Dissipative soliton interaction in Kerr resonators with high-order dispersion
aa r X i v : . [ phy s i c s . op ti c s ] F e b Dissipative soliton interaction in Kerr resonators with high-order dispersion
A. G. Vladimirov a , M. Tlidi b , and M. Taki c a Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany b Département de Physique, Faculté des Sciences, Université Libre de Bruxelles (U.L.B.),CP 231, Campus Plaine, B-1050 Bruxelles, Belgium and c Université de Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France
We consider an optical resonator containing a photonic crystal fiber and driven coherently byan injected beam. This device is described by a generalized Lugiato-Lefever equation with fourthorder dispersion. We use an asymptotic approach to derive interaction equations governing the slowtime evolution of the coordinates of two interacting dissipative solitons. We show that Cherenkovradiation induced by positive fourth-order dispersion leads to a strong increase of the interactionforce between the solitons. As a consequence, large number of equidistant soliton bound states inthe phase space of the interaction equations can be stabilized. We show that the presence of evensmall spectral filtering not only dampens the Cherenkov radiation at the soliton tails and reducesthe interaction strength, but can also affect the the bound state stability.
I. INTRODUCTION . Optical frequency combs generated by micro-cavityresonators have revolutionized many fields of science andtechnology, such as high-precision spectroscopy, metrol-ogy, and photonic analog-to-digital conversion [1]. A par-ticular interest is paid to the soliton frequency combs as-sociated with the formation in the time domain of theso-called temporal cavity solitons – nonlinear localizedstructures of light, which preserve their shape in thecourse of propagation. Temporal dissipative solitons of-ten called cavity solitons were reported experimentally inmode-locked lasers, micro-cavity resonators [2, 3], and incoherently driven fiber cavities [4].In this work we consider a photonic crystal fiber cav-ity driven by a coherent injected beam. When operatingclose to the zero dispersion wavelength, high-order chro-matic dispersion effects could play an important role inthe dynamics of the system. Taking into account theseeffects together with spectral filtering the dimensionlessmodel equation in the mean-field limit reads ∂ t U = S − (1+ iθ ) U + iU | U | +( δ + i ) ∂ τ U + β ∂ τ U + iβ ∂ τ U, (1)where U ( τ, t ) is the complex electric field envelope, τ is time and t is the slow time variable describing thenumber of round trips in the cavity. The parameter S measures the injection rate, θ describes frequency detun-ing, second order dispersion and Kerr nonlinearity coef-ficients are normalized to unity, β and β are the thirdand fourth-order dispersion coefficients, respectively, and < δ ≪ is the small spectral filtering coefficient. Theoptical losses are determined by the mirror transmissionand the intrinsic material absorption. This losses arenormalized to unity.In the absence of high-order dispersion and spectral fil-tering, we recover from Eq. (1) the Lugiato-Lefever equa-tion [5] which is a paradigmatic model to study temporalcavity solitons (see overview [6, 7]). It is widely appliedto describe two important physical systems: passive ringfiber cavity with coherent optical injection and driven optical microcavity used for frequency comb generation[8–11]. The inclusion of the fourth-order dispersion al-lows the modulational instability to have a finite domainof existence delimited by two pump power values [12]. Asa consequence, upper homogeneous steady state solutionbecomes modulationally stable and dark dissipative