PHYSICAL REVIEW A 100, 023816 (2019)

Swift-Hohenberg equation with third-order dispersion for optical fiber resonators

A. Hariz, L. Bahloul, and L. CherbiLaboratoire d’instrumentation, Université des Sciences et de la Technologie Houari Boumediene (USTHB), Bab Ezzouar, Algeria

K. PanajotovVrije Universiteit Brussel, Department of Applied Physics & Photonics, Pleinlaan 2, B-1050 Brussels, Belgium

and G. Nadjakov Institute of Solid State Physics, Bulgarian Academy of Sciences, 72 Tzarigradsko Chaussee Blvd., 1784 Sofia, Bulgaria

M. Clerc and M. A. FerréDepartamento de Fisica and Millennium Institute for Research in Optics, FCFM, Universidad de Chile, Casilla 487-3, Santiago, Chile

B. Kostet, E. Averlant, and M. TlidiFaculté des Sciences, Université Libre de Bruxelles (ULB), CP.231, Campus Plaine, B-1050 Brussels, Belgium

(Received 4 December 2018; published 12 August 2019)

We investigate the dynamics of a ring cavity made of photonic crystal fiber and driven by a coherentbeam working near to the resonant frequency of the cavity. By means of a multiple-scale reduction of theLugiato-Lefever equation with high-order dispersion, we show that the dynamics of this optical device, whenoperating close to the critical point associated with bistability, is captured by a real order parameter equation inthe form of a generalized Swift-Hohenberg equation. A Swift-Hohenberg equation has been derived for severalareas of nonlinear science such as chemistry, biology, ecology, optics, and laser physics. However, the peculiarityof the obtained generalized Swift-Hohenberg equation for photonic crystal fiber resonators is that it possessesa third-order dispersion. Based on a weakly nonlinear analysis in the vicinity of the modulational instabilitythreshold, we characterize the motion of dissipative structures by estimating their propagation speed. Finally, wenumerically investigate the formation of moving temporal localized structures often called cavity solitons.

DOI: 10.1103/PhysRevA.100.023816

I. INTRODUCTION

The control of linear and nonlinear properties of photoniccrystal fibers (PCF) has led to several applications in opto-electronics, sensing, and laser science (see recent overviewon this issue [1] and references therein). The improvementof the fabrication capabilities of high-quality integrated mi-crostructured resonators such as ring, microring, microdisk,or Fabry-Pérot cavities are drawing considerable attentionfrom both fundamental and applied points of view. Amongthose, an important class is the photonic crystal fiber res-onators where high-order chromatic dispersion could play animportant role in the dynamics [2,3], particularly in relationwith supercontinuum generation [4,5]. On the other hand,driven optical microcavities are widely used for the generationof optical frequency combs. They can be modeled by theLugiato-Lefever equation [6] that possesses solutions in theform of localized structures (LSs) [7,8]. Optical frequencycombs generated in high-Q Kerr resonators [9] are in factthe spectral content of the stable temporal pattern occurringin the cavity. Among the possible dissipative structures, thetemporal localized structures often called dissipative solitonsappear in the form of a stable single pulse on top of a lowbackground. They have been theoretically predicted in [7] andexperimentally observed in [8]. The link between temporallocalized structures and Kerr comb generation in high-Qresonators has motivated further interest in this issue.

When inserting a photonic crystal fiber in a cavity whichis driven by a continuous beam, the light inside the fiber iscoherently superimposed with the input beam at each roundtrip. The nonlinear Schrödinger equation supplemented bythe cavity boundary condition leads to a generalized Lugiato-Lefever equation (LLE) [10]. The inclusion of the fourth-order dispersion allows the modulational instability (MI) tohave a finite domain of existence delimited by two pumppower values [10], as well as to stabilize dark temporalLSs [11–13]. In the absence of fourth-order dispersion, withonly second or/and third orders of dispersion, front interac-tion leads to the formation of moving LS in a regime farfrom any modulational instability [14]. An analytical studyof the interaction between LSs under the action of Cherenkovradiation or dispersive waves has been conducted in [15]. Inaddition, it has been shown that the interference between thedispersive waves emitted by the two interacting LSs producesan oscillating pattern responsible for the stabilization of thebound states. A derivation of equations governing the timeevolution of the position of two well-separated LSs interactingweakly via their exponentially decaying tails has been pre-sented in [15].

