite-energy spatiotemporally localized Airy wave packets have been discuss-ed in a recent work by Besieris and Shaarawi [76], which was motivated by an ear-lier study of Porras [77] on Gaussian beams diffracting in time. In a recent work,Sakaguchi [78] has put forward new models for multidimensional stable solitons inboth optical and matter-wave media. By means of both analytical and numericaltechniques, the existence of robust 1D and 2D tail-free self-accelerating solitons andvortices has been shown by Qin et al. [79], in models for either two-component BECscoupled by a microwave field, or optical media with thermal nonlinearities. Therobust propagation of multidimensional dark-soliton wave packets featuring Blochoscillations (BOs) has been recently studied numerically by Driben et al. [80]. Theinterplay of the self-focusing nonlinearity, refractive-index gradient, and normal tem-poral dispersion is able to support robust hybrid waves, which feature BOs of darksolitons in multidimensional photonic settings. The robust evolution of multidimen-sional light bullets of a mixed dark-bright type can also feature BOs, see Ref. [80]for details.

Next we overview a few recent theoretical and experimental works in the areaof spatiotemporal dissipative structures in various physical settings. These uniquestructures were theoretically predicted long time ago in two pioneering works byTuring [81] and by Prigogine and Lefever [82]. A plethora of dissipative structureshave been observed in nonlinear systems far from equilibrium in diverse settings inbiological, chemical, and optical systems; see an overview paper by Rozanov [83],a review by Tlidi et al. [84], and a recent comprehensive overview by Tlidi andPanajotov [85]. We also refer here to an Introduction by Tlidi, Clerc, and Panajotov[86] to a topical issue on dissipative structures in matter out of equilibrium, whichwas dedicated to the memory of Ilya Prigogine. It is worth mentioning that thegeneric dynamical model for describing the unique features of temporal, spatial, andspatiotemporal dissipative structures is the complex cubic-quintic Ginzburg-Landau(CQGL) equation in one, two, and three dimensions.

Veretenov, Fedorov, and Rosanov [87] have recently predicted a new wide classof 3D topological dissipative optical solitons (“hula-hoop solitons“) in homogeneouslaser media with fast saturable absorption, namely topological vortex and knotteddissipative 3D optical solitons that are generated by 2D vortex solitons. In a recentwork, Akhmediev et al. [88] have numerically investigated dissipative solitons withextreme spikes in either normal or anomalous dispersion regimes. It was found that

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such types of solitons exist in large domains of the parameter space of the complexCQGL equation, the variation of any of the five parameters of the model resulting ina rich structure of bifurcations. Grishin et al. [89] have reported experimental ob-servation of self-generation of chaotic dissipative multisoliton complexes supportedby competing nonlinear spin-wave interactions. Descalzi and Brand [90] have nu-merically investigated interaction scenarios for stationary and oscillatory dissipativesolitons in the framework of two coupled complex CQGL equations for counter-propagating waves. Two main interaction scenarios have been identified: (i) theoccurrence of bound states and compound states of exploding dissipative solitons,and (ii) the onset of spatiotemporal disorder due to the creation, interaction, and an-nihilation of dissipative solitons, see Ref. [90]. Milian et al. [91] have introduced anew class of self-sustained states, which may exist as single solitons, or build multi-soliton clusters, in driven passive cylindrical microcavities. These multidimensionalcavity solitons are stabilized by the radiation which they emit and exhibit pronouncedpolychromatic conical tails [91]. Kochetov and Tuz [92] have considered the effectof replication of fundamental dissipative solitons and vortices modeled by a 2D com-plex CQGL equation in the presence of a locally applied potential. It was found thatthe replication of fundamental dissipative solitons remains persistent in the presenceof strong fluctuations of the potential, while the replication of dissipative vortices isonly possible under very weak fluctuations [92].

In a series of works [93–95], the generation of stable vortex clusters, ring-shaped vortices, and multivortex ring beams has been investigated numerically in dis-sipative media with effective inhomogeneous diffusion. In a recent paper, Fedorov,Veretenov, and Rosanov [96] have predicted an irreversible hysteresis of internalstructure of 3D tangle dissipative optical solitons. Using the generic complexGinzburg-Landau model for dissipative localized structures, Kochetov [97] has in-vestigated controllable transitions between stationary and moving dissipative soli-tons, viz., transitions between solitons of the plain, composite, and moving types.We also mention here a recent summary [98] of analytical and numerical studies ofthe existence and stability of dissipative light bullets and diverse light-bullet com-plexes that form in a slew of physical settings, and a recent work by Djoko et al. [99]on robust propagation of optical vortex beams, necklace-ring solitons, soliton clus-ters, and uniform-ring beams that are described by a generalized (3+1)D complexGinzburg-Landau equation including higher-order effects.

Next we briefly overview some relevant recent experimental results reportedin the area of spatiotemporal dynamics of optical solitons in different physical set-tings. Numerous experimental studies of nondiffracting optical wavepackets, e.g.,Airy beams, Bessel beams, Airy-Bessel beams, etc., have been reported in the courseof past few years; see, for example, Refs. [100–104]. We also refer here to anoverview paper by Efremidis et al. [105] on recent advances in the area of Airy

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beams and accelerating optical waves in diverse physical settings.Recently, Liu et al. [106] have experimentally demonstrated the buildup of

soliton molecules in a mode-locked laser. External perturbations, such as a change ofthe light polarization state and the fluctuation of pump power, can strongly influencethe buildup process and the dynamics of soliton molecules, see Ref. [106]. Recently,Salhi et al. [107] have experimentally demonstrated the generation of a broadlytunable dissipative soliton resonance in dual-amplifier figure-of-eight fiber lasers.Asymmetric and single-side splitting of dissipative solitons described by complexCQGL equations with asymmetric wedge-shaped potentials have been reported byLiao et al. [108]. “Molecular complexes” of optical solitons in passively mode-locked fiber lasers have been recently observed by Wang et al. [109]. Ultrashortoptical pulses propagating in dissipative nonlinear systems can also interact and formoptical soliton molecules. Wang et al. [109] have demonstrated that two soliton-pair “molecules” can bind to form a stable complex. The formation of femtoseconddissipative solitons in an enhancement cavity with Kerr nonlinearity and a spectrallytailored finesse has been experimentally demonstrated in a recent work by Lilienfeinet al. [110]. The vast potential for diverse applications of these dissipative opticalsolitons includes spatiotemporal filtering, compression of ultrashort (femtosecond)laser pulses, and cavity-enhanced nonlinear frequency conversion, see Ref. [110] fordetails.

The work by Li et al. [111] has reported experimental realization of 3D versa-tile linear vortex light bullets, which combine a high-order Bessel beam and an Airypulse. These spatiotemporal wave packets propagate without distortion while carry-ing a nonzero orbital angular momentum. In a recent work [112], the creation of one-dimensional pulsed light sheets, which propagate self-similarly in free space in theabsence of nonlinearity or dispersion, has been demonstrated experimentally. Thesediffraction-free space-time light sheets may be useful in nonlinear spectroscopy andmicroscopy, see Ref. [112].

Lahav et al. [113] have recently reported experimental observation of 3D spa-tiotemporal pulse-train solitons in a photorefractive strontium barium niobate crystal,firing 800-nm pulses at it. For this wavelength, the crystal has a normal dispersion,and, as a consequence, the temporal profile of the waveform amounts to a dark pulsealthough the produced beam was a bright one, see also a brief discussion by Wise[114] of this important experimental achievement. The experimental demonstrationof the fact that the electric field can produce 3D solitary waves representing “bul-lets” of oscillating molecular director in nematic liquid crystals, self-trapped alongall three spatial dimensions, has been recently reported by Li et al. [115]. These bul-lets preserve their spatially confined shapes and survive collisions. In a recent work,Shumakova et al. [116] have observed chirp-controlled filamentation in ambient airand formation of ultrashort light bullets in the mid-infrared spectral region.

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2.2. SOLITONS IN CARBON NANOTUBES

The unique dynamics of ultrashort optical pulses and the formation of both 2Dand 3D solitons in carbon nanotubes (CNTs) have been investigated in a series of re-cent works. A theoretical study of the interaction of a 2D electromagnetic pulse withelectron inhomogeneities in arrays of CNTs in the presence of field inhomogeneityhas been reported by Zhukov et al. [117] and the propagation dynamics of 2D ex-tremely short electromagnetic pulses in Bragg media containing immersed arrays ofCNTs has been reported too [118]. Belonenko et al. [119] have studied the forma-tion of discrete solitons in a Bragg environment containing CNTs. The formationof 2D light bullets in Bragg media with harmonically modulated refractive indices,which contain arrays of CNTs, has been investigated by Nevzorova et al. [120].Special opto-acoustic effects in arrays of CNTs have been studied by Zhukov et al.[121]. A comprehensive study of collisions of 3D bipolar optical solitons in arrays ofCNTs has been reported by Zhukov et al. [122]. Three-dimensional ultrashort opticalAiry beams in inhomogeneous media containing CNTs have been also investigated[123]. Such types of “light bullets” exhibit stable propagation, and, by varying thedensity-modulation period of CNTs, one can control the velocity of these 3D opticalmodes, see Ref. [123]. The stable propagation of 2D light bullets in media with in-homogeneous density of CNTs has been studied in a recent work by Dvuzhilov andBelonenko [124]. The propagation of 3D bipolar ultrashort electromagnetic pulsesin inhomogeneous arrays of semiconductor CNTs has been investigated by Fedorovet al. [125]. The heterogeneity of the propagating medium is represented by a planarregion with an increased concentration of conduction electrons. Numerical simula-tions have demonstrated that, after interacting with the higher electron concentrationarea, the ultrashort pulse can propagate steadily, without significant spreading [125].

2.3. ULTRASHORT (FEW-CYCLE) OPTICAL WAVEFORMS

Many experimental and theoretical studies of optical pulses with widths rang-ing from tens of nanoseconds to only a few tens of femtoseconds have been reportedin the course of the last three decades; see, in particular, the review papers on the fastgrowing research area of ultrashort high-power optical pulses [126–136]. The pos-sibility to generate high-intensity ultrashort optical pulses relies on a revolutionaryexperimental achievement, namely the so-called chirped pulse amplification (CPA)technique, which was introduced in 1985 by Strickland and Mourou [137]. In a sub-sequent work published in 1987, Mourou, Strickland and their collaborators had putforward an idea of using large-scale Nd:glass amplifiers to extend the CPA techniqueto the amplification of pulses with the duration of hundreds fs to the hundred-Joulelevel, leading to petawatt-power pulses and to intensities ∼ 1023 W/cm2 [138].