soli-tons sitting this solution can appear [13]. In the presenceof third order dispersion bright and dark dissipative soli-tons become asymmetric and acquire an additional groupvelocity shift associated with this asymmetry [14–17].Being well separated from one another dissipative soli-tons can interact via their exponentially decaying tailsand form bound states characterized by fixed distancesbetween the solitons. This weak interaction can bestrongly affected by different perturbations, such as peri-odic modulation [18, 19] and high-order dispersions [20],which lead to the appearance of the so-called solitonCherenkov radiation at the soliton tails [21]. Soliton in-teraction in the presence of high-order dispersions wastheoretically studied in several works [16, 17, 20, 22, 23].However, they were either focused on the asymmetricsoliton interaction in the presence of third-order disper-sion or based mainly on the numerical calculation of thesoliton interaction potential. Unlike these works, herewe present an analytical theory of the interaction of twodissipative solitons of the Lugiato-Lefever equation withfourth-order dispersion term based on the asymptotic ap-proach developed in [24, 25]. Furthermore, we show thatsimilarly to the case of the interacting oscillatory solitons[19], a small spectral filtering effect can strongly affectthe interaction force and the stability properties of thebound soliton states. II. SINGLE PEAK DISSIPATIVE SOLITON
Without high-order dispersion and spectral filteringterms, β = β = δ = 0 , Eq. (1) supports a single ormultipeak dissipative solitons characterized by dampedoscillatory tails [26]. Stable dissipative solitons have beenfound in a strongly nonlinear regime, where the modula-tional instability is subcritical, i.e., for θ > / . Moreprecisely, they have been found in the pinning region,where the lower stationary homogeneous solution coexistwith a periodic one. The number of dissipative solitonsand their distribution in the cavity are determined by theinitial conditions while their maximum peak power re-mains constant for fixed values of the system parameters[26]. For θ > / Eq. (1) supports a single peak dissi-pative soliton solution in the form U ( t, τ ) = U + u ( τ ) ,where I = | U | = const is the intensity of the station-ary homogeneous solution of Eq. (1) and u ( τ ) decaysexponentially at τ → ±∞ . This solution persists alsoat sufficiently small β , β , and δ . It remains motion-less for β = 0 and becomes uniformly moving other-wise, U ( t, τ ) = U + u ( τ − vt ) . Asymptotic analytictheory of the asymmetric dissipative soliton interactionvia Cherenkov radiation induced by the third-order dis-persion coefficient β was developed in [17]. Below weconsider the case when only small fourth-order disper-sion is present, β = 0 and | β | ≪ . In this case due tothe symmetry property of Eq. (1), τ → − τ , the solitonvelocity is always zero, v = 0 .The dispersion relation for the small amplitude wavesis determined by substituting U ( t, τ ) = U + A e ikτ − i Λ t into Eq. (1) and linearizing the resulting equation at U = U . This yields Λ = − I + i q (1 + δk ) − I + k − β k . The phase velocity of the dispersive waves V = ℜ (Λ) /k isshown in Fig. 1(a) for positive (1(a)) and negative (1(b)) β , as a function of the wave number k . Cherenkov radi-ation appears when the phase velocity V coincides withzero soliton velocity as shown in Fig.1(a). It is seen fromthis figure that the Cherenkov radiation emitted from thesoliton tail occurs only when β is positive. Therefore,below we consider only the case of positive fourth-orderdispersion coefficient < β ≪ when the Cherenkovradiation is present. For negative β the soliton inter-action is only weakly affected