In this paper, we derive a Swift-Hohenberg equation (SHE)with third-order dispersion describing the evolution of pulsespropagating in a photonic crystal fiber resonator. This reduc-tion is performed for small-frequency modes and close to asecond-order critical point marking the onset of a hysteresis

2469-9926/2019/100(2)/023816(8) 023816-1 ©2019 American Physical Society

A. HARIZ et al. PHYSICAL REVIEW A 100, 023816 (2019)

loop (nascent bistability). The dimensionless generalized realSHE reads

∂φ

∂t= y + cφ − φ3 + β

∂2φ

∂τ 2+ β ′ ∂

3φ

∂τ 3− ∂4φ

∂τ 4. (1)

Here φ(t, τ ) is the deviation of the electric field from its valueat the onset of bistability. The variable φ is a scalar real field.The time variable t corresponds to the slow evolution of φ

over successive round trips. τ accounts for the fast dynamicsthat describes how the electric-field envelope changes alongthe fiber. The parameters y and c are deviations of the ampli-tude of the injected field and the cavity detuning from theircritical values at the onset of the modulational instability,respectively. The coefficients β and β ′ account for second-and third-order chromatic dispersion, respectively. Withoutloss of generality, we rescale the fourth-order dispersioncoefficient and the cubic coefficient to unity. The third andthe fourth orders of dispersion are usually neglected in fibercavity models [6]. However, the dispersion characteristicsof photonic crystal resonators impose to consider high-orderdispersion [16–18].

The paper is organized as follows. After an introduction,we present the Lugiato-Lefever equation with high-order dis-persion, and we perform a linear stability analysis of thehomogeneous steady states in Sec. II. In Sec. III, we derive ageneralized real Swift-Hohenberg equation. A weakly nonlin-ear analysis of the real Swift-Hohenberg equation is presentedin Sec. IV. In this section, we also present a derivation of theirlinear and nonlinear velocities. Moving temporal localizedstructures with a single peak or more peaks are demonstratedin Sec. V. Section VI presents a comparison between thegeneralized LLE and the derived SHE models. In the limit ofsmall third-order dispersion, an analytical expression for thespeed of the localized structure is provided in Sec. VII. Weconclude in Sec. VIII.

II. LUGIATO-LEFEVER EQUATION WITHHIGHER-ORDER DISPERSIONS

We consider an optical cavity with a length L filled with aphotonic crystal fiber, and synchronously pumped by a coher-ent injected beam, as described in Fig. 1. A continuous waveEi is injected into the cavity by means of a beam splitter. Thefield E propagates inside the fiber and experiences dispersion,Kerr effect, and dissipation. We neglect the Raman scatteringeffect by assuming that the Kerr time response of fiber isinstantaneous. Indeed, when the pulse width is larger than 1ps, the Raman response can be neglected as discussed in [19].

FIG. 1. Schematic representation of a ring cavity filled with aphotonic crystal fiber (PCF) and driven by a coherent beam.

The linear phase shift accumulated during one cavity roundtrip is denoted by �0. The intensity mirror transmissivity(reflectivity) is T 2 (R2). The second-, third-, and fourth-orderdispersion terms are, respectively, β2,3,4. τ is the time in thereference frame moving at the group velocity of the lightdescribing the fast evolution of the field envelope withinthe cavity; τ ≡ τ (T 2/L)1/2. The time t describes the slowevolution of the field envelope between two consecutive cavityround trips and is scaled such that the decay rate is unity, i.e.,t ≡ tT 2/2tr , where tr is the round trip time. The normalizedintracavity field is E ≡ E

√2γ L/T 2 and the injected field

is S = 2/T (2γ L/T 2)1/2Ei, with γ the nonlinear coefficient.

The normalized cavity detuning is θ = 2�0/T 2. We alsoreplace the coefficients β2,3,4 by β and β ′ through the relations∂/∂τ ≡ β

−1/44 ∂/∂τ , β = β2/

√β4, and β ′ = β3/β

3/44 .

In its dimensionless form, the generalized Lugiato-Lefeverequation studied in [10] reads

∂E

∂t= S − (1 + iθ )E + i|E |2E − iβ

∂2E

∂τ 2

+β ′ ∂3E

∂τ 3+ i

∂4E

∂τ 4. (2)

The homogeneous stationary solutions (HSSs) Es of Eq. (2)are described by S2 = Is[1 + (θ − Is)2] and Is = |Es|2. Thissystem will exhibit a bistable behavior for θ > θc = √

3 anda monostable behavior for θ � θc. The linear stability of thehomogeneous steady states has been performed in [10]. Thepresence of a fourth-order dispersion gives rise to (i) a degen-erate modulational instability where two separate frequenciessimultaneously appear and (ii) appearance of a second MI thatstabilizes the high intensity regime [10]. The linear stabilityanalysis of the homogeneous solutions with respect to finitefrequency perturbation of the form exp(iτ + λt ) yields

λ± = −1 − iβ ′3 ±√

I2s − (θ − 2Is − β2 − 4)2. (3)

This dispersion relation through the conditions ∂λ/∂ =∂2λ/∂2 = 0 yields expressions for the critical frequenciesat the first MI bifurcation which are degenerate:

2l,u = −β ±

√β2 + 4(θ − 2Is)

2. (4)

These two frequencies (l and u) are simultaneously andspontaneously generated at the primary threshold Is = I1m =1. When the two critical frequencies l,u are close to eachother, it has been shown that intrinsic beating frequenciesl ± u appear [20]. Besides this first degenerate modula-tional instability, the fourth-order dispersion allows for thestabilization of the high intensity regime by creating anotherMI. The critical value of the frequency at the upper bifurcationpoint I2m is given by 2

c = −β/4. It is then possible torestabilize the stationary state by driving the system to thelarge intensity regime (I > I2m).