Ursescu et al. [135] have recently described a high-power laser system and

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experiments planned at the Extreme Light Infrastructure–Nuclear Physics (ELI-NP)facility to be built at Magurele Physics Campus, in outskirts of Bucharest. At thisunique experimental facility, ultrashort 10 PW laser pulses with intensities reaching1023 W/cm2 are planned to be delivered; see the recent paper [139] describing phy-sical experiments to be carried out with such 10 PW ultra-intense lasers and with 20MeV high-brilliance gamma beams, and the status of completing the ELI-NP facility.Further, a series of high-field physics and quantum electrodynamics experiments tobe performed at the ELI-NP facility have been surveyed in a comprehensive paperby Turcu et al. [136]. We also refer here to a paper by Jirka et al. [140] on the possi-bility to efficiently generate electron-positron pairs with 10 PW-class laser facilities,by means of a standing wave created by two tightly focused counterpropagating laserpulses, to a recent work by Miron and Apostol [141] on the motion of electric chargesin high-intensity electromagnetic fields, and to a recent paper by Yu et al. [142] on anovel approach to the creation of electron-positron pairs in photon-photon collisionsdriven by 10 PW laser pulses.

In the course of the past few years ultrahigh intensity lasers with peak powers∼ 1 PW have been built at different facilities: (a) at the Lawrence Berkeley Na-tional Laboratory, USA (BELLA high repetition rate PW class laser) [143], (b) at theApollon facility in France it is planned to generate 10 PW peak power pulses withduration of 15 fs at a repetition rate of 1 shot/minute [144], (c) at the laser facilityin South Korea the generation of 4.2 PW laser pulses with duration of 20 fs at a re-petition rate of 0.1 Hz has been demonstrated, using a CPA Ti:sapphire laser system[145], (d) at the laser facility in China the generation of 5.4 PW laser pulses withduration of 24 fs from a CPA Ti:sapphire laser system [146] has been reported, and(e) at the Extreme Light Infrastructure - Beamlines facility in the Czech Republic[147].

The state-of-the-art surface plasma attosource beamlines at Extreme Light In-frastructure Attosecond Light Pulse Source (ELI-ALPS) are currently being designedand developed in Szeged, Hungary, to enable new directions in plasma-based atto-science research, see a recent paper by Mondal et al. [148]. In a recent work, Jahn etal. [149] have investigated the challenging problem of generation of intense isolatedattosecond pulses in the extreme ultraviolet and X-ray spectral ranges, from rela-tivistic surface high harmonics. The measurements and particle-in-cell numericalsimulations to determine the optimum values for the key physical parameters havebeen reported in Ref. [149].

In a recent paper, Turcu et al. [150] have produced an overview of quantum-electrodynamics experiments with colliding PW laser pulses, which are planned tobe performed at the new superhigh-power laser facilities, which are capable to reachintensities of 1023 W/cm2 in the laser focus. Ciappina et al. [151] have discussedrecent results for atomic diagnostics of ultrahigh laser intensities. A method for

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direct and unambiguous measurement of ultrahigh laser intensities in the range of1020− 1024 W/cm2 has been put forward and tested numerically, see Ref. [151].Wilson et al. [152] have investigated theoretically and numerically the dynamics of3D compression of ultrashort intense laser pulses in plasma. Using the variationalmethod, and starting from the slowly-varying envelope model, equations describingthe spatiotemporal evolution of short laser pulses towards collapse have been derived.3D particle-in-cell numerical simulations have been also carried out to study physicalprocesses both preceding and following the pulse collapse, see Ref. [152] for details.Nonlinear Compton scattering of electrons in the presence of an ultraintense few-cycle laser pulse travelling through a plasma has been investigated by Mackenroth etal. [153] in the framework of quantum electrodynamics, and properties of the quan-tum vacuum Cherenkov radiation in strong laser pulses have been recently studiedby Macleod et al. [154].

Next we briefly overview recent theoretical results for ultrashort optical wave-forms. Several research groups [155–162] have investigated a series of topics rang-ing from the important issue of interplay of diffraction and nonlinear effects in thepropagation of very short light pulses [155] to the potential use of focused ultrashortpulses for far-field light nanoscopy [162]. Arkhipov et al. [163] have investigatedthe possibility of the generation of unipolar pulses in Raman-active media, excitedby a series of few-cycle optical pulses, and Pakhomov et al. [164] have put forwarda general framework to produce unipolar half-cycle pulses of controllable form inresonant media with nonlinear field coupling. The possibility of generation of unipo-lar half-cycle pulses via unusual reflection of a single-cycle pulse from thin metallicor dielectric layers has been put forward by Arkhipov et al. [165]. Terniche et al.[166] have investigated the propagation of few-cycle solitons in two parallel opticalwaveguides, in the presence of linear nondispersive coupling, and it was shown thatfew-cycle vector solitons in such coupled waveguides are, in fact, optical breathers.

The propagation dynamics of Gaussian spatiotemporal wavepackets in arraysof parallel optical waveguides, assuming linear nondispersive coupling between adj-acent guides, has been studied by Leblond et al. [167]. The conditions for formationof few-cycle spatiotemporal optical solitons in such waveguide arrays, were also in-vestigated in Ref. [167], including the identification of their time duration, spatialwidths, and energies. A recent work by Leblond and Mihalache [168] has addressedthe propagation of Gaussian spatiotemporal inputs in arrays of parallel optical wave-guides, assuming both linear and nonlinear non-dispersive coupling between adjacentguides. Depending on the numerical values of the key parameters of the spatiotem-poral Gaussian input, the combined effect of linear and nonlinear couplings leads toeither spreading or formation of different types of (2+1)D ultrashort spatiotempo-ral solitons, see Ref. [168] for more details. Note that the formation of relativelybroad semi-discrete spatiotemporal solitons in coupled arrays of nonlinear optical

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waveguides was analyzed in Ref. [169].The problem of wavelength-scaled laser filamentation in solids and plasma-

assisted generation of subcycle light bullets in the long-wavelength infrared regionhas been recently addressed by Grynko et al. [170]. The method put forward in Ref.[170] for the generation of ultrashort (subcycle) light bullets in solids at long wave-lengths may help to open new possibilities in ultrafast nonlinear optics, including thegeneration of high-order harmonics and attosecond pulses. Recently, Sazonov andUstinov [171] have investigated the propagation of few-cycle pulses in nonlinear me-dia modeled by a set of four-level atoms. In this context, an integrable generalizationof the sine-Gordon equation has been reported in Ref. [171] without the use of theslowly-varying envelope approximation. Also, a review of the propagation dynamicsof both resonant and quasi-resonant envelope optical solitons, as well as few-cycleoptical solitons with temporal durations from nanoseconds to femtoseconds, has beenrecently published by Sazonov [172]. Gao et al. [173] have investigated analyti-cally and numerically the stability and interaction scenarios for few-cycle pulses inKerr media, while Konobeeva and Belonenko [174] have studied the propagation of3D few-cycle optical pulses in topological-insulator thin films. Self-compression oflaser pulses in an active system of weakly-coupled waveguides has been investigatedanalytically by Balakin et al. [175], using the variational method based on a spa-tiotemporal Gaussian ansatz. The obtained results agree well with direct numericalsimulations of the corresponding discrete Ginzburg-Landau dynamical model.

The generation of ultrashort (few-cycle) pulses in the terahertz (THz) range isa research topic of great interest during the past decade. Recently, Pakhomov et al.[176] have proposed a new method for generation of subcycle THz pulses with un-usual waveshapes such as rectangular or triangular ones. The method is based oncontrol of the phase delay between different parts of the THz wavefront using lineardiffractive optical elements, see Ref. [176]. Porras [177] has analyzed theoreticallythe effects of the coupling between the orbital angular momentum (OAM) and tem-poral degrees of freedom on few-cycle and sub-cycle optical pulse shapes, in thegeometry used in current experiments involving diffraction and focusing. In anotherrecent work, Porras [178] has found the upper bound for the OAM carried by ultra-short (few-cycle) pulses; note that the single-cycle pulse can carry up to 27 units ofOAM, see Ref. [178] for details.

In the experimental arena of strong laser pulses and their interaction with matterwe refer here to a recent work by Tsatrafyllis et al. [179]. The quantum opticalsignatures in a strong laser pulse after interaction with semiconductors have beenstudied [179] and it has been demonstrated that strongly laser-driven semiconductorslead to the generation of nonclassical states of light having subcycle electric fieldfluctuations that carry the information of the subcycle dynamics of the interaction,see Ref. [179] for details.

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Next, we briefly overview some recent relevant experimental results in the ac-tive research area of high-power ultrashort (femtosecond) lasers and associated spa-tiotemporal phenomena. Spatiotemporal mode-locking in multimode fiber lasers hasbeen addressed by Wright et al. [180]. It was demonstrated in Ref. [180] thatmodal and chromatic dispersions in fiber lasers can be compensated by strong spa-tial and spectral filtering, allowing locking of transverse and longitudinal modes forthe creation of ultrashort laser pulses with a large variety of spatiotemporal profiles.Potential applications of this technique include the realization of super high-powerlasers and efficient control of the emittance of the beam, with the objective to cre-ate specific spatiotemporal patterns. In a recent work, Zheltikov [181] has reviewedthe current status of multioctave supercontinuum generation and the creation of sub-cycle pulses both in air and in solid materials. Applications of these studies rangefrom the frequency-comb metrology, coherence tomography, nonlinear spectroscopy,and chirped-pulse amplifiers to bioimaging [181]. In a recent experimental work,Gonsalves et al. [182] have demonstrated guiding of relativistically intense laserpulses with peak power of 0.85 PW over 15 diffraction lengths, which allows theelectron beam acceleration to 8 GeV in a laser-heated capillary discharge waveguide.

