Two-dimensional matter-wave solitons and vortices in competing cubic-quintic nonlinear lattices
aa r X i v : . [ phy s i c s . op ti c s ] J un Two-dimensional matter-wave solitons and vortices in competing cubic-quintic nonlinear lattices
Xuzhen Gao
1, 2 and Jianhua Zeng ∗ State Key Laboratory of Transient Optics and Photonics,Xi’an Institute of Optics and Precision Mechanics of CAS, Xi’an 710119, China University of Chinese Academy of Sciences, Beijing 100084, China
The nonlinear lattice—a new and nonlinear class of periodic potentials—was recently introduced to generatevarious nonlinear localized modes. Several attempts failed to stabilize two-dimensional (2D) solitons againsttheir intrinsic critical collapse in Kerr media. Here, we provide a possibility for supporting 2D matter-wavesolitons and vortices in an extended setting—the cubic and quintic model—by introducing another nonlinearlattice whose period is controllable and can be different from its cubic counterpart, to its quintic nonlinearity,therefore making a fully ‘nonlinear quasi-crystal’.A variational approximation based on Gaussian ansatz is developed for the fundamental solitons and in par-ticular, their stability exactly follows the inverted
Vakhitov-Kolokolov stability criterion, whereas the vortexsolitons are only studied by means of numerical methods. Stability regions for two types of localized mode—the fundamental and vortex solitons—are provided. A noteworthy feature of the localized solutions is that thevortex solitons are stable only when the period of the quintic nonlinear lattice is the same as the cubic one orwhen the quintic nonlinearity is constant, while the stable fundamental solitons can be created under looser con-ditions. Our physical setting (cubic-quintic model) is in the framework of the Gross-Pitaevskii equation (GPE)or nonlinear Schr¨odinger equation, the predicted localized modes thus may be implemented in Bose-Einsteincondensates and nonlinear optical media with tunable cubic and quintic nonlinearities.
PACS numbers: 05.45.Yv, 03.75.Kk, 03.75.Lm, 42.65.TgKeywords: Soliton, vortex, Bose-Einstein condensate, periodic potential
I. INTRODUCTION
In recent years, the study of Bose-Einstein condensates(BECs) in ultracold alkali gases has attracted considerabletheoretical and experimental interest [1–4]. The signifi-cance of this field is multi-fold. It not only stimulated thedevelopment and application of cutting-edge techniques incondensed-matter physics, and nonlinear atomic and molec-ular physics for atomic cooling, trapping, and manipulations,but also provided a rich and interesting platform for theoret-ical predictions (both analytical and numerical), and a com-parison with their experimental observations. BEC studiesopened a research avenue on the condensation of bosonicatoms and offered new attention and in-depth analyses of otherfascinating degenerate quantum gases, ranging from the con-densation of Fermi gases [5] and Bose-Fermi mixtures to ul-tracold molecules [6, 7]. They gave rise to many revolution-ary implications, e.g., matter-wave interferometry, ultracoldatomic clocks with unprecedented precision, and quantum in-formation processing. Moreover, they shed new light on theinvestigation of collective phenomena related to those pre-dicted many years ago, but never observed in many branchesof physics (especially in solids, fluids, and nuclei). Amongthem are the quantum fluids and quantum-phase transitions[1–4].Within a fundamental mean-field theory, the dynamicsof matter waves in BECs can be described by the Gross-Pitaevskii equation (GPE) [1, 2], which is generally calledthe nonlinear Schr¨odinger equation (NLSE) as they are of ∗ Electronic address: [email protected] or [email protected] the same form. Such theoretical models have had great suc-cess because the GPE can explain many experimental obser-vations, e.g., the shape of the BECs, expansion, and collectiveexcitations [8–10]. The BEC is an intrinsic nonlinear mediadue to the existence of atom-atom collisions, which can berepresented by nonlinear terms in the GPE [1, 