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
1
Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, P. R. China
E-mail∗ : [email protected], [email protected]
2
College of Mathematics Physics and Information Engineering, Jiaxing University,
Jiaxing, Zhejiang, 314001, P. R. China
3
Department of Physics, Pondicherry University, Puducherry 605014, India
4
Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS, Université de Bourgogne
Franche-Comté, 9 Av. A. Savary, B.P. 47870, 21078 Dijon Cedex, France
5
Horia Hulubei National Institute for Physics and Nuclear Engineering,
Reactorului 30, RO-077125, P.O.B. MG-6, Bucharest-Magurele, Romania
6
Department of Physical Electronics, Faculty of Engineering, Tel Aviv University,
Tel Aviv 69978, Israel
7
Laboratory of Nonlinear-Optical Informatics, ITMO University, St. Petersburg 197101, Russia
8
Department 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 nonlinearity can generate critical effects of localization of energy, which are potentially
harmful for the transmission of data. Using the nonlinear Schrödinger equation as the
universal transmission model, we establish the existence of ultrashort light pulses, in
the form of breathers on top of the continuous-wave (CW) background, and of structural discontinuities (SDCs), in the form of jumps of the breathers’ phase and group
velocities (i.e., the SDC is a variety of an optical shock). We produce exact analytical
solutions, which demonstrate that, passing the SDC point, the breathers are converted
into rogue waves (RWs), which is a potentially penalizing nonlinear effect in optical
telecommunications. On the other hand, numerical simulations demonstrate that the
modulational instability of the underlying CW effectively replaces the abrupt transition
by a smooth one, and makes the breathers and RWs strongly unstable close to the SDC
point. This dynamical scenario, which may be effectively controlled by a frequency
shift of the optical signal from the CW background, opens a way to mitigate the strong
nonlinear 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 dramatically increase capacities of fiber-optic data-transmission channels [1]. However,
such modulation formats come with their drawbacks. It is a well known fact that
increasing the number of symbols of the modulation beyond four makes the transmission systems highly prone to nonlinear effects, because of distortions that such
Romanian Journal of Physics 62, 203 (2017)
v.2.0*2017.3.13#8ca1cc14
Article no. 203
Jingsong He et al.
2
effects induce in the signal’s phase structure. Therefore, at high bit rates, one needs
to develop methods for manipulations of these effects in combination with linear
ones, so as to reduce the distortion of signals at high bit-rates. On the other hand, in
optical fibers, the combined effects of dispersion and nonlinearity may give rise to
critical effects of the localization of energy, such as the creation of breathers, rogue
waves (RWs), and shocks, which are still more potentially harmful for the transmission systems. Thus, the nonlinearity drives various processes of degradation of the
data transmission quality. This understanding suggests to elaborate schemes for periodic compensation of the accumulated nonlinear phase shift in long-haul telecom
lines [2–4].
In more general contexts, techniques allowing one to control light propagation
under the action of diverse linear and nonlinear factors have been drawing a steadily
growing interest [5]-[14]. In particular, the propagation of optical breathers was recently considered in a two-level atomic medium interacting with an electromagnetic
field, 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 the
breather transforms into a RW, i.e., a temporarily existing peak on top of a flat background [16]. The two-level atomic system considered in Ref. [15] was chosen for
its fundamental significance and relative simplicity, which allows a fully analytical
consideration 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 (in
particular, optical fibers). In this context, we resort to a potentially effective method
of controlling the velocity of light by adjusting the carrier frequency to the injection
beam. This setting also allows us to propose the control of nonlinearity in spatially
inhomogeneous optical patterns.
We consider the commonly known model of a dispersive nonlinear medium
with the cubic self-focusing nonlinearity, which is described, both in optics [17–19]
and in the general context [20, 21], by the ubiquitous nonlinear Schrödinger (NLS)
equation for amplitude ψ of the field envelope:
∂ψ iβ2 ∂ 2 ψ
+
− iγ|ψ|2 ψ = 0,
∂z
2 ∂t2
(1)
where ψ(z, t) is the complex envelope amplitude of the electric field at position z in
the system, and t is time in the moving reference frame. Parameters β2 and γ designate the chromatic dispersion and Kerr nonlinearity coefficients, respectively. In the
case of β2 γ < 0, this equation gives rise to commonly known bright solitons, which
were experimentally created in nonlinear optical fibers as temporal pulses [18], and
in planar waveguides as self-trapped beams [19] (in the latter case, t is replaced by
transverse coordinate x, the second-derivative term representing the paraxial diffrac(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
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 tends
to split the CW into chains of solitons [22–24]. Therefore, unlike their bright counterparts, solitons built on top of the CW background are always subject to instability; nevertheless, such solitons, especially RWs, which have recently drawn much
interest in nonlinear optics [25]-[32], following their study in other fields, such as
ocean waves [16], [33], may be physically meaningful if the characteristic propagation distance necessary for the development of the instability essentially exceeds
the distance relevant to the experiment. In addition to solitons and RWs, other exact
solutions of the NLS equation have been studied in detail by means of the inverse
scattering transform, Hirota’s bilinear method, Bäcklund transform, and other techniques [21] based on the exact integrability of the equation. In particular, periodic
breather 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]. Different
additional effects, such as the third-order dispersion, self-steepening, stimulated Raman scattering, birefringence etc., have been incorporated into the NLS equation and
investigated in detail too [22]- [24].
