Handling shocks and rogue waves in optical fibers · HANDLING SHOCKS AND ROGUE WAVES IN OPTICAL FIBERS JINGSONG HE1,, SHUWEI XU2, KAPPUSWAMY PORSEZIAN3, PATRICE TCHOFO DINDA4, DUMITRU

HANDLING SHOCKS AND ROGUE WAVES IN OPTICAL FIBERS

JINGSONG HE1,∗, SHUWEI XU2, KAPPUSWAMY PORSEZIAN3, PATRICE TCHOFO DINDA4,DUMITRU MIHALACHE5, BORIS A. MALOMED6,7, EDWIN DING8

1Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, P. R. ChinaE-mail∗: [email protected], [email protected]

2College of Mathematics Physics and Information Engineering, Jiaxing University,Jiaxing, Zhejiang, 314001, P. R. China

3Department of Physics, Pondicherry University, Puducherry 605014, India4Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS, Universite de Bourgogne

Franche-Comte, 9 Av. A. Savary, B.P. 47870, 21078 Dijon Cedex, France5Horia Hulubei National Institute for Physics and Nuclear Engineering,

Reactorului 30, RO-077125, P.O.B. MG-6, Bucharest-Magurele, Romania6Department of Physical Electronics, Faculty of Engineering, Tel Aviv University,

Tel Aviv 69978, Israel7Laboratory of Nonlinear-Optical Informatics, ITMO University, St. Petersburg 197101, Russia

8Department of Mathematics and Physics, Azusa Pacific University, Azusa, CA97102-700, USA

Received December 11, 2016

Abstract. In standard optical fibers, combined effects of dispersion and non-linearity can generate critical effects of localization of energy, which are potentiallyharmful for the transmission of data. Using the nonlinear Schrodinger equation as theuniversal transmission model, we establish the existence of ultrashort light pulses, inthe form of breathers on top of the continuous-wave (CW) background, and of struc-tural discontinuities (SDCs), in the form of jumps of the breathers’ phase and groupvelocities (i.e., the SDC is a variety of an optical shock). We produce exact analyticalsolutions, which demonstrate that, passing the SDC point, the breathers are convertedinto rogue waves (RWs), which is a potentially penalizing nonlinear effect in opticaltelecommunications. On the other hand, numerical simulations demonstrate that themodulational instability of the underlying CW effectively replaces the abrupt transitionby a smooth one, and makes the breathers and RWs strongly unstable close to the SDCpoint. This dynamical scenario, which may be effectively controlled by a frequencyshift of the optical signal from the CW background, opens a way to mitigate the strongnonlinear effects. On the other hand, we also consider possibilities to stabilize the RWs,for their possible use in other settings.

Key words: Optical fibers, rogue waves, breather solitons, ultrashort light pulses.

1. INTRODUCTION

In recent years, the use of multi-level modulation formats has permitted to dra-matically increase capacities of fiber-optic data-transmission channels [1]. However,such modulation formats come with their drawbacks. It is a well known fact thatincreasing the number of symbols of the modulation beyond four makes the trans-mission systems highly prone to nonlinear effects, because of distortions that such

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Article no. 203 Jingsong He et al. 2

effects induce in the signal’s phase structure. Therefore, at high bit rates, one needsto develop methods for manipulations of these effects in combination with linearones, so as to reduce the distortion of signals at high bit-rates. On the other hand, inoptical fibers, the combined effects of dispersion and nonlinearity may give rise tocritical effects of the localization of energy, such as the creation of breathers, roguewaves (RWs), and shocks, which are still more potentially harmful for the transmis-sion systems. Thus, the nonlinearity drives various processes of degradation of thedata transmission quality. This understanding suggests to elaborate schemes for pe-riodic compensation of the accumulated nonlinear phase shift in long-haul telecomlines [2–4].

In more general contexts, techniques allowing one to control light propagationunder the action of diverse linear and nonlinear factors have been drawing a steadilygrowing interest [5]-[14]. In particular, the propagation of optical breathers was re-cently considered in a two-level atomic medium interacting with an electromagneticfield, whose amplitude and frequency are controlled by a high-intensity laser source[15]. In that case, the existence of a critical frequency was demonstrated, at which thebreather transforms into a RW, i.e., a temporarily existing peak on top of a flat back-ground [16]. The two-level atomic system considered in Ref. [15] was chosen forits fundamental significance and relative simplicity, which allows a fully analyticalconsideration of the mechanism of the generation of RWs.

In the present work, we examine the generation of RWs in a different system,which is closer to the practical situation, namely, a nonlinear dielectric material (inparticular, optical fibers). In this context, we resort to a potentially effective methodof controlling the velocity of light by adjusting the carrier frequency to the injectionbeam. This setting also allows us to propose the control of nonlinearity in spatiallyinhomogeneous optical patterns.