2]. Many in-teresting nonlinear phenomena that arise in such ultracold de-generate quantum gases were predicted theoretically (by theGPE) and subsequently observed experimentally. These in-clude the generation of dark solitons [11–15] (ref. a recent re-view [16]), fundamental bright [17–20] and gap solitons [21],vortices (solitons with embedded vorticity), and related struc-tures [22–26], to name a few. It is commonly known that thebalanced interplay of dispersion/diffraction and nonlinearitycan create various localized modes, including dark and brightsolitons, which exist under defocusing and self-focusing non-linearities, respectively [27]. By tuning the atom-atom inter-actions in BECs, the nonlinearity can be readily set to defo-cusing or self-focusing [1, 2].In the one-dimensional (1D) case, the GPE is an analyti-cal solvable model and thus permits exact solutions for bothdark and bright solitons [4]. In uniform media, the 1D brightsolitons, supported by the cubic self-focusing nonlinearity,are exceptionally stable and robust. However, the stability oftheir multidimensional (two- and three-dimensional, 2D and3D) counterparts is fragile, as the cubic self-focusing leads tothe well-known phenomena of wave collapse (also known as“blowing up” in mathematical literature) or catastrophic self-focusing, making the fundamental multidimensional localizedstates unstable in free space; see e.g., some closely correlatedreviews [28–35] and the more comprehensive books [36, 37].It is well known that the 2D localized modes ( viz. , the
Townes solitons [38]), supported by the cubic self-focusingnonlinearity, are restricted by the instability introduced bythe critical collapse, and accordingly, have not yet been ob-served in experiments. Under the same background, morecomplex multidimensional localized states—vortical solitons( alias vortex tori)—would again be subject to a still strongerazimuthal instability, making them split into fragments, whichfinally suffer an intrinsic collapse [28, 30]. It is, therefore, anissue of significant importance to the stabilization of multidi-mensional localized states, which, accordingly, has attractedmuch attention in the past years.To this end, many approaches were tried to stabilize multi-dimensional fundamental localized states—solitons and soli-tary vortices. A general method is to use trapping poten-tials; i.e. , the external harmonic-oscillator trapping potentials[39, 40], which are axially and spherically symmetric, wereutilized to trap, respectively, 2D and 3D fundamental matter-wave solitons and solitary vortices (with embedded vorticityS = 1), in self-attractive BECs. A more popular and promis-ing method is the application of periodic potentials—opticallattices, which can be readily realized in experiments via theinterference of multiple counter-propagating laser beams [1–4, 30, 31, 41–45]—rather than the normal harmonic-oscillatorpotentials.Since the introduction of periodic structures, the creation oflocalized states [2, 4, 30, 41–46, 48] has been endowed withmore prolific meanings; e.g., in addition to the fundamentalsolitons that exist under the self-focusing nonlinearity in thesemi-infinite gap of the underlying linear spectrum, a type ofgap soliton supported by defocused nonlinearity can also ex-ist in the finite band gaps of the spectrum. The stabilization ofmultidimensional localized states (both 2D and 3D) were pre-dicted in the forms of fundamental and gap solitons, as well astheir vortical counterparts—solitary and gap vortices [30, 41–48]. Furthermore, the combination of a harmonically confin-ing magnetic trap and an optical lattice was also used to createvarious localized modes [49]. It is necessary to point out that1D gap solitons were realized experimentally in self-repulsiveBECs [21].In addition to the atomic BECs, other physical realizationsof multidimensional localized states, supported by periodicpotentials, were extended to several optical structures, rang-ing from photonic crystals [27, 50] to semiconductor micro-cavities and photorefractive optically induced photonic