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III. GENERALIZED SWIFT-HOHENBERG EQUATION FORPHOTONIC CRYSTAL FIBER RESONATORS

Derivation of a generalized Swift-Hohenberg equation

The purpose of this section is to present the derivationof the generalized Swift-Hohenberg for a photonic crystalfiber resonator. For this purpose, we use the multiple scalemethod. To this aim, we explore the space-time dynamics inthe vicinity of the critical point associated with the nascentbistability. More precisely, for θ = θc, there exists a second-order critical point marking the onset of a hysteresis loop.This transition point is defined by ∂S/∂Is = ∂2S/∂I2

s = 0. Thecoordinates of this critical point are

Ec =(

3

4− i

√3

4

)Sc, S2

c = 8√

3

9. (5)

We decompose the electrical field in real and imaginary partsas E = X1 + iX2, replace it in Eq. (2), and get

∂X1

∂t= S − X1 + θX2 − X2X 2

1 − X 32

+β∂2X2

∂τ 2+ β ′ ∂

3X1

∂τ 3− ∂4X2

∂τ 4, (6)

∂X2

∂t= −X2 − θX1 + X1X 2

2 + X 31

−β∂2X1

∂τ 2+ β ′ ∂

3X2

∂τ 3+ ∂4X1

∂τ 4. (7)

We introduce the excess variables U and V : X1(τ, t ) = x1s +U (τ, t ) and X2(τ, t ) = x2s + V (τ, t ) with x1s and x2s being,respectively, the real and imaginary parts of the homogeneousstationary solution given by

S − x1s + θx2s − x2sx21s − x3

2s = 0, (8)

−x2s − θx1s + x1sx22s + x3

1s = 0. (9)

At the critical point associated with bistability, the real andimaginary parts are x1c = 3Sc/4 and x2c = −√

3Sc/4 [cf.Eq. (5)]. In order to explore the vicinity of the nascenthysteresis, we introduce a small parameter ε which measuresthe distance from the critical point associated with bistability

θ =√

3 + ε2δ. (10)

Indeed, δ accounts for the separation of detuning with respectto the critical one. We then expand in powers of ε the excessvariables and the injected field as

U = εu0 + ε2u1 + ε3u2 + · · · , (11)

V = εv0 + ε2v1 + ε3v2 + · · · , (12)

S = Sc + εS1 + ε2S2 + ε3S3 + · · · , (13)

and also introduce the slow and the fast time scales, t ≡t/ε2 and τ ≡ τ/

√ε. A preliminary analysis indicates that we

need to consider that β is small β ≡ εβ. This is because weconsider a low-frequency (or large period) regime. In this way,the MI instability threshold is close to the critical point. Wenow replace all the above scalings and expansions in Eq. (6)

and Eq. (7), and make an expansion in series of ε up to theorder ε3. To O(ε), one gets(

S1

0

)=

[1 + 2x1cx2c x2

1c + 3x22c − θc

−x22c − 3x2

1c + θc 1 − 2x1cx2c

](u0

v0

).

By replacing the values of θc and x1c,2c in these equations, thesolvability condition yields S1 = 0 and u0(τ, t ) = √

3v0(τ, t ).To the next order O(ε2), we obtain[

1 + 2x1cx2c x21c + 3x2

2c − θc

−x22c − 3x2

1c + θc 1 − 2x1cx2c

](u1

v1

)

=(

s2 + x2cδ

−x1cδ

)+

(−x2cu20 − 3x2cv

20 − 2x1cu0v0

x1cv20 + 3x1cu2

0 + 2x2cu0v0

).

In these equations, we replace the values of θc, x1c,2c, S1 = 0,u0(τ, t ) = √

3v0(τ, t ), and the solvability condition leading to

S2 = δ

21/231/4,

(14)

u1 =√

3v1 +(

3

4

)3/4

δ − 23/23−1/2u20.

Finally, to the next order O(ε3), through the solvability condi-tion, we get

√3∂u0

∂t=

√3S3 − 4

3u3

0 + δu0 + β∂2u0

∂τ 2+

√3β ′ ∂

3u0

∂τ 3−∂4u0

∂τ 4.