From these results, it follows that one of major problems of the soliton transmission is maintaining the exact balance between the group-velocity dispersion and
self-phase modulation throughout the entire transmission network. In particular, in
a recent work [15] it was reported that a configuration containing structural discontinuities (SDC), i.e., jumps of group and phase velocities (in other words, a variety
of an optical shock), may transform the breather into an RW, which is an obviously
detrimental nonlinear effect. To find possibilities for the mitigation of such effects, it
is necessary to analyze how group and phase velocities of breathers, built on top of
the CW background, can be controlled by varying the frequency shift of the injection
field, with respect to the background. This issue, which was not addressed in previous works dealing with the integrable NLS equation, is the subject of the present
work.
The rest of this paper is organized as follows. Exact analytical solutions for
breathers existing on top of the CW background, which feature the velocity jump
while transforming into RWs, are reported and discussed in Sec. 2. The corresponding 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 replaces
the abrupt jump by a gradual transformation of breathers into RWs, which may be
then quickly destroyed by the MI. This finding suggests a straightforward possibility
of mitigation of the potentially harmful nonlinear effects, using the above-mentioned
frequency shift as a control parameter. The paper is concluded by Sec. 4.
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
Article no. 203
Jingsong He et al.
4
2. EXACT SOLUTIONS
2.1. ANALYTICAL RESULTS
To obtain exact analytical solutions for breathers and RWs, the parameters in
Eq. (1) are scaled to be β2 = −2 and γ = 2. We start the analysis with an exact
breather solution obtained by means of the Darboux transform [46, 47]:
δ2
ψ [1] = c + 2η
exp{i[at + (2c2 − a2 )z]},
(2)
δ1
where
δ1 = r2 cosh(M1 ) + r1 cos(M2 ),
δ2 = r1 cosh(M1 ) + r2 cos(M2 ) + r3 sinh(M1 ) + r4 sin(M2 ),
M1 = 2R1 ηz + (t + 2zξ − za)R2 ,
M2 = 2R2 ηz − (t + 2zξ − za)R1 ,
r1 = −4c(R2 + 2η),
r2 = R12 + 4c2 + 2R1 (2ξ + a) + (2ξ + a)2 + (R2 + 2η)2 ,
r3 = 4ic(R1 + 2ξ + a),
r4 = i[R12 − 4c2 + 2R1 (2ξ + a) + (2ξ + a)2 + (R2 + 2η)2 ],
p
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 in
Fig. 1. For a fixed value of the background-CW amplitude c, the parameter a is a
modulation frequency which, in the context of fiber optics, is a frequency shift between a signal and the pump [48, 49]. In these experiments, the injected power is
P0 = c2 . The analytical expression for the breather given by Eq. (2) features periodic oscillations along the spatial and temporal coordinates, represented by the term
cos M2 . Indeed, M2 can be rewritten as M2 = Kz − Ωt, where K and Ω correspond
to 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, which
yields
s
r
a2
a4
K = aR1 + 2R2 , Ω = R1 =
+
+ 4a2 , R2 = 2a/R1 .
(4)
2
4
Further, Eq. (4) gives the following expressions for the spatial and the temporal
periods of the oscillations of the breather’s intensity profile :
π
π
λb =
, Tb =
.
(5)
aR1 + 2R2
R1
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
5
Handling shocks and rogue waves in optical fibers
Article no. 203
Thus, Eqs. (4) and (5) demonstrate that the internal frequency of the breather can be
controlled 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 along
a straight trajectory, which is defined by the condition M1 = 0 and denoted as line
L1 :
p
p
−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 solution
shown in Fig. 1. Therefore, this solution is more general than both the Akhmediev
breather [38], which is time-periodic, and the Kuznetsov-Ma breather [34, 35], which
is spatially periodic, being a novel solution in this sense. Several trajectories L1 are
plotted 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] :
Vg−1 = −
2R1 η
− 2ξ + a,
R2
(7)
2R2 η
− 2ξ + a.
(8)
R1
It follows from these expressions that, in the general case, there exists a jump in
Vg 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 breather
solutions, as shown below. This jump corresponds to the above-mentioned SDC, or,
in other words, a shock induced by the imbalanced optical nonlinearity.
Vp−1 =
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
Article no. 203
Jingsong He et al.
6
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 of
the usual space- (Kuznetsov-Ma) and time- (Akhmediev) periodic breathers, i.e., solution |ψ [1] | under
condition ξ = −a/2.
A better understanding of this case can be obtained by setting η = c, ξ = 0, and
c = 1, then Vg and Vp become two simple functions of parameter a, viz.,
8
√
+ a,
(9)
−a + sgn(a) a2 + 16
√
−a + sgn(a) a2 + 16
−1
Vp =
+ a.