lat-tices (called photonic lattices, for simplicity, in many studies)[51, 52]. With the aid of spatially periodic potentials (aliaslattice potentials), the stabilized mechanisms of multidimen-sional (both 2D and 3D) fundamental and vortical solitonswere predicted in various nonlinear optics settings (mainly inphotonic crystals and photonic lattices) [27, 50–52]. Experi-mental realizations were, however, only demonstrated in the2D cases, which included 2D optical vortex solitons in pho-tonic lattices [53, 54] and 2D plasmon-polariton gap solitons[55], which are in the form of polariton condensates (aliasexciton-polariton BECs) in semiconductor microcavities witha lattice structure.The periodic potentials are recently extended to their non-linear counterparts (sometimes called pseudopotentials), alias nonlinear lattices (NLs), which are characterized by nonlin- ear potentials with spatially periodic modulations of the signand/or local strength of the nonlinearity (see a comprehensiveinvestigation in a recent review [44] and references therein).The 1D NLs [56, 57] and the combined linear-nonlinear lat-tices [58, 59] were widely used for studying various local-ized states. However, stabilizing 2D solitons against a crit-ical collapse using purely NLs is still a challenging work.Although nonlinear shapes with sharp edges ( e.g. , circles orstripes) can support stable 2D solitons [60–64], which are es-sentially equal to the cases supported by a single circle, andthe periodicity (of the NLs) does not play a dominant role inthe stabilization [61]. With regard to this, a noteworthy workis on the experimental observation of NL-supported opticalsolitons formed at the interface between two lattices (solitonsof this type are known as surface solitons) [65].This work is focused on a stabilized mechanism for the for-mation of 2D fundamental matter-wave and vortical solitonsusing purely NLs. Our physical setting is based on the com-monly used cubic-quintic model by introducing NLs in bothcubic and quintic nonlinearities, thereby forming a nonlinear quasi-lattice . Although the study of soliton properties in thecubic-quintic model was widely reported [66–73], the case fora spatially periodic modulation of both cubic and quintic non-linear terms has not yet been investigated well (a recent workin [74] confirmed that the 1D version of the cubic-quintic NLscan stabilize solitons against the critical collapse to some ex-tent).In the combined cubic and quintic NLs model, we showhow 2D localized states can be created and stabilized via com-peting self-focusing cubic and self-defocusing quintic nonlin-earities. The fundamental solitons are studied by means of nu-merical methods and a variational approximation. Merely nu-merical methods are used for vortex solitons, as taking an an-alytical approach is evidently an extremely difficult task. Sta-bility regions for the fundamental and vortex solitons are iden-tified. In particular, the Gaussian ansatz can match up with itsnumerical counterpart for fundamental solitons, whose stabil-ity condition is found to obey the inverted
Vakhitov-Kolokolov (anti-VK) stability criterion [58, 59, 75], dµ/dN > , asshown below.For physical realizations, the quintic nonlinearity in theGPE arises from three-body interactions in a dense BEC [1–4]. In diverse nonlinear optical media [74], e.g., glass, liq-uids, and ferroelectric films, the quintic nonlinearity usuallyappears together with the cubic term. It should be noted that2D spatial fundamental solitons in liquid carbon disulfide (abulk optical media) with a competing cubic-quintic nonlinear-ity have recently been generated in experiments [76]. There-fore, in addition to the matter waves in BECs, the theoreticalresults predicted here can also be realized in other physicalsettings— e.g., nonlinear optical media—by filling the holesof photonic crystals with index-matching materials such asliquids.The paper is organized as follows. After introducing thetheoretical model (GPE/NLSE) and its variational approxima-tion grounded on the usual Gaussian ansatz in Sec. II, thenumerical results for the relevant 2D localized modes—bothfundamental and vortical solitons—and their stability regions,obtained by direct simulations of thus-found stationary solu-tions under weak perturbations, are presented in Sec. III. Fi-nally, the paper is summarized in Sec. IV. II. OUR MODEL AND ITS VARIATIONALAPPROXIMATIONA. Gross-Pitaevskii equation