3. NONLINEAR WAVES IN PARITY-TIME-SYMMETRIC SYSTEMS

In the course of the last two decades, the concept of the space-time reflec-tional symmetry, alias parity-time (PT ) symmetry, first proposed in the quantumtheory by Bender and Boettcher [183], has been thoroughly explored in diverse areasof fundamental and applied physics, such as optics and photonics, atomic physicsand Bose-Einstein condensates, superconductivity, electronic circuits, acoustics, etc.Unique features of linear or nonlinear waves in PT -symmetric physical systemshave been investigated in detail. PT -symmetric quantum mechanics is a naturalextension of conventional quantum theory, dealing with non-Hermitian Hamiltoni-ans characterized by complex-valued external potentials obeying the PT -symmetry:the spatially odd (antisymmetric) imaginary part of the complex potential representssymmetrically placed and mutually balanced gain and loss elements. The respectivenon-Hermitian Hamiltonians exhibit all-real, i.e., physically relevant, energy spectra,provided that the strength of gain-loss terms does not exceed a certain critical values,above which the PT symmetry breaks down. In combination with the usual intrin-sic nonlinearity of PT -symmetric media, non-Hermitian Hamiltonians give rise to aplethora of new phenomena that have no counterparts in common dissipative physi-cal systems, see two comprehensive reviews published by Konotop et al. [13] andSuchkov et al. [14]. In particular, unlike normal dissipative systems, in which stablesolitons may exist as attractors, separated from all other solutions, PT -symmetricmodels give rise to continuous families of solitons, sharing the latter property with

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conservative models. In optics the PT symmetry can be relatively easily imple-mented, e.g. in planar waveguides [184] and in coupled optical structures [185, 186];see two relevant experimental works [187, 188].

Theoretical studies of dynamics of spatial solitons in PT -symmetric opticallattices, reported in the course of the past several years, have been recently sur-veyed by He and Malomed [189] and by He et al. [190]. PT -symmetric opticalcouplers with competing cubic self-focusing and quintic self-defocusing nonline-arities have been theoretically investigated by Burlak et al. [191]. Also, a PT -symmetric dual-core system with the sine-Gordon nonlinearity and derivative cou-pling has been studied by Cuevas-Maraver et al. [192] as a relevant extension ofthe class of nonlinear PT -symmetric physical models. Barashenkov et al. [193]have studied the problem of integrability and PT -symmetry restoration in nonlinearSchrodinger (NLS) dimers with gain and loss. The problem of nonlinear parity-time-symmetry breaking in optical waveguides with complex-valued Gaussian-type poten-tials has been addressed in Ref. [194], and the existence and stability of both sym-metric and asymmetric solitons that form in self-focusing optical waveguides withPT -symmetric double-hump Scarf-II potentials have been analyzed in Ref. [195].Three-dimensional solitons supported by different types of PT -symmetric complex-valued external potentials have been also investigated numerically in recent papers[37, 196, 197]. Kartashov et al. [37] have reported a first example, to the best ofour knowledge, of continuous families of stable 3D propagating solitons supportedby PT -symmetric complex-valued external potentials.

Continuous families of one- and two-dimensional stable solitons in PT -sym-metric NLS equations with position-dependent effective masses have been investi-gated by Chen et al. [198]. A combination of the PT -symmetric gain-loss structurewith inhomogeneous repulsive nonlinearity, growing fast enough from the center toperiphery, makes it possible to predict both 1D [199] and 2D [200] solitons (includingvortex solitons, in the 2D case), with unbreakable PT symmetry, which extends toarbitrarily large values of the strength of the gain-loss terms. The effects of the third-order dispersion on constant-amplitude solutions of the NLS equation with complex-valued potentials have been considered by Liu et al. [201]. Lombard and Mezhoud[202] have performed a detailed analytical and numerical study of the spectrum of thePT -symmetric complex-valued linear potential, and in a subsequent work, Lombardet al. [203] have investigated the relationship between complex potentials with realeigenvalues and the inverse scattering problem.

Li et al. [204] have studied the properties of optical solitons in models ofwaveguides with focusing or defocusing saturable nonlinearity and a PT -symmetriccomplex-valued external potential of the Scarf-II type. Fundamental and multi-pole solitons for both focusing and defocusing signs of the saturable nonlinearityin such PT -symmetric waveguides have been studied by Li et al. [204]. The exis-

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tence, stability, and dynamics of optical solitons that form in media with competingcubic-quintic nonlinearity and PT -symmetric complex-valued external potentialshave been investigated in Ref. [205]. The effects of periodically-modulated third-order dispersion on periodic solutions of the NLS equation with complex-valuedpotentials have been studied by Liu et al. [206]. Wang et al. [207] have investi-gated numerically the evolution dynamics of continuous families of dipole solitons inmedia with self-defocusing Kerr nonlinearity and partial PT -symmetric optical po-tentials. This type of complex-valued external potentials can support one-parameterfamilies of diagonal and adjacent dipole solitons with two humps, which feature ei-ther the out-of-phase or in-phase structure. Both diagonal out-of-phase and diagonalin-phase dipole solitons can propagate stably at moderate values of their power. Itwas pointed out in Ref. [207] that only adjacent in-phase dipole solitons can be stablein a certain region of their existence domain.

Analytical and numerical studies of the optical-beam dynamics in single activewaveguides and in a dimer of active waveguides with quadratic nonlinearities havebeen recently reported by Tsoy et al. [208]. Stationary solutions for a dimer withidentical waveguides and for a PT -symmetric dimer were obtained, and the respec-tive stability analysis has demonstrated that stable optical beams exist in a wide rangeof parameters of the model. It is worth mentioning that the active monomer can beused as an amplitude filter or as an optically controllable switch, see Ref. [208]. Liet al. [209] have investigated numerically the stability, dynamics, and bifurcationscenarios of both symmetric and asymmetric solitons supported by saturable nonli-near media in the presence of a PT -symmetric external potential. In a recent work,Scheel and Szameit [210] have studied the impact of gain and loss on the evolutionof photonic quantum states, concluding that PT symmetric states do not exist in theframework of the quantum field theory.

Kartashov and Vysloukh [211] have recently addressed the formation of edgeand bulk dissipative solitons in modulated PT -symmetric continuous waveguide ar-rays composed of waveguides with amplifying and absorbing sections, whose densitygradually increases either towards the center of the array or towards its edges. It wasfound that in such modulated PT -symmetric waveguide arrays, the symmetry break-ing phenomenon may occur via collision of eigenvalues of compact edge or bulkmodes, which are localized in the domains where the waveguides have the smallestseparation, see Ref. [211] for more details. Wang et al. [212] have investigatedanalytically and numerically the families of solitons pinned to a defect described bya PT -symmetric potential given by a δ-functional dipole, which is embedded in a1D uniform medium with a general self-focusing nonlinearity. A physical schemefor creation of combined linear and nonlinear optical potentials with PT -symmetry,with the aim to address scattering of optical solitons by such types of potentials incoherent atomic gases was put forward by Qin et al. [213].

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Yang [214] has recently reported a new type of symmetry-breaking bifurca-tion of solitons in optical media with cubic-quintic or saturable nonlinearities andPT -symmetric complex-valued external potentials. The two bifurcated branches ofasymmetric solitons exhibit opposite stability, which contrasts all previous symmetry-breaking bifurcations in either conservative or dissipative physical systems, see Ref.[214] for details.

The results briefly outlined in this Section pave the way for exploiting and bet-ter understanding the unique properties of non-Hermitian systems in many physicalsettings.

4. ROGUE WAVES

In this Section we discuss recent theoretical and experimental advances inthe area of rogue waves (RWs), alias “freak waves”. These special wave struc-tures emerge in a broad class of physical systems ranging from optics to Bose-Einstein condensates, see the comprehensive reviews [215–218]. In the course ofthe past years, theoretical and experimental studies of such high-amplitude waveshave mainly focused on optical and hydrodynamic settings; see, for example, Refs.[219–232]. The plethora of theoretical and experimental papers published in thepast decade in the area of RWs were strongly influenced by the seminal works ofPeregrine [233], Kuznetsov and Ma [234, 235], and Akhmediev et al. [236], whereexact rational solutions of the integrable NLS equation were reported; see also thepapers published in 1993 by Mihalache and Panoiu [237] and by Mihalache et al.[238] where the above-mentioned exact rational solutions of the NLS equation andother types of “first-order” exact solutions of this nonlinear equation were obtainedin the context of the pulse propagation in optical fibers, for both the anomalous- andthe normal-dispersion regime. The multiparameter families of first-order exact solu-tions of the NLS equation in the normal-dispersion regime have been independentlyreported in 1993 by Akhmediev and Ankiewicz [239] and by Gagnon [240]. Thereare also other integrable nonlinear partial differential equations admitting Peregrine-type solitons, as well as Akhmediev and Kuznetsov-Ma breathers [234, 235]; see, forexample, the recent review [218].

In what follows we will overview theoretical and experimental works per-formed in the past two years in this vast research area. Ankiewicz and Akhme-diev [241] reported a plethora of RW solutions of the integrable NLS hierarchy withan infinite number of terms, namely higher-order dispersion and nonlinear terms.Some mechanisms of generation of fundamental RW spatiotemporal structures inN -component coupled systems have been considered by Ling et al. [242]. Chowduryand Krolikowski [243] have presented analytical one- and two-breather solutions ofthe fourth-order NLS equation in the degenerate, soliton, and RW limits. The pos-

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sibility of the generation of higher-order RWs in optical fibers from multibreathersby dint of a double-degeneracy procedure has been investigated recently by Wanget al. [244]. Baronio [245] has investigated the formation of Akhmediev breathersand Peregrine solitary waves in quadratic nonlinear media in the regime of cascad-ing second-harmonic generation. In a subsequent work, Baronio et al. [246] havereported the formation and key properties of quadratic Peregrine solitary waves, inthe presence of significant group-velocity mismatch between the waves (or beamwalk-off, in terms of the Poynting vector).

Ankiewicz and Akhmediev [247] have studied the relation between the numberof “elementary units” of multi-rogue waves and the triangular numbers. It is wellknown that multi-rogue-wave solutions of integrable nonlinear evolution equationshave a specific number of elementary components within their structures. Ankiewiczand Akhmediev [247] have revealed that, for the n-th-order rogue-wave solutions,these numbers are given by triangular numbers.

Chan and Chow [248] have performed a numerical investigation of the genera-tion and dynamics of “hot spots” as models of dissipative RWs. Effects of gain or losson the dynamics of RWs have been investigated in the framework of the generic com-plex Ginzburg-Landau equation and the robustness of the Peregrine soliton in suchdissipative systems has been studied by means of numerical methods, see Ref. [248].The effects of a seed pulse on RW formation for mid-infrared supercontinuum gener-ation in chalcogenide photonic-crystal fibers have been studied numerically by Zhaoet al. [249]. Yang et al. [250] have studied, analytically and numerically, possibili-ties of obtaining controllable optical RWs via nonlinearity management. Bright-darksolitons and semi-rational RW solutions for the coupled Sasa-Satsuma equations havebeen obtained by Liu et al. [251]. Zhang and Yan [252] have systematically inves-tigated, with the aid of the Darboux-transformation method, the dark-bright mixedhigher-order semi-rational solutions of n-component nonlinear Schrodinger equa-tions. Peregrine solitons beyond the threefold-amplitude magnification limit havebeen recently studied by Chen et al. [253]. Within the framework of coupled Fokas-Lenells model, the Peregrine solitons can reach peak amplitudes as high as five timesthe background level, see Ref. [253]. El-Tantawy and Wazwaz [254] have studied,analytically and numerically, the formation of RWs and dark-soliton collisions industy plasmas.