(10)
2
The velocity jump is made obvious by the approximate forms of Eqs. (9) and (10) at
a → 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−1 = −
Vg (|a| → ∞) ≈ −a/4, Vp (|a| → ∞) ≈ 1/a.
(12)
These results provide a unique mechanism for adjusting the velocity of the
optical breather by controlling Vp and Vg through tuning the frequency shift of the
injection beam. Figure 3 shows the group and phase velocities as functions of the
frequency-shift parameter, a, which agrees with the asymptotic approximations given
by Eqs. (11) and (12). The curve for Vg features two branches located in the second
and 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
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
7
Handling shocks and rogue waves in optical fibers
Article no. 203
Fig. 3 – Plots showing the jump in the group velocity Vg (red solid lines) and phase velocity Vp (blue
dotted 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 governed
by the NLS equation (1) remains centered in the moving reference frame. When
β2 6= 0, any frequency shift (a 6= 0) is converted, by the second-order dispersion, into
a continual temporal shift of the soliton, with respect to the moving frame [50]. The
soliton 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 the
direction of the soliton’s motion in the moving frame. Hence, the negative branch of
Vg in Fig. 3 corresponds to the reversal of the direction of the soliton’s propagation
in the moving frame. In fact, the most striking feature revealed by Fig. 3 is that the
soliton’s velocity does not vanish gradually at a → 0, but makes an abrupt jump to
zero at a = 0, which may be interpreted as an optical shock. We refer to critical point
a = 0, where the velocity jump occurs, as the SDC. Recall that the jump happens at
a = 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 breather
solution (2) for a → ajump = −2ξ and η → c yields the first-order RW solution of the
NLS equation as (see Fig. 4)
!
δ˜2
[1]
ψr = c + 2c
exp{i[at + (2c2 − a2 )z]},
(13)
˜
δ1
where
δ˜2
2 + 8ic2 z
= −1 + 2 2
.
4c t − 16ac2 tz + 16a2 c2 z 2 + 1 + 16c4 z 2
δ˜1
The first-order RW of the NLS equation was obtained in Refs. [36, 37]. It
has been observed experimentally in optical fibers [48] and in water tanks [52]. It
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
Article no. 203
Jingsong He et al.
8
[1]
Fig. 4 – The first-order rogue wave solution of the NLS equation, |ψr |, for a = 0 and c = 1.
is also relevant to mention several recent works on optical RWs: the study of RWs
in normal-dispersion fiber lasers [25], the RW statistics in optical systems due to
caustics produced by focusing of random coherent spatial fields [26], the study of
the base-band MI as the origin of the RW formation [27], the optical RW dynamics
in parametric three-wave mixing [28], and the first experimental observation of dark
optical RWs [29]. A series of relevant theoretical and experimental results in this
broad 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 mechanism
of controlling the velocity of light-wave patterns by varying the frequency shift of the
injection beam with respect to the background. This control mechanism gives rise to
the effect of the velocity jump through the SDC, and makes it possible to clarify two
significant characteristics of the RWs. First, it is observed that a condition facilitating
the existence of an RW is that the system should feature the SDC, which forces the
group 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 the
breather into an RW, which has zero velocity in the present notation. Second, the
group 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 a
breather with values of the parameters adopted above and a taking values in the range
of [−0.5, 0.5], the limit of a → 0 may be presented as follows. Define line L2 , z = t/2
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
9
Handling shocks and rogue waves in optical fibers
Article no. 203
Fig. 5 – Evolution following the creation of the first-order rogue wave solution of the NLS equation,
[1]
|ψr |, 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 when
a → 0+ . For the clarity’s sake, we did not plot L2 and L3 (which are very close to the
lines 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 trajectories
L2 and L3 , producing a single RW. In other words, the two aforementioned rotations
of the breather solutions generate a single RW. Thus, we cannot know through which
[1]
rotation (clockwise or counter-clockwise) this single RW, ψr , is generated from the
two breathers (B1 and B2 ) in the limit of a → 0. In that sense, it is possible to say
that 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 t
axis. In terms of the limit procedure, the t axis cannot be thought of as a trace of
[1]
RW ψr , since the RW considered above can also be obtained from the KuznetsovMa breather [34, 35] sitting on the z-axis. To summarize, the breather has a definite
trace, and loses it in the limit of a → 0, reducing to the traceless RW.
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
Article no. 203
Jingsong He et al.