We consider the commonly known model of a dispersive nonlinear mediumwith the cubic self-focusing nonlinearity, which is described, both in optics [17–19]and in the general context [20, 21], by the ubiquitous nonlinear Schrodinger (NLS)equation for amplitude ψ of the field envelope:

∂ψ

∂z+iβ22

∂2ψ

∂t2− iγ|ψ|2ψ = 0, (1)

where ψ(z, t) is the complex envelope amplitude of the electric field at position z inthe system, and t is time in the moving reference frame. Parameters β2 and γ desig-nate the chromatic dispersion and Kerr nonlinearity coefficients, respectively. In thecase of β2γ < 0, this equation gives rise to commonly known bright solitons, whichwere experimentally created in nonlinear optical fibers as temporal pulses [18], andin planar waveguides as self-trapped beams [19] (in the latter case, t is replaced bytransverse coordinate x, the second-derivative term representing the paraxial diffrac-

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3 Handling shocks and rogue waves in optical fibers Article no. 203

tion, rather than temporal dispersion).The existence of solitons is closely related to the modulational instability (MI)

of continuous-wave (CW) states, i.e., constant-amplitude solutions. The MI tendsto split the CW into chains of solitons [22–24]. Therefore, unlike their bright coun-terparts, solitons built on top of the CW background are always subject to instabil-ity; nevertheless, such solitons, especially RWs, which have recently drawn muchinterest in nonlinear optics [25]-[32], following their study in other fields, such asocean waves [16], [33], may be physically meaningful if the characteristic propa-gation distance necessary for the development of the instability essentially exceedsthe distance relevant to the experiment. In addition to solitons and RWs, other exactsolutions of the NLS equation have been studied in detail by means of the inversescattering transform, Hirota’s bilinear method, Backlund transform, and other tech-niques [21] based on the exact integrability of the equation. In particular, periodicbreather solutions sitting on top of the CW state were found in the exact form as well,for both NLS equation and other nonlinear evolution equations [34]-[45]. Differentadditional effects, such as the third-order dispersion, self-steepening, stimulated Ra-man scattering, birefringence etc., have been incorporated into the NLS equation andinvestigated in detail too [22]- [24].

From these results, it follows that one of major problems of the soliton trans-mission is maintaining the exact balance between the group-velocity dispersion andself-phase modulation throughout the entire transmission network. In particular, ina recent work [15] it was reported that a configuration containing structural discon-tinuities (SDC), i.e., jumps of group and phase velocities (in other words, a varietyof an optical shock), may transform the breather into an RW, which is an obviouslydetrimental nonlinear effect. To find possibilities for the mitigation of such effects, itis necessary to analyze how group and phase velocities of breathers, built on top ofthe CW background, can be controlled by varying the frequency shift of the injectionfield, with respect to the background. This issue, which was not addressed in pre-vious works dealing with the integrable NLS equation, is the subject of the presentwork.

The rest of this paper is organized as follows. Exact analytical solutions forbreathers existing on top of the CW background, which feature the velocity jumpwhile transforming into RWs, are reported and discussed in Sec. 2. The correspond-ing numerical results, and their application to the fiber optics, are presented in Sec. 3.In particular, the simulations reveal an important fact that the MI of the CW replacesthe abrupt jump by a gradual transformation of breathers into RWs, which may bethen quickly destroyed by the MI. This finding suggests a straightforward possibilityof mitigation of the potentially harmful nonlinear effects, using the above-mentionedfrequency shift as a control parameter. The paper is concluded by Sec. 4.

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2. EXACT SOLUTIONS

2.1. ANALYTICAL RESULTS

To obtain exact analytical solutions for breathers and RWs, the parameters inEq. (1) are scaled to be β2 = −2 and γ = 2. We start the analysis with an exactbreather solution obtained by means of the Darboux transform [46, 47]:

ψ[1] =

(c+ 2η

δ2δ1

)expi[at+ (2c2−a2)z], (2)

where δ1 = r2 cosh(M1) + r1 cos(M2),

δ2 = r1 cosh(M1) + r2 cos(M2) + r3 sinh(M1) + r4 sin(M2),

M1 = 2R1ηz+ (t+ 2zξ−za)R2,

M2 = 2R2ηz− (t+ 2zξ−za)R1,

r1 =−4c(R2 + 2η),

r2 =R21 + 4c2 + 2R1(2ξ+a) + (2ξ+a)2 + (R2 + 2η)2,

r3 = 4ic(R1 + 2ξ+a),

r4 = i[R21−4c2 + 2R1(2ξ+a) + (2ξ+a)2 + (R2 + 2η)2],√

4c2 + 4(ξ+ iη)2 + 4(ξ+ iη)a+a2 ≡R1 + iR2,

with four real parameters a, c, η, and ξ. The typical breather solution is plotted inFig. 1. For a fixed value of the background-CW amplitude c, the parameter a is amodulation frequency which, in the context of fiber optics, is a frequency shift be-tween a signal and the pump [48, 49]. In these experiments, the injected power isP0 = c2. The analytical expression for the breather given by Eq. (2) features peri-odic oscillations along the spatial and temporal coordinates, represented by the termcosM2. Indeed, M2 can be rewritten as M2 =Kz−Ωt, where K and Ω correspondto the spatial and temporal frequencies:

K = 2ηR2−2ξR1 +aR1, Ω =R1. (3)

In order to get simpler expressions for K and Ω, one can set η = c= 1, ξ = 0, whichyields

K = aR1 + 2R2, Ω =R1 =

√a2

2+

√a4

4+ 4a2, R2 = 2a/R1. (4)

Further, Eq. (4) gives the following expressions for the spatial and the temporalperiods of the oscillations of the breather’s intensity profile :

λb =π

aR1 + 2R2, Tb =

π

R1. (5)

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Thus, Eqs. (4) and (5) demonstrate that the internal frequency of the breather can becontrolled via the frequency-shift parameter, a, as we discuss below.

