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State Transition Induced by Self-Steepening and Self Phase-Modulation

01 Jan 2014-Chinese Physics Letters (IOP Publishing)-Vol. 31, Iss: 1, pp 010502

Abstract: We present a rational solution for a mixed nonlinear Schrodinger (MNLS) equation. This solution has two free parameters, a and b, representing the contributions of self-steepening and self phase-modulation (SPM) of an associated physical system, respectively. It describes five soliton states: a paired bright-bright soliton, a single soliton, a paired bright-grey soliton, a paired bright-black soliton, and a rogue wave state. We show that the transition among these five states is induced by self-steepening 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 MNLS equation.
Topics: Soliton (61%), Rogue wave (54%), Self-phase modulation (51%)

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Article:
Jing-Song, H., Shu-Wei, X., Ruderman, M.S. et al. (1 more author) (2014) State Transition
Induced by Self-Steepening and Self Phase-Modulation. Chinese Physics Letters, 31 (1).
ARTN 010502. ISSN 0256-307X
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State transition induced by self-steepening and self phase-modulation
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 self-steepenin g and self phase-
modulation (SPM) of an associated physical system. It describes five soliton states: a paired
bright-bright soliton, single soliton, a paired bright-grey soliton , a paired bright-black soliton, and a
rogue wave state. We show that the transition among these five states is induced by self-steepening
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, ultra-short
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, solar-terrestrial, space and astrophysics[2, 3].
Later it has been shown [46] that the MNLS equation
also provides an accurate mo d el l i ng of ultra-short 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 non-negative
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 phase-modul 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-
Konno-Ichikawa (WKI) spectral problem and the first
non-trivial 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 [811]. 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 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 long-standing
problem to find unique phe nom en a related to the simulta-
neous effects of self-st 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 bright-bright soliton
state, a single soliton state, a paired bright-grey soliton
state, a paired bright-dark soliton state, and a rogue wave
state. We also show that the transition between these five
states is induced by the self-steepening 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 Taylor-expansion, according to a similar
procedure demonstrated by [2630] for constructing the
rational rogue wave (RW) of the NLS, DNLS, Hirota and
NLS-Maxwell-Bloch 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 non-zero 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
upper-right 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 bright-bright
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 bright-grey 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 bright-dark 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 self-steepening 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 bright-bright, a single, a paired
bright-grey, a paired bright-dark 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. I-III) 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. I-IV) states, and it passes th r ough the
paired bright-grey soliton state twice; finally, for b < b
c
,
the system passes through all five states and will be in
paired bright-grey and paired bright-dark 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 long-standing 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 bright-bright
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
bright-grey 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
205
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
bright-dark 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 ght-dark 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 first-order 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 self-steepening and
self phase-modulation, 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
bright-bright, single, paired bright-grey, and a paired
bright-dark soliton, and a rogue wave. We have found
that the state transition among these five states is in-
duced by tuning the effects of self-steepening 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
self-steepening 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 [3437].
Finally, we discuss briefly th e novelty of the s olu t i on
describing the paired bright-bright 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
explode-decay soli t on [38], a two-peak soliton [39, 40],
a W-shape soliton [41], a dark-in-bright soliton [42, 43],
a rogue wave [44] and a two-peak rogue wave [45, 46],
in addition to the well-known classical soliton, breather,
and kink. In particular, the paired bright-bright soliton
is not a travellin g wave. The non-autonomous solitons
[4750] have the p r operties similar to those of the paired
bright-bright 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 ght-grey 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 2012-2013
and for many useful discussions .
[1] K. Mio, T. Ogino, K. Minami and S. Ta keda, J. Phys.
Soc. Jpn. 41(1976), 265-271.
[2] M. V. Medvedev, P. H. Diamond, V. I Shevchenko and
V. L. Galinsky Phys. Rev. Lett. 78(1997), 4934-4937.
[3] D. Laveder, T. Passot, P.L. Sulem, Phys. Lett.
A377(2013),1535-1441.
[4] N. Tzoar an d M Jain, Phys. Rev. A 23(1981),1266-127 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), 500-502.
[7] M. Wadati, K. Konno a n d Y.H. Ichikawa, J. Phys. Soc.
Jpn. 46(1979), 1965-1966.
[8] A. Kundu, J. Math. Phys. 25(1984),3433-3438.
[9] K. Porsezian, P. Seenuvasakumaran and K. Sara-
vanan,Chao s, Solitons and Fractals. 11(2000), 2223-2231.
[10] Q. Ding and Z.N. Zhu,Phys. Lett. A. 295 (2 0 0 2 ) , 192-197.
[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), 1371-1379.
[13] A. R. Chowdhury, S. Paul and S. Sen, Phys. Rev. D.
32(1985), 3233-3237.
[14] D.Mihalache, N.Truta, N.C.Panoiu and D.M..Baboi u ,
Phys. Rev. A. 47(1993), 3190-3194.
[15] E. V. Doktorov, Eur. P hys. J. B. 29(2002), 227-231.
[16] A.A.Rangwala and J.A. Rao, J. Math. Phys. 31(1990),
1126-1132.
[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), 87-91.
[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), 115-129.
[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), 405-406.
[33] D. B. Jess, M. Mathioudakis, R. Erd´elyi, P. J.
Crockett, F. P. Keenan and D. J. Christian, Science
323(2009),1582-1585.
[34] C. E. Parnell and I. De Moortel, Phil. Trans. R. Soc. A
370(2012),3217-3240
[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. Wedemeyer-B¨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),505-508.
[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),19-20.
[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), 4096-4099.
[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), 364-367.
[44] D. H. Peregrine, J. Aust. Math. Soc. Ser. B: Appl. Math.
25(1983), 16-43.
[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. Belmonte-Beitia, V. M. Perez-Garcia,V. Vekslerchik
and P. J. Torres, Phys. Rev. Lett. 98 (2007),064102 .
[49] J. Belmonte-Beitia, V. M. Perez-Garcia, 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),1-15.
Fig. 1: (Colour online) Five regions in the upper-right
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 bright-bright 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 bright-grey soliton
(left) and its trajectories (right).
Fig. 5: (Colour online) The paired bright-black soliton
(left) and its trajectories (right).
Fig. 6: (Colour online) The first-order rogue wave (left)
and it s densi ty plot (right).
Citations
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Abstract: We construct an analytical and explicit representation of the Darboux transformation (DT) for the KunduEckhaus (KE) equation. Such solution and n-fold DT Tn are given in terms of determinants whose...