10
The breather and RW are unstable around the jump point, ajump = 0. We have
clearly observed this instability at fixed values of c = η = 1 and ξ = 0, by numerically solving the NLS equation by means of the fourth-order Runge-Kutta method
in variable z, and the Fourier transform in variable t. The computational domain
was chosen to be sufficiently large (t = −200 to t = +200) to avoid boundary effects (the expanding field did not hit edges in the course of the simulations). We
have found that the numerical solutions exhibit strong instability of the breather near
the velocity-jump point, i.e., when a is very small. On the contrary, the breather is
weakly unstable when a is large. In the presence of small noise acting as a strong
perturbation, the instability generates a series of peaks of intensity in the temporal
domain, which are located far from the initial breather for large values of a, whereas
the peaks are close to the initial breather for small a. On the other hand, when a
is very small, the first-order RW features strong instability. The peaks generated by
the perturbation strongly interact with the main peak of the RW, causing a reduction
of the height of the main peak of the RW. Thus, the RW may last for a very short
time, before being broken by the instability. The strong instabilities of the breathers
at small values of a, and the instability of the RWs are consistent, because the firstorder RW is a limit form of the breather at a → 0. We have carried out systematic
simulations of the instability dynamics of the breathers and first-order RWs for various values of a. Here we just provide, in Fig. 5, a specific numerical simulation of
the RW pattern for a = 3, shown in the temporal domain. The main peak in the right
panel of this figure, which is associated with the peak of the first-order RW in the left
panel, strongly interacts with other peaks generated by the perturbation.
The breather and RW considered in this work, appear in the anomalous dispersion regime of the fiber ( i.e., β2 < 0), in which the usual MI may have a destructive
impact. However, it is also well known that the MI requires a minimum propagation
length in the fiber to grow significantly. Therefore, a natural question is if controllable generation of breathers and RWs is possible over the propagation length at
which MI effects are still weak. To address this question, it is necessary to perform
simulations that include the analysis of the dynamics of breathers and RW not only in
the temporal domain, but in the frequency domain as well (via the Fourier transform
of the fields of those signals) that should be performed in a sufficiently wide spectral
domain. 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 simulated over a distance for which effects of the dispersion and nonlinearity fully man(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
11
Handling shocks and rogue waves in optical fibers
Article no. 203
ifest themselves. To estimate this distance, one may use a conventional bright soliton with
√ as the breather. The temporal profile of the soliton is
√ the same energy
ϕS = PS sech[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. Parameters
of the soliton that have the same energy are PS = 3.531 and ∆t = 0.938. The corresponding nonlinearity length [scaled as perhEq. (1)] is LNL ≡
(2PS )−1 = 0.1416,
i
√ 2
the same as the dispersion length, LD ≡ ∆t/ 2 ln(1 + 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, the
breather features exactly the same internal dynamics. However, this internal dynaa=0.5
3
1
2.5
−20
2
−10
t
1.5
0
1
10
20
a=−0.5
(a2)
|ψ(t)|
|ψ(t)|
(a1)
3
2
1
2.5
−20
2
−10
z
t
0.5
1.5
0
(b1)
1
10
20
0
z
0.5
0
0.5
0.3
0.1
|ψ̃(ω)|
|ψ̃(ω)|
(b2)
0.5
0.3
0.1
2.5
−2
2
−1
ω
0
1.5
1
1
2
0.5
0
z
2.5
−2
2
−1
ω
0
1.5
1
1
2
z
0.5
0
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 the
temporal 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 temporal
profiles shown in panels (a1) and (a2), respectively. The noise power is taken to be Pnoise ≈ 10−6 at
z = 0.
mics is accompanied by a continual temporal shift of the breather (with respect to the
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
Article no. 203
Jingsong He et al.
12
center of its rest frame), which takes place in opposite directions depending on the
sign of a. In this regard, it is well known that any frequency shift a of a light structure
in the fiber is converted, through the fiber’s dispersion, into a temporal shift δt, with
δt ∝ aβ2 z [50]. Consequently, inversion of the sign of a results in the inversion of
the sign of δt. More importantly, Figs. 6(a1) and 6(a2) demonstrate that the breather
propagates in a relatively stable manner over a distance for which the dispersion and
nonlinearity are fully in action. This propagation is accompanied by periodic internal
dynamics, in which the breather executes two full cycles of oscillations.
Given that the dynamical behavior of the breather for a > 0 is the mirror image
of 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 have
performed numerical simulations over different propagation distances for a = 0.5,
with and without the photon noise. The obtained results are shown in Fig. 7. Figure
7 (a1), which shows the result of the simulation conducted over a relatively short
distance, without taking into account the photon noise, reveals that when the breather
is injected into the fiber, it enters the first stage of its evolution, from z = 0 up to
z ∼ 4, where it propagates in quite a stable manner. This indicates that in this first
section of the fiber, the perturbation induced by MI is still in a latent stage of the
development. But beyond z ∼ 4, under the effect of the MI, the breather’s profile
gradually changes and generates an oscillatory structure on one side of the breather’s
profile, as can be seen in Fig. 7 (a2). In the spectral domain, this oscillatory structure
generates initial MI sidebands, that are highlighted on each side of the breather’s
central frequency in Fig. 7 (b2). The key point to note in Figs. 7 (a2)-(b2) is that the
photon noise is not necessary to trigger the MI, because it is readily initiated by the
leading or trailing edge of the light structure already present in the system (i.e., the
breather). To account for the photon noise present in the real fiber, we have performed
the simulation represented in Figs. 7 (a3)-(b3), which clearly show that the combined
effects of the photon noise and MI generate a perturbation that is only slightly greater
than 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 modulation
which, as it develops, participates in the destruction of the breather, as can be seen in
Figs. 7 (a3)-(b3). Another important note is that the photon noise that we have added
to the initial profile of the breather in Figs. 7 (a3)-(b3), Pnoise ≈ 10−6 , is actually
very strong. Nevertheless, in this strongly perturbed environment, the breather is
capable to propagate over an appreciable distance, while executing three full cycles
of its internal dynamics, before starting to be destroyed.