A theoretical study of extreme events occurring in phononic lattices has beenreported by Charalampidis et al. [255]. This work illustrates the potential of dyna-mical lattices towards the experimental observation of phononic (acoustic) RWs. Wealso mention here the work of Randoux et al. [256] on nonlinear spectral analysis ofdata recorded in optics and in hydrodynamic experiments that reported the pioneeringobservation of nonlinear waves with spatiotemporal localization similar to Peregrinesolitons. Related results were reported by Chen et al. [257], in the form of some

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special types of exact two- and three-soliton solutions of the Kadomtsev-PetviashviliII equation, written in terms of hyperbolic cosines. The obtained exact solutions ex-hibit rich intriguing interaction patterns on a finite background, see Ref. [257]. Acomprehensive theoretical study of RWs in ultracold bosonic gases has been recentlyreported by Charalampidis et al. [258]. We also point out that the analytic forms ofloop RW solutions for the Wadati-Konno-Ichikawa equation have been reported byZhang et al. [259] using the Darboux transformation and inverse hodograph trans-formation. RWs and hybrid solutions of the Davey-Stewartson I equation have beenobtained recently by Liu et al. [260] using the Hirota bilinear method. Zhang etal. [261] have recently studied, by means of analytical and numerical methods, themodulational instability and higher-order vector RW and multi-dark soliton struc-tures in the framework of the general coupled Hirota equations, which describe thepropagation of two ultrashort pulses in an optical fiber.

Different types of first-order and second-order rational solutions and periodicsolutions of the Sasa-Satsuma equation have been obtained by Guo et al. [262] us-ing the Darboux-transform method. Unique dynamics of dust-acoustic and dust-cyclotron freak waves in a magnetized dusty plasma has been recently studied ana-lytically and numerically by Akhtar et al. [263]. Families of semi-rational solutionsto the Fokas system, named lump-soliton modes, have been reported by Rao et al.[264], using the Hirota bilinear method. First-order and second-order rogue waves ona periodic background for the Kaup-Newell equation have been obtained analyticallyby Liu et al. [265], also using the Darboux transform. Chen and Yan [266] have re-cently investigated the matrix Riemann-Hilbert problem for the Hirota equation withnon-zero boundary conditions. Using the n-fold Darboux transform, higher-orderrogue waves for the Hirota equation have been obtained in a closed form, see Ref.[266]. The complex dynamics of vector rogue waves on a double-plane wave back-ground has been investigated by Zhao et al. [267]. Degasperis et al. [268] haveexplored a connection between the instability properties of the plane-wave states andthe existence and key features of rogue-wave solutions of integrable versions of asystem of two coupled NLS equations. Using the Darboux transform, rogue wavesin the nonlocal PT -symmetric NLS model have been investigated recently by Yangand Yang [269]. The obtained exact solutions for these types of rogue waves exhibitrich patterns, most of which having no counterparts in the local NLS model. Re-cently, Ye et al. [270] have obtained the general rogue wave solutions of the coupledFokas-Lenells equations using a non-recursive Darboux transformation technique.

Chen et al. [271] have recently investigated both analytically and numericallysuperchirped rogue waves in optical fibers within the framework of a generalizednonlinear Schrodinger equation including the group-velocity dispersion, cubic andquintic nonlinearities, and self-steepening effect. These super-rogue waves involve afrequency chirp, localized in both time and space, and are robust against white-noise

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perturbations. The possibility of generating them in a turbulent field suggests theiraccessibility to experimental observation, see Ref. [271].

It is necessary to stress that all RWs are unstable states, as they are supportedby a continuous-wave background that is subject to the modulational instability. Inthis connection, it is relevant to mention a scenario for stabilization of these states bymeans of dispersion and nonlinearity management [272].

As concerns experiments, we mention the work by Klein et al. [273] on theobservation of ultrafast RW patterns in the form of single-peak, twin-peak, and triple-peak states in fiber laser systems. In this connection, it is worth mentioning that thefast dynamics of fiber lasers makes them very convenient tools for studying rare,extreme events, such as RWs [273]. In a recent work, Coulibaly et al. [274] haveinvestigated, both experimentally and theoretically, turbulence-induced rogue wavesin Kerr resonators. The existence of rogue waves as extreme events characterized bylong-tail statistics has been unveiled, see Ref. [274] for details.

5. MATTER-WAVE SOLITONS, VORTICES, AND QUANTUM DROPLETS

In this Section we briefly overview recent theoretical and experimental studiesof physical properties of matter-wave fundamental solitons and vortices in atomicBECs; see also a viewpoint on multidimensional soliton structures and their legacyin contemporary atomic, molecular, and optical physics [9], and a recent brief review[275] of the creation of matter-wave solitons by means of spin-orbit coupling, and areview [276] on 2D vortex solitons in spin-orbit-coupled dipolar BECs.

In a recent work, Adhikari [277] has studied, by means of a variational me-thod and numerical simulations, the possibility of creating self-bound stable 3Dmatter-wave spherical boson-fermion quantum balls, in the presence of an attractiveboson-fermion interaction and weak repulsive three-boson interaction. Stable andmetastable bright vortex solitons in 3D spin-orbit-coupled three-component BECswith spin F = 1 have been studied by Gautam and Adhikari [278], using the nu-merical solution of mean-field equations and a Gaussian variational approximationfor bright vortex solitons. Li et al. [279] have obtained 3D stable solitons in bi-nary atomic BECs with spin-orbit coupling and out-of-phase linear and nonlinearBessel optical lattices, employing a variational method and direct numerical sim-ulations of a coupled system of Gross-Pitaevskii (GP) equations. Stable solitonswith the semivortex and mixed-mode structures have been found, depending on thestrength of the intra- and intercomponent spatial modulation of the nonlinearity andspin-orbit coupling. Solitary-vortex evolution in 2D harmonically trapped BECs hasbeen investigated by Wang and Zhou [280], using a variational method based on aGaussian ansatz and taking asymmetric perturbation effects into regard. Wang andYang [281] have investigated, analytically and numerically, the dynamics of 2D vor-

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tex solitons described by the GP model that incorporates a harmonic trapping poten-tial and higher-order nonlinear interactions. The stability of a trapless dipolar BECwith temporal modulation of the short-range contact interaction has been studied bySabari and Dey [282]. Analytical and numerical considerations have shown that theoscillatory contact interaction prevents the collapse of the trapless dipolar BEC, seeRef. [282]. A spectral and dynamical analysis of single vortex rings in anisotropicharmonically trapped 3D BECs has been performed by Ticknor et al. [283]. Ina recent work, Salerno and Baizakov [284] have studied normal-mode oscillationsof nonlocal composite matter-wave solitons by means of the variational approachand numerical simulations. The dynamics of traveling dark-bright solitons in spin-orbit-coupled BECs has been investigated by D’Ambroise et al. [285]. Sakaguchiet al. [286] have reviewed the properties of 2D matter-wave solitons, governed bythe spinor system of GP equations with cubic nonlinearity, including the spin-orbitcoupling and the Zeeman splitting.

Recently, Sakaguchi [287] has investigated, both analytically (using a varia-tional method) and numerically, soliton lattices in a physical setting described bythe GP equation with nonlocal and repulsive coupling. Also, effects of the rotat-ing harmonic potential and spin-orbit coupling have been studied numerically in thesame work. A numerical study of collision of two vortex dipoles propagating in op-posite directions along parallel lines in 2D BECs has been reported by Yang et al.[288]. Three generic scenarios of the interaction of vortex dipoles in homogeneousBECs have been identified by varying the impact parameter: (i) the vortex recombi-nation mode, (ii) the encircling mode, and (iii) the flyby mode. The axial-symmetry-breaking instability of leapfrogging vortex rings in BECs has been investigated nu-merically by Ikuta et al. [289], using both the GP equation and the vortex-filamentmodel. It has been revealed that three coaxial quantized vortex rings exhibit ape-riodic leapfrogging dynamics, see Ref. [289] for details. Kartashov and Zezyulin[290] have numerically identified families of stable multiring and rotating solitons in2D spin-orbit-coupled BECs with radially periodic potentials, by solving a systemof Gross-Pitaevskii equations with the Rashba spin-orbit coupling, see Ref. [290]for a detailed discussion of the physical model. A theoretical study of Bose-Einsteincondensation of light via laser cooling of an ensemble of two-level atoms inside aFabry-Perot cavity has been recently reported by Wang et al. [291].

A true breakthrough in the creation of stable multidimensional soliton-likestates, in the form of “quantum droplets”, which are stabilized by Lee-Huang-Yangcorrections to the mean-field dynamics, has been reported recently. These correc-tions take effects of quantum fluctuations into regard [292]. As demonstrated byPetrov [35] for two-component BECs with contact repulsive intra-component and at-tractive inter-component interactions, the accordingly modified GP equation for thecondensate’s wave function, ψ, in 3D acquires an additional repulsive (hence, stabi-

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lizing) quartic term∼ |ψ|3ψ (an effect of the Rabi coupling between the componentswas additionally considered in Ref. [293]).

In the 2D setting, the effective nonlinearity is different, ∼ ln(|ψ|2/ψ2

0

)|ψ|2ψ,

with some density constant, ψ20 [294]. In the 1D limit, in which the quantum-

fluctuation correction is represented by an attractive term ∼ |ψ|ψ, the structure anddynamics of localized states was studied in Ref. [295]. Two different types of 1Ddroplets were identified, namely small ones with an approximately Gaussian shapeand broad flat-top states. Three-dimensional droplets with embedded vorticity havebeen studied by Kartashov et al. [296]. By means of numerical and approximate ana-lytical methods, families of self-trapped vortex tori, carrying the topological chargesm1 =m2 = 1 or m1 =m2 = 2 in their components, were constructed and their sta-bility was investigated numerically. The stability regions of the vortex droplets arebroad for m1,2 = 1 and are narrower for m1,2 = 2, whereas the droplets with hiddenvorticity (that is, when m1 = −m2 = 1 in the two components) are completely un-stable in the relevant parameter domain. In this connection, Gautam and Adhikari[297] have recently investigated, by means of a Gaussian variational approximation,the formation of stable self-trapped spherical quantum balls in the model of binaryBECs with the above-mentioned quartic terms and additional repulsive quintic ones.Collisions between such spherical balls have been also investigated by numericalmethods in Ref. [297].