Fig. 1 – Breather solution (2), shown by means of∣∣∣ψ[1]

∣∣∣, for η = c, ξ = 0, c= 1, and a= 1/2.

The above breather solution may move in any direction in the (t,z)-plane alonga straight trajectory, which is defined by the condition M1 = 0 and denoted as lineL1: (

−8−a2 + |a|√a2 + 16

)z+(a− sgn(a)

√a2 + 16

)t= 0. (6)

The trajectory may be realized as a line connecting local maxima of the solutionshown in Fig. 1. Therefore, this solution is more general than both the Akhmedievbreather [38], which is time-periodic, and the Kuznetsov-Ma breather [34, 35], whichis spatially periodic, being a novel solution in this sense. Several trajectories L1 areplotted in Fig. 2 by setting c = 1, η = c, and ξ = 0 in the above expression for M1.A simple calculation gives the group velocity Vg and phase velocity Vp of ψ[1]:

V −1g =−2R1η

R2−2ξ+a, (7)

V −1p =2R2η

R1−2ξ+a. (8)

It follows from these expressions that, in the general case, there exists a jump inVg and Vp at a→ −2ξ and c→ η, because R1 = R2 → 0 under these conditions.These are the same conditions that are necessary to obtain optical RWs from breathersolutions, as shown below. This jump corresponds to the above-mentioned SDC, or,in other words, a shock induced by the imbalanced optical nonlinearity.

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Fig. 2 – Trajectories (red, yellow, blue, and gray lines) of breather∣∣∣ψ[1]

∣∣∣ are shown for a = −0.5,−0.01, 0.01, and 0.5, respectively. The green and black lines are the corresponding trajectories ofthe usual space- (Kuznetsov-Ma) and time- (Akhmediev) periodic breathers, i.e., solution |ψ[1]| undercondition ξ =−a/2.

A better understanding of this case can be obtained by setting η = c, ξ = 0, andc= 1, then Vg and Vp become two simple functions of parameter a, viz.,

V −1g =− 8

−a+ sgn(a)√a2 + 16

+a, (9)

V −1p =−a+ sgn(a)

√a2 + 16

2+a. (10)

The velocity jump is made obvious by the approximate forms of Eqs. (9) and (10) ata→ 0:

Vg(a→ 0)≈−(1/2)sgn(a), Vp(a→ 0)≈ (1/2)sgn(a). (11)In the opposite limit of a→±∞, the asymptotic values of the velocities are

Vg(|a| →∞)≈−a/4, Vp(|a| →∞)≈ 1/a. (12)

These results provide a unique mechanism for adjusting the velocity of theoptical breather by controlling Vp and Vg through tuning the frequency shift of theinjection beam. Figure 3 shows the group and phase velocities as functions of thefrequency-shift parameter, a, which agrees with the asymptotic approximations givenby Eqs. (11) and (12). The curve for Vg features two branches located in the secondand fourth quadrants of the parameter plane, namely, [0<Vg ≤ 5] and [−5≤ Vg < 0],respectively. The presence of these two branches is not a surprising fact. Indeed, it is

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Fig. 3 – Plots showing the jump in the group velocity Vg (red solid lines) and phase velocity Vp (bluedotted lines) of breather ψ[1]. The right panel is a zoom of the left one around a= 0.

well known that, in the absence of any frequency shift (a= 0), the light field governedby the NLS equation (1) remains centered in the moving reference frame. Whenβ2 6= 0, any frequency shift (a 6= 0) is converted, by the second-order dispersion, intoa continual temporal shift of the soliton, with respect to the moving frame [50]. Thesoliton velocity in the moving frame is proportional to a and β2. In other words,inversion of the sign of a, without a change of the sign of β2, causes a reversal of thedirection of the soliton’s motion in the moving frame. Hence, the negative branch ofVg in Fig. 3 corresponds to the reversal of the direction of the soliton’s propagationin the moving frame. In fact, the most striking feature revealed by Fig. 3 is that thesoliton’s velocity does not vanish gradually at a→ 0, but makes an abrupt jump tozero at a= 0, which may be interpreted as an optical shock. We refer to critical pointa = 0, where the velocity jump occurs, as the SDC. Recall that the jump happens ata= 0 in the case of ξ = 0 in Eq. (2), otherwise, the jump point is ajump =−2ξ.