Thus, after completing the first stage of its evolution in which the breather is
quite stable, it enters a second stage, where the noise and MI come into the play
and strongly affect the breather’s dynamics. Then, after a sufficiently long propagation distance, the MI becomes the dominant mechanism, eventually converting the
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
13
Handling shocks and rogue waves in optical fibers
Article no. 203
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.
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
Article no. 203
Jingsong He et al.
14
ANALYTICAL CALCULATIONS
10
(a1)
a=0.5
10
(b1)
a=0.5
z=0
(d1)
a=0.05
z=0
(f1)
a=5×10−3
z=0
|ψ(t)|2
|ψpeak |2
8
6
5
4
2
>
<
λb
0
0
a=0.05
<
λ
8
10
>
|ψpeak |2
b
|ψ(t)|2
10
(c1)
6
5
4
2
0
0
a=5×10−3
(e1)
<
10
>
λb
|ψpeak |2
8
|ψ(t)|2
10
6
5
4
2
0
0
5
10
0
−20
15
−10
0
z
10
20
t
NUMERICAL SIMULATIONS
a=0.5
<
|ψpeak |2
8
λb
10
>
(b2) a=0.5
<
|ψ(t)|2
10
(a2)
6
δt
z=3λb=4.43
>
5
TMI
< >< >
4
2
0
0
(c2)
<
a=0.05
>
λ
8
10
|ψpeak |2
b
(d2) a=0.05
z=λb=4.93
|ψ(t)|2
10
6
5
4
2
0
10
0
(e2)
−3
a=5×10
10
(f2) a=5×10−3
z=5
|ψ(t)|2
|ψpeak |2
8
6
5
4
2
0
0
1
2
3
z
4
5
6
0
−20
−10
0
10
20
t
Fig. 8 – Plots showing the transformation of breather |ψ [1] |2 into the RW, for η = 1, ξ = 0, c = 1, and
different values of a (see the main text for explanation).
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
15
Handling shocks and rogue waves in optical fibers
Article no. 203
3.2. INTERNAL DYNAMICS
The stability analysis in Fig. 7 shows that, although the breather evolves in a
highly unstable system, it is able to maintain itself over a relatively long propagation
distance, featuring several full cycles of its internal dynamics. In what follows, we
will show that the frequency-shift parameter a plays a crucial role in the dynamical
behavior of the breathers.
As we have already outlined it in Sec. 2, the procedure that we envisage for
the generation of the RWs is, first, to generate a breather by applying the frequency
shift 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, the
breather undergoes internal vibrations whose frequency is determined by parameter
a. To examine the vibrations, we follow the evolution of the breather’s peak power
as a function of the propagation distance, z. Figure 8 shows that the breather features
internal 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) are
directly 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 initial
conditions as those corresponding to the analytical solution. These figures show the
evolution 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 propagation
distance z. The panels (b1), (d1), and (f1) show the input profile of the breather
injected into the fiber while the panels (b2), (d2), and (f2) show the temporal profile
of the breather after passing the considered distance.
To provide an overview of the impact of the perturbation induced by the MI on
the internal dynamics of the breather, we have compared the analytical results obtained 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 conditions
corresponding 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 perturbation, exhibit the following fundamental features. The breather vibrates with spatial
period λb , which strongly depends on a, decreasing with the increase of a, as can
also be seen in Figs. 8 (a1), 8 (c1) and 8 (e1). On the other hand, by carefully
inspecting Fig. 8 (a1), we note that, in the course of the internal dynamics of the
breather, the minimum value of its peak power always lies above the value corresponding to the CW background (i.e., P0 = 1). In other words, when the value of
a is sufficiently high, the light intensity associated with the breather varies periodically 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
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
Article no. 203
Jingsong He et al.
16
progressively decreases, getting closer to the CW background. When the value of a
is close enough to the SDC, the minimum power of the breather attains the value of
the CW background, as can be seen in Figs. 8 (c1) and 8 (e1). In other words, the
light intensity associated with the breather varies leading to the periodic vanishing
of the breather on top of the CW background. Thus, when a is sufficiently close to
the SDC point, the light intensity of the breather starts to flash, with a frequency that
decreases 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 distinguishable
from that of the RW, the only remaining difference between the breather and the RW
being 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 a
constant level, as long as a = 0. The breather is thus transformed into a RW. From
the practical point of view, the RW state may be considered as being reached when
the 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 the
SDC point to observe the RW. At this stage, it should be emphasized that the general
features that we have just explained pertain to the internal dynamics of the breather in
the 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 to
the MI, with dramatic consequences to the breather stability, as we discuss below.