In terms of physical setting, our theoretical model is basedon the normalized form of the underlying GPE (or NLSE) forthe mean-field wave function (or the amplitude of an electro-magnetic wave flowing in nonlinear optical media), ψ ( r , z ) : iψ t = −
12 ( ∂ x ψ + ∂ y ψ ) + ǫ [cos(2 x ) + cos(2 y )] | ψ | ψ + g [cos( qx ) + cos( qy )] | ψ | ψ, (1)where time parameter t is replaced by propagation distance z for electromagnetic-wave propagation in nonlinear optics; ε and g , separately, are the strengths of the cubic-quintic NLs.At g = 0 , Eq. (1) is equal to the cubic nonlinear model withthe spatially periodic nonlinearity, i.e., NL, where 2D solitonscannot be stabilized at all. Considering that the center of thesoliton would be placed at point x = y = 0 , ε < cor-responds to the self-focusing cubic nonlinearity, with g > being the defocusing quintic term. Throughout this paper, un-less otherwise specifically mentioned, we set ε ≡ − .The remaining variable q in the quintic NL will be givenbelow. It is worth mentioning that the cubic NL and quinticNL are commensurate at q = 2 ; the subharmonic commensu-rability appears at q = 1 , while it is incommensurate when q is at other values. Therefore, the cubic NL and quintic NL arein different periods (spatial arrangements) with the variationof q , making the medium virtually equivalent to a purely non-linear quasi-lattice (a quasi-crystal for nonlinear excitations).Obviously, the quintic nonlinearity is uniform at q = 0 .Stationary solutions to Eq. (1) with chemical potential µ (or propagation constant − µ , for nonlinear optical waves) aresought in the form of ψ ( x, y, t ) = φ ( x, y ) exp( − iµt ) , withwave function φ ( x, y ) yielding the following stationary equa-tion: µφ = −
12 ( ∂ x φ + ∂ y φ ) + ǫ [cos(2 x ) + cos(2 y )] | φ | φ + g [cos( qx ) + cos( qy )] | φ | φ. (2)The above equation can be directly derived from its La-grangian form, L = 12 Z + ∞−∞ { µ | φ | −
12 ( | ∂φ∂x | + | ∂φ∂y | ) − ǫ x ) +cos(2 y )] | φ | − g qx ) + cos( qy )] | φ | } dxdy. (3) B. Variational approximation
Variational approaches are generally used to study sta-tionary solutions, particularly for fundamental modes, sincethey can predict the shape and even the stability condi-tion (used in combination with the stability criterion of Ref.[75]), which can supplement, perfect, and verify the re-sults obtained by direct simulations. To implement a vari-ational approximation, we take a Gaussian ansatz as usual, φ ( x, y ) = A exp (cid:2) − (cid:0) x + y (cid:1) / (cid:0) W (cid:1)(cid:3) , with width W ;the corresponding norm (alias number of atoms) N ≡ R R φ ( x, y ) dxdy = π ( AW ) (or the total power, in termsof optics). The substitution of such ansatz into LagrangianEq. (3), after simplification, leads to the following expression,written with variables N and W : L eff = N (cid:20) µ − W − εN πW e − W − gN π W e − q W (cid:21) , (4)and the corresponding variational equations, ∂L eff /∂N = ∂L eff /∂W = 0 : εN π (1 + W ) e − W + 8 gN π W (1 + q W
24 ) e − q W = − , W + εNπW e − W + 2 gN π W e − q W = µ. (5)In the following, the variational equations (5) based on theGaussian ansatz will be solved numerically for fundamentalsoliton studies. III. NUMERICAL RESULTS FOR TWO-DIMENSIONALLOCALIZED STATES
Before proceeding with the numerical computations, we in-troduce our numerical methods. Specifically, the localized sta-tionary modes (both fundamental and vortex solitons) wereconstructed numerically by means of the imaginary time-integration method [77] applied to Eq. (1). The stability ofthe stationary solutions thus found against small perturbationswas valuated through direct simulations of Eq. (1) (in realtime) using the finite-difference time-domain method [77]. Itis relevant to stress that the localized stationary modes canalso be found as numerical solutions of stationary equation(2) using Newton’s method. The numerical calculations wereperformed in a × domain on a grid of × points. A. Fundamental solitons