Furthermore, similar to other systems [33, 51], the Taylor expansion of breathersolution (2) for a→ ajump =−2ξ and η→ c yields the first-order RW solution of theNLS equation as (see Fig. 4)

ψ[1]r =

(c+ 2c

δ2

δ1

)expi[at+ (2c2−a2)z], (13)

whereδ2

δ1=−1 +

2 + 8ic2z

4c2t2−16ac2tz+ 16a2c2z2 + 1 + 16c4z2.

The first-order RW of the NLS equation was obtained in Refs. [36, 37]. Ithas been observed experimentally in optical fibers [48] and in water tanks [52]. It

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Fig. 4 – The first-order rogue wave solution of the NLS equation, |ψ[1]r |, for a= 0 and c= 1.

is also relevant to mention several recent works on optical RWs: the study of RWsin normal-dispersion fiber lasers [25], the RW statistics in optical systems due tocaustics produced by focusing of random coherent spatial fields [26], the study ofthe base-band MI as the origin of the RW formation [27], the optical RW dynamicsin parametric three-wave mixing [28], and the first experimental observation of darkoptical RWs [29]. A series of relevant theoretical and experimental results in thisbroad area were summarized in recent review papers [30, 31].

2.2. DISCUSSION OF THE ANALYTICAL RESULTS

As shown above, in this work we propose a previously unexplored mechanismof controlling the velocity of light-wave patterns by varying the frequency shift of theinjection beam with respect to the background. This control mechanism gives rise tothe effect of the velocity jump through the SDC, and makes it possible to clarify twosignificant characteristics of the RWs. First, it is observed that a condition facilitatingthe existence of an RW is that the system should feature the SDC, which forces thegroup and phase velocities to perform a sharp jump (shock) at point ajump (see Fig.3 for ajump = 0). Specifically, Eq. (13) demonstrates that the SDC transforms thebreather into an RW, which has zero velocity in the present notation. Second, thegroup velocity Vg plays the role of the slope of the breather’s trajectory, L1, in the(t,z)-plane. Various trajectories for different breathers are shown in Fig. 2. For abreather with values of the parameters adopted above and a taking values in the rangeof [−0.5,0.5], the limit of a→ 0 may be presented as follows. Define lineL2, z= t/2

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Fig. 5 – Evolution following the creation of the first-order rogue wave solution of the NLS equation,|ψ[1]

r |, for a= 3 and c= 1.

with slope Vg|a→0− , and line L3, z=−t/2 with slope Vg|a→0+ . This implies that L1

rotates clockwise to L2 when a→ 0−, and L1 rotates counter-clockwise to L3 whena→ 0+. For the clarity’s sake, we did not plot L2 and L3 (which are very close to thelines associated with a=−0.01 and a= 0.01) in Fig. 2. Two breathers B1 and B2,such as the ones corresponding to a > 0 and a < 0 in solution (2), follow trajectoriesL2 and L3, producing a single RW. In other words, the two aforementioned rotationsof the breather solutions generate a single RW. Thus, we cannot know through whichrotation (clockwise or counter-clockwise) this single RW, ψ[1]

r , is generated from thetwo breathers (B1 and B2) in the limit of a→ 0. In that sense, it is possible to saythat this RW does not have a trace [16], i.e., it does not keep its history.

Additionally, the Akhmediev breather [38] generates the same RW on the taxis. In terms of the limit procedure, the t axis cannot be thought of as a trace ofRW ψ

[1]r , since the RW considered above can also be obtained from the Kuznetsov-

Ma breather [34, 35] sitting on the z-axis. To summarize, the breather has a definitetrace, and loses it in the limit of a→ 0, reducing to the traceless RW.

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The breather and RW are unstable around the jump point, ajump = 0. We haveclearly observed this instability at fixed values of c = η = 1 and ξ = 0, by numer-ically solving the NLS equation by means of the fourth-order Runge-Kutta methodin variable z, and the Fourier transform in variable t. The computational domainwas chosen to be sufficiently large (t = −200 to t = +200) to avoid boundary ef-fects (the expanding field did not hit edges in the course of the simulations). Wehave found that the numerical solutions exhibit strong instability of the breather nearthe velocity-jump point, i.e., when a is very small. On the contrary, the breather isweakly unstable when a is large. In the presence of small noise acting as a strongperturbation, the instability generates a series of peaks of intensity in the temporaldomain, which are located far from the initial breather for large values of a, whereasthe peaks are close to the initial breather for small a. On the other hand, when ais very small, the first-order RW features strong instability. The peaks generated bythe perturbation strongly interact with the main peak of the RW, causing a reductionof the height of the main peak of the RW. Thus, the RW may last for a very shorttime, before being broken by the instability. The strong instabilities of the breathersat small values of a, and the instability of the RWs are consistent, because the first-order RW is a limit form of the breather at a→ 0. We have carried out systematicsimulations of the instability dynamics of the breathers and first-order RWs for vari-ous values of a. Here we just provide, in Fig. 5, a specific numerical simulation ofthe RW pattern for a= 3, shown in the temporal domain. The main peak in the rightpanel of this figure, which is associated with the peak of the first-order RW in the leftpanel, strongly interacts with other peaks generated by the perturbation.

