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JingSong, H., ShuWei, X., Ruderman, M.S. et al. (1 more author) (2014) State Transition
Induced by SelfSteepening and Self PhaseModulation. Chinese Physics Letters, 31 (1).
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State transition induced by selfsteepening and self phasemodulation
J.S. He
1,2
, S.W. Xu
3
, M.S. Ruderman
4
and R. Erd´elyi
4
1
Depar tmen t of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, P.R.China.
2
DAMTP, University of Cambridge, Cambridge CB3 0WA, UK
3
Schoo l of Mathematics, USTC, Hefei, Anhui 230026, P.R.China
4
Solar Physics and Space Plasma Research Centre, University of Sheﬃeld, Sheﬃeld, S3 7RH, UK
We present a rat io n a l solution for a mixed nonlinear Schr¨odinger (MNLS) equation. This solution
has two free parameters a and b representing the contributions of selfsteepenin g and self phase
modulation (SPM) of an associated physical system. It describes ﬁve soliton states: a paired
brightbright soliton, single soliton, a paired brightgrey soliton , a paired brightblack soliton, and a
rogue wave state. We show that the transition among these ﬁve states is induced by selfsteepening
and SPM through tuning the values of a and b. This is a unique and potentially fundamentally
important phenomenon in a physical system described by the M N LS equ a t io n .
KEYWORDS: Mixed nonlinear Schr¨odinger equati o n , state transfer, Alfv´e n wave, ultrashort
light pulse, rational solution.
PACS numbers: 05.45.Yv, 42.65.Re, 52.35.Sb,02.30.Ik
Introduction. The mixed nonli ne ar Schr¨odinger
(MNLS) equation [1],
q
t
− iq
xx
+ a(q
∗
q
2
)
x
+ ibq
∗
q
2
= 0, (1)
has been derived by the mod iﬁ ed reductive pe rt u r bat i on
method to describe the propagation of e.g. the Alfv´en
waves wit h small but ﬁnite amplitude along the magnetic
ﬁeld i n the cold plasma approximation widely applica
ble in solar, solarterrestrial, space and astrophysics[2, 3].
Later it has been shown [4–6] that the MNLS equation
also provides an accurate mo d el l i ng of ultrashort light
pulse propagation in optical ﬁbers. In eq. (1) the complex
quantity q represents th e magnetic ﬁeld perturbat i on in
the case of Alfv´en waves, and the electrical ﬁeld envelope
in the case of waves in optical ﬁbres. The asterisk de
notes comp le x conjugate, a and b are two nonnegative
constants de t er mi ne d by the unperturbed state, and the
subscript x (t) denotes the partial derivative with respect
to the spatial coordinate x (time t). T he MNLS equation
reduces to the Nonlinear Schr¨odinger (NLS) equation
when a = 0, and to the Derivative Non li n ear Schr¨odinger
(DNLS) equation when b = 0. Because the MNLS equa
tion is applicable even in the case where the lengths of the
envelope wave and the carrier wave are comparable [1] ,
it is more general than the nonlinear Schr¨odinger equa
tion in the modelling of waves in optical ﬁbres. The last
three terms in eq. (1) describe the group velocity disp e r
sion (GVD), s el f st e epening and self phasemodul at i on
(SPM), respectively.
The Lax pair for th e MNLS equation provides the
mathematical basis for the solvability of this equation
by the inverse scattering method and D arboux transfor
mation (DT), the latter being deﬁned by the Wadati
KonnoIchikawa (WKI) spectral problem and the ﬁrst
nontrivial ﬂow [7]
∂
x
ψ = (−aJλ
2
+ Q
1
λ + Q
0
)ψ ≡ Uψ,
∂
t
ψ = (−2a
2
Jλ
4
+ V
3
λ
3
+ V
2
λ
2
+ V
1
λ + V
0
)ψ ≡ V ψ,
with the reduction condition r = −q
∗
. Here the complex
quantity λ is the eigenvalue (or spectral parameter), and
ψ = (φ, ϕ)
T
is the eigenfunction associated with λ. The
superscript T denotes transposition. There are also other
methods to show the integrability of the M NLS equation
and to obtain its exact solutions [8–11]. Various types of
solutions of the MNLS equation, including a soliton and
a breather have alr ead y been obtained in [12–21]. The
decay of soliton solution for a perturbed MNLS system
has been demonstrated numerically [22]. Small pertur
bations of the MNLS equation have been studied either
by a direct method [23], or using the inve rs e scattering
transform [24, 25].
Up to now, all known solutions of the MNLS eq uat i on
represent rather common nonlinear wave solutions, like
solitons and breathers. These are similar corresponding
ones to their ancestors — the NLS and DNLS equation,
so they do not describe any speciﬁc properties of either
the Alf v´en waves in magnetised plasmas, or ult r ash ort
light pu l se s in op t ic al ﬁbres. Thus, it is a longstanding
problem to ﬁnd unique phe nom en a related to the simulta
neous eﬀects of selfst e epening and SPM that can be de
scribed by the exp li c it analytical solution s of the MNLS
equation.
In this letter we p r ese nt novel rational soluti ons of the
MNLS equation. These solutions describe ﬁve states of
the associated sy st e m s: a paired brightbright soliton
state, a single soliton state, a paired brightgrey soliton
state, a paired brightdark soliton state, and a rogue wave
state. We also show that the transition between these ﬁve
states is induced by the selfsteepening and SPM through
tuning the values of a and b determi n ed by the phys i ca l
properties of the background state.
Analytical form of rational solution. By the one
fold DT and Taylorexpansion, according to a similar
procedure demonstrated by [26–30] for constructing the
rational rogue wave (RW) of the NLS, DNLS, Hirota and
NLSMaxwellBloch equations, a novel rational solution
of the MNLS equation is given as follows
q
[1]
= −exp[i(−x − tb − t + ta) ]
r
1
r
2
r
∗
1
2
, (2)
r
1
= X + 1 + 2ia[(3a − 2)t − x],
r
2
= X − 3 + 2i[(−6a + 3a
2
+ 4b)t − ax],
X = 2(a − b)[(−2b + 3a
2
− 6a + 4)t
2
+ 4(1 − a)xt + x
2
].
We omit the tedious calculation of t h is solution. The
validity of this solution has been conﬁrmed by symbolic
computation. By letting x → ∞ and t → ∞ it is easy to
arXiv:1305.1977v2 [nlin.SI] 10 May 2013
2
show that q
[1]