We begin with the full commensurability case between theself-attractive cubic and self-repulsive quintic NLs (with thesame period π ), which is to say q = 2 for the quintic term inEq. (1) and all the others.Figure 1 depicts the relations µ ( N ) for the fundamentalsolitons based on the variational approximation, produced by N µ VANumerical
StableC DA B
FIG. 1: Chemical potential ( µ ) vs. the norm ( N ) for 2D fundamen-tal solitons in the model with combined cubic and quintic periodicpotentials (nonlinear lattices), produced by the variational approxi-mation (VA, red curve), and found from numerical solutions of Eq.(2) (blue curve), at ε = − , g = 1 , and q = 2 . Stable sectionsof the soliton families are within the marked stripe. Typical exam-ples of stable and unstable fundamental solitons, corresponding tothe marked points (B, C and A, D) are respectively displayed in Figs.(2) and (3). -10 0 10 x (y) | φ | (a) -10 0 10 x (y) | φ | (b) FIG. 2: Typical examples of stable 2D fundamental solitons foundin the model with combined cubic and quintic nonlinear lattice po-tentials, at ε = − , g = 1 , and q = 2 . Only contour plots of themodes (the modules of the stationary wave functions) are shown byprojecting them onto a 2D plane. The left (a) and right (b) panelscorrespond, respectively, to the marked points B and C in Fig. (1).Hereinafter, the blue solid and red-dashed curves are numerical sta-tionary solutions and their relevant Gaussian ansatz, respectively. a numerical solution of variational equations (5). They werecombined with their fully numerical counterparts produced bynumerical stationary solutions of the stationary equation (2),and checked through direct simulations of the perturbed solu-tions in nonlinear evolution equation (1), at ε = − , g = 1 ,and q = 2 .Examples of stable 2D fundamental solitons, supported by -10 0 10 x (y) | φ | (b) -10 0 10 x (y) | φ | (a) FIG. 3: Typical examples of unstable 2D fundamental solitons sup-ported by a model with combined cubic and quintic nonlinear latticepotentials, at ε = − , g = 1 , and q = 2 . The contour plots shownhere are only the modes of the stationary wave functions in the 2Dplane. The left (a) and right (b) panels correspond, respectively, tothe points A and D in Fig. (1). g N q N (a) (b) Stable solitons Stable solitons
FIG. 4: Stability borders for the entire set of 2D fundamental soli-tons supported by cubic and quintic nonlinear lattice potentials: (a)curve N ( g ) with different values of quintic nonlinear strength g at ε = − and q = 2 ; (b) curve N ( q ) with different values of quinticnonlinear lattice structure variable q at ε = − and g = 1 . Portionsof the stable solitons are confined to the respective stability borders(the magenta areas). the competing cubic-quintic nonlinear lattices, are shown inFig. 2. These stable localized modes can match well with theirGaussian ansatz, and are quasi-isotropic and highly localizedwithin a single cell (recall that the period here is π , for bothcubic and quintic NLs). The former feature is natural as oursystem meets spatial-inversion symmetry: e.g., equation (1) orequation (2) is invariant under the symmetry operations x →− x and y → − y . The latter case may be explained by thefact that, to arrest a 2D critical collapse, the localized modesshould reside themselves into a single well.However, as seen from Fig. 3, although broad solitons—their widths are much bigger than the NLs relevant period-icity π (recall that both the cubic and quintic NLs have thesame period at q = 2 )— also exist, direct simulations verifiedthat they are totally unstable, in contrast to their 1D counter- N -4-3.5-3-2.5-2-1.5 µ StableStable q=0q=2
E F