by the small fourth-orderdispersion term.Linear stability of the dissipative soliton solution u ( τ ) is determined by calculating the eigenvalue spectrum λ of the operator ˆ L ( u ) = ˆ L + ˆ L ( u ) , (2)obtained by linearization of Eq. (1) around the solitonsolution. Here u = (cid:18) u u ∗ (cid:19) , ˆ L = ˆ L (0) is the lineardifferential operator evaluated at the stationary homoge-neous solution U = U : ˆ L = (cid:18) − − iθ + 2 iI + ( i + δ ) ∂ τ + iβ ∂ τ iU − iU ∗ − iθ − iI − ( i − δ ) ∂ τ − iβ ∂ τ (cid:19) , and ˆ L ( u ) = (cid:18) iU ∗ u + 2 iU u ∗ + 2 i | u | iU u + iu − iU ∗ u ∗ − iu ∗ − iU ∗ u + 2 iU u ∗ − i | u | (cid:19) . We have calculated numerically the soliton solution andthe eigenvalue spectrum λ of the operator ˆ L ( u ) by dis-cretizing Eq. (1) on an uniform grid of points onthe interval τ ∈ [0 , with periodic boundary condi-tions. The result is shown in Fig. 2 for β = β = δ = 0 .The continuous spectrum lies on the line ℜ ( λ ) = − ,while the discrete spectrum of the soliton is symmetricwith respect to this line [27]. For the parameter values ofFig. 2 apart from two real eigenvalues: zero eigenvalue, λ = 0 , associated with the translational symmetry of theLugiato-Lefever equation and symmetric one, λ = − ,soliton has two symmetric pairs of complex conjugatedeigenvalues. The right pair of these complex eigenvaluesis responsible for an Andronov-Hopf bifurcation takingplace with the increase of the injection parameter S . Thedecay rates of the soliton tails depend on the eigenvalues µ satisfying the characteristic equation β µ + 2 β µ + (cid:2) δ + 2 β (2 I − θ ) (cid:3) µ + 2(2 I − θ − δ ) µ + (cid:2) − I + (2 I − θ ) (cid:3) = 0 . (3)obtained by lineariazation of the Eq. (1) with ∂ t U = 0 at the homogeneous steady state solution U = U .In the case when the high-order dispersion and spectral filtering are absent β = β = δ = 0 , Eq. (3) gives two Figure 1. Phase velocity V of small dispersive waves with pos-itive (a) and negative (b) fourth-order dispersion coefficient β , and β = 0 . Solid line corresponds to β = 0 . (a) and β = − . (b). Dashed line corresponds to β = 0 . Theparameter values are S = 1 . , θ = 3 . , and δ = 0 . .Figure 2. Soliton solution of the Lugiato-Lefever equation(1) with β = β = δ = 0 (a) and eigenvalue spectrum ob-tained by numerical linear stability analysis of this solution(b). Other parameters are the same as in Fig. 1. pairs of complex conjugate eigenvalues: µ (0)1 , = ± r θ − I + i q − I (4)and µ (0) ∗ , , which determine the decay and oscillation ratesof the soliton tails. For example, for S = 2 . and θ =3 . we have µ (0)1 , = ± (1 . . i ) , which meansthat in the absence of high-order dispersions the solitontail oscillations are strongly damped. This might explainthe fact that without soliton Cherenkov radiation it ishardly possible to observe soliton bound state formationexperimentally [4].For nonzero but sufficiently small fourth-order disper-sion coefficient, < β ≪ , the eigenvalues (4) of Eq.(3) are only slightly perturbed. However, in addition to(4) two more pairs of complex conjugate eigenvalues, µ , and µ ∗ , , appear. For zero spectral filtering coefficient, δ = 0 , they are given by µ , = ∓ i vuut β " s − β (cid:18) I − θ + i q − I (cid:19) . (5) It is seen that real (imaginary) parts of µ , in Eq. (5)vanish (diverge) in the limit β → . When the spectralfiltering coefficient is nonzero, δ > , analytical expres-sions for the eigenvalues µ , become very cumbersome.However, in the limit β = O ( δ ) ≪ we get: µ , = ∓ p β q (1 + δ/β ) − I i (cid:18) β + θ − I (cid:19) + O ( δ ) (cid:21) . (6)Due to the presence of the eigenvalues µ , and µ ∗ , thetails of the soliton of Eq. (1) with β = 0 and < β ≪ become weakly decaying