In the 2D geometry, vortex droplets were predicted in Ref. [298], which maybe stable up to m1 =m2 = 5. Furthermore, 2D hidden-vorticity states have their sta-bility region too. Ring-shaped clusters of 2D droplets were very recently predictedin Ref. [299]. In the presence of spin-orbit coupling, 2D droplets of the semi-vortextype were considered in Ref. [300]). The problem of spontaneous symmetry break-ing (SSB) of QDs in a dual-core trap has been recently investigated by Liu et al.[301]. One-dimensional QDs in a binary bosonic gas loaded in a symmetric double-core cigar-shaped potential have been addressed in Ref. [301]. We note that unlikethe usual SSB mechanism for matter-wave solitons, which is induced by mean-fieldinteractions, the SSB of QDs in this physical system is driven by the interplay of themean-field and Lee-Huang-Yang quadratic attractive terms in the linearly coupledGP equations. Also, collisions between moving two-components QDs have beenconsidered by Liu et al. [301], which demonstrate inelastic interactions, leading tothe merger of QDs into breathers.

In the experiment, nearly two-dimensional [302, 303] and fully three-dimensio-nal [304, 305] quantum droplets have been created in a condensate of 39K atoms, withtwo components represented by different hyperfine states. Another prediction and re-alization of stable 3D quantum droplets was demonstrated in single-component BECwith long-range dipole-dipole interactions between atoms carrying permanent mag-netic dipole moments [306–308]. However, unlike the above-mentioned droplets,

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supported by the contact interactions, with embedded vorticity, vortex droplets indipolar condensates are completely unstable [309].

Recently, Kartashov et al. [310] have introduced an exactly integrable non-linear model describing the dynamics of spinor solitons in space-dependent matrixgauge potentials of rather general types. As an example of this dynamical model,a self-attractive BEC with random spin-orbit coupling has been considered in Ref.[310]. If the Zeeman splitting is also included in the model, the nonlinear systembecomes nonintegrable. However, for a strong Zeeman splitting, the integrability isrestored, see Ref. [310].

The results surveyed in this Section are largely based on the broad arsenal ofanalytical and numerical frameworks currently available for solving the GP equa-tions and its siblings in nonlinear optics settings. Among relevant models that arecurrently available in this context, we mention 1D and 2D equations used to approxi-mately describe 3D systems [311–317] (in addition to the above-mentioned dimen-sional reduction for quantum droplets), and a large family of numerical solvers fornonlinear partial differential equations, such as the time-dependent GP equation andits generalizations [318–322].

On the experimental side, we refer here to a work by Danaila et al. [323] inwhich an experimental evidence for the existence, stability, and dynamics of dark-antidark solitary waves in two-component BECs, confined in elongated traps, hasbeen reported. Theoretical and numerical investigation of vector dark-antidark statesin the 2D setting have been also reported in Ref. [323]. Experimental demonstra-tion of the existence of robust three-component soliton states in spinor F = 1 BECs,of the dark-bright-bright and dark-dark-bright types, has been recently reported byBersano et al. [324]. These multicomponent states have lifetimes on the order ofhundreds of milliseconds. Kang et al. [325] have recently reported the observationof spin domain walls bounded by half-quantum vortices in spin-1 BEC with antifer-romagnetic interactions. In a recent work, Nguyen et al. [326] have investigated,both experimentally and theoretically, the response of an elongated BEC to directmodulations of the interaction, induced by the Feshbach resonance. The parametricexcitation of BEC leads to two distinct modulation regimes: formation of Faradaypatterns, and the formation of granular states featuring irregular fluctuating spatialpatterns.

6. CONCLUSION

The objective of this article has been to provide a brief overview of recentlyobtained results in theoretical and experimental studies of nonlinear-wave phenome-nology in the areas of nonlinear optics and matter waves in BEC. The article hasbeen structured, essentially, as a resource text, which provides a large number of

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references to relevant publications reporting recent advances in the research field ofnonlinear waves in optical and matter-wave media. In particular, included are newfindings concerning few-cycle (ultra-narrow) optical pulses, parity-time-symmetricnonlinear modes, rogue waves in various media, quantum droplets recently predictedand experimentally created in bosonic condensates, multidimensional (2D and 3D)and multicomponent (spinor and vector) settings, and some other topics.

The survey of the currently available results clearly demonstrates that thereremains a vast room for further theoretical and experimental studies of solitons andrelated self-trapped modes in photonics, BEC, and other quickly developing areas ofphysics.

Acknowledgements. A part of the recent research work overviewed in this resource paper hasbeen carried out in collaboration with Fabio Baronio, Mikhail B. Belonenko, Roland Bouffanais, ShihuaChen, Eduard G. Fedorov, A. S. Fokas, Dimitri J. Frantzeskakis, Philippe Grelu, Jingsong He, Ying-JiHe, Yaroslav V. Kartashov, Panayotis G. Kevrekidis, Yuri S. Kivshar, Herve Leblond, Lu Li, Pengfei Li,Yongyao Li, K. Porsezian, Nikolay N. Rosanov, Vladimir Skarka, Jose M. Soto-Crespo, Lluis Torner,Frank Wise, Zhenya Yan, and Alexander V. Zhukov. We are deeply indebted to them for the fruitfulcollaboration and for many insightful discussions.

REFERENCES

1. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic,San Diego, 2003.

2. L. P. Pitaevskii and S. Stringari, Bose-Einstein condensation, Oxford University Press, Oxford,2003.

3. C. J. Pethick and H. Smith, Bose-Einstein condensates in dilute gases, Cambridge UniversityPress, Cambridge, 2008.

4. B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Opt. B 7, R53 (2005).5. A. S. Desyatnikov, L. Torner, and Yu. S. Kivshar, Prog. Optics 47, 291 (2005).6. Y. V. Kartashov, B. A. Malomed, and L. Torner, Rev. Mod. Phys. 83, 247 (2011).7. Z. Chen, M. Segev, and D. N. Christodoulides, Rep. Prog. Phys. 75, 086401 (2012).8. V. S. Bagnato, D. J. Frantzeskakis, P. G. Kevrekidis, B. A. Malomed, and D. Mihalache, Rom.

Rep. Phys. 67, 5 (2015).9. B. Malomed, L. Torner, F. Wise, and D. Mihalache, J. Phys. B: At. Mol. Opt. Phys. 49, 170502

(2016).10. P. G. Kevrekidis and D. J. Frantzeskakis, Reviews in Physics 1, 140 (2016).11. B. A. Malomed, Eur. Phys. J. Special Topics 225, 2507 (2016).12. D. Mihalache, Rom. Rep. Phys. 69, 403 (2017).13. V. V. Konotop, J. Yang, and D. A. Zezyulin, Rev. Mod. Phys. 88, 035002 (2016).14. S. V. Suchkov, A. A. Sukhorukov, J. H. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, Laser

Photonics Rev. 10, 177 (2016).15. L. Salasnich, Opt. Quant. Electron. 49, 409 (2017).16. Y. Kartashov, G. Astrakharchik, B. Malomed, and L. Torner, Nature Reviews Physics 1, 185

(2019).

(c) RJP64(Nos. 5-6), ID 106-1 (2019) v.2.1r20190102 *2019.7.8#e65077d9


17. B. A. Malomed, Vortex solitons: Old results and new perspectives, Physica D (2019);https://doi.org/10.1016/j.physd.2019.04.009.

18. Y. Silberberg, Opt. Lett. 15, 1282 (1990).19. N. Akhmediev and J. M. Soto-Crespo, Phys. Rev. A 47, 1358 (1993).20. R. McLeod, K. Wagner, and S. Blair, Phys. Rev. A 52, 3254 (1995).21. D. Mihalache, D. Mazilu, J. Dorring, and L. Torner, Opt. Commun. 159, 129 (1999).22. L. C. Crasovan, B. A. Malomed, and D. Mihalache, Phys. Lett. A 289, 59 (2001).23. D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, Phys. Rev. E 67, 056608

(2003).24. D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, Opt. Express 15, 589 (2007).25. D. Mihalache, D. Mazilu, F. Lederer, and Y. S. Kivshar, Opt. Lett. 32, 3173 (2007).26. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, Phys. Rev. A 77, 033817

(2008).27. P. Genevet et al., Phys. Rev. Lett. 101, 123905 (2008).28. D. Mihalache, Rom. J. Phys. 57, 352 (2012).29. B. A. Malomed, J. Opt. Soc. Am. B 31, 2460 (2014).30. D. Mihalache, Rom. J. Phys. 59, 295 (2014).31. C. Fernandez-Oto et al., Phys. Rev. A 89, 055802 (2014).32. V. Skarka et al., Phys. Rev. A 90, 023845 (2014).33. N. N. Rosanov et al., Phil. Trans. R. Soc. A 372, 20140012 (2014).34. W. H. Renninger and F. W. Wise, Optica 1, 101 (2014).35. D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015).36. D. A. Zezyulin and V. V. Konotop, Rom. Rep. Phys. 67, 223 (2015).37. Y. V. Kartashov, C. Hang, G. X. Huang, and L. Torner, Optica 3, 1048 (2016).38. P. Grelu and N. Akhmediev, Nat. Photon. 6, 84 (2012).39. A. V. Husakou and J. Herrmann, Phys. Rev. Lett. 87, 203901 (2001).40. J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135 (2006).41. H. Leblond, Ph. Grelu, and D. Mihalache, Phys. Rev. A 90, 053816 (2014).42. H. Leblond, Ph. Grelu, D. Mihalache, and H. Triki, Eur. Phys. J. Special Topics 225, 2435 (2016).43. F. Leo et al., Nature Photon. 4, 471 (2010).44. T. Herr et al., Nature Photon. 8, 145 (2014).45. X. Yi, Q.-F. Yang, K. Y. Yang, M. G. Suh, and K. Vahala, Optica 2, 1078 (2015).46. T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L. Gorodetsky, Science 361, eaan8083 (2018).47. A. D. Martin and J. Ruostekoski, New J. Phys. 14, 043040 (2012).48. G. D. McDonald et al., Phys. Rev. Lett. 113, 013002 (2014).49. H. Sakaguchi and B. Malomed, New J. Phys. 18, 025020 (2016).50. L. Berge, Phys. Rep. 303, 260 (1998).51. G. Fibich, The Nonlinear Schrodinger Equation: Singular Solutions and Optical Collapse,

Springer, Heidelberg, 2015.52. W. J. Firth and D. V. Skryabin, Phys. Rev. Lett. 79, 2450 (1997).53. L. Torner and D. V. Petrov, Electr. Lett. 33, 608 (1997).54. L. Torner and D. V. Petrov, J. Opt. Soc. Am. B 14, 2017 (1997).55. D. V. Skryabin and W. J. Firth, Phys. Rev. E 58, 3916 (1998).56. E. L. Falcao-Filho et al., Phys. Rev. Lett. 110, 013901 (2013).57. A. S. Reyna, G. Boudebs, B. A. Malomed, and C. B. de Araujo, Phys. Rev. A 93, 013840 (2016).