The breather and RW considered in this work, appear in the anomalous disper-sion regime of the fiber ( i.e., β2 < 0), in which the usual MI may have a destructiveimpact. However, it is also well known that the MI requires a minimum propagationlength in the fiber to grow significantly. Therefore, a natural question is if control-lable generation of breathers and RWs is possible over the propagation length atwhich MI effects are still weak. To address this question, it is necessary to performsimulations that include the analysis of the dynamics of breathers and RW not only inthe temporal domain, but in the frequency domain as well (via the Fourier transformof the fields of those signals) that should be performed in a sufficiently wide spectraldomain. This is done in the next Section.

3. NUMERICAL RESULTS AND THE APPLICATION TO OPTICAL FIBERS

3.1. STABILITY ANALYSIS

To properly assess the stability of the breather, the propagation must be simu-lated over a distance for which effects of the dispersion and nonlinearity fully man-

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ifest themselves. To estimate this distance, one may use a conventional bright soli-ton with the same energy as the breather. The temporal profile of the soliton isϕS =

√PSsech[2 ln(1 +

√2)t/∆t], where PS and ∆t represent the peak power and

the temporal width (FWHM) of the soliton, respectively. For c= 1, ξ = 0, and η = c,the energy of the breather (without the continuous background) is 3.758. Parametersof the soliton that have the same energy are PS = 3.531 and ∆t = 0.938. The cor-responding nonlinearity length [scaled as per Eq. (1)] is LNL ≡ (2PS)−1 = 0.1416,the same as the dispersion length, LD ≡∆t/

[2(ln(1 +

√2))2]

.We injected into the fiber a signal corresponding to the breather solution (2)

obtained in the previous section, ψ[1] at z = 0, for two values of the frequency shift,a = ±0.5. The propagation was then simulated over distance z = 2.93 ≈ 20LNL.The results, which are shown in Fig. 6, exhibit a completely symmetric behavior,with respect to the sign of a. Indeed we observe that, whatever be the sign of a, thebreather features exactly the same internal dynamics. However, this internal dyna-

00.5

11.5

22.5

−20−10

010

20

1

3

z

a=0.5

t

(a1)

|ψ(t

)|

00.5

11.5

22.5

−2−1

01

2

0.10.30.5

zω

(b1)

|ψ(ω

)|

00.5

11.5

22.5

−20−10

010

20

123

z

a=−0.5

t

(a2)

|ψ(t

)|

00.5

11.5

22.5

−2−1

01

2

0.10.30.5

zω

(b2)

|ψ(ω

)|

Fig. 6 – Dynamical behavior over distance z = 2.93, obtained by launching the analytical profile ψ[1]

from Eq. (2), for η= c, ξ = 0, c= 1, and a=±0.5. Figures (a1) and (a2) represent the evolution of thetemporal profile of the breather as a function of the propagation distance z, for a= 0.5 and a=−0.5,respectively. Panels (b1) and (b2) show the evolution of the spectra corresponding to the temporalprofiles shown in panels (a1) and (a2), respectively. The noise power is taken to be Pnoise ≈ 10−6 atz = 0.

mics is accompanied by a continual temporal shift of the breather (with respect to the

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center of its rest frame), which takes place in opposite directions depending on thesign of a. In this regard, it is well known that any frequency shift a of a light structurein the fiber is converted, through the fiber’s dispersion, into a temporal shift δt, withδt ∝ aβ2z [50]. Consequently, inversion of the sign of a results in the inversion ofthe sign of δt. More importantly, Figs. 6(a1) and 6(a2) demonstrate that the breatherpropagates in a relatively stable manner over a distance for which the dispersion andnonlinearity are fully in action. This propagation is accompanied by periodic internaldynamics, in which the breather executes two full cycles of oscillations.

Given that the dynamical behavior of the breather for a > 0 is the mirror imageof its behavior at a < 0, from now on we mainly focus on the breathers with a > 0,examining their behavior under the action of noise and MI. To this end, we haveperformed numerical simulations over different propagation distances for a = 0.5,with and without the photon noise. The obtained results are shown in Fig. 7. Figure7 (a1), which shows the result of the simulation conducted over a relatively shortdistance, without taking into account the photon noise, reveals that when the breatheris injected into the fiber, it enters the first stage of its evolution, from z = 0 up toz ∼ 4, where it propagates in quite a stable manner. This indicates that in this firstsection of the fiber, the perturbation induced by MI is still in a latent stage of thedevelopment. But beyond z ∼ 4, under the effect of the MI, the breather’s profilegradually changes and generates an oscillatory structure on one side of the breather’sprofile, as can be seen in Fig. 7 (a2). In the spectral domain, this oscillatory structuregenerates initial MI sidebands, that are highlighted on each side of the breather’scentral frequency in Fig. 7 (b2). The key point to note in Figs. 7 (a2)-(b2) is that thephoton noise is not necessary to trigger the MI, because it is readily initiated by theleading or trailing edge of the light structure already present in the system (i.e., thebreather). To account for the photon noise present in the real fiber, we have performedthe simulation represented in Figs. 7 (a3)-(b3), which clearly show that the combinedeffects of the photon noise and MI generate a perturbation that is only slightly greaterthan that generated in the absence of the photon noise, cf. Figs. 7 (a2)-(b2). Indeed,the photon noise acts mainly on the continuous background, generating a modulationwhich, as it develops, participates in the destruction of the breather, as can be seen inFigs. 7 (a3)-(b3). Another important note is that the photon noise that we have addedto the initial profile of the breather in Figs. 7 (a3)-(b3), Pnoise ≈ 10−6, is actuallyvery strong. Nevertheless, in this strongly perturbed environment, the breather iscapable to propagate over an appreciable distance, while executing three full cyclesof its internal dynamics, before starting to be destroyed.

