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Ring dark and anti-dark solitons in nonlocal media Theodoros P. Horikis1 and Dimitrios J. Frantzeskakis2 1Department of Mathematics, University of Ioannina, Ioannina 45110, Greece 2Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece Ringdarkandanti-darksolitonsinnonlocalmediaarefound. Thesestructureshave,respectively, the form of annular dips or humps on top of a stable continuous-wave background, and exist in a weak or strong nonlocality regime, defined by the sign of a characteristic parameter. It is demon- strated analytically that these solitons satisfy an effective cylindrical Kadomtsev-Petviashvilli (aka Johnson’s) equation and, as such, can be written explicitly in closed form. Numerical simulations showthattheypropagateundistortedandundergoquasi-elasticcollisions,attestingtotheirstability properties. 6 1 0 In nonlinear optics, spatial dark solitons are known to edge, RDSs in nonlocal media have not been considered 2 be intensity dips, with a phase-jump across the intensity so far. minimum, on top of a continuous-wave (cw) background It is the purpose of this article to study RDSs and n beam. These structures may exist in bulk media and ring anti-dark solitons (RASs) in nonlocal media. These a J waveguides, due to the balance between diffraction and structures have, respectively, the form of annular dips or 6 defocusing nonlinearity, and have been proposed for po- humps on top of a stable cw background, and exist in 2 tentialapplicationsinphotonicsasadjustablewaveguides a weak or strong nonlocality regime, defined by the sign for weak signals [1]. of a characteristic parameter. Using a multiscale asymp- ] totic expansion technique, we find that RDSs and RASs s In the two-dimensional (2D) geometry, spatial dark c obey an effective Johnson’s equation, that models ring- solitons, in the form of stripes, are prone to the trans- i shaped waves in shallow water [30]. We also perform di- t versemodulationinstability(MI)[2],whichleadstotheir p rectsimulationstoshowthatRDSsandRASspropagate bending and their eventual decay into vortices [3]. How- o undistorted and undergo quasi-elastic collisions. . ever,theinstabilitybandofthedarksolitonstripes,may s Lightpropagationinnonlocalmediaisgovernedbythe be suppressed if the stripe is bent so as to form a ring c following dimensionless model [13, 14, 21, 24, 26]: i of particular length. This idea led to the introduction s y of ring dark solitons (RDSs) [4], whose properties have ∂u 1 h been studied both in theory [5, 6] and in experiments i + ∇2u−2ηu=0, (1a) ∂z 2 p [7], and potential applications of RDS to parallel guid- [ ing of signal beams were proposed [8]. RDSs have also ν∇2η−2η =−2|u|2, (1b) been predicted to occur in other physically relevant con- 1 with the transverse Laplacian in cylindrical geometry v texts, such as atomic Bose-Einstein condensates [9] and ∂2 1 ∂ 1 ∂2 3 polariton superfluids [10, 11]. being: ∇2 = + + . Here, u = u(z,r,θ) ∂r2 r∂r r2∂θ2 8 While the above results rely on the study of nonlinear is the complex electric field envelope, η =η(z,r,θ) is the 0 Schrödinger(NLS)modelswithalocalnonlinearity,there optical refractive index, and the parameter ν stands for 7 0 exist many physical settings where the use of NLS mod- the strength of nonlocality. Notice that two interesting . els with a nonlocal nonlinearity are more appropriate. limits are possible: the local limit, with ν small, where 1 Thisoccurs, e.g., inmediafeaturingstrongthermalnon- (1) reduce to a NLS-type equation