(15)

Introducing the following change of variables and param-eters φ ≡ νu0, y ≡ √

3S3, c ≡ (δ/ν), t ≡ (νt/√

3), β ≡ νβ,and τ ≡ ντ , with ν = (4/3)1/3, we obtain the real Swift-Hohenberg equation with the third-order dispersion

∂φ

∂t= y + cφ − φ3 + β

∂2φ

∂τ 2+ β ′ ∂

3φ

∂τ 3− 4

3

∂4φ

∂τ 4. (16)

Trivially, by normalizing the time τ , and the dispersion co-efficients, we recover Eq. (1). The SHE is a well-knownparadigm in the study of pattern formation and localizedstructures. Generically, it applies to systems that undergoa symmetry breaking modulational instability (often calledTuring instability [6]) close to the critical point associatedwith bistability (nascent optical bistability). It has been de-rived first under these conditions in hydrodynamics [21], andlater on in chemistry [22], plant ecology [23], and nonlinearoptics [24,25]. Other real order parameter equations in theform of nonvariational Swift-Hohenberg model have been alsoderived for spatially extended systems [26].

In the absence of the third-order dispersion, the Swift-Hohenberg equation (16) is variational, i.e., there exists aLyapunov functional guaranteeing that evolution proceedstowards the state for which the functional has the smallestpossible value which is compatible with the system boundaryconditions. Without the third-order dispersion in the Swift-Hohenberg model Eq. (16), localized structures do not move.However, when a nonvariational term such as φ∂2φ/∂τ 2 isconsidered, localized structures can become propagative [27].The conditions under which periodic patterns and localizedstructures appear are closely related. Dynamically speaking,a subcritical modulational instability underlies the pinning

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A. HARIZ et al. PHYSICAL REVIEW A 100, 023816 (2019)

phenomena responsible for the generation of temporal local-ized structures [28].

The homogeneous steady states φs of Eq. (16) satisfy thecubic equation y = φs(φ2

s − c). The monostable regime willbe for c < 0 and the bistable regime will occur when c > 0.The linear stability analysis with respect to finite frequencyperturbation of the form φ = φs + δφ eλt−iωτ + c.c., wherec.c. denotes the complex conjugate and with δφ � 1, yieldsthe characteristic equation

λ = c − βω2 − iβ ′ω3 − 4

3ω4 − 3φ2

s . (17)

In the absence of the third-order dispersion, i.e., β ′ =0, the homogeneous steady states undergo a modulationalinstability at

φM± = ±√

β2

16+ c

3, yM± =

(β2

16− 2c

3

)φM±. (18)

At both thresholds associated with MI, the critical frequencyis ω2

M = −3β/8.

IV. WEAKLY NONLINEAR ANALYSIS

To calculate the nonlinear solutions bifurcating from thethreshold associated with modulational instability, we use aweakly nonlinear analysis. To this end, we introduce an excessvariable as φ ≡ φs + ψ . We expand φs, ψ , and y in termsof a small parameter μ that measures the distance from themodulational instability threshold:

φs = φM± + μφ1 + μ2φ2 + μ3φ3 + · · · ,

ψ = μψ1 + μ2ψ2 + μ3ψ3 + · · · , (19)

y = yM± + μy1 + μ2y2 + μ3y3 + · · · .

The coordinates of the thresholds associated with the modula-tional instability φM± and sM± are explicitly given by Eq. (18).We introduce a slow time

∂

∂t= ∂

∂t1+ μ2 ∂

∂t2. (20)

The solution to the homogeneous linear problem obtained atthe leading order in μ is

ψ1 = W exp [i(ωMτ + κt1)] + c.c. (21)

The quantities ψi and φ1 (i = 1, 2) can be calculated byinserting Eqs. (19) into the real Swift-Hohenberg Eq. (16) andequating terms with the same powers of μ.

At third order in μ, the solvability condition yields thefollowing amplitude equation:

∂A/∂t = −6αA + ( f + ig)|A|2A, (22)

where A = μW , α = φs − φM± measures the distance fromthe second instability threshold, and

f =[

36φ2M±( 27

8 β2)72964 β4 + 256κ2

+ 192

β2φ2

M± − 3

],

(23)

g = 576φ2M±κ

72964 β4 + 256κ2

.

(a)

(b)

FIG. 2. Linear (a) and nonlinear (b) speed as a function of theparameter β ′ computed from Eqs. (26) for β = −0.6 and c = −0.06.

The amplitude equation (22) admits the following solution:A = |A| exp[i(qt + ωMτ )]. The third-order dispersion adds anew nonlinear phase q. By replacing this solution in Eq. (22),we obtain

|A|2 = 6φs − φM±

f, q = g|A|2. (24)

The nonlinear phase q is caused by the third-order disper-sion. When taking into account the nonlinear correction, thevelocity takes the following form:

v = vL + vNL, (25)

where the linear and the nonlinear velocities are

vL = ∂ Im(λ)

∂ωM= −3β ′ω2, vNL = ∂q

∂ωM. (26)

In the linear regime, the critical frequency, as well as thethreshold associated with modulational instability, are notaffected by the third-order dispersion. The linear and non-linear speeds as a function of the parameter β ′ are shownin Fig. 2 for β = −0.6 and c = −0.06. In the absenceof the third-order dispersion, the transition from super- tosubcritical modulational instability occurs when c = csub =−87β2/38 [29]. The modulational bifurcation is subcriticalwhen c > csub. The subcritical nature of the bifurcation canoccur even in the monostable regime csub < c < 0.

