Spatiotemporal vortex solitons in hexagonal arrays of waveguides
SSpatiotemporal vortex solitons in hexagonal arrays of waveguides
Herv´e Leblond , Boris A. Malomed , , and Dumitru Mihalache , , Laboratoire de Photonique d’Angers, EA 4464, Universit´e d’Angers, 2 Bd Lavoisier, 49000 Angers, France Department of Physical Electronics, School of Electrical Engineering,Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain Horia Hulubei National Institute for Physics and Nuclear Engineering (IFIN-HH),407 Atomistilor, Magurele-Bucharest, 077125, Romania Academy of Romanian Scientists, 54 Splaiul Independentei, Bucharest 050094, Romania
By means of a systematic numerical analysis, we demonstrate that hexagonal lattices of parallellinearly-coupled waveguides, with the intrinsic cubic self-focusing nonlinearity, give rise to threespecies of stable semi-discrete complexes (which are continuous in the longitudinal direction), withembedded vorticity S : triangular modes with S = 1, hexagonal ones with S = 2, both centeredaround an empty central core, and compact triangles with S = 1, which do not not include the emptysite. Collisions between stable triangular vortices are studied too. These waveguiding lattices canbe realized in optics and BEC. PACS numbers: 42.65.Tg, 42.81.Dp, 03.75.Lm, 05.45.Yv
I. INTRODUCTION
Lattice solitons are a topic of great interest to on-going studies of nonlinear dynamics in photonic mediaand BECs (Bose-Einstein condensates) [1]. These local-ized modes are produced by the interplay of the intrin-sic nonlinearity of the medium with an effective periodicpotential induced in it by permanent or virtual latticepatterns. In fact, the lattice may itself be a nonlinearstructure if it is induced by a spatially periodic mod-ulation of the local nonlinearity [2]. In the limit of adeep periodic potential, the fundamental models of lat-tice media reduce to various versions of the discrete non-linear Schr¨odinger (DNLS) equation [3]. The realizationof the one-dimensional (1D) DNLS model in arrayed op-tical waveguides was originally proposed in Ref. [4]. Thesame model was later applied to BECs loaded into deepoptical-lattice potentials [5] (see Ref. [6] for a brief re-view). A physical realization of the DNLS model is alsopossible in the form of lattices of microcavities whichserve as traps for polaritons [7].Lattice solitons take the form of discrete solitons interms of the DNLS equations, which correspond to quasi-discrete solitons in the respective experimental settings.Such solitons were created in a set of semiconductorwaveguides built on top of a slab substrate [8], and alsoin arrays of optical fibers [9]. In addition to using perma-nent photonic structures, quasi-discrete solitons were alsomade in virtual waveguiding arrays, using the versatiletechnique of inducing interference lattices in photorefrac-tive crystals [10]. The latter method was used to createthe first examples of 2D quasi-discrete fundamental soli-tons [11], which was followed by the making of vortexsolitons [12], i.e., localized lattice excitations with embed-ded vorticity, that were predicted in Ref. [13]. Anothersignificant contribution to this area was the creation of2D solitons in a bundle of fiber-like waveguides writtenin bulk silica [14]. Such arrays and bundles are created by means of tightly focused femtosecond laser pulses [15].Following the analysis of the fundamental localized dis-crete vortices with topological charge S = 1 [13], theirhigher-order counterparts, with S >
1, and multipolediscrete solitons, such as quadrupoles, were predicted inRefs. [16, 17]. Many other objects were studied in thisarea, including supervortices (circular chains of compactvortices with an imprinted overall topological charge,which is independent of the vorticity of the individual ed-dies [17]), necklace-shaped patterns [18], discrete solitonsin hexagonal and honeycomb lattices [19, 20], composite semidiscrete spatial solitons in arrays of waveguides withquadratic and cubic nonlinearities [21], quasi-discretetopological solitons in photonic-crystal fibers [22], etc.Nonstationary soliton effects were studied too. These in-clude the mobility of discrete solitons [23, 24], collisionsbetween traveling ones [24, 25], and