(c) RJP64(Nos. 5-6), ID 106-1 (2019) v.2.1r20190102 *2019.7.8#e65077d9

https://doi.org/10.1016/j.physd.2019.04.009


58. M. A. Porras, M. Carvalho, H. Leblond, and B. A. Malomed, Phys. Rev. A 94, 053810 (2016).59. Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, M. R. Belic, and L. Torner, Opt. Lett. 42, 446

(2017).60. L. Zeng, J. Zeng, Y. V. Kartashov, and B. A. Malomed, Opt. Lett. 44, 1206 (2019).61. V. Biloshytskyi, A. Oliinyk, P. Kruglenko, A. Desyatnikov, and A. Yakimenko, Phys. Rev. A 99,

043835 (2019).62. G. Liang, Opt. Express 27, 14667 (2019).63. F. Zang, Y. Wang, and L. Li, Opt. Express 27, 15079 (2019).64. X. Peng et al., Laser Phys. Lett. 14, 126001 (2017).65. T. Mayteevarunyo and B. A. Malomed, Opt. Lett 41, 2919 (2016).66. Z. Wu et al., Opt. Express 25, 30468 (2017).67. Y. I. Salamin, Opt. Express 25, 28990 (2017).68. X. Peng et al., Sci. Reports 8, 4174 (2018).69. W. P. Zhong, M. Belic, and Z. P. Yang, Annalen der Physik 2018, 1800059 (2018).70. Z. Huang and D. Deng, J. Opt. Soc. Am. A 35, 536 (2018).71. Y. I. Salamin, Sci. Reports 8, 11362 (2018).72. M. A. Porras, Opt. Express 26, 19606 (2018).73. M. Yessenov et al., Phys. Rev. A 99, 023856 (2019).74. M. Yessenov et al., Optica 6, 598 (2019).75. O. V. Shtyrina, M. P. Fedoruk, Y. S. Kivshar, and S. K. Turitsyn, Phys. Rev. A 97, 013841 (2018).76. I. M. Besieris and A. M. Shaarawi, Opt. Express 27, 792 (2019).77. M. A. Porras, Opt. Lett. 42, 4679 (2017).78. H. Sakaguchi, Frontiers of Physics 14, 12301 (2019).79. J. Qin, Z. Liang, B. A. Malomed, and G. Dong, Phys. Rev. A 99, 023610 (2019).80. R. Driben, X. Ma, S. Schumacher, and T. Meier, Opt. Lett. 44, 1327 (2019).81. A. M. Turing, Philos. Trans. R. Soc. London B 237, 37 (1952).82. I. Prigogine and R. Lefever, J. Chem. Phys. 48, 1695 (1968).83. N. N. Rozanov, J. Opt. Technol. 76, 187 (2009).84. M. Tlidi et al., Phil. Trans. R. Soc. A 372, 20140101 (2014).85. M. Tlidi and K. Panajotov, Rom. Rep. Phys. 70, 406 (2018).86. M. Tlidi, M. G. Clerc, and K. Panajotov, Phil. Trans. R. Soc. A 376, 20180114 (2018).87. N. A. Veretenov, S. V. Fedorov, and N. N. Rosanov, Phys. Rev. Lett. 119, 263901 (2017).88. N. Akhmediev, J. M. Soto-Crespo, P. Vouzas, N. Devine, and W. Chang, Phil. Trans. R. Soc. A

376, 20180023 (2018).89. S. V. Grishin et al., Phys. Rev. E 98, 022209 (2018).90. O. Descalzi and H. R. Brand, Chaos 28, 075508 (2018).91. C. Milian, Y. V. Kartashov, D. V. Skryabin, and L. Torner, Phys. Rev. Lett. 121, 103903 (2018).92. B. A. Kochetov and V. R. Tuz, Phys. Rev. E 98, 062214 (2018).93. H. Li, S. Lai, Y. Qui, X. Zhu, J. Xie, D. Mihalache, and Y. He, Opt. Express 25, 27948 (2017).94. S. Lai, H. Li, Y. Qui, X. Zhu, D. Mihalache, B. A. Malomed, and Y. He, Nonl. Dyn. 93, 2159

(2018).95. Y. Qiu, B. A. Malomed, D. Mihalache, X. Zhu, J. Peng, and Y. He, Chaos, Solitons & Fractals

117, 30 (2018).96. S. V. Fedorov, N. A. Veretenov, and N. N. Rosanov, Phys. Rev. Lett. 122, 0233903 (2019).97. B. A. Kochetov, Physica D 393, 47 (2019).

(c) RJP64(Nos. 5-6), ID 106-1 (2019) v.2.1r20190102 *2019.7.8#e65077d9


98. M. Djoko and T. C. Kofane, Commun. Nonl. Sci. and Numerical Simulation 68, 169 (2019).99. M. Djoko, C. B. Tabi, and T. C. Kofane, Phys. Scr. 94, 075501 (2019).

100. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901(2007).

101. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Opt. Lett. 33, 207 (2008).102. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nature Photon. 4, 103 (2010).103. P. Panagiotopoulos et al., Nature Commun. 4, 2622 (2013).104. P. Piksarv et al., J. Phys.: Conf. Series 497, 012003 (2014).105. N. K. Efremidis, Z. Chen, M. Segev, and D. N. Christodoulides, Optica 6, 686 (2019).106. X. Liu, X. Yao, and Y. Cui, Phys. Rev. Lett. 121, 023905 (2018).107. M. Salhi, G. Semaan, F. Ben Braham, A. Niang, F. Bahloul, and F. Sanchez, Rom. Rep. Phys. 70,

402 (2018).108. Y. C. Liao, B. Liu, J. Liu, and J. Chen, Chin. Phys. Lett. 36, 014203 (2019).109. Z. Q. Wang, K. Nithyanandan, A. Coillet, P. Tchofo-Dinda, and Ph. Grelu, Nature Commun. 10,

830 (2019).110. N. Lilienfein et al., Nature Photon. 13, 214 (2019).111. H. Li et al., Chinese Opt. Lett. 15, 030009 (2017).112. H. Esat Kondakci and A. F. Abouraddy, Nature Photon. 11, 733 (2017).113. O. Lahav, O. Kfir, P. Sidorenko, M. Mutzafi, A. Fleischer, and O. Cohen, Phys. Rev. X 7, 041051

(2017).114. F. W. Wise, Nature (London) 554, 179 (2018).115. Bing-Xiang Li et al., Nature Commun. 9, 2912 (2018).116. V. Shumakova et al. Opt. Lett. 44, 2173 (2019).117. A. V. Zhukov, R. Bouffanais, H. Leblond, D. Mihalache, E. G. Fedorov, and M. B. Belonenko,

Eur. Phys. J. D 69, 242 (2015).118. A. V. Zhukov et al., Eur. Phys. J. D 69, 129 (2015).119. M. B. Belonenko, J. V. Nevzorova, and E. N. Galkina, Modern Phys. Lett. B 29, 1550041 (2015).120. J. V. Nevzorova, M. B. Belonenko, and E. N. Galkina, Bulletin Russ. Acad. Sci.: Physics 80, 837

(2016).121. A. V. Zhukov, R. Bouffanais, N. N. Konobeeva, and M. B. Belonenko, J. Appl. Phys. 120, 134307

(2016).122. A. V. Zhukov et al., Phys. Rev. A 94, 053823 (2016).123. A. V. Zhukov, R. Bouffanais, M. B. Belonenko, I. S. Dvuzhilov, Phys. Lett. A 381, 931 (2017).124. I. S. Dvuzhilov and M. B. Belonenko, Physics of the Solid State 60, 357 (2018).125. E. G. Fedorov et al., Phys. Rev. A 97, 043814 (2018).126. A. M. Zheltikov, Phys.-Usp. 50, 705 (2007).127. S. V. Sazonov, Bull. Russ. Acad. Sci.: Physics 75, 157 (2011).128. H. Leblond and D. Mihalache, Phys. Rep. 523, 61 (2013).129. H. Leblond, H. Triki, and D. Mihalache, Rom. Rep. Phys. 65, 925 (2013).130. G. Mourou, S. Mironov, E. Khazanov, and A. Sergeev, Eur. Phys. J. Special Topics, 223, 1181

(2014).131. M. Kolesik and J. V. Moloney, Rep. Prog. Phys. 77, 016401 (2014).132. D. J. Frantzeskakis, H. Leblond, and D. Mihalache, Rom. J. Phys. 59, 767 (2014).133. K. Homma et al., Rom. Rep. Phys. 68, S233 (2016).134. F. Negoita et al., Rom. Rep. Phys. 68, S37 (2016).