and fast oscillating, which fa-vors the formation of soliton bound states, and can bereferred to as the soliton Cherenkov radiation [21]. Note,that when β is sufficiently small, the term δ/β describ-ing in Eq. (6) the contribution of spectral filtering intothe real part of µ , can lead to a considerable increaseof the decay rate of the soliton tails without significantchange of their oscillation frequency. For example, for S = 2 . , θ = 3 . , β = 0 . , and δ = 0 . we get µ = − . − . i , while for the same parameter setand δ = 0 one obtains µ = 0 . − . i . Numeri-cally calculated intensity profile of the soliton solutionof Eq. (1) with small fourth-order dispersion coefficient β = 0 . is depicted in Fig. 3 together with the cor-responding eigenvalue spectrum of the operator ˆ L ( u ) defined by Eq. (2).Note, that the proof of the reflectional symmetry prop-erty of the discrete soliton spectrum with respect to the ℜ λ = − line given in [27] is trivially generalized to thecase when even high order dispersions are present. Nev-ertheless, the soliton spectrum shown in Fig. 3 does notpossess this symmetry property due to the presence ofnonzero spectral filtering coefficient δ = 0 . . Further-more, as it is seen from Fig. 3, for δ = 0 . real partsof the complex conjugate eigenvalues, responsible for theAndronov-Hopf bifurcation of the soliton, are shifted tothe left from the imaginary axis as compared to thoseshown in Fig. 2 obtained for δ = 0 .Sufficiently far away from the soliton core its trailingtail can be represented in the form u ( τ ) ∼ a e µ τ + a e µ ∗ τ , when τ → + ∞ , (7)where the Cherenkov radiation amplitude a is exponen-tially small in the limit β → [21, 28], a = p a a ∗ , andfor β = O ( δ ) ≪ we get p a = i − p − I A ∗ (cid:18) δβ √ − I + 1 (cid:19) + O ( δ ) , (8)where p a is independent of β at δ = 0 . Numerically for S = 2 . , θ = 3 . , δ = 0 . , and β = 0 . we obtain p a ≈ . . i . Figure 3. Soliton solution of the Lugiato-Lefever equation(1) with β = 0 . and δ = 0 . (a); eigenvalue spectrumobtained by numerical linear stability analysis of this solution(b). Other parameters are the same as in Fig. 1. III. INTERACTION BETWEEN DISSIPATIVESOLITONS . Two or more solitons will interact through their over-lapping oscillatory tails when they are sufficiently closeto one another. In what follows, we investigate the inter-action between two dissipative solitons. We consider thelimit of weak overlap when the solitons are well separatedfrom each other and derive the interaction equations de-scribing the slow time evolution of the soliton coordinatesdenoted by τ , . To this end, let us first rewrite Eq. (1)in a general form: ∂ t U = ˆ F U , (9)where U = (cid:18) UU ∗ (cid:19) , ˆ F U = (cid:18) ˆ f U ˆ f ∗ U ∗ (cid:19) , and ˆ f is the dif-ferential operator defined by the RHS of Eq. (1). Welook for the solution of Eq. (9) in the form U ( τ, t ) = U + u + u + ∆ u ( τ, t ) . (10)Here u , = u ( τ − τ , ) are two unperturbed soliton so-lutions, with slowly evolving in time coordinates τ , ( εt ) , ∆ u ( τ, t ) = O ( ε ) is a small correction to the superposi-tion of two solitons, and small parameter ε describes theweakness of the overlap of the two solitons. SubstitutingEq. (10) into the model equation (9) and collecting firstorder terms in ε we obtain the following linear inhomo-geneous equation for ∆u = (cid:18) ∆ u ∆ u ∗ (cid:19) : ˆ L ( u + u ) ∆u = − ∂ x u ∂ t τ − ∂ x u ∂ t τ − ˆ F ( u + u ) , (11)where the linear operator ˆ L ( u ) is defined by Eq. (2).Due to the transnational invariance of Eq. (1) this lin-ear operator evaluated at the soliton solution u haszero eigenvalue corresponding to the so-called transna-tional neutral (or Goldstone) mode v = (cid:18) v v ∗ (cid:19) with