with saturable non- 0 6 linearity[12]orinnematicliquidcrystals[13], wherethe linearity [31], and the nonlocal limit, with ν large. Here, 1 nonlinearcontributiontotherefractiveindexdependson we will treat ν as an arbitrary parameter. : the intensity distribution in the transverse plane. It has We start by expressing functions u and η as [19, 32]: v been shown that dark solitons in one-dimensional (1D) i X settings exist in media with a defocusing nonlocal non- u=ub(z)v(r,θ,z)=u0e−2iu20zv(r,θ,z), r linearity [14–19] while, in the case of stripes, transverse η =η (z)w(r,θ,z)=u2w(r,θ,z), a MI may be suppressed due to the nonlocality [20]. The b 0 smoothing effect of the nonlocal response was shown to where u is an arbitrary real constant, while u (z) = 0 b occur even in the case of shock wave formation [20–23], u exp(−2iu2z) and η (z)=u2 form the cw background 0 0 b 0 or give rise to stable 2D solitons [24]. Here we should solution of (1) so that v and w satisfy: note that, generally, pertinent nonlocal models do not possess soliton solutions in explicit form (other than the ∂v 1 i + ∇2v−2η (w−1)v =0, (2a) weakly nonlinear limit [25]). As such, various techniques ∂z 2 b have been used to analyze soliton dynamics and interac- ν∇2w−2w =−2|v|2. (2b) tions, with the most common one being the variational approximation,whereaparticularformofthesolutionis By doing so, we have now fixed constant unit boundary chosen [13, 26–29]. However, to the best of our knowl- conditionsatinfinityandtheasymptoticanalysisforthe 2 determination of v and w may be directly applied. How- ever, before proceeding further, it is relevant to investi- gateifthecwbackgroundissubjecttoMI.Wethusper- form a standard MI analysis, assuming small perturba- tionsofu (z)andη (z)behavinglikeexp[i(k z+k ·r )]. b b z ⊥ ⊥ Then, it is found that the longitudinal and transverse perturbation wavenumbers k and k obey the disper- z ⊥ sion relation: k2 = 2u2k2 (cid:2)1+(1/2)k2(cid:3)−1 +(1/4)k4. z 0 ⊥ ⊥ ⊥ Thisequationshowsthatk isalwaysrealand, thus, the z cw solution is modulationally stable for the considered model (note that, generally, for certain response functions, nonlocality could possibly lead to MI even in the defocusing case [33]). Next, we use the Madelung transformation v = ρeiφ (where real functions ρ and φ denote the amplitude and phase of v), and obtain from (2) the following system: ∂φ 1 1 (cid:18)∂φ(cid:19)2 1 (cid:18)∂φ(cid:19)2 ρ − ∇2ρ+ ρ + +2η (w−1)ρ=0, ∂z 2 2 ∂r 2r2 ∂θ b (3a) ∂ρ 1 ∂ρ∂φ 1 ∂ρ∂φ + ρ∇2φ+ + =0, (3b) FIG.1: (ColorOnline)TypicalRDS(top)andRAS(bottom), ∂z 2 ∂r ∂r r2∂θ ∂θ for u =α=1 and ν =1 (ν =1/3) for the RAS (RDS). 0 ν∇2w−2w =−2ρ2. (3c) Seek, now, small-amplitude solutions on top of the cw Next, solve (5)-(6) for ρ and substitute above to ob- 4 background in the form of the asymptotic expansions: tain the following nonlinear evolution equation for ρ : 2 ρ=(cid:88)∞ ε2jρ , φ=(cid:88)∞ ε2j+1φ , w =(cid:88)∞ ε2jw , ∂ (cid:18)∂ρ2 + 3Cρ ∂ρ2 + SC ∂3ρ2 + 1 ρ (cid:19)+ 1 ∂2ρ2 =0, 2j 2j+1 2j ∂R ∂Z 2 2∂R 8 ∂R3 2Z 2 2CZ2 ∂Θ2 j=0 j=0 j=0 (7) where the unknown functions depend on the slow vari- where parameter S is given by: ables R = ε(r − Cz) (where C is the wave velocity), Θ=θ/ε, and Z =ε3z. Substituting these expansions to 2C2ν−1 4u2ν−1 (3)weobtainahierarchyofcoupledsystems. Toleading S = = 0 . C2 2u2 orderinε,i.e.,forO(ε−4)andO(ε−3),asystemoflinear 0 equations is obtained: Equation (7) is a cylindrical Kadomtsev-Petviashvili 2u20w2−C∂∂φR1 =0,   ∂∂φR1 = 