Thus, after completing the first stage of its evolution in which the breather isquite stable, it enters a second stage, where the noise and MI come into the playand strongly affect the breather’s dynamics. Then, after a sufficiently long propaga-tion distance, the MI becomes the dominant mechanism, eventually converting the

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breather into a train of bright solitons, as shown in Figs. 7 (a4) - (b4). Figure 7 (a4)shows the train of solitons in the course of the formation.

Fig. 7 – Evolution of the breather solution |ψ[1]| for η = c, ξ = 0, c= 1, and a= 0.5.

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0

2

4

6

8

10

|ψ pea

k|2

(a1) a=0.5

<>λb

ANALYTICAL CALCULATIONS

0

5

10(b1)

|ψ(t)|2

a=0.5 z=0

0

2

4

6

8

10

|ψ pea

k|2

(c1) a=0.05

< >λ

b

0

5

10(d1)

|ψ(t)|2

a=0.05 z=0

0 5 10 150

2

4

6

8

10

|ψ pea

k|2

(e1) a=5×10−3

< >λ

b

z

−20 −10 0 10 200

5

10(f1)

|ψ(t)|2

a=5×10−3

t

z=0

0

2

4

6

8

10

|ψ pea

k|2

(a2) a=0.5

< >λ

b

NUMERICAL SIMULATIONS

0

5

10(b2)

|ψ(t)|2

a=0.5 z=3λb=4.43

<δt

>

< >

TMI

< >

0

2

4

6

8

10

|ψ pea

k|2

(c2) a=0.05< >

λb

0

5

10(d2)

|ψ(t)|2

a=0.05 z=λb=4.93

0 1 2 3 4 5 60

2

4

6

8

10

|ψ pea

k|2

(e2) a=5×10−3

z

−20 −10 0 10 200

5

10(f2)

|ψ(t)|2

t

a=5×10−3z=5

Fig. 8 – Plots showing the transformation of breather |ψ[1]|2 into the RW, for η = 1, ξ = 0, c= 1, anddifferent values of a (see the main text for explanation).

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3.2. INTERNAL DYNAMICS

The stability analysis in Fig. 7 shows that, although the breather evolves in ahighly unstable system, it is able to maintain itself over a relatively long propagationdistance, featuring several full cycles of its internal dynamics. In what follows, wewill show that the frequency-shift parameter a plays a crucial role in the dynamicalbehavior of the breathers.

As we have already outlined it in Sec. 2, the procedure that we envisage forthe generation of the RWs is, first, to generate a breather by applying the frequencyshift a. Then, we decrease a progressively to the point where the SDC emerges.In this respect, we have seen in Sec. 2 that, in the course of its propagation, thebreather undergoes internal vibrations whose frequency is determined by parametera. To examine the vibrations, we follow the evolution of the breather’s peak poweras a function of the propagation distance, z. Figure 8 shows that the breather featuresinternal vibrations, and, accordingly, its peak power oscillates with a spatial period,which is denoted λb in Figs. 8. The panels (a1), (b1), (c1), (d1), (e1), and (f1) aredirectly produced by analytical formula (2) while the panels (a2), (b2), (c2), (d2),(e2), and (f2) are results of numerical simulations of Eq. (1) with the same initialconditions as those corresponding to the analytical solution. These figures show theevolution of the breather with the variation of parameter a. The panels (a1), (a2), (c1),(c2), (e1), and (e2) show the peak power of the breather as a function of propagationdistance z. The panels (b1), (d1), and (f1) show the input profile of the breatherinjected into the fiber while the panels (b2), (d2), and (f2) show the temporal profileof the breather after passing the considered distance.

To provide an overview of the impact of the perturbation induced by the MI onthe internal dynamics of the breather, we have compared the analytical results ob-tained directly from profile (2), which are represented in Figs. 8 (a1)-(b1)-(c1)-(d1)-(e1)-(f1), with the results obtained by simulating Eq. (1) with the initial conditionscorresponding to analytical profile (2), which are displayed in Figs. 8 (a2)-(b2)-(c2)-(d2)-(e2)-(f2).