2
→ 1, and q
[1]

2
(0, 0) = 9. So, q
[1]
de
notes a rational solution on a nonzero background wi t h a
unit height. This is a new and wide class of solutions for
the MNLS equation because it can encompass ﬁve kinds
of solut i on .
Transition between ﬁve states. We are n ow i n the
position to explore the properties of q
[1]
with more de
tails. The trajectory of q
[1]

2
in the (x, t)plane is de
ﬁned by the location of the ridge (or val e ) of its proﬁle.
A good approx i mat i on of the trajectory for q
[1]

2
is gi ven
through a simple equation X + 1 = 0 in general. By a
straightforward but tedious calculation of the stationary
point of q
[1]

2
in the (x, t)plane we ﬁnd, that, q
[1]
de
scribes ﬁve solutions as sociated with ﬁve regions in the
upperright quadrant on the (a, b)plane (see Figure 1),
as foll ows
I ( b > a). There is only one saddle point of q
[1]

2
at
(0, 0), and there are two simultaneous trajectories X
1
and
X
2
on the (x, t)plane. Note that the trajectories are not
two straight lines as usu al in the case of double solitons.
If the height of soliton q
[1]

2
is increasing as it evolves
along X
1
, then it will decrease as it evolves on X
2
. The
asymptotic height of q
[1]

2
is H
1
= H
1
(a, b) as t = −∞,
and H
2
= H
2
(a, b) as t = +∞ on X
1
; q
[1]

2
approaches
to H
2
as t = −∞ and to H
2
as t = +∞ on X
2
. Thus,
when (a, b) ∈ region I, q
[1]
is called a paired brightbright
soliton because H
2
> H
1
> 1 and the appearance of two
peaks (i.e. an increasing peak and a decreasing one).
Obviously, there exists energy e x change between the two
bright peaks propagating along X
1
and X
2
in accordance
with the en er gy conservation. In particular, the distance
between the two peaks is proportional to
√
δ
1
t
2
+ δ
2
in
contrast to a linear function of t for the known two peaks
in the case of a double soliton. Here δ
1
and δ
2
are two
real f un ct i ons of a and b.
II (b = a). q
[1]