the onset of the spa-tiotemporal collapse in self-focusing arrayed waveguides[26].Most works on lattice solitons dealt with the spatial-domain settings. In particular, optically-induced latticesin photorefractive crystals do not makes it possible toobserve the evolution in the temporal domain because ofa very large response time in these materials. However,the spatiotemporal dynamics can be realized in waveg-uiding arrays written in bulk silica [15], where the spa-tially localized quasi-discrete patterns in the transverseplane can be combined with the temporal self-trapping inthe longitudinal direction. Recently, the creation of thecorresponding quasi-discrete “light bullets” was reportedin this system [27] (for a review of spatiotemporal solitonsin nonlinear optics and BEC, see Ref. [28]). Previously,a number of manifestations of the spatiotemporal self-trapping in similar settings were studied theoretically,including the related modulational instability [29], for-mation of “bullets” in fiber arrays [30] and photonic wires[31], and self-compression [32] and steering [33] of pulsedbeams. Continuing the work in this direction, semi- a r X i v : . [ n li n . PS ] M a y discrete spatiotemporal surface solitons were introduced,as surface modes , in models of semi-infinite waveguidearrays [34], and in a system with an interface betweendifferent arrays [35]. Also analyzed were spatiotemporalsolitons in waveguide arrays with the quadratic nonlin-earity [36].Once stable fundamental spatiotemporal soliton com-plexes in bundled arrays of waveguides are available, itis natural to seek for vortex solitons in the same set-ting. A systematic analysis of spatiotemporal vorticesand quadrupoles in the model based on the square lat-tice of discrete waveguides was reported in Ref. [37].A vast stability area was found for the solitary vorticeswith S = 1 and quadrupoles, which are built as rhom-buses, alias on-site-centered modes, with respect to theunderlying lattice (the rhombus is built as a set of four“bright” cores, with a nearly “dark” one at the cen-ter). The stability region is much smaller for the off-site-centered modes of the “square” type, without an emptypivotal site in the middle (the reduced stability domainof square-shaped vortices and quadrupoles, in compari-son with their rhombic counterparts, in a generic featureof topological solitons in lattice media [38]). All the spa-tiotemporal vortex solitons with S = 2 were found to beunstable unstable in the same model. Further, collisionsbetween stable vortices and quadrupoles (with identicalor opposite topological charges), propagating along thebundle in opposite directions, were analyzed in Ref. [39].Four different outcomes of the collisions were identified:rebound of slowly moving solitary vortices, fusion, split-ting, and quasi-elastic interactions between fast ones.Hexagonal lattices may be created by means of thesame techniques which were used for the building thesquare-shaped structures. On the other hand, the changeof the underlying geometry may essentially alter funda-mental properties of topological lattice solitons [3, 19]. Inparticular, it was predicted theoretically and conformedin an experiment that spatial solitons in the form of dou-ble vortices (with S = 2) in hexagonal lattices may bestable, while their unitary counterparts (with S = 1) areunstable.The objective of the present work is to study spa-tiotemporal vortex solitons in hexagonal lattices of dis-crete waveguides. The model is formulated in Section 2,and at the end of it we also briefly consider fundamen-tal solitons, driven by a temporally self-trapped pulsein a single waveguiding core. In Section 3, we demon-strate, also in a brief form, that a straightforward inputin the form of a hexagon-shaped spatiotemporal vortexwith S = 1 always leads to a decay. Nevertheless, threedifferent species of stable spatiotemporal complexes withthe embedded vorticity are revealed by a systematic nu-merical analysis. In Section 4, we demonstrate that aspatiotemporal input of a triangular shape generates self-trapped vortices in the form of triangles with an emptycore in the middle. Further, in Section 5 it is shownthat a hexagonally shaped input with S = 2 producesstable spatiotemporal hexagons with the same ( double ) topological charge. Finally, a modified (shifted) inputansatz gives rise to stable densely packed triangular vor-tices with S = 1, without an empty central core, as shownin Section 6. In addition to the study of these speciesof spatiotemporal vortex solitons, in Section 7 collisionsbetween counterpropagating triangular ones are studied.The paper is concluded by Section 8. II. THE MODEL AND FUNDAMENTALSOLITONS