4Cu20ρ2, (iTnchtKreoPred)ueecxqeiudstatitniroantn,hseafolscromonkatnteioxowtnnsof[a3ss4h]Jaolilhnlonkwsinognw’atshteeiqsrumwaaotivdoeensl,w[fi3ir0ts]ht. −2C∂∂wρR22 =+2∂∂ρ2R2φ,21 =0,⇔ Cw22 ==22ρu220., (4) tgheoemmeotrreyc[o3m5]m, wonhliychknalolwownsKfoPrecqounasttirounctinionthoefCsaorlutetisoianns of cKP from solutions of KP. Although —obviously— Notice that the velocity C, determined by (4), may have thereexistotherchoices, herewefocusonsolutionswith twosigns,correspondingtooutwardorinwardpropagat- radial symmetry, which do not depend on Θ. In this ing ring solitons (see below). Next, at O(ε1) and O(ε−2) case, the system reduces to the cylindrical Korteweg- we get: de Vries (cKdV) equation, which possesses cylindrical, sech2-shaped soliton solutions, on top of a rational back- ∂φ ∂φ ∂2ρ 2C2w +4C2ρ2+2 1 −2C 3 − 2 =0, (5) ground [36]: 4 2 ∂Z ∂R ∂R2 w4 =ρ22+2ρ4+ν∂∂2Rρ22, (6) ρ2(R,Z)= 3CRZ + SZα2sech2(cid:18)S√CZα3 + √αRZ +R0(cid:19), (8) and at O(ε−1): where, α is an arbitrary real parameter [of order O(1)]. ∂ρ ∂2φ ∂ρ ∂ρ ∂2φ 2C2Z2 2 + 1 −4C2RZ 2 −2C3Z2 4 +2CRZ 1Note that the characteristics of the solitons’ core, i.e., ∂Z ∂Θ2 ∂R ∂R ∂R2amplitude, power, velocity, and inverse width, scale as: +C2Zρ (cid:18)2+4CZ∂ρ2 +Z∂2φ1 +C2Z2∂2φ3(cid:19)=0. α2, α4, α2, and α, respectively, similarly to the case of 2 ∂R ∂R2 ∂R2 the usual KdV solitons [35]. 3 initial rings expand outwards, keeping their shapes dur- ing the evolution – at least for relatively short propagation distances. This fact, however, does not ensure stability of solitons, especially against azimuthal perturbations. Nevertheless, information regarding the RDS and RAS stability can be inferred from the cKP: in fact, (7)includesbothmodels,so-called[34]cKP-I(forS <0) andcKP-II(forS >0). Then,similarlytothecaseofthe KP equation, where lower-dimensional line (KdV-type) solitons of KP-I (KP-II) are unstable (stable) against transverse perturbations [35], we can infer the following: ring(cKdV-type)solitonsofcKP-I(cKP-II),i.e.,the RDS (for S < 0) and RAS (for S > 0) respectively, are unstable (stable) against azimuthal perturbations. Nev- ertheless,inoursimulationswehavenotobservedthein- stabilityofRDS,forpropagationdistancesuptoz ≈70. FIG. 2: (Color Online) Evolution of a RDS (left) and a RAS Second, we find that the soliton velocities are C = √ (right). Parameter values are as in Fig. 1. (cid:112) 2/3 for the RDS and C = 2 for the RAS, with a deviation less than 2% from the analytical prediction. It is also observed in Fig. 2 that, indeed, the RAS’s radius Clearly,thesignofparameterS determinesthenature is larger than that of RDS at the same propagation dis- ofthesoliton: ifS <0,thesolitonsaredepressionsoffof tance. NotethattheamplitudesofRDSandRASdepend thecwbackgroundandare,hence,darksolitons;ifS >0, on S, but do not depend on C (the sign of C determines the solitons are humps on top of the cw background and if the soliton will contract inwards or expand outwards). are, thus, anti-dark solitons (note that if S →0, modifi- Thus, according to these results, the RDS and the RAS cationoftheasymptoticanalysisandinclusionofhigher- cannot coexist. However, we note that in the presence order terms is needed as, e.g., in the shallow water wave of a competing quintic nonlinearity, it would be in prin- problem [37]). Examples of these RDS and RAS solu- ciple possible to find parameter regimes where RDS and tions, as introduced above, are shown in Fig. 1. Notice RASdocoexist,aswasthecaseinRefs.[39,40](seealso that, havingdeterminedtheformofthesoliton[(8)], the Refs.