2
takes its maximum value 9 when (x, t) is
on the line x = (− 2+3a)t. This line is also the trajectory
of q
[1]

2
. This is a single soliton solution.
III (a > b > max[0, a −
3
8
a
2
]). There is only one ex
tremum of q
[1]

2
at (0, 0) on the (x, t)plane. Most fea
tures of the obtained solution are similar to those of
region I except H
2
> 1 > H
1
> 0. Thus, in this re
gion, q
[1]
is called a paired brightgrey soliton. Note that
a −
3
8
a
2
< 0 if a >
8
3
.
IV (b = a −
3
8
a
2
, a ∈ [0,
8
3
]). There is only one extremum
of q
[1]

2
at (0, 0) in the (x, t)plane. The solution resem
bles that of region III except that H
2
> 1 > H
1
≥ 0.
Thus, in region IV of t he (a, b)plane, q
[1]
is call e d a
paired brightdark soliton.
V (0 < b < a −
3
8
a
2
). For q
[1]

2
, there is only one
maximum at (0,0), where q
[1]

2
= 9, and two minima,
where q
[1]

2
= 0. The two minima are located with the
coordinates given by
x = ∓
−6a + 3a
2
+ 4b
a − b
s
3
32((a −
3
8
a
2
) − b)
,
t = ±
a
a − b
s
3
32((a −
3
8
a
2
) − b)
at the points in th e (x, t)plane. When (a, b) is r egi on V,
q
[1]

2
is localised both in t h e x and t direction, and thus
q
[1]
is a rogue wave solution of the MNLS equation.
The main diﬀerence between the grey and dark soliton
is that the minimum of the solution may or may not reach
zero [31] for the grey one.
For a physical system modelled by the MNLS equation,
each solution pre se nted above gives a particular phase
state. So, it is rather interesting to observe in Figu re 1
that there exists the state transition induced by tuning
the selfsteepening and SPM, which c an be realised by
adjusting the values of a and b. For example, by setting
a =
1
2
and varying b, the system will evolve consecu
tively through a paired brightbright, a single, a paired
brightgrey, a paired brightdark soliton states, an d the
rogue wave st at e as b de cr ease s f rom a value larger than
1
2
to one that is smaller than
13
32
. Moreover, for a given
b > b
c
(=
2
3
), the system passes through the ﬁrst three
(i.e. IIII) states as a increases from zero to a suﬃciently
large value; for b = b
c
, the system now pass es through
the ﬁrst four (i.e. IIV) states, and it passes th r ough the
paired brightgrey soliton state twice; ﬁnally, for b < b
c
,
the system passes through all ﬁve states and will be in
paired brightgrey and paired brightdark soliton states
twice. This newly discovered unique stat e transition phe
nomenon is not desc ri bed by the rational solutions of th e
NLS and DNLS equations becaus e they do not describe
two diﬀerent nonlin ear eﬀects simultaneously. Thus, we
have now solved the longstanding problem mentioned in
the Introduction.
Particular cases. In what follows we use two me t hods
to visualise f un ct i on q
[1]

2
. The ﬁrst method consists of
plotting the proﬁle of q
[1]

2
which is the graph of t h i s
function of two variables i n three dimensions. The sec
ond method is similar to drawing the level lines of this
function, however, not using the lines but various colours
instead. The ﬁgure obtained th is way is called the den
sity plot of the function q
[1]

2
. In this plot each colour
corresponds to a d eﬁ n it e value of q
[1]

2
.
To illustrate the general results we investigate the evo
lution of the rational solution for a ﬁxed a =
1
2
and vary
ing b. In Figure 2 the proﬁle (left panel) and density plot
(right panel) of q
[1]

2
are shown for a paired brightbright
soliton with b = 1. In Figure 3 the energy exchange (left
panel) for two peaks along the two trajectories (right
panel) is plotted. The limit heights in Figures 2 and 3 are
H
2
= 21 + 4
√
5 ≈ 29.9 and H
1
= 21 − 4
√
5 ≈ 12.1. Note
that the height of the background in Figure 2 is equal
to 1. On the right panel of Figure 3, the explicit equa
tions of trajectories X
1
(red line) and X
2
(green line) are
−t+
1
2
√
5t
2
+ 4 = x and −t−
1
2
√
5t
2
+ 4 = x, r e spectively.
The distance between the two peaks at a given time is
√
5t
2
+ 4 unlike the case of the two p eak s in a double
soliton solution. It is straightforward to see in Figure 3
(left panel) that the energy is transmitted gradually from
a bright soliton (green line) moving along X
2
to the other
bright soliton (red line) moving along X
1
. The reﬂective
symmetry of Figure 3 (left panel) refers to the energy
conservation . By comparing Figu re 2 (right panel) w it h
Figure 3 (right panel), we can now see that the trajectory
gives a very good approximation of the ridge location in
the proﬁle of q
[1]