FIG. 5: Numerically found curves µ ( N ) for families of 2D vortexsolitons with topological charge 1, supported by the cubic and quinticnonlinear lattice potentials ( q = 2 ) and constant quintic term ( q =0 ), at ε = − and g = 1 . The stable vortex solitons are limitedto the corresponding marked stripes. Typical examples of the stableand unstable vortex solitons, marked by points E and F at q = 2 , arerespectively depicted in Fig. 6. parts, where the broad solitons and solitons with symmetricside peaks are found to be stable; cf. Figs. 5 and 7 in ref.[74].From Fig. 1, we can see that the variational approximationprovides a relatively reasonable accuracy for 2D fundamentalsolitons. Direct simulations of the perturbed solutions fur-ther demonstrated that they are stable only roughly around thenorm N ∈ [3 , , and their stability follows the anti-VK cri-terion [58, 59, 75], viz., dµ/dN > . The shapes of unstable2D fundamental solitons (we call them broad solitons) in Fig.3 cover several lattice cells, making them unstable after somefinite evolution time.A vast number of direct simulations, made by changingvariable q and nonlinear coefficient g , demonstrate that thestable 2D fundamental solitons are only limited to the norm N ∈ [ N L , N H ] (ref. the stable region marked in Fig. 1). Thiscan be understood since, at the lower limit N L , the stationarysolutions are too weak to form stable localized modes, whileabove the threshold—the upper limit N H —the stationary so-lutions are naturally unstable because of the critical collapse.Fig. 4(a) summarizes the stability of the 2D fundamen-tal solitons, supported by the combined cubic and quinticNLs, for various values of the quintic coefficient g . It is ob-served that, for the given cubic nonlinear strength ε = − ,the range of g is within [0 . , . when allowed by thestable 2D fundamental solitons. This is natural since thecubic-focusing nonlinearity is overwhelming when its quintic-defocusing term g is small; only above some certain value canthe quintic nonlinearity help to arrest the critical collapse. Thequintic coefficient g , however, cannot be too large either; oth-erwise, the quintic defocusing nonlinearity dominates, finallydelocalizing the solitons.Fig. 4(b) shows the stability region for the 2D fundamen- FIG. 6: Examples of stable (left panels) and unstable (right panels)2D vortex solitons with topological charge 1, in the competing fo-cusing cubic and defocusing quintic nonlinear lattices. The first, sec-ond, and third rows represent, respectively, the contour plots of wavefunction | φ | , the phase distribution that carries the vorticity, and theircross-section profiles at y = π . The stable and unstable vortex soli-tons correspond respectively to points E and F in Fig. (5). tal solitons under various conditions, including the aforemen-tioned incommensurate, subcommensurate, and commensu-rate relations between the cubic NL and quintic NL. The fun-damental solitons are stable at q < q max ≈ . , for mod-erate values of N . When above such a threshold ( q max ), theincommensurability of the competing cubic-quintic NLs ex-pands quickly, allowing the quintic NL to play a inappreciablerole in stabilizing the localized modes, and in overcoming thecritical collapse rooted in the cubic-focusing nonlinearity. B. Vortex solitons
In our analysis, the ansatz for vortex solitons is takenas φ ( x, y ) = A exp (cid:2) − (cid:0) x + y (cid:1) / (cid:0) W (cid:1)(cid:3) e iSθ ( x,y ) , withwidth W , topological charge S , and azimuthal coordinate θ .Fig. 5 displays the numerically found relations µ ( N ) for gen-eral 2D vortex solitons with topological charge 1, supportedby the combined cubic and quintic NLs, at ε = − and g = 1 .Both the constant quintic nonlinearity ( q = 0 ) and quinticNL ( q = 2 , commensurate cubic-quintic NLs) are shown inthe figure. Direct simulations demonstrated that the stable2D vortex solitons are within limited scopes and at moderatenorm N .The left column of Fig. 6 shows a typical example of astable vortex soliton with topological charge 1, which wascreated as hollow four-peak complexes, with a separation be- N Stable vortex solitons