[41–43]forthesameeffectinasettingincorporating refractive index can readily be found in terms of ρ : in third-order dispersion). 2 fact,uptoO((cid:15)2),itisgivenbyn=nbw ≈u20(1+2(cid:15)2ρ2), Third, although RDS and the RAS cannot coexist thus having the form of an annular well (barrier) for the and, thus, cannot interact with each other, it is possible RDS (RAS). to study interactions of two solitons of the same type, Here,recallingthatν definesthedegreeofnonlocality, namelyRDS-RDSorRAS-RAS:thiswouldbeanimpor- it is important to observe that S < 0 ⇒ ν < (1/2u )2, tanttestontheirrobustnessandsolitoniccharacter—at 0 while S > 0 ⇒ ν > (1/2u )2. These inequalities indi- least up to relatively short propagation distances, as ex- 0 cate that RDS (RAS) are supported in a regime of weak plained above. In Fig. 3, we show the interaction of two (strong) nonlocality, as defined by the sign of S. Indeed, solitonsofunequalamplitudes,namelyα=1(α=2)for in the local limit with ν = 0 the NLS does not exhibit the inner (outer) soliton; other parameters are as above. these RASs. The velocities are also chosen as before, but with a dif- To numerically investigate the propagation properties ferent sign, so that the solitons will undergo a head-on of RDS and RAS, we evolve an initial (z = 1) profile collision. Asseen,thecollisionisquasielastic: afterpass- for both cases. Note that the rational background is not ing through each other, the solitons restore their shapes. shown in Cartesian coordinates —see Ref. [38] for a dis- Thisbehaviorisinagreementwiththeperturbationthe- cussion on the asymptotics for z → 0. In Fig. 2, we ory of Ref. [6], which predicts that the head-on collision evolve this initial condition under (1) for u = α = 1 is elastic up to the second-order. 0 and ν = 1 (ν = 1/3) for the RAS (RDS); note that in It should be mentioned that the numerical results ob- the simulations we use a high accuracy spectral integra- tained above refer to the collision between concentric tor in Cartesian coordinates. We find that the role of RDSorRAS.However, thereexiststhepossibilityofthe nonlinearity is crucial for the soliton formation: indeed, collisionbetweenslightlymismatchedrings. Inthiscase, in the linear regime, the electric field envelope features a it is expected that the collision will produce small oscil- diffraction-induced broadening, while when nonlinearity lations of the rings, that will be oscillating between two is present a strong localization is observed (results not elliptic configurations with small positive and negative shown here), and solitons are formed. Other interesting eccentricities. features, directly connected with the soliton form, (8), To conclude, we have found and analyzed ring dark are reported below. solitons (RDSs) and ring anti-dark solitons (RASs) in First, the two solitons propagate undistorted, i.e., the nonlocal media. These structures were found as special, 4 radiallysymmetric,solutionsofacylindricalKdVmodel, which is a lower-dimensional reduction of an underly- ingcylindricalKP(aliasJohnson’s)equation. RDSsand RASs are supported, respectively, in a weak or strong nonlocal regime, as defined by the sign of a characteristic parameter. The same parameter controls the stability of these structures: in particular, RDSs (RASs) are predicted to be unstable (stable) against azimuthal perturbations. These facts highlight the role of nonlocality, which, not only support RASs that do not exist in the local limit, but also renders them stable in the higher-dimensionalsetting. 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