V. MOVING TEMPORAL LOCALIZED STRUCTURES

Examples of a single, two, or three peaks stationary sym-metric temporal localized structures are shown in Figs. 3–5.They have been obtained numerically by using a periodicboundary condition compatible with the ring geometry of the

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SWIFT-HOHENBERG EQUATION WITH THIRD-ORDER … PHYSICAL REVIEW A 100, 023816 (2019)

(a) (b)

(c)

FIG. 3. Space-time map (a) and temporal profile (b) of onelocalized structure, integrated for 1024 cells, c = 0.5, y = −0.05,β = −1, and β ′ = 0. Panel (c) shows the corresponding Fourierspectra, calculated using the integration of 500 round trips in thecavity.

optical resonator depicted in Fig. 1. However, the third-orderdispersion breaks the reflection symmetry and allows for themotion of localized structures. Examples of a single, two, orthree peaks moving temporal localized structures are shownin Figs. 6–8. The frequency resolution for the discrete Fouriertransform is proportional to the sample rate and inverselyproportional to the length of the time series. We increasethe frequency resolution by increasing the length of our timeseries, i.e., by taking 500 round trips in the cavity [30].

Localized structures occur in the regime where the ho-mogeneous steady state coexists with a spatially periodicstructure. In addition, the system exhibits a high degree ofmultistability in a finite range of the control parameter valuesoften called the pinning region [31]. The number of LSs andtheir temporal distribution along the longitudinal direction

(a) (b)

(c)(arb

. uni

ts)

(arb. units) (arb. units)

(arb. units)

(arb

. uni

ts)

(arb

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ts)

FIG. 4. Space-time map (a) and temporal profile (b) of twolocalized structures, integrated for 1024 cells, c = 0.5, y = −0.05,β = −1, and β ′ = 0. Panel (c) shows the corresponding Fourierspectra, calculated using the integration of 500 round trips in thecavity.

(a) (b)

(c)(arb

. uni

ts)

(arb. units) (arb. units)

(arb

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ts)

(arb

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ts)

(arb. units)

FIG. 5. Space-time map (a) and temporal profile (b) of threelocalized structures, integrated for 1024 cells, c = 0.5, y = −0.05,β = −1, and β ′ = 0. Panel (c) shows the corresponding Fourierspectra, calculated using the integration of 500 round trips in thecavity.

within the cavity are determined by the initial conditions [28].Two or more localized structures interact through their over-lapping oscillatory tails when they are close to one another.In the case where β ′ = 0, an analytical expression of thepotential that describes such interaction in the case of weakoverlap is derived in [32].

The interaction between the LSs then leads to the formationof clusters or LS complexes. Dissipative structures have beenobserved in all areas of nonlinear science, such as chemistry,biology, ecology, optics, and physics (see recent overview onthis issue [33,34].

A quantitative comparison between the stationary andmoving LS obtained from the SHE and from the LLE willbe discussed in the next section.

(a) (b)

(c)(arb

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ts)

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(arb

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ts)

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ts)

(arb. units)

FIG. 6. Space-time map (a) and temporal profile (b) of onelocalized structure, integrated for 1024 cells, c = 0.5, y = −0.05,β = −1, and β ′ = 0.5. Panel (c) shows the corresponding Fourierspectra, calculated using the integration of 500 round trips in thecavity.

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A. HARIZ et al. PHYSICAL REVIEW A 100, 023816 (2019)

(a) (b)

(c)(arb

. uni

ts)

(arb. units) (arb. units)

(arb. units)

(arb

. uni

ts)

(arb

. uni

ts)

FIG. 7. Space-time map (a) and temporal profile (b) of twolocalized structures, integrated for 1024 cells, c = 0.5, y = −0.05,β = −1, and β ′ = 0.5. Panel (c) shows the corresponding Fourierspectra, calculated using the integration of 500 round trips in thecavity.

VI. COMPARISON BETWEEN THE LLE ANDTHE SHE MODELS

The real order parameter description leading to the deriva-tion of a Swift-Hohenberg equation is rather generic and itis a well-known paradigm in the study of spatial periodicor localized patterns. Generically, it applies to systems thatundergo a symmetry-breaking instability close to a second-order critical point marking the onset of a hysteresis loop(nascent bistability). The reduction from the generalized LLEto a SHE type of model equation with a third dispersionhas been performed in Sec. III A. The SHE captures severalbehaviors of the LLE such as optical bistability, modulationalinstability, and moving localized structures.