12 14 16 18 20 22 24 - - (cid:1) - (cid:1) G - G Figure 4. RHS of Eq. (16) as a function of the soliton sep-aration τ − τ . Black (red) dots indicate the separations ofthe two solitons in stable (unstable) bound states calculatednumerically. Parameter values: S = 2 . , θ = 3 . , δ = 0 . and β = 0 . .Figure 5. Stable bound state of two dissipative solitons, δ =0 . . (a) – intensity distribution, (b) – eigenvalue spectrum.Other parameter values are the same as for Fig. 4. v = du /dτ , ˆ L ( u ) v = 0 . The adjoint linear op-erator ˆ L † ( u ) obtained from ˆ L ( u ) by transposition andcomplex conjugation also has zero eigenvalue with theeigenfunction w = (cid:18) w w ∗ (cid:19) , which is referred below asthe “adjoint neutral mode”, ˆ L † ( u ) w = 0 . Below wewill assume that w satisfies the normalization condition h w · u i = ´ ∞−∞ ( w · u ) dτ = 2 ´ ∞−∞ ℜ ( w ∗ u ) dτ = 1 .Far away from the soliton core the asymptotic behaviorof adjoint neutral mode is given by: w ( τ ) ∼ b e µ ∗ τ + b e µ τ , τ → + ∞ , (12)with b = p b b ∗ , where asymptotic expression for p b coin-cides with that of p a given by Eq. (8).When the two interacting solitons are located suffi-ciently far away from one another the solvability con-ditions of Eq. (11) can be written as ∂ t τ , ≈ G , , G , = D w , · ˆ F ( u + u ) E , (13)where we approximated the adjoint neutral modes of the Figure 6. Unstable bound state of two dissipative solitons, δ =0 . . (a) – intensity distribution, (b) – eigenvalue spectrum.Other parameter values are the same as for Fig. 4..Figure 7. The same bound bound state as shown in Fig. 5,but calculated for δ = 0 . (a) – intensity distribution, (b) –eigenvalue spectrum. Bound state is unstable with respect toan Andronov-Hopf bifurcation. operator ˆ L † ( u + u ) by the adjoint neutral modes w , = w ( τ − τ , ) of the operators ˆ L † ( u , ) . In order to derive the soliton interaction equations weneed to calculate G , in Eq. (13). To this end we splitthe integral in Eq. (13) into two parts and using the rela-tions ˆ L † ( u , ) w , = 0 together with the fact that u and w ( u and w ) are small for τ ∈ [0 , + ∞ ) ( τ ∈ ( −∞ , ),where the origin of coordinates τ = 0 corresponds to thecentral point between two solitons, ( τ + τ ) / , weget G , = D w , · ˆ F ( u + u ) E , + D w , · ˆ F ( u + u ) E , ≈ D w , · ˆ F ( u + u ) E , ≈ D w , · ˆ L ( u , ) u , E , − D ˆ L † ( u , ) w , · u , E , = ( δ + i ) h(cid:10) w , ∂ τ u , (cid:11) , − (cid:10) u , ∂ τ w , (cid:11) , i + iβ h(cid:10) w , ∂ τ u , (cid:11) , − (cid:10) u , ∂ τ w , (cid:11) , i + c.c. (14)with h w · u i = ´ −∞ ( w · u ) dτ , h w · u i = ´ ∞ ( w · u ) dτ and ˆ L † ( u , ) w , = 0 .Next, performing integration by parts and using thesymmetry properties of the soliton and its neutral modes, u ( τ ) = u ( − τ ) , ∂ τ u ( τ ) = − ∂ τ u ( − τ ) , w ( τ ) = − w ( − τ ) , and ∂ τ w ( τ ) = ∂ τ w ( τ ) we get: G , ≈ ± (cid:2) ( δ + i ) (cid:0) w ∗ , ∂ τ u , − u , ∂ τ w ∗ , (cid:1) + iβ (cid:0) w ∗ , ∂ τ u , − u , ∂ τ w ∗ , − ∂ τ w ∗ , ∂ τ u , + ∂ τ w ∗ , ∂ τ u , (cid:1)(cid:3) τ =0 + c.c. = ± (cid:2) ( δ + i ) ∂ τ ( w ∗ u ) − iβ (cid:0) w ∗ ∂ τ u + u ∂ τ w ∗ + ∂ τ ( ∂ τ w ∗ ∂ τ u ))] τ =( τ − τ ) / + c.c. (15)Finally, substituting into Eq. (15) the asymptotic re-lations (7) and (12) we obtain: d ( τ − τ ) dt ≈ − √ β e − γ ( τ − τ ) ℜ (cid:20)(cid:18) − i δ (cid:19) (cid:16) a b ∗ e − i Ω( τ − τ ) − p a p ∗ b a ∗ b e i Ω( τ − τ ) (cid:17)(cid:21) , (16) d ( τ + τ ) dt = 0 , (17)where γ = ℜ ( µ ) ≈ (cid:0) √ β / (cid:1) p (1 + δ/β ) − I , Ω = −ℑ ( µ ) ≈ / √ β + √ β ( θ − I ) , and the Cherenkovradiation coefficients a and b are exponentially small in the limit β → . For S = 2 . , θ = 3 . , d = 0 . ,and β = 0 . numerically we get a ≈ − .