The analytical results, which correspond to an ideal system without any pertur-bation, exhibit the following fundamental features. The breather vibrates with spatialperiod λb, which strongly depends on a, decreasing with the increase of a, as canalso be seen in Figs. 8 (a1), 8 (c1) and 8 (e1). On the other hand, by carefullyinspecting Fig. 8 (a1), we note that, in the course of the internal dynamics of thebreather, the minimum value of its peak power always lies above the value corres-ponding to the CW background (i.e., P0 = 1). In other words, when the value ofa is sufficiently high, the light intensity associated with the breather varies periodi-cally but never vanishes. In contrast, as a decreases (i.e., one approaches the SDC),the spatial period λb increases, and the minimum value of the breather’s peak power

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progressively decreases, getting closer to the CW background. When the value of ais close enough to the SDC, the minimum power of the breather attains the value ofthe CW background, as can be seen in Figs. 8 (c1) and 8 (e1). In other words, thelight intensity associated with the breather varies leading to the periodic vanishingof the breather on top of the CW background. Thus, when a is sufficiently close tothe SDC point, the light intensity of the breather starts to flash, with a frequency thatdecreases as parameter a gets closer to the SDC, as illustrated in Figs. 8 (c1) and 8(e1). It should also be stressed that, when the value of a is sufficiently close to SDC,the breather’s profile [Figs. 8 (d1) and 8 (f1)] is no longer virtually distinguishablefrom that of the RW, the only remaining difference between the breather and the RWbeing breathing itself, with a low but nonzero frequency. In the limit case of a = 0,the breathing ceases. Accordingly, the light intensity stops flashing and remains at aconstant level, as long as a = 0. The breather is thus transformed into a RW. Fromthe practical point of view, the RW state may be considered as being reached whenthe breather enters the region of flashing of its light intensity, as in the case of Fig.8 (e1). This indicates that it is not actually necessary for a to be exactly fixed at theSDC point to observe the RW. At this stage, it should be emphasized that the generalfeatures that we have just explained pertain to the internal dynamics of the breather inthe ideal system (without any perturbation). However, as we have already mentioned,the conditions of the existence of our breathers coincide with those that give rise tothe MI, with dramatic consequences to the breather stability, as we discuss below.

While in the ideal system the breather exactly recovers its input profile [dis-played in Figs. 8 (b1), 8 (d1) and 8 (f1)] after each period of its internal dynamics, itcan be clearly observed in the simulations, that, for the three values of a consideredin Fig. 8, the breather is no longer able to exactly retrieve its initial profile after eachoscillation cycle. Indeed, as it enters the fiber, the breather is subject to a destabi-lization process, which, however, does not lead to its immediate destruction. Figure8 (a2), which is obtained for a = 0.5, shows that the breather executes three full cy-cles of oscillations, while undergoing the destabilization process which ends up bysignificantly distorting its profile. This destabilization is the result of the underlyingMI, which competes with the breather dynamics by forcing the envelope of the elec-tric field to perform temporal modulations at frequency ΩMI =

√2. Indeed, one can

clearly observe in Fig. 8 (b2), which shows the breather’s profile at z = 3λb, a largeoscillatory structure developing on one side of the breather, with several peaks thatappear with a period that coincides exactly with the MI period, TMI = 4.4 = 2π/ΩMI.In this simulation, we have found that the breather recovers its input profile only afterthe first cycle of oscillations (z = λb), and that at z = 2λb the MI is still in a latentstage of the development, while its effect on the breather’s profile begins to be visi-ble. More generally, in the numerical simulations displayed in Figs. 8, for any valueof a considered, it was found that the breather enters a stage of complete destruction

(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14


at z > zc ∼ 4.5. Consequently, for a 0.5 we have λb zc, and in that case, thebreather can perform a large number of full cycles of oscillations, showing excellentquantitative agreement with the predictions of our analytical consideration, beforebeing destroyed by the MI. But from a= 0.5 downward, the breather executes only afew oscillations (three at most) before its total destruction by the MI. We found that,even in this situation, when the breather is able of executing at least a complete cycleof oscillation, the period of oscillations obtained numerically is in excellent agree-ment with the analytical prediction given by formula (5), as illustrated in Fig. 9,which shows the evolution of λb as a function of a, for 0.005≤ a≤ 0.5. In particular,we notice in Fig. 9 that for 0.075 ≤ a ≤ 0.5, the agreement between the numeri-cal simulation and the analytical result is excellent. But given that λb increases asa decreases, from a = 0.075 downward, the value of λb exceeds the length of 4.5for which the MI effect becomes totally destructive. Consequently, for the valuesof a lower than 0.075, a disagreement appears between the analytical result and thesimulation (which converges toward 4.5), as Fig. 9 shows. Thus for a . 0.075, wehave λb > zc, and in that case the breather can no longer complete a full oscillationcycle before its destruction by the MI. This is also what we observe in Figs. 8 (c2)and 8 (e2), which pertain, respectively, to a = 0.05 and a = 5× 10−3. We clearlyobserve the destruction of the breather’s profile in Figs. 8 (d2) and 8 (f2), where theMI generates peaks within the MI period TMI.

