Breather-to-soliton and rogue wave-to-soliton transitions in a resonant erbium-doped fiber system with higher-order effects

2.2 MB·

by  Lei Wang

Noname manuscript No.
(will be inserted by the editor)
Breather-to-soliton and rogue wave-to-soliton transitions
in a resonant erbium-doped fiber system with higher-order
effects
Lei Wang · Shen Li · Feng-Hua Qi
6
1 Received:date/Accepted:date
0
2
Abstract Underinvestigationinthispaperisthehigher- (NLS) equation [2,3], the mechanism of which is based
n
a ordernonlinearSchr¨odingerandMaxwell-Bloch(HNLS- onthebalancebetweenthesecond-orderdispersionand
J MB)systemwhichdescribesthewavepropagationinan Kerr nonlinearity. This type of soliton is called the

0 erbium-doped nonlinear fiber with higher-order effects NLS soliton. The other type of lossless pulse propa-
3
including the fourth-order dispersion and quintic non- gation is the self-induced transparency (SIT) soliton in

Kerr nonlinearity. The breather and rogue wave (RW) the two-level resonance medium, the dynamic response
]
S solutions are shown that they can be converted into of which is governed by the Maxwell-Bloch (MB) sys-
P
various soliton solutions including the multipeak soli- tem [4]. When erbium is doped with the core of the
.
n ton, periodic wave, antidark soliton, M-shaped soliton, optical fibers, the NLS soliton can exist with the SIT
li and W-shaped soliton. In addition, under different val- soliton. The propagation of such pluse can be mod-
n
ues of higher-order effect, the locus of the eigenvalues eledbytheNLS-MBsystem[5,6,7,8,9].However,inan
[
onthe complexplanewhich convertsbreathers orRWs optical-fiber transmission system, one always needs to

1 into solitons is calculated. increase the intensity of the incident light field to pro-
v
duce ultrashort optical pulses [10,11,12,13,14]. Thus,
7 PACS 42.65.Tg, 42.65.Sf, 05.45.Yv, 02.30.Ik
7 suchhigher-ordereffectsasthehigher-orderdispersion,
0 Keywords higher-order nonlinear Schr¨odinger and higher-ordernonlinearity,self-steepening,andself-frequency
0 Maxwell-Bloch system · breather-soliton dynamics · shift, should be included [10,11,12,13,14].
0
rogue wave-soliton dynamics · asymmetric rogue waves Inthispaper,wewillworkonthehigher-orderNLS-
.
2 MB system as follows [15,16,17,18,19],
0
6
1 1 Introduction Ez =i(Ett+2|E|2E)+iτ(Etttt+8|E|2Ett
v: +2E2Et∗t+6E∗Et2+4|Et|2E
Xi Optical solitons have attracted considerable interest +6|E|4E)+2p, (1.1)
ofscientistsaroundtheworldduringthepastdecades[1].
r p =2iωp+2Eη,
a In optical fibers, there are two different types of soli- t
tons. One is described by the nonlinear Schr¨odinger ηt =−(Ep∗+E∗p).
LeiWang∗ where the subscripts z, t are partial derivatives with
Department of Mathematics and Physics, North China Electric
respect to the distance and time, the asterisk symbol
PowerUniversity,Beijing102206,P.R.China
∗Correspondingauthor;E-mail:[email protected] stands for the complex conjugate. E denotes the nor-
malized slowly varying amplitude of the complex field
ShenLi
envelope, p = v v∗ represents as the polarization, and
School of Electrical and Electronic Engineering, North China 1 2
ElectricPowerUniversity,Beijing102206,P.R.China η = |v1|2 −|v2|2 means the population inversion with
v and v being the wave functions of the two energy
Feng-HuaQi 1 2
School of Information, Beijing Wuzi University, Beijing 101149, levels of the resonant atoms, ω is the frequency, τ is a
P.R.China small dimensionless real parameter.