2
. Setting a = b =
1
2
, so that ( a, b) is
3
in re gi on II, we obtain q
[1]

2
=
36 + ( t + 2x)
2
4 + ( t + 2x)
2
, which is
a sin gl e soliton with the height 9 and the exact trajec
tory t + 2x = 0. We do not show the proﬁle i n this case
because it is a standard solit on.
In region III, by setting b =
9
20
, we obtain a paired
brightgrey soliton. Its proﬁle is plotted in Figure 4 (left
panel). The l i mi t height of the bright soliton is H
2
=
8−
√
15
20−5
√
15
≈ 6.5, and the limit height of the grey soliton is
H
1
=
8+
√
15
20+5
√
15
≈ 0.3. The X
1
(red line) and X
2
(green
line) in Figu r e 4 (right panel) give good approximation
of the trajectories for t > 8.
Let us set a =
1
2
and b = a−
3
8
a
2
≈ 0.41, the point (a, b)
is now in region IV, and the rational solution describes a
brightdark soliton. The proﬁle of this sol u t ion s is shown
in Figure 5 (left p ane l ) . The limit heights are H
2
= 4
for the bright soliton and H
1
= 0 for the dark one. The
trajectories X
1
and X
2
are shown on the left panel of
Figure 5 by the red and green lines, resp ect i vely. The
analysis of the two solutions corres ponding to the bright
grey and b ri ghtdark solitons shows that: (i) There is
energy exchange between the bright and grey (dark) soli
tons; (ii ) the bright soliton is transformed into a grey
(dark) one because its energy is lost during t he interac
tion, while the grey (dark) soliton is transformed into a
bright on e for t ≫ 1.
Finally, by setting a =
1
2
and b =
1
3
, we put the point
(a, b) in region V. In this case the rational solution de
scribes the ﬁrstorder rogue wave, which is plotted in
Figure 6.
Conclusions. We have presented a range of new types
of rational solution of the MNLS equati on that describe
the propagation of e.g. Alfv´en waves in magnetised plas
mas and the femtosecond light pulses in optical ﬁbres.
The obt ai ne d solutions have two free paramet e rs , a and
b, representing the contributions of selfsteepening and
self phasemodulation, and, depending on the values of
these parameters, these solutions describ e s ﬁve types of
novel solitons corresponding to ﬁve st at es of an asso
ciated physical system. These solutions are: A paired
brightbright, single, paired brightgrey, and a paired
brightdark soliton, and a rogue wave. We have found
that the state transition among these ﬁve states is in
duced by tuning the eﬀects of selfsteepening and SPM.
We urge that this novel phenomenon may be obs er ved in
laboratory or in magnetised p l asma in nature i n order to
demonstrate an intricate bal anc e between the eﬀects of
selfsteepening and SPM in an associ at ed physical sys
tem. Furthermore, because of the recent discovery of
Alfv´en waves (see [32, 33] and references therein) in the
magnetised solar atmosphere, it is now paramount inter
est to ﬁnd these novel s t at es and the state trans fe r in
space plasmas, and , to establish their connection with
the l ong st an di n g coronal heating problem [34–37].
Finally, we discuss brieﬂy th e novelty of the s olu t i on
describing the paired brightbright soliton. Three of its
characteristics, i.e. the existence of two peaks, decreas
ing or increasing amplit ud e, and the curved trajectories,