(a) (b)

(c)(arb

. uni

ts)

(arb. units) (arb. units)

(arb. units)

(arb

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ts)

(arb

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ts)

FIG. 8. Space-time map (a) and temporal profile (b) of threelocalized structures, integrated for 1024 cells, c = 0.5, y = −0.05,β = −1, and β ′ = 0.5. Panel (c) shows the corresponding Fourierspectra, calculated using the integration of 500 round trips in thecavity.

(a) (b)

(arb

. uni

ts)

(arb

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ts)

(arb. units) (arb. units)

FIG. 9. Comparison of profiles obtained from the Lugiato-Lefever model (dark blue) and from the Swift-Hohenberg model(light red). Parameters are θ = 1.73196, S = 1.2407, β = 0.145for the LL model, y = 0.0578245, c = −0.00630608, β = 1.15 forthe SH model, and for both (a) motionless localized structuresobtained for β ′ = 0 and (b) traveling localized structures obtainedfor β ′ = 0.04.

In order to perform a quantitative comparison betweenthe LLE and a SHE, we fix the detuning parameter to θ =1.73196. In this case, the transmitted intensity as a function ofthe input intensity is bistable. The distance from this criticalpoint is provided by Eq. (10). This equation fixes the valueof ε = 0.12 for δ = −0.00573. The relation between injectedfield S in the LLE and y in the SHE is given by Eq. (13), withS1 = 0 and S2 = δ/(21/231/4) [cf. Eq. (14)]. y ≡ √

3S3. Thisrelation reads

S = Sc + δ

21/231/4ε2 + ε3y/

√3 + · · · . (27)

The numerical value of the injected field is S = 1.2407. Thevalues of coefficients of the second derivative in τ in the LLEand the SHE are β = 0.145 and β = 1.266, respectively. Tocompare the localized solutions of both models, we use therelation between the real order parameter φ and the real partof the intracavity field X1 = Re(E )

X1 = x1s + ε

(3

4

)1/3

φ + ε2

[(3

4

)5/12

c − 61/6φ2

]+ · · · .

(28)

The results of the comparison between the numerical simula-tion of LLE and SHE are shown in Fig. 9. In the absence ofthe third-order dispersion, the profiles of stationary solutionsobtained by numerical simulation of the LLE and the SHEare shown in Fig. 9(a). The gap between the homogeneousdomains scales as ε2. In the presence of the third-orderdispersion, the profiles of moving localized structures areasymmetric. The obtained numerical simulation of the LLEand the SHE are shown in Fig. 9(b). Both profiles (of eitherstationary or moving localized solutions) demonstrate that thepick intensities and the widths of the LSs calculated by LLEand SHE are in good agreement. Next, we compare the speedof a moving LS obtained from the LLE and the SHE. To dothat, we fix all parameters and vary the third-order dispersioncoefficient. The results are shown in Fig. 10.

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FIG. 10. Comparison of the speed of localized structures ob-tained from the Lugiato-Lefever model (dark blue) and from theSwift-Hohenberg model (light red). Parameters are θ = 1.73196,S = 1.2407, β = 0.145 for the LL model and y = 0.0578245, c =−0.00630608, β = 1.15 for the SH model.

VII. PULSE SOLUTIONS FOR SMALL THIRD-ORDERDISPERSION

Another strategy to understand the effect of the third-orderchromatic dispersion is to consider small β ′. Damped oscilla-tions towards its flanks characterize the localized structure.Analytical expressions of this motionless solution are notaccessible as a consequence of the chaos theory [35]. Let usintroduce φ0(τ − τ0) as the localized structure of model (16)with β ′ = 0, where τ0 accounts for the temporal position ofthe global maximum of the localized structure. Hence thetemporal variation of τ0 accounts for the speed of movinglocalized structure. In order to calculate the speed of the LSs,we consider the following ansatz:

φ(t, τ ) = φ0[τ − τ0(t )] + w(t, τ ), (29)

where τ0 is promoting to a temporal variable, which hasvariation of the order β ′ [τ̇0(t ) ∼ β ′], and w(t, τ ) is a smallcorrection function of the order of third chromatic dispersion.Introducing the ansatz (29) in the generalized real Swift-Hohenberg equation (16), linearizing in w, and imposing thesolvability conditions, after straightforward calculations, weobtain

τ̇0 = v ≡ β ′∫ (

∂2φ0

∂2τ

)2dτ∫ (

∂φ0

∂τ

)2dτ

. (30)

Note that the speed of propagation of the pulse is proportionalto the third-order chromatic dispersion. From Eq. (30) wesee that, if β ′ is positive (negative), the localized structurepropagates towards the positive (negative) flank. Figure 11shows a comparison between the analytical expression of thespeed of single peak LSs, Eq. (30), and numerical simulationsof the governing equation (16). From this figure, we can infera quite good agreement when β ′ is small.

(arb

. uni

ts)

(arb. units)

FIG. 11. Localized structure speed as function of β ′. Points areobtained from numerical simulations of Eq. (16). The continuous linecorresponds to the plot of formula (30). Parameters are c = 0.5, y =−0.05, and β = −1.