While in the ideal system the breather exactly recovers its input profile [displayed in Figs. 8 (b1), 8 (d1) and 8 (f1)] after each period of its internal dynamics, it
can be clearly observed in the simulations, that, for the three values of a considered
in Fig. 8, the breather is no longer able to exactly retrieve its initial profile after each
oscillation cycle. Indeed, as it enters the fiber, the breather is subject to a destabilization process, which, however, does not lead to its immediate destruction. Figure
8 (a2), which is obtained for a = 0.5, shows that the breather executes three full cycles of oscillations, while undergoing the destabilization process which ends up by
significantly distorting its profile. This destabilization is the result of the underlying
MI, 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 large
oscillatory structure developing on one side of the breather, with several peaks that
appear 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 after
the first cycle of oscillations (z = λb ), and that at z = 2λb the MI is still in a latent
stage of the development, while its effect on the breather’s profile begins to be visible. More generally, in the numerical simulations displayed in Figs. 8, for any value
of 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
17
Handling shocks and rogue waves in optical fibers
Article no. 203
at z > zc ∼ 4.5. Consequently, for a 0.5 we have λb zc , and in that case, the
breather can perform a large number of full cycles of oscillations, showing excellent
quantitative agreement with the predictions of our analytical consideration, before
being destroyed by the MI. But from a = 0.5 downward, the breather executes only a
few 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 cycle
of oscillation, the period of oscillations obtained numerically is in excellent agreement 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 numerical simulation and the analytical result is excellent. But given that λb increases as
a decreases, from a = 0.075 downward, the value of λb exceeds the length of 4.5
for which the MI effect becomes totally destructive. Consequently, for the values
of a lower than 0.075, a disagreement appears between the analytical result and the
simulation (which converges toward 4.5), as Fig. 9 shows. Thus for a . 0.075, we
have λb > zc , and in that case the breather can no longer complete a full oscillation
cycle 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 clearly
observe the destruction of the breather’s profile in Figs. 8 (d2) and 8 (f2), where the
MI generates peaks within the MI period TMI .
15
λb
10
5
0
0.1
0.2
0.3
0.4
0.5
a
Fig. 9 – Plot showing the evolution of the spatial period of the internal vibrations of the breather, as
a function of parameter a, for the same parameter set and operating conditions as in Fig. 8. The solid
curve shows the analytical result obtained from formula (5). The small circles represent the results of
numerical simulations.
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
Article no. 203
Jingsong He et al.
18
Thus, if we ignore the effects of the MI (assuming that it can be suppressed, as
we discuss below), it is clear that the main features of the evolution of the breather
towards the RW, as predicted analytically, are qualitatively present in the results of
the simulations. Indeed, we see in Figs. 8 (a2), 8 (c2), and 8 (e2) that the breather
passes through minima of the peak power whose values (indicated by the horizontal
dashed lines) progressively approach the CW background, while a approaches the
SDC 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 possibility
of controlling the evolution of the breather towards the RW.
On the other hand, in comparison with the analytical results that predict the
conversion of the breather into the RW (at the SDC point), accompanied by the abrupt
velocity jump in Fig. 3 and Eqs. (9)-(11), as a continuously decreases to zero, in the
numerical simulations we have observed a gradual transformation of the breather into
the RW. Here also, this difference is explained by the concomitant growth of the MI
of the CW background, which supports both the breather and the RW.
In this Section, we have shown that the conditions for the existence of our
breather in the standard optical fiber coincide with the conditions of the development
of the spontaneous MI. The breather is thus embedded in the environment destabilized by the MI. We have found that, in spite of this hostile environment, the breather
is able to propagate over an appreciable distance, while featuring several full periods of oscillations. Its further propagation is obviously limited by the growth of the
MI. Further, our observations indicate that, in the course of the transformation of the
breather into the RW, the fundamental features of the RW behavior start to become
visible in the close proximity of the SDC, before the exact SDC is reached. This
situation suggests a possibility of the controllable generation of the RW from the
breather. Nevertheless, Figs. 8 and 9 clearly indicate that the MI is a major obstacle
preventing the experimental observation of the RW and breather modes. On the other
hand, the same instability-dominated scenario may be quite useful in telecommunication networks, where it is necessary to eliminate harmful RWs.
If the objective of the experiment is to generate a well-defined RW, suppression
of 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-processing
schemes. One way to achieve the MI suppression would be to use spectral filtering,
with a bandwidth that can eliminate any sideband perturbation without altering the
breather’s spectrum. For the parameter set considered
in Figs. 8 and 9, the MI
√
sidebands 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
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
19
Handling shocks and rogue waves in optical fibers
Article no. 203
a band-pass filter that may suppress the MI sidebands without significantly affecting
the breather’s profile.
4. CONCLUSION
The results reported in the present work help to understand conditions for the
existence, robustness, and, on the other hand, possibilities for effective suppression
of rogue waves (RWs) in models of nonlinear optical media (including optical fibers)
based on the nonlinear Schrödinger equation. The results suggest a possibility for
controlling the main features of such waves, adjusting the frequency shift (denoted a
above) of the control optical signal and the pump beam. The route to the formation
of 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 which
the 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. In
the same time, direct simulations demonstrate that the modulational instability of
the carrier CW easily suppresses this potentially harmful effect, replacing the abrupt
jump by a gradual transition, and quickly destroying the emergent RW. On the other
hand, we have also outlined a possibility to stabilize the RWs, in case they may be of
interest as bit carriers in all-optical data-processing schemes.