2
Various solutions of System (1.1) have been dis- 16,17,18,19] of System (1.1), the first-order symmetric
cussed. For example, based on the Lax pair and Dar- breather solution reads as
boux transformation (DT), the multi-soliton solutions
of System (1.1) have been presented in Ref. [15]. The G[1]+iH[1]
E[1] =d(1+8n B B )eiρ,
breather,roguewave(RW)andhybridsolutionsofSys- B D[1]
B
tem (1.1) have been obtained by means of the general-
1
izedDTinRefs.[16,17,18,19].Solitons,breathers,and p[B1] =2((EB[1])z−i((EB[1])tt+2|EB[1]|2EB[1])−iτ((EB[1])tttt
RWs are different types of nonlinear localized waves
+8|E[1]|2(E[1]) +2(E[1])2(E[1])∗ +6(E[1])∗
andarecentralobjectsindiversenonlinearphysicalsys- B B tt B B tt B
tems[20,21,22,23,24,25,26,27,28,29,30].Thehybridso- (E[1])2+4|(E[1]) |2E[1]+6|E[1]|4E[1])),
B t B t B B B
lutions,whichareexpressedinformsofthemixedrational-
−2iω(p[1])[1]+(p[1])[1]
exponential functions, describe the nonlinear superpo- η[1] = B B t .
sition of the RW and breather [31]. Recently, Liu et al. B 2EB[1]
haveshownthatthebreatherscanbeconvertedintodif- (2.1)
ferenttypesofnonlinearwavesinthecoupledNLS-MB
where
system, including the multipeak soliton, periodic wave,
antidarksoliton,andW-shapedsoliton[32].Thesecon- G[1] =2k k n cos(th +z(cid:36) )+2n cosh(th +z(cid:36) )
B 1 2 R B I A
versions occur under a special condition in which the
−k k h sin(th +z(cid:36) )+h sinh(th +z(cid:36) ),
soliton and a periodic wave in the breather have the 1 2 R R B I I A
same velocity [32]. In addition, the similar transitions HB[1] =k1k2(b+2m) cos(thR+z(cid:36)B)+(b+2m) cosh(thI
have also been reported in some higher-order nonlinear +z(cid:36) )+k k h sin(th +z(cid:36) )+h sinh(th
A 1 2 I R B R I
equation of evolutions. Akhmediev et al. have found
+z(cid:36) ),
A
that the breather solutions of the Hirota [33] equation
andfifth-orderNLS[34]equationcanbeconvertedinto DB[1] =k1k2(|h|2−χ) cos(thR+z(cid:36)B)−(|h|2+χ) cosh(thI
soliton solutions on the constant background, which +z(cid:36) )−k k Γ sin(th +z(cid:36) )−Γ sinh(th
A 1 2 A R B B I
does not exist in the standard NLS equation. Liu et
+z(cid:36) ),
A
al. have revealed that the transition between the soli-
with
tonandRWoccursasaresultoftheattenuationofMI
growth rate to vanishing in the zerofrequency pertur- ρ= az+bt, λ=m+in,
bation region [35,36]. 4
a=(b4−12b2d2+6d4)τ −(b2−2d2)+ ,
It is natural to ask: Can the transition between 2ω−b
the breather (or RW) and soliton occur in the higher- χ=4m2+4n2+4mb+b2+4d2 ,
ordercoupledsystem,e.g.,System(1.1)?Inthispaper,
Γ =2bh +4mh −4nh ,
A I I R
we show that System (1.1) does have such transition
Γ =2bh +4mh +4nh ,
when the eigenvalues are located at the special locus B R R I
onthecomplexplane.Further,wedemonstratethatthe (cid:36)A =2(κRhI +κIhR),
breather can be converted into the multipeak soliton, (cid:36) =2(κ h −κ h ),
B R R I I
periodic wave, antidark soliton, and M-shaped soliton 1
while the RW can be transformed into the W-shaped κ= (−b+2λ+(b3−6bd2+(4d2−2b2)λ+4bλ2
2
soliton. 1
−8λ3)τ)− =κ +iκ ,
The outline of this paper will be as follows: The (b−2ω)(λ+ω) R I
breather-to-solitonconversionandseveraltypesoftrans-
(cid:114)
formednonlinearwaveswillbestudiedinSection2.The b
h=2 d2+(λ+ )2 =h +ih .