158 + 0 . i and b ≈ .
017 + 0 . i . Finally, neglecting O ( δ ) termsand taking into account that in the leading order in δ wehave p a = p b ≡ p , Eq. (16) can be rewritten in the form: d ( τ − τ ) dt ≈ √ β e − γ ( τ − τ ) | a b | (cid:0) | p | − (cid:1) cos [Ω ( τ − τ ) + arg ( b /a )] . (18)The RHS of Eq. (18) is plotted in Fig. 4, where the intersections of the black solid line with axis of abscis-sas correspond to the soliton bound states. Examplesof stable and unstable soliton bound states calculatednumerically are shown in Figs. 5 and 6, respectively, to-gether with the most unstable eigenvalues of the operator ˆ L evaluated on the bound state solutions.Finally, in Fig. 7 we present the same soliton boundstate as the one shown in Fig. 5, but calculated for δ = 0 . It is seen that the eigenvalue spectrum of this statecontains many discrete eigenvalues, which split from thecontinuous spectrum, and that it is oscillatory unstabledue to the presence of two complex conjugate eigenvalueswith positive real parts. Therefore, we can conclude thatin the absence of spectral filtering the one-dimensionalasymptotic equations (16)-(18) can be insufficient to de-scribe the soliton interaction. The derivation of the inter-action equations taking into account an Andronov-Hopfbifurcation of the soliton bound states in the presence offourth-order dispersion is beyond the scope of this study.A related problem concerning the effect of oscillatory in-stability on the soliton interaction was studied in [19]. IV. CONCLUSIONS . We have considered an all fiber photonic crystal cav-ity coherently driven by an injected field. The intracavityfield inside the fiber experiences self-phase modulation,dispersion, optical injection, and optical losses. Its space-time evolution can be described by the Lugiato-Lefeverequation with high order dispersion, where, in addition,we have taken into account small spectral filtering term.We have first discussed the properties of a single dis-sipative soliton and derived asymptotic expressions forthe soliton Cherenkov radiation amplitudes. We havefocused our analysis on the regime, where the forth or-der dispersion and the spectral filtering coefficients are small, < β , δ ≪ . Second, we have investigated theinteraction between two dissipative solitons in the casewhen they are well separated from each other. Assum-ing a weak overlap of soliton tails, we have establishedanalytically the interaction law (Eqs. 18) governing theslow time evolution of the coordinates of two interactingsolitons. We have shown that although the Cherenkovradiation due to the small fourth-order dispersion canstrongly enhance the soliton interaction and thus lead tothe formation of a large number of soliton bound states,in the absence of spectral filtering these states can beunstable with respect to an oscillatory instability evenwhen single soliton is well below the Andronov-Hopf bi-furcation threshold. This means that one-dimensionalequation (18) can be insufficient to describe the interac-tion of solitons in the generalized Lugiato-Lefever model(1) with zero spectral filtering coefficient, δ = 0 . Onthe other hand, the inclusion of small but sufficientlylarge spectral filtering, < δ ≪ , allows to stabilizeoscillatory unstable bound states and validate the one-dimensional interaction equation (18). ACKNOWLEDGMENTS