are essentially diﬀerent from the similar characteristics
of the recently found solition solutions that include an
explodedecay soli t on [38], a twopeak soliton [39, 40],
a Wshape soliton [41], a darkinbright soliton [42, 43],
a rogue wave [44] and a twopeak rogue wave [45, 46],
in addition to the wellknown classical soliton, breather,
and kink. In particular, the paired brightbright soliton
is not a travellin g wave. The nonautonomous solitons
[47–50] have the p r operties similar to those of the paired
brightbright soliton but only when they are solutions to
the variable coeﬃc i ent soliton equations. This discus si on
can also be applied to the paired br i ghtgrey and bright
dark s oli t on soluti on s.
Acknowledgements This work is supported by the
NSF of China Grant No.10971109 and 11271210, and
K.C. Wong Magna Fund in Nin gbo University. J. He
is also s up ported by the Natural Science Foundation of
Ningbo, Grant No. 2011A610179. R.E. acknowledges M.
K´eray for patient encou rage ment and is also grateful to
NSF, Hungary (OTKA, Ref. No. K83133) for the sup
port received. J. He thank since re l y Prof. A.S. Fokas for
arranging the visit to Cambridge University in 20122013
and for many useful discussions .
[1] K. Mio, T. Ogino, K. Minami and S. Ta keda, J. Phys.
Soc. Jpn. 41(1976), 265271.
[2] M. V. Medvedev, P. H. Diamond, V. I Shevchenko and
V. L. Galinsky Phys. Rev. Lett. 78(1997), 49344937.
[3] D. Laveder, T. Passot, P.L. Sulem, Phys. Lett.
A377(2013),15351441.
[4] N. Tzoar an d M Jain, Phys. Rev. A 23(1981),1266127 0 .
[5] D. Anderson and M . Lisak, Phys.Rev. A 27 ( 1 9 8 3 ), 1 3 9 3 
1398
[6] A. A. Zabolotskii,Phys. Lett. A. 124(1987), 500502.
[7] M. Wadati, K. Konno a n d Y.H. Ichikawa, J. Phys. Soc.
Jpn. 46(1979), 19651966.
[8] A. Kundu, J. Math. Phys. 25(1984),34333438.
[9] K. Porsezian, P. Seenuvasakumaran and K. Sara
vanan,Chao s, Solitons and Fractals. 11(2000), 22232231.
[10] Q. Ding and Z.N. Zhu,Phys. Lett. A. 295 (2 0 0 2 ) , 192197.
[11] A.Kundu, S ymm et ry, Integrability and Geometry: Meth
ods and Applications. 2(2006), 078(12pp).
[12] T.Kawata, J.I.Sakai and N.Ko b ayashi, J.Phys. Soc. Jp n .
48(1980), 13711379.
[13] A. R. Chowdhury, S. Paul and S. Sen, Phys. Rev. D.
32(1985), 32333237.
[14] D.Mihalache, N.Truta, N.C.Panoiu and D.M..Baboi u ,
Phys. Rev. A. 47(1993), 31903194.
[15] E. V. Doktorov, Eur. P hys. J. B. 29(2002), 227231.
[16] A.A.Rangwala and J.A. Rao, J. Math. Phys. 31(1990),
11261132.
[17] O.C. Wrigh, Chaos, Solitons and Fractals. 20(2004), 735
749.
[18] T.C. Xia, X.H.Chen and D.Y.Ch en , Chaos, Solitons and
Fractals . 26( 2 0 0 5 ), 889 8 9 6 .
[19] H.Q.Zhang , B.G.Zhai a n d X.L. Wang, Phys. Scr.
85(2012), 015007(8pp).
[20] S.L. Liu and W.Z.Wang, Phys. Rev. E. 48(1993), 3054
3059.
[21] M.Li, B. Tian, W.J.Liu, H.Q.Zhang and P.Wang, Phys.
Rev. E. 81(2010), 046606(8pp).
[22] E.A.Golovchenko,E.M.Di a n ov,A.M.Prokhorov and
V.N.Serkin, JETP Lett.42(1985), 8791.
[23] X. J. Chen and J. K. Yan,Phys. Rev. E.65(2002 ) , 06 6 6 08
(12pp).
[24] V.S. Shchesnovich and E.V. Doktorov, Physica. D. 129(
1999), 115129.
[25] V. M. Lashkin, Phys. Rev. E. 6 9( 2 0 0 4 ), 01 6 6 1 1 (1 1p p ) .