RW-to-solitonconversionwillbediscussedinSection3. 2 R I
Finally,ourconclusionswillbeaddressedinSection4.
Solution(2.1)includesthehyperbolicfunctionssinh
(th +z(cid:36) ) ( or cosh(th +z(cid:36) )) and the trigono-
I A I A
2 Breather-to-soliton conversions metricfunctionssin(thR+z(cid:36)B)(orcos(thR+z(cid:36)B)),
where κR+ κIhhIR and κR− κhIRhI are the corresponding
In this section, we present different types of non- velocities. In this case, the hyperbolic functions and
linear wave solutions on constant background for Sys- trigonometric functions, respectively, characterize the
tem (1.1). We omit the discussions of the components localization and the periodicity of the transverse dis-
p and η because the types of them have the similar tribution t of those waves. The nonlinear structure de-
characteristicsasE.BasedonthelaxpairandDT[15, scribed by Solution (2.1) can be seen as a nonlinear
3
combination of a soliton with the velocity κR + κIhhIR main peaks, which is referred to as the M-shaped soli-
awnedwailpledriiospdliacywvaavreiowuisthntohnelinveealorciwtyavκeRs−truκhcIRhtuIr.eNsedxet-, tFoingsi.n3∼th5isapreaptehre. lNoocateliztehdatsttrhuectmuureltsi-aploeankgstohleittonasxiisn.
pending on the values of velocity difference, namely, However,changingthevaluesoftherealandimaginary
κI(h2R+h2I). parts of eigenvalue, we display another type of solu-
hRhI tion that shows a type of nonlocal structure, i.e., the
If the velocity difference is not equal to zero, i.e.,
κI(hh2RR+hhI2I)(cid:54)=0(orkI (cid:54)=0),Solution(2.1)characterizes pthearitodthicewstarvuect(useree F(piega.k6n).umThbeerrefaonrde,liotciasliczoanticolnu)deodf
thelocalizedwaveswithbreathingbehavioronaplane-
themulti-peaksolitoncanbecontrolledbytherealand
wave background (i.e., the breathers and RWs). Fur-
imaginary parts of eigenvalue, namely, m and n.
ther,ifm=−b,wehavetheAkhmedievbreatherswith
2 Inordertobetterunderstandthismulti-peakstruc-
|n|<|d|,theKuanetsov-Masolitonswith|n|>|d|,and
tureofSystem(1.1),wewillextractseparatelythesoli-
thePeregrinesolitonwith|n|=|d|.Suchsolutionshave
ton and periodic wave from the mixed solution (2.1).
been derived in Refs. [16,17,18,19] (also see Fig. 1).
Specifically,thesolitonexistsinisolationwhenh van-
Instead, if κ = 0, the wave described by Solu- R
I
ishes,whiletheperiodicwaveindependentlyexistswhen
tion(2.1)iscomposedofasolitonandaperiodicwave,
h vanishes. Correspondingly, the analytic expressions
whereeachhasthesamevelocityκ .Itshouldbenoted I
R
read, for the antidark soliton,
that the case κ =0 is equivalent to
I
(cid:36)hIA = (cid:36)hRB , (2.2) ES[1] = d(1+8nG[S1]D+S[1i]HS[1])eiρ (2.4)
i.e., with
Θ G[1] =2n cos(2zκ h )+2n cosh(th +2zκ h )
τ = 1 , (2.3) S I I I R I
Θ2 +hI sinh(thI +2zκRhI),
where H[1] = (b+2m) cos(2zκ h )+(b+2m) cosh(th
S I I I
Θ =(b−2ω)(cid:0)(m+ω)2+n2(cid:1)+1, +2zκ h )− h sin(2zκ h ),
1 R I I I I
Θ2 =(b−2ω)(cid:0)(m+ω)2+n2(cid:1)(b2−4bm−2(d2− D[1] = (h2−χ) cos(2zκ h )−(h2+χ) cosh(th
S I I I I I
6m2+2n2)).