We also acknowledge the support from the DeutscheForschungsgemeinschaft (DFG-RSF project No.445430311), French National Research Agency (LABEXCEMPI, Grant No. ANR-11- LABX-0007) as well asthe French Ministry of Higher Education and Research,Hauts de France council and European Regional Devel-opment Fund (ERDF) through the Contrat de ProjetsEtat-Region (CPER Photonics forSociety P4S). M.Tlidi is a Research Director at the Fonds National dela Recherche Scientifique (Belgium). A. Vladimirov andM. Taki acknowledge the support from Invited ResearchSpeaker Programme of the Lille University. [1] T. Fortier and E. Baumann, Communications Physics ,1 (2019).[2] T. J. Kippenberg, A. L. Gaeta, M. Lipson, and M. L.Gorodetsky, Science (2018).[3] T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kon-dratiev, M. L. Gorodetsky, and T. J. Kippenberg, NaturePhotonics , 145 (2014).[4] F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, andM. Haelterman, Nature Photonics , 471 (20103).[5] L. A. Lugiato and R. Lefever, Phys. Rev. Lett. , 2209(1987).[6] Y. K. Chembo, D. Gomila, M. Tlidi, and C. R. Menyuk,The European Physical Journal D , 198 (2017).[7] L. Lugiato, F. Prati, M. Gorodetsky, and T. Kippen-berg, Philosophical Transactions of the Royal Society A:Mathematical, Physical and Engineering Sciences ,20180113 (2018).[8] M. Haelterman, S. Trillo, and S. Wabnitz, Optics com-munications , 401 (1992).[9] L. Maleki, V. Ilchenko, A. Savchenkov, W. Liang, D. Sei-del, and A. Matsko, in (IEEE, 2010), pp. 558–563.[10] A. Matsko, A. Savchenkov, W. Liang, V. Ilchenko, D. Sei-del, and L. Maleki, Optics letters , 2845 (2011).[11] Y. K. Chembo and C. R. Menyuk, Phys. Rev. A ,053852 (2013).[12] M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G.Vladimirov, and M. Taki, Optics letters , 662 (2007).[13] M. Tlidi and L. Gelens, Optics letters , 306 (2010).[14] N. Akhmediev, V. Korneev, and N. Mitskevich, Radio-physics and quantum electronics , 95 (1990).[15] M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, andS. Coulibaly, Physical Review A , 035802 (2013).[16] C. Milián and D. Skryabin, Opt. Express , 3732 (2014).[17] A. G. Vladimirov, S. V. Gurevich, and M. Tlidi, PhysicalReview A , 013816 (2018).[18] J. M. Soto-Crespo, N. Akhmediev, P. Grelu, and F. Bel-hache, Optics Letters , 1757 (2003).[19] D. Turaev, A. G. Vladimirov, and S. Zelik, Phys. Rev.Lett. , 263906 (2012).[20] M. Olivier, V. Roy, and M. Piché, Optics Letters , 580(2006). [21] N. Akhmediev and M. Karlsson, Phys. Rev. A , 2602(1995).[22] P. Parra-Rivas, E. Knobloch, D. Gomila, and L. Gelens,Physical Review A , 063839 (2016).[23] P. Parra-Rivas, D. Gomila, P. Colet, and L. Gelens, TheEuropean Physical Journal D , 198 (2017).[24] K. Gorshkov and L. Ostrovsky, Physica D: NonlinearPhenomena , 428 (1981). [25] V. Karpman and V. Solov’ev, Physica D: Nonlinear Phe-nomena , 487 (1981).[26] A. Scroggie, W. Firth, G. McDonald, M. Tlidi,R. Lefever, and L. A. Lugiato, Chaos, Solitons & Fractals , 1323 (1994).[27] I. Barashenkov and Y. S. Smirnov, Physical Review E , 5707 (1996).[28] V. I. Karpman, Phys. Rev. E47