[26] J.S.He, H.R. Zh a n g , L.H. Wang, K. Porsezian and
A.S.Fokas, A generating mechanism for higher order
4
rogue waves(arXiv:1209.3742v4).
[27] S.W. Xu, J.S.He and L.H.Wang, J. Phys. A: Math.
Theor. 44(2011), 305203 (22pp).
[28] S.W. Xu and J.S.He, J. Math. Phys. 53(2012), 063507
(17pp)
[29] Y.S.Tao and J.S.He, Phys.Rev.E 85(2012), 026601(7pp).
[30] J.S.He, S.W.Xu and K.Porsezian, Phys. Rev.E 86
(2012),066603(17pp).
[31] M.J. Ablowitz,2011, Nonlinear Dispersive Waves:
Asymptotic Analysis and Solitons ( Cambridge Univer
sity Press, Cambridge )p.153.
[32] H. Alfv´en, Nature 150(1942), 405406.
[33] D. B. Jess, M. Mathioudakis, R. Erd´elyi, P. J.
Crockett, F. P. Keenan and D. J. Christian, Science
323(2009),15821585.
[34] C. E. Parnell and I. De Moortel, Phil. Trans. R. Soc. A
370(2012),32173240
[35] M. Mathiou d a ki s, D. B. Jess and R. Erd´elyi, Alfv´en
Waves in the Solar Atmosphere: From Theory to Ob
servations (arXiv:12 1 0 .3 6 2 5 ) 2 9 p p .
[36] S. WedemeyerB¨ohm, E. Scullion, O. Stei n er, L. R. van
der Voort, J. de la Cruz Rodrigu e z, V. Fedun and R.
Erd´elyi, Nature 528(2012),505508.
[37] R. Morton, G. Verth, D.B. Jess, D. Kuridze, M.S. Rud
erman, M. Mathioudakis, and R. Erd´elyi, Nature Comm,
3(2012), i.d.1315.
[38] A. Nakamura, J. Phys. Soc. Jpn. 50 ( 1 9 81 ) , 2469
2470;ibid., 51(1982),1920.
[39] N.Sasa and J.Satsuma, J. Phys. S oc . Jp n . 6 0 (1 9 9 1 ) , 409
417.
[40] Y.S.Li and W.T.Han, Chin.Ann.Math . 2 2B ( 2 00 1 ), 1 7 1 
176.
[41] Z.H.Li, L.Li, H.P.Tian and G.S.Zhou, Phys. Rev. Lett.84
(2000), 40964099.
[42] P. G. Kevrekidis, D. J. Frantzeskakis, B. A. Malomed,A.
R. Bishop and I. G. Kevrekidis, New.J.Phys.5(2003),64.
[43] A. Choudhuri and K. Porsezian, Optics Communications
285(2012), 364367.
[44] D. H. Peregrine, J. Aust. Math. Soc. Ser. B: Appl. Math.
25(1983), 1643.
[45] J.S.He, S.W.Xu and K.Porsezian, J. Phys. S oc. Jpn.
81(2012), 033002.
[46] U.Bandelow and N.Akhmediev, Phys. Rev. E 86 (2012),
026606.
[47] V.N. Serkin, A. H a seg awa and T.L. Belyaeva, Phys. Rev.
Lett.98 (2007), 074102.
[48] J. BelmonteBeitia, V. M. PerezGarcia,V. Vekslerchik
and P. J. Torres, Phys. Rev. Lett. 98 (2007),064102 .
[49] J. BelmonteBeitia, V. M. PerezGarcia, V. Vekslerchik
and V. V. Konotop, Phys. Rev. Lett.100(2008),164102.
[50] J.S. He a n d Y.S. Li, Stud. in Appl. Math.126(2011),115.
Fig. 1: (Colour online) Five regions in the upperright
quadrant on the (a, b)plane. In each region, q
[1]
gives a
new ki nd of solution for the MNLS equation. The
straight green l i ne is a = b, the red curve is deﬁned by
b = a −
3
8
a
2
for a ≤
8
3
, b = 0 for a ≥
8
3
.
Fig. 2: (Colour online) The proﬁle (left) and density
plot ( r i ght) for a paired brightbright soliton.
Fig. 3: (Colour online) Left: The energy transmission
from a bright soliton (green line) to another one (red
line). Ri ght: The traject ori e s of the solitons, X
2
(green
line) and X
1
(red line).
Fig. 4: (Colour online) The paired brightgrey soliton
(left) and its trajectories (right).
Fig. 5: (Colour online) The paired brightblack soliton
(left) and its trajectories (right).
Fig. 6: (Colour online) The ﬁrstorder rogue wave (left)
and it s densi ty plot (right).