We consider the hexagonal array of nonlinear waveg-uides, with cells in the transverse lattice numbered asshown in Fig. 1. The transmission of waves in the ar-ray is described by the following system of coupled NLSequations, written in the scaled form, similar to that usedin many earlier works [29]-[37], [27]: i∂ z u m,n + [ u m − ,n − + u m − ,n + u m,n − + u m,n +1 + u m +1 ,n + u m +1 ,n +1 − (6 + µ ) u m,n ]+ (1 / ∂ t u m,n + u m,n | u m,n | = 0 . (1)In terms of the optical setting, z and t are, respectively,the propagation distance and reduced time, assumingthat each guiding core features the anomalous chromaticdispersion and cubic self-focusing, while − µ is the propa-gation constant of the localized solution to be sought for.In terms of the corresponding BEC model, Eqs. (1) isa system of coupled discrete Gross-Pitaevskii equations[40], with z and t playing the roles of the scaled time andaxial coordinate, respectively, while µ is the chemical po-tential.Simulations of Eq. (1) were carried out in the Fourierdomain, with the help of the standard fourth-orderRunge-Kutta scheme, the nonlinear term being evalu-ated by means of the combination of inverse and directfast Fourier transforms at each sub-step of the scheme.We used an 11 ×
12 matrix in the plane of ( m, n ), 512points for variable t in the computation window of width∆ t = 20, and the stepsize in the propagation direction dz = 5 × − . The use of the Fourier transform impliesperiodic boundary conditions in t , which make sense if acharacteristic temporal size of the localized objects willbe essentially smaller than ∆ t = 20. As concerns theboundary conditions for the discrete coordinates m and n , the values of u m,n corresponding to the coordinateswhich fall outside of the computation box are replacedby zeros.Before proceeding to the search for complex spatiotem-poral vortical patterns, it makes sense to test the prop-agation of fundamental solitons, which are carried, es-sentially, by a temporal pulse in a single core. For thispurpose, the simulations were initiated with obvious ini-tial conditions, u , ( z = 0) = η sech ( ηt ) , (2) η = (cid:112) µ ) , (3) FIG. 1. (Color online) The setup and notation: We con-sider the hexagonal array of cylindrical waveguides, as shownin the upper left corner. The transverse distribution of thelight intensity of the propagating waves in the guiding coresis displayed symbolically in the bottom left corner. Each coreis assumed to be a single-mode waveguide, represented bywave function u m,n , with discrete coordinates ( m, n ) definedas shown in the figure. The map of integers m, n into theCartesian coordinates in the transverse plane, ( x, y ), is per-formed as per Eq. (5), a being the width of the hexagonalcell. setting u m,n ( z = 0) = 0 at | m | + | n | (cid:54) = 0. The simula-tions were run in interval 1 ≤ µ ≤
16 of values of thepropagation constant.It has been concluded that input (2) decays, underthe action of the lattice diffraction, at µ < .
6, and astable fundamental soliton, concentrated in the centralcore, is formed at µ ≥ .
6. The temporal pulse which liesat the core of the so created fundamental soliton is notquite stationary, but rather features regular pulsations,as shown in Fig. 2.One may surmise that the oscillations of the funda-mental soliton could be a result of its interaction withthe radiation background, which was generated by theinput field in the course of self-trapping into the funda-mental soliton. To check this possibility, the backgroundaround the soliton was explicitly removed, at a particu-lar step of the simulations. Nevertheless, the oscillationsremain virtually unaffected by the “cleaning”, i.e., theyseem to be a genuine feature of the dynamics of the soli-ton, possibly representing its intrinsic mode.