Acknowledgements. This work is supported by the National Natural Science Foundation of
China under Grant No. 11271210 and K. C. Wong Magna Fund at the Ningbo University. J.S. He thanks
Prof. 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 through
major projects. The work of B.A.M. is partly supported by grant No. 2015616 from the joint program in
physics between the National Science Foundation (US) and Binational Science Foundation (US-Israel).
REFERENCES
1. T. Rahman, D. Rafique, A. Napoli, and H. de Waardt, Ultralong haul 1.28 TB/s pm-16QAM WDM
transmission employing hybrid amplification, J. Lightwave Technol. 33, 1794–1804 (2014).
2. C. Paré, A. Villeneuve, P.-A. Bélanger, and N. J. Doran, Compensating for dispersion and the
nonlinear Kerr effect without phase conjugation, Opt. Lett. 21, 459–461 (1996).
3. C. Paré, A Villeneuve, and S. LaRochelle, Split compensation of dispersion and self-phase modulation in optical communication systems, Opt. Commun. 160, 130–168 (1999).
4. R. Driben, B. A. Malomed, M. Gutin, and U. Mahlab, Implementation of nonlinearity management
for Gaussian pulses in a fiber-optic link by means of second-harmonic-generating modules, Opt.
Commun. 218, 93–104 (2003).
5. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Light speed reduction to 17 metres per
second in an ultracold atomic gas, Nature 397, 594–598 (1999).
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
Article no. 203
Jingsong He et al.
20
6. Q. H. Park and R. W. Boyd, Modification of self-induced transparency by a coherent control field,
Phys. Rev. Lett. 86, 2774–2777 (2001).
7. J. B. Khurgin, Optical buffers based on slow light in electromagnetically induced transparent
media and coupled resonator structures: comparative analysis, J. Opt. Soc. Am. B 22, 1062–1074
(2005).
8. G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, and R. W. Boyd, Observation of backward
pulse propagation through a medium with a negative group velocity, Science 312, 895–897 (2006).
9. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, Dispersionless slow light using gap
solitons, Nature Phys. 2, 775–780 (2006).
10. R. W. Boyd and D. J. Gauthier, Controlling the velocity of light pulses, Science 326, 1074–1077
(2009).
11. R. W. Boyd, Slow and fast light: fundamentals and applications, J. Mod. Opt. 56, 1908–1915
(2009).
12. J. B. Khurgin, Slow light in various media: a tutorial, Adv. Opt. Phot. 2, 287-318 (2010).
13. S. Fu, Y. Liu, Y. Li, L. Song, J. Li, B. A. Malomed, and J. Zhou, Buffering and trapping ultrashort
optical pulses in concatenated Bragg gratings, Opt. Lett. 38, 5047–5050 (2013).
14. D. Giovannini, J. Romero, V. Potoček, G. Ferencz, F. Speirits, S. M. Barnett, D. Faccio, and M.
J. Padgett, Spatially structured photons that travel in free space slower than the speed of light,
Science 347, 857–860 (2015).
15. J. S. He, S. W. Xu, K. Porsezian, Y. Cheng, and P. Tchofo Dinda, Rogue wave triggered at a critical
frequency of a nonlinear resonant medium, Phys. Rev. E 93, 062201 (2016).
16. N. Akhmediev, A. Ankiewicz, and M. Taki, Waves that appear from nowhere and disappear without a trace, Phys. Lett. A 373, 675–678 (2009).
17. R. Y. Chiao, E. Garmire, and C. H. Townes, Self-trapping of optical beams, Phys. Rev. Lett. 13,
479–482 (1964).
18. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Experimental Observation of Picosecond Pulse
Narrowing and Solitons in Optical Fibers, Phys. Rev. Lett. 45, 1095–1098 (1980).
19. S. Maneuf, R. Desailly, and C. Froehly, Stable self-trapping of laser beams: Observation in a
nonlinear planar waveguides, Opt. Commun. 65, 193–198 (1988).
20. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: the Inverse
Scattering Method, Nauka Publishers, Moscow, 1980.
21. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991.
22. A. Hasegawa and Y. Kodama, Solitons in Optical Communications, Oxford University Press, Oxford, 1995.
23. Y. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic
Press, San Diego, 2003.
24. G. Agrawal, Nonlinear Fiber Optics, 5th Edition, Academic Press, Oxford, 2013.
25. Z. W. Liu, S. M. Zhang, and F. W. Wise, Rogue waves in a normal-dispersion fiber laser, Opt. Lett.
40, 1366–1369 (2015).
26. A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty, and J. M. Dudley, Caustics and Rogue Waves
in an Optical Sea, Sci. Rep. 5, 12822 (2015).
27. F. Baronio, S. H. Chen, P. Grelu, S. Wabnitz, and M. Conforti, Baseband modulation instability as
the origin of rogue waves, Phys. Rev. A 91, 033804 (2015).