VIII. CONCLUSIONS

Employing a multiple-scale reduction near the bifurcationthreshold associated with the modulational instability andclose to a second-order critical point marking the onset of ahysteresis loop (nascent bistability), we have derived a realSwift-Hohenberg equation describing the evolution of theenvelope of the electric field circulating inside an optical Kerrresonator. The presence of third-order dispersion renders theobtained Swift-Hohenberg equation nonvariational meaningthere is no free energy or Lyapunov functional to minimizein the instability problem we have considered. A linear and aweakly nonlinear analysis has been performed to identify theconditions under which a transition from super- to subcriticalmodulational instability takes place. More importantly, wehave shown that the third-order dispersion allows for temporallocalized structures to move with a constant speed as a resultof the broken reflexion symmetry. We have characterizedthis motion by estimating the speed associated with it. Wehave also performed a quantitative comparison between theresults obtained from the Lugiato-Lefever model and theSwift-Hohenberg equation. This comparison includes the tem-poral profile and the speed of the LS demonstrating a verygood agreement between the two models. Finally, we havederived a simple formula for the speed of the moving local-ized structures in the limit of a small third-order dispersioncoefficient.

ACKNOWLEDGMENTS

K.P. acknowledges the Methusalem foundation for finan-cial support and the Fonds Wetenschappelijk Onderzoek-Vlaanderen under Project No. GOE5819N. M.T. acknowl-edges support as a Research Director with the Fonds de laRecherche Scientifique F.R.S.-FNRS, Belgium. M.G.C. andM.F. acknowledge the financial support of the MillenniumInstitute for Research in Optics (Miro).

[1] C. Markos, J. C. Travers, A. Abdolvand, B. J. Eggleton, and O.Bang, Rev. Mod. Phys. 89, 045003 (2017).

[2] S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, andA. S. Gouveia-Neto, Phys. Rev. A 43, 6162 (1991);

S. Pitois and G. Millot, Opt. Commun. 226, 415 (2003);J. D. Harvey, R. Leonhardt, S. Coen, G. Wong, J. C. Knight,W. J. Wadsworth, and P. St. J. Russell, Opt. Lett. 28, 2225(2003).

023816-7

A. HARIZ et al. PHYSICAL REVIEW A 100, 023816 (2019)

[3] N. Y. Joly, F. G. Omenetto, A. Efimov, A. J. Taylor, J. C. Knight,and P. St. J. Russell, Opt. Commun. 248, 281 (2005).

[4] A. V. Yulin, D. V. Skryabin, and P. St. J. Russell, Opt. Lett.29, 2411 (2004); W. Wadsworth, N. Joly, J. Knight, T. Birks,F. Biancalana, and P. Russell, Opt. Express 12, 299 (2004); A.Demircan and U. Bandelow, Opt. Commun. 244, 181 (2005).

[5] J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys. 78, 1135(2006); M. Kues, N. Brauckmann, T. Walbaum, P. Groß, and C.Fallnich, Opt. Express 17, 15827 (2009).

[6] L. A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).[7] A. J. Scroggie et al., Chaos Solitons Fractals 4, 1323 (1994).[8] F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M.

Haelterman, Nat. Photon. 4, 471 (2010).[9] T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, Science

332, 555 (2011); F. Ferdous, H. Miao, D. E. Leaird, K.Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner,Nat. Photon. 5, 770 (2011); Y. K. Chembo and C. R. Menyuk,Phys. Rev. A 87, 053852 (2013).

[10] M. Tlidi, A. Mussot, E. Louvergneaux, G. Kozyreff, A. G.Vladimirov, and M. Taki, Opt. Lett. 32, 662 (2007).

[11] M. Tlidi and L. Gelens, Opt. Lett. 35, 306 (2010).[12] M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly,

Phys. Rev. A 88, 035802 (2013).[13] L. Bahloul, L. Cherbi, A. Hariz, and M. Tlidi, Philos. Trans. R.

Soc. A 372, 20140020 (2014).[14] S. Coen, M. Tlidi, P. Emplit, and M. Haelterman, Phys. Rev.

Lett. 83, 2328 (1999); P. Parra-Rivas, D. Gomila, and L. Gelens,Phys. Rev. A 95, 053863 (2017); J. H. Talla Mbé, C. Milián, andY. K. Chembo, Eur. Phys. J. D 71, 196 (2017); P. Parra-Rivas,D. Gomila, M. A. Matias, and P. Colet, Phys. Rev. Lett. 110,064103 (2013); V. Odent, M. Tlidi, M. G. Clerc, P. Glorieux,and E. Louvergneaux, Phys. Rev. A 90, 011806(R) (2014).

[15] A. G. Vladimirov, S. V. Gurevich, and M. Tlidi, Phys. Rev. A97, 013816 (2018).

[16] F. Biancalana, D. V. Skryabin, and P. S. J. Russell, Phys. Rev. E68, 046603 (2003).