III. THE HEXAGONAL INPUT: ATRANSITION TO INSTABILITY
First, we attempted to create hexagonal vortical modeswith S = 1, which seems a natural approach to thesystem based on a hexagonal lattice. To this end, weused the following input, based on an ansatz factorizedin the longitudinal (temporal) and transverse (spatial) (a) m a x ( | u | ) µ (b) FIG. 2. (Color online) (a) Oscillations of the fundamentalsoliton generated by input (2), (3) with µ = 8. (b) The up-per and lower curves illustrate the oscillations by showing,respectively, the largest and smallest values of the soliton’samplitude, i.e., max z (max t ( | u | )) and min z (max t ( | u | )), asfunctions of the propagation constant, µ . directions, cf. Ref. [41]: u m,n = ( x m,n + iy m,n ) a exp (cid:104) − α (cid:16)(cid:113) x m,n + y m,n − a (cid:17)(cid:105) × η sech ( ηt ) , (4)with η taken as per Eq. (3), and x m,n ≡ a (cid:16) m − n (cid:17) , y m,n ≡ √ an, (5) α = ln (2 (6 + µ )) , (6) a being the width of the hexagonal cell, see Fig. 1.The model’s scale is fixed by setting a ≡
1. Factor( x m,n + iy m,n ) in Eq. (4) obviously corresponds to vor-ticity S = 1, and the exponential factor with α takenas per Eq. (6) is determined as in the 2D spatial soli-ton with propagation constant − µ . The choice of η as π π /30 π /3 2 π /3 5 π /3 FIG. 3. (Color online) The phase and energy ( (cid:82) + ∞−∞ | u m,n | dt )patterns corresponding to input ansatz (4) which generates an(unstable) hexagonal vortex with µ = 350. per Eq. (3) implies that, simultaneously, the wave fieldin the factorized ansatz is localized in the longitudinaldirection, in each core, as in the temporal soliton corre-sponding to the same propagation constant, cf. the struc-ture of the fundamental solitons considered above. Thephase and energy structure of ansatz (4) is illustrated, ina schematic form, by Fig. 3.Direct simulations of Eq. (1) with this input have beenrun in a broad range of values of the propagation con-stant, 7 < µ < S = 1 have never emerged. In fact, the evolutionof the input organized as the ansatz of this type neverleads to formation of any stable pattern. In the intervalof 7 ≤ µ ≤
13, the system makes an attempt to gener-ate a robust pattern of a triangular shape, as shown inFig. 4: at three sites belonging to the original hexagon,the field quickly decays, while at three others it survives,for a while. However, the largest amplitude of the tem-porarily emerging triangular set is (cid:39) µ = 13), while triangular vortices may be stable for am-plitudes above a threshold value of the amplitude whichis (cid:39)
18 (see below), therefore the triangles developingfrom the unstable hexagons are also subject to an insta-bility, eventually splitting into uncorrelated single-coreexcitations which separate in the longitudinal direction,see Fig. 4.Further, in the interval of 13 < µ ≤
70, the instabilitysplits the original hexagon into a set of separating single-core excitations, the number of which varies randomlybetween 2 and 6 (not shown here in detail). In an adja-cent interval, 100 ≤ µ < ≤ µ ≤ FIG. 4. (Color online) The evolution of the six main compo-nents of the hexagon with initial propagation constant µ = 9.In this case, the simulations demonstrate the decay of thehexagon into a transient triangle, which is followed by a lon-gitudinal instability (splitting). ent. The six sites forming the hexagon keep their posi-tions and amplitudes for a while, but loose the mutualphase coherence. Then, instabilities of amplitudes andpositions set in, but they manifest themselves on a muchlonger scale of the propagation distance, with z rangingfrom 10 to a few hundreds, instead of z ∼ µ ,cf. Fig. 4. The separation between excitations in indi-vidual cores grows very slowly too, in comparison withthe quick split of the transient triangle observed in Fig.4. IV. THE GENERATION OF STABLETRIANGULAR VORTICES