28. S. H. Chen, F. Baronio, J. M. Soto-Crespo, P. Grelu, M. Conforti, and S. Wabnitz, Optical rogue
waves in parametric three-wave mixing and coherent stimulated scattering, Phys. Rev. A 92,
033847 (2015).
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
21
Handling shocks and rogue waves in optical fibers
Article no. 203
29. B. Frisquet, B. Kibler, P. Morin, F. Baronio, M. Conforti, G. Millot, and S. Wabnitz, Optical dark
rogue waves, Sci. Rep. 6, 20785 (2016).
30. M. Onorato, S. Residori, U. Bortolozzo, A. Montinad, and F. T. Arecchi, Rogue waves and their
generating mechanisms in different physical contexts, Phys. Rep. 528, 47–89 (2013).
31. J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, Instabilities, breathers and rogue waves in
optics, Nature Photonics 8, 755–764 (2014).
32. M. Onorato, S. Resitori, and F. Baronio, Rogue and shock waves in nonlinear dispersive media,
Lecture Notes in Physics, vol. 926, Springer, 2016.
33. J. S. He, L. J. Guo, Y. S. Zhang, and A. Chabchoub, Theoretical and experimental evidence of
non-symmetric doubly localized rogue waves, Proc. R. Soc. A 470, 20140318 (2014).
34. E. A. Kuznetsov, Solitons in a parametrically unstable plasma, Sov. Phys. Doklady 22, 507–508
(1997).
35. Y. C. Ma, The perturbed plane-wave solutions of the cubic Schrödinger equation, Stud. Appl.
Math. 60, 43–58 (1979).
36. D. H. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, J. Austral.
Math. Soc. B 25, 16–43 (1983).
37. N. Akhmediev, M. M. Eleonskii, and N. E. Kulagin, Generation of a Periodic Sequence of Picosecond Pulses in an Optical Fibre: Exact solutions, Sov. Phys. JETP 62, 894–899 (1985).
38. N. Akhmediev and V. I. Korneev, Modulation instability and periodic solutions of the nonlinear
Schrödinger equation, Theor. Math. Phys. 69, 1089–1093 (1986).
39. J. S. He, M. Ji, and Y. S. Li, Solutions of two kinds of non-isospectral generalized nonlinear
Schrödinger equation related to Bose-Einstein condensates, Chinese Phys. Lett. 24, 2157–2160
(2007).
40. S. H. Chen and D. Mihalache, Vector rogue waves in the Manakov system: diversity and compossibility, J. Phys. A: Math. Theor. 48, 215202 (2015).
41. D. Mihalache, Localized structures in nonlinear optical media: A selection of recent studies, Rom.
Rep. Phys. 67, 1383–1400 (2015).
42. F. Yuan, J. Rao, K. Porsezian, D. Mihalache, and J. S. He, Various exact rational solutions of the
two-dimensional Maccari’s system, Rom. J. Phys. 61, 378–399 (2016).
43. S. H. Chen, P. Grelu, D. Mihalache, and F. Baronio, Families of rational solutions of the
Kadomtsev-Petviashvili I equation, Rom. Rep. Phys. 68, 1407–1424 (2016).
44. Y. B. Liu, A. S. Fokas, D. Mihalache, and J. S. He, Parallel line rogue waves of the third-type
Davey-Stewartson equation, Rom. Rep. Phys. 68, 1425–1446 (2016).
45. S. H. Chen, J. M. Soto-Crespo, F. Baronio, P. Grelu, and D. Mihalache, Rogue-wave bullets in a
composite (2+1)D nonlinear medium, Opt. Express 24, 15251–15260 (2016).
46. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer-Verlag, Berlin,
1991.
47. J. S. He, L. Zhang, Y. Cheng, and Y. S. Li, Determinant representation of Darboux transformation
for the AKNS system, Science in China Series A: Mathematics 49, 1867–1878 (2006).
48. B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhmediev, and J. M. Dudley, The
Peregrine soliton in nonlinear fibre optics, Nat. Phys. 6, 790–795 (2010).
49. K. Hammani, B. Kibler, C. Finot, P. Morin, J. Fatome, J. M. Dudley, and G. Millot, Peregrine soliton generation and breakup in standard telecommunications fiber, Opt. Lett. 36, 112–114 (2011).
50. P. Tchofo Dinda, A. Labruyere, and K. Nakkeeran, Theory of Raman effect on solitons in optical
fibre systems: impact and control processes for high-speed long-distance transmission lines, Opt.
Commun. 234, 137–151 (2004).
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14
Article no. 203
Jingsong He et al.
22
51. J. S. He, H. R. Zhang, L. H. Wang, K. Porsezian, and A. S. Fokas, Generating mechanism for
higher-order rogue waves, Phys. Rev. E 87, 052914 (2013).
52. A. Chabchoub, N. P. Hoffmann, and N. Akhmediev, Rogue Wave Observation in a Water Wave
Tank, Phys. Rev. Lett. 106, 204502 (2011).
(c) RJP 62(Nos. 1-2), id:203-1 (2017) v.2.0*2017.3.13#8ca1cc14