(c) RJP64(Nos. 5-6), ID 106-1 (2019) v.2.1r20190102 *2019.7.8#e65077d9


135. D. Ursescu et al., Rom. Rep. Phys. 68, S11 (2016).136. I. C. E. Turcu et al., Rom. Rep. Phys. 68, S145 (2016).137. D. Strickland and G. Mourou, Opt. Commun. 56, 219 (1985).138. P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, Revue Phys. Appl. 22, 1657 (1987).139. S. Gales et al., Rep. Prog. Phys. 81, 094301 (2018).140. M. Jirka, O. Klimo, M. Vranic, S. Weber, and G. Korn, Sci. Rep. 7, 15302 (2017).141. C. Miron and M. Apostol, Rom. J. Phys. 62, 117 (2017).142. J. Q. Yu et al., Phys. Rev. Lett. 122, 014802 (2019).143. K. Nakamura et al., IEEE J. Quantum Electron. 53, 1200121 (2017).144. D. N. Papadopoulos et al., High Power Laser Sci. and Eng. 4, e34 (2016).145. J. H. Sung et al., Opt. Lett. 42, 2058 (2017).146. Z. Gan et al., Opt. Express 25, 5169 (2017).147. F. Batysta et al., Opt. Lett. 43, 3866 (2018).148. S. Mondal et al., J. Opt. Soc. Am. B 35, A93 (2018).149. O. Jahn et al., Optica 6, 280 (2019).150. I. C. E. Turcu et al., High Power Laser Science and Engineering, 7, e10 (2019).151. M. F. Ciappina et al., Phys. Rev. A 99, 043405 (2019).152. T. C. Wilson et al., J. Phys. B: At. Mol. Opt. Phys. 52, 055403 (2019).153. F. Mackenroth, N. Kumar, N. Neitz, and C. H. Keitel, Phys. Rev. E 99, 033205 (2019).154. A. J. Macleod, A. Noble, and D. A. Jaroszynski, Phys. Rev. Lett. 122, 161601 (2019).155. C. L. Korpa, G. Toth, and J. Hebling, J. Phys. B: At. Mol. Opt. Phys. 49, 035401 (2016).156. S. V. Sazonov, Opt. Commun. 380, 480 (2016).157. D. V. Novitsky, Opt. Commun. 358, 202 (2016).158. R. M. Arkhipov et al., Laser Phys. Lett. 13, 046001 (2016).159. A. A. Voronin and A. M. Zheltikov, Phys. Rev. A 94, 023824 (2016).160. A. A. Voronin and A. M. Zheltikov, J. Opt. 18, 115501 (2016).161. Y. A. Kapoyko et al., Phys. Rev. A 94, 033803 (2016).162. P. Maurer et al., Phys. Rev. Lett. 117, 103602 (2016).163. R. M. Arkhipov, A. V. Pakhomov, I. V. Babushkin, M. V. Arkhipov, Yu. A. Tolmachev, and N. N.

Rosanov, J. Opt. Soc. Am. B 33, 2518 (2016).164. A. V. Pakhomov, R. M. Arkhipov, I. V. Babushkin, M. V. Arkhipov, Yu. A. Tolmachev, and N. N.

Rosanov, Phys. Rev. A 95, 013804 (2017).165. M. V. Arkhipov et al., Opt. Lett. 42, 2189 (2017).166. S. Terniche, H. Leblond, D. Mihalache, and A. Kellou, Phys. Rev. A 94, 063836 (2016).167. H. Leblond, D. Kremer, and D. Mihalache, Phys. Rev. A 95, 043839 (2017).168. H. Leblond and D. Mihalache, J. Phys. A: Math. Theor. 51, 435202 (2018).169. R. Blit and B. A. Malomed, Phys. Rev. A 86, 043841 (2012).170. R. I. Grynko, G. C. Nagar, and B. Shim, Phys. Rev. A 98, 023844 (2018).171. S. V. Sazonov and N. V. Ustinov, Phys. Rev. A 98, 063803 (2018).172. S. V. Sazonov, Rom. Rep. Phys. 70, 401 (2018).173. Z. J. Gao, H. J. Li, and J. Lin, Opt. Express 26, 9027 (2018).174. N. N. Konobeeva and M. B. Belonenko, Rom. Rep. Phys. 70, 403 (2018).175. A. A. Balakin, A. G. Litvak, V. A. Mironov, S. A. Skobelev, and L. A. Smirnov, Laser Phys. 28,

105401 (2018).

(c) RJP64(Nos. 5-6), ID 106-1 (2019) v.2.1r20190102 *2019.7.8#e65077d9


176. A. V. Pakhomov, R. M. Arkhipov, M. V. Arkhipov, A. Demircan, U. Morgner, N. N. Rosanov,and I. Babushkin, Sci. Reports 9, 7444 (2019).

177. M. A. Porras, Opt. Lett. 44, 2538 (2019).178. M. A. Porras, Phys. Rev. Lett.122, 123904 (2019).179. N. Tsatrafyllis et al., Phys. Rev. Lett. 122, 193602 (2019).180. L. G. Wright, D. N. Christodoulides, and F. W. Wright, Science 358, 94 (2017).181. A. Zheltikov, J. Opt. Soc. Am. B 36, A168 (2019).182. A. J. Gonsalves et al., Phys. Rev. Lett. 122, 084801 (2019).183. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).184. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A: Math. Gen. 38, L171 (2005).185. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632

(2007).186. R. Driben and B. A. Malomed, Opt. Lett. 36, 4323 (2011).187. A. Guo et al., Phys. Rev. Lett. 103, 093902 (2009).188. C. E. Rutter et al., Nature Phys. 6, 192 (2010).189. Y. J. He and B. A. Malomed, in Progress in Optical Science and Photonics, Volume 1, pp. 125-

148, Springer, Berlin, 2013.190. Y. J. He, X. Zhu, and D. Mihalache, Rom. J. Phys. 61, 595 (2016).191. G. Burlak, S. Garcia-Paredes, and B. A. Malomed, Chaos 26, 113103 (2016).192. J. Cuevas-Maraver, B. A. Malomed, and P. G. Kevrekidis, Symmetry 8, 39 (2016).193. I. V. Barashenkov, D. E. Pelinovsky, and P. Dubard, J. Phys. A: Math. Theor. 48, 325201 (2015).194. P. F. Li, B. Li, L. Li, and D. Mihalache, Rom. J. Phys. 61, 577 (2016).195. P. F. Li, D. Mihalache, and L. Li, Rom. J. Phys. 61, 1028 (2016).196. Si-Liu Xu et al., EPL 115, 14006 (2016).197. Li-Hua Wang et al., Eur. Phys. J. Plus 131, 211 (2016).198. Y. Chen, Z. Yan, D. Mihalache, and B. A. Malomed, Sci. Rep. 7, 1257 (2017).199. Y. V. Kartashov, B. A. Malomed, and L. Torner, Opt. Lett. 39, 5641 (2014).200. E. Luz, V. Lutsky, E. Granot, and B. A. Malomed, Sci. Rep. 9, 4483 (2019).201. B. Liu, L. Li, and B. A. Malomed, Eur. Phys. J. D 71, 140 (2017).202. R. J. Lombard and R. Mezhoud, Rom. J. Phys. 62, 112 (2017).203. R. J. Lombard, R. Mezhoud, and R. Yekken, Rom. J. Phys. 63, 101 (2018).204. P. Li, D. Mihalache, and B. A. Malomed, Phil. Trans. R. Soc. A 376, 20170378 (2018).205. P. Li, L. Li, and D. Mihalache, Rom. Rep. Phys. 70, 408 (2018).206. B. Liu, L. Li, and D. Mihalache, Rom. Rep. Phys. 70, 409 (2018).207. H. Wang, J. Huang, X. Ren, Y. Weng, D. Mihalache, and Y. He, Rom. J. Phys. 63, 205 (2018).208. E. N. Tsoy, F. Kh. Abdullaev, and V. E. Eshniyazov, Phys. Rev. A 98, 043854 (2018).209. P. Li, C. Dai, R. Li, and Y. Gao, Opt. Express 26, 6949 (2018).210. S. Scheel and A. Szameit, EPL 122, 34001 (2018).211. Y. V. Kartashov and V. A. Vysloukh, Opt. Lett. 44, 791 (2019).212. L. Wang, B. A. Malomed, and Z. Yan, Phys. Rev. E 99, 052206 (2019).213. L. Qin, C. Hang, and G. Huang, Phys. Rev. A 99, 043832 (2019).214. J. Yang, Opt. Lett. 44, 2641 (2019).215. N. Akhmediev et al., J. Opt. 18, 063001 (2016).216. J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, Nature Photon. 8, 755 (2014).217. M. Onorato, S. Residori, U. Bortolozzo, A. Montina, and F. T. Arecchi, Phys. Rep. 528, 47 (2013).

(c) RJP64(Nos. 5-6), ID 106-1 (2019) v.2.1r20190102 *2019.7.8#e65077d9


218. S. Chen, F. Baronio, J. M. Soto-Crespo, P. Grelu, and D. Mihalache, J. Phys. A: Math. Theor. 50,463001 (2017).

219. M. Onorato, D. Proment, G. Clauss, and M. Klein, PLOS ONE 8, e54629 (2013).220. F. Baronio, S. Wabnitz, and Y. Kodama, Phys. Rev. Lett. 116, 173901 (2016).221. J. M. Soto-Crespo, N. Devine, and N. Akhmediev, Phys. Rev. Lett. 116, 103901 (2016).222. A. Ankiewicz et al., Phys. Rev. E 93, 012206 (2016).223. S. W. Xu, K. Porsezian, J. S. He, and Y. Cheng, Rom. Rep. Phys. 68, 316 (2016).224. S. Chen, J. M. Soto-Crespo, F. Baronio, Ph. Grelu, and D. Mihalache, Opt. Express 24, 15251

(2016).225. S. Chen, F. Baronio, J. M. Soto-Crespo, Y. Liu, and Ph. Grelu, Phys. Rev. A 93, 062202 (2016).226. F. Yuan, J. Rao, K. Porsezian, D. Mihalache, and J. S. He, Rom. J. Phys. 61, 378 (2016).227. Y. Liu, A. S. Fokas, D. Mihalache, and J. S. He, Rom. Rep. Phys. 68, 1425 (2016).228. S. Chen, P. Grelu, D. Mihalache, and F. Baronio, Rom. Rep. Phys. 68, 1407 (2016).229. W. P. Zhong, M. Belic, and B. A. Malomed, Phys. Rev. E 92, 053201 (2015).230. S. Chen and D. Mihalache, J. Phys. A: Math. Theor. 48, 215202 (2015).231. H. N. Chan, B. A. Malomed, K. W. Chow, and E. Ding, Phys. Rev. E 93, 012217 (2016).232. W. Liu, Rom. J. Phys. 62, 118 (2017).233. D. H. Peregrine, J. Aust. Math. Soc., Ser. B, Appl. Math. 25, 16 (1983).234. E. A. Kuznetsov, Sov. Phys.-Dokl. 22, 507 (1977).235. Y. C. Ma, Stud. Appl. Math. 60, 43 (1979).236. N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, Theor. Math. Phys. 72, 809 (1987).237. D. Mihalache and N. C. Panoiu, J. Phys. A: Math. Gen. 26, 2679 (1993).238. D. Mihalache, F. Lederer, and D. M. Baboiu, Phys. Rev. A 47, 3285 (1993).239. N. Akhmediev and A. Ankiewicz, Phys. Rev. A 47, 3213 (1993).240. L. Gagnon, J. Opt. Soc. Am. B 10, 469 (1993).241. A. Ankiewicz and N. Akhmediev, Phys. Rev. E 96, 012219 (2017).242. L. Ling et al., Phys. Rev. E 96, 022211 (2017).243. A. Chowdury and W. Krolikowski, Phys. Rev. E 96, 042209 (2017).244. L. Wang, J. He, H. Xu, J. Wang, and K. Porsezian, Phys. Rev. E 95, 042217 (2017).245. F. Baronio, Opt. Lett. 42, 1756 (2017).246. F. Baronio, S. Chen, and D. Mihalache, Opt. Lett. 42, 3514 (2017).247. A. Ankiewicz and N. Akhmediev, Rom. Rep. Phys. 69, 104 (2017).248. H. N. Chan and K. W. Chow, Appl. Sci. 2018, 1223 (2018).249. S. Zhao, H. Yang, and Y. Xiao, Phys. Rev. A 98, 043817 (2018).250. Z. Yang et al., Opt. Express 26, 7587 (2018).251. L. Liu et al., Phys. Rev. E 97, 052217 (2018).252. G. Zhang and Z. Yan, Proc. R. Soc. A 474, 20170688 (2018).253. S. Chen, Y. Ye, J. M. Soto-Crespo, P. Grelu, and F. Baronio, Phys. Rev. Lett. 121, 104101 (2018).254. S. A. El-Tantawy and A. M. Wazwaz, Phys. of Plasmas 25, 092105 (2018).255. E. G. Charalampidis, J. Lee, P. G. Kevrekidis, and C. Chong, Phys. Rev. E 98, 032903 (2018).256. S. Randoux et al., Phys. Rev. E 98, 022219 (2018).257. S. Chen, Y. Zhou, F. Baronio, and D. Mihalache, Rom. Rep. Phys. 70, 102 (2018).258. E. G. Charalampidis, J. Cuevas-Maraver, D. J. Frantzeskakis, and P. G. Kevrekidis, Rom. Rep.