The next step is an attempt to generate a triangu-lar vortical structure, which is suggested by the emer-gence of a transient one in the course of the evolution ofthe unstable hexagon (Fig. 4). For this purpose, weused the same input as defined by Eqs. (4)-(6), butwith three main peaks suppressed, which was done byreplacing the fields at the corresponding sites by thosefrom adjacent sites in the outer layer: u − , → u − , , u , → u , , u , − → u , − , as shown in Fig. 5. The soconstructed triangular ansatz keeps the vorticity of theoriginal hexagon, S = 1.The evolution of this input was simulated in a broadrange of values of the propagation constant, 4 ≤ µ ≤ µ ≤ , the three main peaks formingthe triangle merge into a single-core fundamental soli-ton, which may be localized at the central site, or atany one belonging to the original triangle. Thus, stablevortices do not emerge in this case. For 7 ≤ µ ≤ µ ).Finally, the same input generates stable triangular vor-tices at µ ≥ FIG. 5. (Color online) The reduction of the hexagonal in-put to the triangular-vortex one: three of the six main peaksare replaced by fields taken from the surrounding layer, asshown by arrows (the replacement makes the amplitudes atthe corresponding sites much smaller, but does not alter theirphases). long propagation distances, e.g., z = 986 for µ = 182.The temporal pulses in the cores which represent ver-tices of the triangle remain well phase-locked, keeping thephase circulation of 2 π , which corresponds to vorticity S = 1. The excitations at secondary sites (between thevertices) are phase-locked to the primary ones, but fea-turing some oscillations. The oscillations enhance with µ , but the overall vortical phase pattern always persists.On the other hand, the amplitudes of excitations at thesecondary sites feature fast irregular oscillations, whichalso become stronger at larger µ (variations of these am-plitudes by a factor ∼ µ = 182). These amplitude oscillationsare coupled to small variations of amplitudes at the pri-mary sites, as shown in Fig. 7. It has been checked thatthe oscillations were not induced by reflection of small-amplitude radiation waves from edges of the integrationdomain (absorbers installed at the edges do not suppressthe oscillations).Figure 8 displays the total energy, E = (cid:80) m,n (cid:82) + ∞−∞ | u m,n ( t ) | dt , and amplitude of the tri-angular vortices, both stable and unstable ones, asfunctions of the effective propagation constant, µ + δµ, where the contribution δµ from oscillations of thefields is computed as follows. For each vertex of thetriangle, ( m, n ), peak time t m,n is defined, such that | u m,n ( t m,n ) | = max t | u m,n ( t ) | , and the correspondingphase, φ m,n ( t m,n ), is identified. Next, we compute δµ = (cid:28) ddz φ m,n (cid:29) , (7)where the average is taken over the three vertices of thetriangle (or six ones for stable hexagonal vortices with S = 2, see the next section), and, for the stable trian-gular modes, over z . For unstable triangles, the latter π /3 π /30 (a) | u m n | t (b) FIG. 6. (Color online) (a) The transverse energy( (cid:82) + ∞−∞ | u m,n | dt ) and phase profile of the stable triangular vor-tex generated by the input with µ = 200. (b) The longitudinal(temporal) profile of excitations in the cores representing thevertices of the triangle. average was taken over a short interval ∆ z , within whichthe pattern was not disturbed by the instability. As con-cerns the amplitude shown in Fig. 8, it was defined as | u m,n ( t m,n ) | , averaged over z and over the three vertices,to smooth effects of small persistent oscillations of the lo-cal amplitudes. A small gap between the unstable andstable portions of the amplitude plot in Fig. (7) is dueto the difference in wavenumber shift (7), as computedfor the stable and unstable solutions. V. HEXAGONAL VORTICES WITH THEDOUBLE TOPOLOGICAL CHARGE