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Abstract: Recently, Anjan Kundu, Abhik Mukherjee and Tapan Naskar (KMN) have introduced a (2+1)-dimensional equation as a new extension of the well-known nonlinear Schrodinger (NLS) equation, which is called KMN equation in this paper. We provide a triplet Lax pair of the KMN equation. Basing on triplet Lax pair, we present the first-order rogue wave (RW) solution of the KMN equation by the one-fold Darboux transformation from a non-zero “seed” solution. This RW solution also satisfies the complex modified Korteweg–de Vries (mKdV) and the NLS equation by two different transformations of variables. Moreover, we discuss the localization of rogue wave of KMN equation, observing that the area of rogue wave is a constant which is just related to the amplitude of seed solution.

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Abstract: We consider a next-higher-order extension of the Chen–Lee–Liu equation, i.e., a higher-order Chen–Lee–Liu (HOCLL) equation with third-order dispersion and quintic nonlinearity terms. We construct the n-fold Darboux transformation (DT) of the HOCLL equation in terms of the n × n determinants. Comparing this with the nonlinear Schrodinger equation, the determinant representation Tn of this equation is involved with the complicated integrals, although we eliminate these integrals in the final form of the DT, so that the DT of the HOCLL equation is unusual. We provide explicit expressions of multi-rogue wave (RW) solutions for the HOCLL equation. It is concluded that the rogue wave solutions are likely to be crucial when considering higher-order nonlinear effects.

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Abstract: We study the AB system describing marginally unstable baroclinic wave packets in geophysical fluids and also ultrashort pulses in nonlinear optics. We show that the breather can be converted into different types of stationary nonlinear waves on constant backgrounds, including the multi-peak soliton, M-shaped soliton, W-shaped soliton and periodic wave. We also investigate the nonlinear interactions between these waves, which display some novel patterns due to the nonpropagating characteristics of the solitons: (1) Two antidark solitons can produce a W-shaped soliton instead of a higher-order antidark one; (2) the interaction between an antidark soliton and a W-shaped soliton can not only generate a higher-order antidark soliton, but also form a W-shaped soliton pair; and (3) the interactions between an oscillation W-shaped soliton and an oscillation M-shaped soliton show the multi-peak structures. We find that the transition occurs at a modulational stability region in a low perturbation frequency region.

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References
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Journal ArticleDOI
Abstract: Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

1,088 citations


Journal ArticleDOI
01 Oct 1942-Nature
Abstract: IF a conducting liquid is placed in a constant magnetic field, every motion of the liquid gives rise to an E. M. F. which produces electric currents. Owing to the magnetic field, these currents give mechanical forces which change the state of motion of the liquid. Thus a kind of combined electromagnetic-hydro-dynamic wave is produced which, so far as I know, has as yet attracted no attention.

879 citations


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TL;DR: Novel soliton solutions for the nonautonomous nonlinear Schrödinger equation models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonaut autonomous solitONS that interact elastically and generally move with varying amplitudes, speeds, and spectra.
Abstract: Novel soliton solutions for the nonautonomous nonlinear Schr\"odinger equation models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds, and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. The nonautonomous soliton concept can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter waves, and offer many opportunities for further scientific studies.

524 citations


Journal ArticleDOI
20 Mar 2009-Science
TL;DR: The detection of oscillatory phenomena associated with a large bright-point group that is 430,000 square kilometers in area and located near the solar disk center suggests that these torsional Alfvén oscillations are induced globally throughout the entire brightening.
Abstract: The flow of energy through the solar atmosphere and the heating of the Sun's outer regions are still not understood. Here, we report the detection of oscillatory phenomena associated with a large bright-point group that is 430,000 square kilometers in area and located near the solar disk center. Wavelet analysis reveals full-width half-maximum oscillations with periodicities ranging from 126 to 700 seconds originating above the bright point and significance levels exceeding 99%. These oscillations, 2.6 kilometers per second in amplitude, are coupled with chromospheric line-of-sight Doppler velocities with an average blue shift of 23 kilometers per second. A lack of cospatial intensity oscillations and transversal displacements rules out the presence of magneto-acoustic wave modes. The oscillations are a signature of Alfven waves produced by a torsional twist of ±22 degrees. A phase shift of 180 degrees across the diameter of the bright point suggests that these torsional Alfven oscillations are induced globally throughout the entire brightening. The energy flux associated with this wave mode is sufficient to heat the solar corona.

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Journal ArticleDOI
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Abstract: For nonlinear short-pulse propagation in long optical fibers, the conventional static approximation of the nonlinear terms in the wave equation must be extended to include the derivative of the pulse envelope. As a result, an initially symmetric pulse will develop an asymmetric self-phase modulation and a self-steepening, which ultimately lead to shock formation unless balanced by dispersion. This effect may be responsible for the pulse asymmetries observed in recent experiments.

357 citations


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