Phys. 70, 504 (2018).259. Y. Zhang, D. Qiu, D. Mihalache, and J. He, Chaos 28, 103108 (2018).

(c) RJP64(Nos. 5-6), ID 106-1 (2019) v.2.1r20190102 *2019.7.8#e65077d9


260. Y. Liu, C. Qian, D. Mihalache, and J. He, Nonl. Dyn. 95, 839 (2019).261. G. Zhang, Z. Yan, and L. Wang, Proc. R. Soc. A 475, 20180625 (2019).262. L. Guo, Y. Cheng, D. Mihalache, and J. He, Rom. J. Phys. 64, 104 (2019).263. N. Akhtar et al., Rom. Rep. Phys. 71, 403 (2019).264. J. Rao, D. Mihalache, Y. Cheng, J. He, Phys. Lett. A 383, 1138 (2019).265. W. Liu, Y. Zhang, and J. He, Rom. Rep. Phys. 70, 106 (2018).266. S. Chen and Z. Yan, Appl. Math. Lett. 95, 65 (2019).267. Li-Chen Zhao, Liang Duan, Peng Gao, and Zhan-Ying Yang, EPL 125, 40003 (2019).268. A. Degasperis, S. Lombardo, and M. Sommacal, Fluids 4, 57 (2019).269. B. Yang and J. Yang, Lett. Math. Phys. 109, 945 (2019).270. Y. L. Ye, Y. Zhou, S. H. Chen, F. Baronio, and P. Grelu, Proc. R. Soc. A 475, 20180806 (2019).271. S. Chen, Y. Zhou, L. Bu, F. Baronio, J. M. Soto-Crespo, and D. Mihalache, Opt. Express 27,

11370 (2019).272. J. Cuevas-Maraver, B. A. Malomed, P. G. Kevrekidis, and D. J. Frantzeskakis, Phys. Lett. A 382,

968 (2018).273. A. Klein et al., Optica 5, 774 (2018).274. S. Coulibaly et al., Phys. Rev. X 9, 011054 (2019).275. B. A. Malomed, EPL 122, 360001 (2018).276. W. Pang et al., Appl. Sci. 2018, 1771 (2018).277. S. K. Adhikari, Laser Phys. Lett. 15, 095501 (2018).278. S. Gautam and S. K. Adhikari, Phys. Rev. A 97, 013629 (2018).279. H. Li et al., Phys. Rev. A 98, 033827 (2018).280. Y. Wang and S. Y. Zhou, Commun. Theor. Phys. 70, 132 (2018).281. Y. Wang and Y. Yang, AIP Advances 8, 095317 (2018).282. S. Sabari and B. Dey, Phys. Rev. E 98, 042203 (2018).283. C. Ticknor, W. Wang, and P. G. Kevrekidis, Phys. Rev. A 98, 033609 (2018).284. M. Salerno and B. B. Baizakov, Phys. Rev. E 98, 062220 (2018).285. J. D’Ambroise, D. J. Frantzeskakis, and P. G. Kevrekidis, Rom. Rep. Phys. 70, 503 (2018).286. H. Sakaguchi, B. Li, E. Ya. Sherman, and B. A. Malomed, Rom. Rep. Phys. 70, 502 (2018).287. H. Sakaguchi, Phys. Lett. A 383, 1132 (2019).288. G. Yang, S. Zhang, and J. Jin, J. Phys. B: At. Mol. Opt. Phys. 52, 065201 (2019).289. M. Ikuta, Y. Sugano, and H. Saito, Phys. Rev. A 99, 043610 (2019).290. Y. V. Kartashov and D. A. Zezyulin, Phys. Rev. Lett. 122, 123201 (2019).291. Chiao-Hsuan Wang, M. J. Gullans, J. V. Porto, W. D. Phillips, and J. M. Taylor, Phys. Rev. A 99,

031801 (R) (2019).292. T. D. Lee, K. Huang, and C. N. Yang, Phys. Rev. 106, 1135 (1957).293. A. Cappellaro, T. Macrı, G. F. Bertacco, and L. Salasnich, Sci. Rep. 7, 13358 (2017).294. D. S. Petrov and G. E. Astrakharchik, Phys. Rev. Lett. 117, 100401 (2016).295. G. E. Astrakharchik and B. A. Malomed, Phys. Rev. A 98, 013631 (2018).296. Y. V. Kartashov, B. A. Malomed, L. Tarruell, and L. Torner, Phys. Rev. A 98, 013612 (2018).297. S. Gautam and S. K. Adhikari, J. Phys. B: At. Mol. Opt. Phys. 52, 055302 (2019).298. Y. Li, Z. Chen, Z. Luo, C. Huang, H. Tan, W. Pang, and B. A. Malomed, Phys. Rev. A 98, 063602

(2018).299. Y. V. Kartashov, B. A. Malomed, and L. Torner, Phys. Rev. Lett. 122, 193902 (2019).

(c) RJP64(Nos. 5-6), ID 106-1 (2019) v.2.1r20190102 *2019.7.8#e65077d9


300. Y. Li, Z. Luo, Y. Liu, Z. Chen, C. Huang, S. Fu, H. Tan, and B. A. Malomed, New J. Phys. 19,113043 (2017).

301. Bin Liu, Hua-Feng Zhang, Rong-Xuan Zhong, Xi-Liang Zhang, Xi-Zhou Qin, Chunqing Huang,Yong-Yao Li, and Boris A. Malomed, Phys. Rev. A 99, 053602 (2019).

302. C. R. Cabrera et al., Science 359, 301 (2018).303. P. Cheiney et al., Phys. Rev. Lett. 120, 135301 (2018).304. G. Semeghini et al., Phys. Rev. Lett. 120, 235301 (2018).305. G. Ferioli et al., Phys. Rev. Lett. 122, 090401 (2019).306. H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. Ferrier-Barbut, and T. Pfau, Nature 530,

194 (2016).307. I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Phys. Rev. Lett. 116, 215301

(2016).308. L. Chomaz et al., Phys. Rev. X 6, 041039 (2016).309. A. Cidrim, F. E. A. dos Santos, E. A. L. Henn, and T. Macrı, Phys. Rev. A 98, 023618 (2018).310. Y. V. Kartashov, V. V. Konotop, M. Modugno, and E. Ya. Sherman, Phys. Rev. Lett. 122, 064101

(2019).311. L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 65, 043614 (2002).312. L. Salasnich and B. A. Malomed, Phys. Rev. A 74, 053610 (2006).313. A. Munoz Mateo and V. Delgado, Phys. Rev. A 77, 013607 (2008).314. A. I. Nicolin and M. C. Raportaru, Physica A 389, 4663 (2010).315. A. Munoz Mateo, V. Delgado, and B. A. Malomed, Phys. Rev. A 82, 053606 (2010).316. A. Munoz Mateo and V. Delgado, Phys. Rev. E 88, 042916 (2013).317. A. I. Nicolin, M. C. Raportaru, and A. Balaz, Rom. Rep. Phys. 67, 143 (2015).318. D. Vudragovic, I. Vidanovic, A. Balaz, P. Muruganandam, and S. K. Adhikari, Comput. Phys.

Commun. 183, 2021 (2012).319. B. Sataric, V. Slavnic, A. Belic, A. Balaz, P. Muruganandam, and S. K. Adhikari, Comput. Phys.

Commun. 200, 411 (2016).320. L. E. Young-S., D. Vudragovic, P. Muruganandam, S. K. Adhikari, and A. Balaz, Comput. Phys.

Commun. 204, 209 (2016).321. V. Loncar, L. E. Young-S., S. Skrbic, P. Muruganandam, S. K. Adhikari, and A. Balaz, Comput.

Phys. Commun. 209, 190 (2016).322. L. E. Young-S., P. Muruganandam, S. K.Adhikari, V. Loncar, D. Vudragovic, and A. Balaz,

Comput. Phys. Commun. 220, 503 (2017).323. I. Danaila, M. A. Khamehchi, V. Gokhroo, P. Engels, and P. G. Kevrekidis, Phys. Rev. A 94,

053617 (2016).324. T. M. Bersano, V. Gokhroo, M. A. Khamehchi, J. D’Ambroise, D. J. Frantzeskakis, P. Engels,

and P. G. Kevrekidis, Phys. Rev. Lett. 120, 063202 (2018).325. S. Kang, Sang Won Seo, H. Takeuchi, and Y. Shin, Phys. Rev. Lett. 122, 095301 (2019).326. J. H. V. Nguyen, M. C. Tsatsos, D. Luo, A. U. J. Lode, G. D. Telles, V. S. Bagnato, and R. G.

Hulet, Phys. Rev. X 9, 011052 (2019).

(c) RJP64(Nos. 5-6), ID 106-1 (2019) v.2.1r20190102 *2019.7.8#e65077d9

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