As said above, the hexagonal input based on Eqs. (4)-(6) could not produce any stable pattern. However, thesame initial ansatz, but with inverse signs of three of itsmain peaks—say u − , , u , , u , − —can give rise to sta-ble hexagonal spatiotemporal patterns carrying vorticity S = − S = +1), see an example of the stable mode inFig. 9. Note that no changes were made to the hexagonalinput at sites in outer layers.Simulations with this input were also run in a broad FIG. 7. (Color online) The evolution of the longitudinal (tem-poral) profiles of one primary and one secondary componentsof a stable triangular spatiotemporal vortex, generated by thetriangular input with µ = 450.FIG. 8. (Color online) The amplitude and energy of the stabletriangular vortex vs. the effective propagation constant, µ + δµ , see Eq. (7). Blue (dashed) and red (solid) segmentsdesignate unstable and stable solution families, respectively. range of values of the propagation constant, 5 ≤ µ ≤ µ < ≤ µ ≤
10, the pattern forms a transient triangularstructure, which eventually splits, and in a broad inter-val of 20 ≤ µ ≤
222 the initial hexagon fissions into twotriangles, which also turn out to be unstable—essentially,because the amplitudes of the triangular patterns fall be-low the stability threshold.
Stable double ( S = 2) hexagonal vortices emerge at µ ≥ µ , an instabilityisland was revealed around µ = 300. In that case, thesix temporal pulses remain locked to their positions, butthe phase structure is lost at z (cid:38) µ .It is possible that other narrow intervals of the instabilitymay be found inside the stability region.Figure 10 presents the energy and amplitude of thedouble vortices as functions of the effective wavenumber, π π /30 10 π /3 8 π /3 2 π /3 (a) | u m n | t (b) FIG. 9. (Color online) The same as in Fig. 6, but for a stablehexagonal vortex with topological charge S = 2, generated bythe modified input with µ = 223.FIG. 10. (Color online) The same as in Fig. 8, but for thehexagon-shaped vortices with the double topological charge.Only stable modes are presented in these plots. similar to Fig. 8 for the triangular vortices with S = 1.However, only the stable family of the hexagonal vorticesis shown here, as we were not able to measure characteris-tics of unstable ones at µ < VI. COMPACT TRIANGULAR VORTICES
Still another type of stable spatiotemporal patternscan be produced by the input taken as per Eqs. (4)-(6),but centered at an edge of the original hexagon, i.e., with m and n replaced by m − / n − /
3, respectively. In
20 40 60 80 100 π /3 2 π /30 (a) | u m n | t (b) FIG. 11. The same as in Fig. 6, but for a stable compact triangular vortex with topological charge S = 1, generatedby the shifted input with µ = 15. this case, the results were collected for 5 ≤ µ ≤ S = 1, shaped as a densely packed trian-gle, without an empty site in the center, cf. Fig. 6. Thisstructure is stable for µ ≥
11. It is relevant to stress thatthis stabilization threshold is more than an order of mag-nitude lower than its counterparts for the triangular andhexagonal spatiotemporal vortices reported in the previ-ous sections (recall those thresholds were µ triangle = 182and µ hexagon = 223, respectively).In fact, direct simulations initiated by the above-mentioned shifted input ansatz generate the compact tri-angle which seems “noisy”. The noise can be removedby means of the “temporal filtering”, setting the fieldequal to zero outside of the main pulse in each core, andrunning the additional propagation over ∆ z = 10. Fur-thermore, for 31 ≤ µ ≤
60, direct simulations startingfrom the shifted input ansatz lead to a phase instability.For instance, at µ = 32 and 40, the phases of the threevertices would take values 0, π/ π , instead of thosedisplayed in Fig. 6. Actually, this instability is caused bythe fact that the input is far from the shape of the sta-ble mode, giving rise to several temporal peaks in eachcore. If the initial data are “cleaned up” by nullifyingthe field outside of the main temporal pulse, the simula-tions converge to stable compact triangular vortices. The -8 -4 0 4 8 0 1 2 3 4 5 6 t z FIG. 12. Trajectories of the motion of colliding triangularvortices rotated by angle π/ µ =250 and velocities ± k = ± amplitude of stable compact triangular vortices evolvesslowly and almost linearly versus the effective propaga-tion constant µ + δµ : the computed values of the latterrange from (cid:39)
22 to 78, then the amplitude goes from 40.3to 41.2.