FIG. 7: Stability border, shown as the curve N ( q ) , for the familiesof 2D vortex solitons, supported by the cubic and quintic nonlinearlattice potentials with ε = − and q = 2 , and under the differentfifth-order (quintic) nonlinearity g . The vortex solitons are stablebetween the relevant stability borders (the magenta areas). tween the peaks of two times the period of the NL potentials,and an empty site at the center. As reported previously in otherperiodic potentials [30, 44, 59], such vortex types with innervoids can be stable because of the weak interaction betweenthe peaks. Such a hollow vortex structure was constructed,rather than the normal isotropic vortex that resides within asingle cell of the nonlinear lattice, since our detailed numer-ical examination found that the latter one cannot exist in thecurrent model.As can be seen from the right column of Fig. 6, similar totheir unstable fundamental counterparts in Fig. 3, the station-ary solutions of unstable vortex solitons occupy more than asingle well, which makes them hard to station there. It is rele-vant to note that the wave structures of the stable and unstablevortex solitons at q = 0 are very similar to their counterpartsat q = 2 , as displayed in Fig. 6.We obtained the stability areas of such hollow four-peakcomplexes (vortex soliton at topological charge 1) for differ-ent values of fifth-order nonlinear coefficient g of the modelthrough mass numerical simulations, which are shown in Fig.7. Our calculations verified that the formation condition ofstable vortex solitons is strictly under the full commensura-bility of the combined cubic NL and quintic NL ( q = 2 ) orunder the model with constant quintic nonlinearity ( q = 0 ).Any case deviating from these two scenarios would be unsta-ble; i.e. , neither q = 0 . nor q = 1 . can generate stable vortex solitons. IV. CONCLUSIONS
We explored the 2D matter-wave fundamental and vor-tex solitons in the general cubic-quintic model, which is theGross-Pitaevskii equation or nonlinear Schr¨odinger equation,by introducing nonlinear lattices with tunable periods (incom-mensurate, subcommensurate, and commensurate) to bothnonlinear terms, which can be recognized as a fully nonlinearquasi-crystal. The physical setting is the dense Bose-Einsteincondensates under both two- and three-body interactions, withperiodic changes of the scattering lengths of interatomic col-lisions by means of the Feshbach resonance. The cubic NL ofthis model is fixed, while both the strength and period of thequintic NL are variable.A variational approximation was developed for the fun-damental solitons and they were found to obey the anti-VKstability criterion, while the vortex solitons were studied bysimply relying on numerical methods. Stability regions forboth localized modes—fundamental and vortex solitons—were identified. In particular, stable vortex solitons existedas long as the quintic NL had the same period as its cubicterm or the quintic nonlinearity was constant; in contrast, theformation conditions for the fundamental solitons were con-siderably more relaxed.The physical model considered here can be extended to atwo-component model—the coupled Gross-Pitaevskii equa-tions [68]—to consider the existence of multidimensional lo-calized modes therein. Considering that the experimental ob-servation of 2D optical solitons in a cubic-quintic-septimalmedia was reported very recently [78], it would be interest-ing, at least on a theoretical level, to add nonlinear lattices tothese combined cubic-quintic-septimal nonlinearities (to one,two, or three nonlinear terms) and study the possible localizedmodes [79].
V. ACKNOWLEDGMENTS
This work was supported by the NSFC, China (project Nos.61690224, 61690222, 11204151), by the Youth InnovationPromotion Association of the Chinese Academy of Sciences(project No. 2016357) and the CAS/SAFEA InternationalPartnership Program for Creative Research Teams, and par-tially by the Initiative Scientific Research Program of the StateKey Laboratory of Transient Optics and Photonics. [1] L. Pitaevskii and S. Stringari,
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