0.1 0.2 0.3 0.4 0.50

5

10

15

λb

a

Fig. 9 – Plot showing the evolution of the spatial period of the internal vibrations of the breather, asa function of parameter a, for the same parameter set and operating conditions as in Fig. 8. The solidcurve shows the analytical result obtained from formula (5). The small circles represent the results ofnumerical simulations.

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Thus, if we ignore the effects of the MI (assuming that it can be suppressed, aswe discuss below), it is clear that the main features of the evolution of the breathertowards the RW, as predicted analytically, are qualitatively present in the results ofthe simulations. Indeed, we see in Figs. 8 (a2), 8 (c2), and 8 (e2) that the breatherpasses through minima of the peak power whose values (indicated by the horizontaldashed lines) progressively approach the CW background, while a approaches theSDC point (a→ 0). This is a manifestation of flashing of the breather’s light intensity,as predicted by the analytical result. On the other hand, we also see in Figs. 8 (a2),8 (c2), and 8 (e2) that the breathing period increases as a approaches the SDC point(a→ 0). Thus, both analytical results and numerical simulations predict a possibilityof controlling the evolution of the breather towards the RW.

On the other hand, in comparison with the analytical results that predict theconversion of the breather into the RW (at the SDC point), accompanied by the abruptvelocity jump in Fig. 3 and Eqs. (9)-(11), as a continuously decreases to zero, in thenumerical simulations we have observed a gradual transformation of the breather intothe RW. Here also, this difference is explained by the concomitant growth of the MIof the CW background, which supports both the breather and the RW.

In this Section, we have shown that the conditions for the existence of ourbreather in the standard optical fiber coincide with the conditions of the developmentof the spontaneous MI. The breather is thus embedded in the environment destabi-lized by the MI. We have found that, in spite of this hostile environment, the breatheris able to propagate over an appreciable distance, while featuring several full peri-ods of oscillations. Its further propagation is obviously limited by the growth of theMI. Further, our observations indicate that, in the course of the transformation of thebreather into the RW, the fundamental features of the RW behavior start to becomevisible in the close proximity of the SDC, before the exact SDC is reached. Thissituation suggests a possibility of the controllable generation of the RW from thebreather. Nevertheless, Figs. 8 and 9 clearly indicate that the MI is a major obstaclepreventing the experimental observation of the RW and breather modes. On the otherhand, the same instability-dominated scenario may be quite useful in telecommuni-cation networks, where it is necessary to eliminate harmful RWs.

If the objective of the experiment is to generate a well-defined RW, suppressionof the instability should be a decisive step. In addition to the fundamental studies,one may try to use sufficiently robust RWs as bit carries in all-optical data-processingschemes. One way to achieve the MI suppression would be to use spectral filtering,with a bandwidth that can eliminate any sideband perturbation without altering thebreather’s spectrum. For the parameter set considered in Figs. 8 and 9, the MIsidebands are located at frequencies ΩMI =±

√2, that are too close to the breather’s

spectrum. As ΩMI ∝√P0, raising power P0 of the CW background, one can push

the MI sidebands sufficiently far away from the breather’s spectrum, before applying

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a band-pass filter that may suppress the MI sidebands without significantly affectingthe breather’s profile.

4. CONCLUSION

The results reported in the present work help to understand conditions for theexistence, robustness, and, on the other hand, possibilities for effective suppressionof rogue waves (RWs) in models of nonlinear optical media (including optical fibers)based on the nonlinear Schrodinger equation. The results suggest a possibility forcontrolling the main features of such waves, adjusting the frequency shift (denoted aabove) of the control optical signal and the pump beam. The route to the formationof the RWs from breathers existing on top of the continuous wave (CW) background,revealed by the exact analytical solutions reported here, is that, varying its frequency,the breather transforms itself into a RW at the structural discontinuity point at whichthe group velocity of wave excitations features a jump (a kind of an optical shock).This dynamical scenario may be controlled by means of the frequency shift a. Inthe same time, direct simulations demonstrate that the modulational instability ofthe carrier CW easily suppresses this potentially harmful effect, replacing the abruptjump by a gradual transition, and quickly destroying the emergent RW. On the otherhand, we have also outlined a possibility to stabilize the RWs, in case they may be ofinterest as bit carriers in all-optical data-processing schemes.

Acknowledgements. This work is supported by the National Natural Science Foundation ofChina under Grant No. 11271210 and K. C. Wong Magna Fund at the Ningbo University. J.S. He thanksProf. A.S. Fokas for arranging a visit to the Cambridge University and for many useful discussions. K.P.thanks the IFCPAR, DST, NBHM, and CSIR, Government of India, for the financial support throughmajor projects. The work of B.A.M. is partly supported by grant No. 2015616 from the joint program inphysics between the National Science Foundation (US) and Binational Science Foundation (US-Israel).

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