VII. COLLISIONS BETWEEN MOVINGVORTEX SOLITONS
The availability of stable solutions for the vortex spa-tiotemporal solitons, and the obvious Galilean invarianceof Eq. (1) suggest to study collisions between movingvortices. In particular, it is interesting to simulate colli-sions between stable triangular modes shown in Fig. 6,rotated by angle π/ t = 16, for values of µ > ± ik t ), which, obviously, lends the solitons veloci-ties ± k (in terms of the optical waveguides, these areshifts of the inverse velocities).In fact, the fusion of colliding triangles into a hexagonwas never observed. Instead, slowly moving trianglesdemonstrate a long-range repulsion and stop at finite dis-tance (but do not bounce back), as shown in Fig. 12. Atintermediate velocities, the colliding triangular vorticesdo bounce back, and eventually they get destroyed by thelongitudinal instability (splitting into uncorrelated tem-poral pulses in different cores), as shown in Fig. 13. Athigh velocities, the solitons, quite naturally, pass througheach other, loosing some kinetic energy. There is a sharpthreshold between the rebound regime and the passage.Just above this threshold, the passing vortices get de-stroyed by the longitudinal instability shortly after the -8 -4 0 4 8 0 1 2 3 4 5 6 t z FIG. 13. The collision of two triangular vortices at µ = 250and k = 12. Trajectories of individual pulses forming thevortices are displayed.FIG. 14. (Color online) Regions of different outcomes of col-lisions between two mutually symmetric triangular vortices,rotated by angle π/ collision. Domains corresponding to different outcomesof the collisions in the ( µ, k ) plane are shown in Fig. 14.While the collisions seem elastic with the increase ofthe velocity, it was not possible to conclude if the vortices remain stable indefinitely long after such quasi-elasticcollisions. Indeed, since the numerical box has a finitelength, and periodic boundary conditions in t are used,the moving vortices undergo repeated collisions, loos-ing some velocity each time. Eventually, they would bedestabilized by a collision occurring at a lower speed. VIII. CONCLUSION
We have introduced a system of parallel waveguideswith the linear coupling between nearest neighbors, basedon the hexagonal lattice in the transverse plane. Eachguiding core features the cubic self-attractive nonlinear-ity. The system can find straightforward realizationsin nonlinear optics, and in BEC trapped in the corre-sponding optical lattice. Systematic simulations, start-ing with a natural input ansatz for vortical hexagons,reveal three distinct species of stable semi-discrete spa-tiotemporal complexes, which are discrete in the trans-verse plane and continuous in the longitudinal direction.These are triangular modes with vorticity S = 1 andhexagonal ones with S = 2, both built with an emptycore at the center, and compact triangles carrying S = 1,without the central empty core. Collisions between stabletriangular vortices were also studied by means of simula-tions, demonstrating the stoppage of the slowly movingvortex solitons, destabilizing rebounds, and quasi-elasticpassage, depending on the collision velocity.More complex structure of the arrayed waveguides canbe considered as a generalization of this work (in partic-ular, quasi-periodic lattices). It may also be interestingto study vortex complexes in two-component models ofthe same type. ACKNOWLEDGEMENTS
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