[17] F. Biancalana and D. V. Skryabin, J. Opt. A: Pure Appl. Opt. 6,301 (2004).

[18] M. J. Schmidberger, D. Novoa, F. Biancalana, P. St. J. Russell,and N. Y. Joly, Opt. Express 22, 3045 (2014).

[19] G. P. Agrawal, Nonlinear Fiber Optics, Optics and PhotonicsSeries, 2nd ed. (Academic Press, San Diego, 1995).

[20] G. Kozyreff, M. Tlidi, A. Mussot, E. Louvergneaux, M. Taki,and A. G. Vladimirov, Phys. Rev. Lett. 102, 043905 (2009).

[21] J. Swift and P. C. Hohenberg, Phys. Rev. A 15, 319 (1977).[22] M’F Hilali, G. Dewel, and P. Borckmans, Phys. Lett. A 217,

263 (1996).[23] R. Lefever et al., J. Theor. Biol. 261, 194 (2009).[24] P. Mandel, Theoretical Problems in Cavity Nonlinear Optics

(Cambridge University Press, Cambridge, UK, 1997).

[25] K. Staliunas, Phys. Rev. A 48, 1573 (1993); J. Lega, J. V.Moloney, and A. C. Newell, Phys. Rev. Lett. 73, 2978 (1994);S. Longhi and A. Geraci, Phys. Rev. A 54, 4581 (1996); M.Tlidi et al., Opt. Lett. 25, 487 (2000); S. Hachey, C. R. Jones,and J. N. Kutz, Opt. Commun. 283, 2588 (2010).

[26] M. G. Clerc, A. Petrossian, and S. Residori, Phys. Rev. E71, 015205(R) (2005); C. Durniak, M. Taki, M. Tlidi, P. L.Ramazza, U. Bortolozzo, and G. Kozyreff, ibid. 72, 026607(2005); G. Kozyreff, S. J. Chapman, and M. Tlidi, ibid. 68,015201(R) (2003); G. Kozyreff and M. Tlidi, Chaos 17, 037103(2007); M. Tlidi et al., Lect. Notes Phys. 751, 381 (2008).

[27] A. J. Alvarez-Socorro, M. G. Clerc, and M. Tlidi, Chaos 28,053119 (2018).

[28] M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640(1994); A. G. Vladimirov, R. Lefever, and M. Tlidi, Phys. Rev.A 84, 043848 (2011).

[29] M. Tlidi, M. Georgiou, and P. Mandel, Phys. Rev. A 48, 4605(1993); M. Tlidi and P. Mandel, Chaos Solitons Fractals 4, 1475(1994).

[30] W. H. Press, B. P. Flannery, and S. A. Teukolsky, T VW: Numer-ical Recipes in C: The Art of Scientific Computing (CambridgeUniversity Press, Cambridge, UK, 2002).

[31] Y. Pomeau, Physica D 23, 3 (1986).[32] M. Tlidi, A. G. Vladimirov, and P. Mandel, IEEE J. Quantum

Electron. 39, 216 (2003).[33] M. Tlidi, K. Staliunas, K. Panajotov, A. G. Vladimirov, and

M. G. Clerc, Philos. Trans. R. Soc. A 372, 20140101 (2014); L.Lugiato, F. Prati, and M. Brambilla, Nonlinear Optical Systems(Cambridge University Press, Cambridge, UK, 2015); Y. K.Chembo et al., Eur. Phys. J. D 71, 299 (2017); E. Meron,Nonlinear Physics of Ecosystems (CRC Press, Taylor & FrancisGroup, London, 2015); M. Tlidi, M. Clerc, and K. Panajotov,Philos. Trans. R. Soc. A 376, 20140014 (2018).

[34] K. Staliunas and V. Sanchez-Morcillo, Transverse Patternsin Nonlinear Optical Resonators, Springer Tracts in ModernPhysics (Springer, Berlin, 2003); Dissipative Solitons: fromOptics to Biology and Medicine, edited by N. Akhmediev andA. Ankiewicz, Lecture Notes in Physics Vol. 751 (Springer,New York, 2008); N. N. Rosanov, in I. Transverse Patterns inWide-aperture Nonlinear Optical Systems, edited by E. Wolf,Progress in Optics Vol. 35 (Elsevier, Amsterdam, 1996), pp. 1–60; M. Tlidi, M. Taki, and T. Kolokolnikov, Chaos 17, 037101(2007); O. Descalzi, M. Clerc, S. Residori, and G. Assanto,Localized States in Physics: Solitons and Patterns (SpringerScience & Business Media, New York, 2011); H. Leblond andD. Mihalache, Phys. Rep. 523, 61 (2013); J. D. Murray, Math-ematical Biology, Biomathematics Vol. 19 (Springer, Berlin,1989).

[35] D. A. C. Contreras and M. G. Clerc, Phys. Rev. E 91, 032922(2015).

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