+2zκ h )+ (2bh +4mh ) sin(2zκ h )
R I I I I I
Eq.(2.2)[i.e.,Eq.(2.3)]impliestheextremaoftrigo- −4nhI sinh(thI +2zκRhI),
nometric and hyperbolic functions in Solution (2.1) is
and for the periodic wave,
locatedalongthesamestraightlinesinthe(z,t)-plane,
which leads to the transformation of the breather into E[1] = d(1+8nG[P1]+iHP[1])eiρ (2.5)
a continuous soliton. Choosing different values of τ in P D[1]
P
Eq.(2.3),wedisplaythelocusoftherealandimaginary
with
partsoftheeigenvaluesonthecomplexplaneinFig.2.
G[1] =−2n cos(th +2zκ h )+2n cosh(2zκ h )
It is found that the higher-order effect τ can alter the P R R R I R
shapeofthelocus.Bydecreasingthevalueofτ,wefind + h sin(th +2zκ h ),
R R R R
that the quantity of branches of the locus is reduced H[1] =−(b+2m) cos(th +2zκ h )+(b+2m)
from three to two. P R R R
cosh(2zκ h )+h sinh(2zκ h ),
When the five parameters m, n, ω, τ, b and d sat- I R R I R
isfy Eq. (2.3), Expressions (2.1) describe the soliton on D[1] =−(h2 −χ) cos(th +2zκ h )−(h2 +χ)
P R R R R R
constant background, as depicted in Fig. 3. This type
cosh(2zκ h )−4nh sin(th +2zκ h )
I R R R R R
of multi-peak structure is because of the mixture of
−(2bh +4mh ) sinh(2zκ h ).
a soliton and a periodic wave. The similar multi-peak R R I R
structures have been observed analytically in the NLS- Fig. 7 describes a soliton on the constant back-
MB equations [32] and numerically in the AC-driven ground, which propagates along t direction. This kind
damped NLS system [37,38]. Further, to exhibit the of soliton is called the antidark soliton firstly reported
effect of the real part (m) of the eigenvalue on the in the scalar NLS system with the third-order disper-
peak number, we plot Fig. 4 where the soliton has sig- sion [39,40]. Futher, as d is approaching to zero, this
nificantly less peaks than the one in Fig. 3. On the soliton will turn into a standard bright soliton. Fig. 8
other hand, due to the different choices of the value isplottedfortheperiodicwavespropagatingintdirec-
of the eigenvalue, Fig. 5 shows the soliton with two tion with the period P = π .
hR
4
3 RW-to-soliton conversion 800d2z2b6+18d5tb6+3360d6z2b5+96d7zb5+192
d8tzb5−216d6b4−432d8t2b4+1152d10z2b4−4800
In this section, we investigate the transition of first-
d4z2b4−432d7tb4−1560d5zb4−3120d6tzb4+5760
order localized wave |E (t,z)|2 from the RW to the W-
1
shaped traveling wave. Taking λ → −b +id in Solu- d8z2b3+576d9zb3+1152d10tzb3+144d8b2+288d10
2
tion (2.1), we obtain the first-order RW which is asym- t2b2+7200d6z2b2+288d9tb2+1440d7zb2+2880d8
metric with respect to z-axis in Fig. 9. Further, to tzb2−192(9(b−d)2(2dt(dt+1)+1)(b+d)2+8b2d2
convert the asymmetric RW into a soliton, the param-
(2b2+3d2)2z2−12d(2b5+d2b3−3d4b)(2dt+1)z)ω5
eters ω, τ and b have to satisfy Eq. (2.3). Unlike the
b−32(9(b−d)2(5b2+8d2)(2dt(dt+1)+1)(b+d)2
cases in Section 2, such transition is only controlled by
threefreeparametersduetothefixedvaluesofthereal −12(b−d)d(24bd4+(31b3+12)d2+b2(10b3−7))
(m = −b) and imaginary (n = d) parts of the eigen- (2dt+1)z(b+d)+8bd2(2b2+3d2)(24bd4+(31b3+
2
value. In addition, Expression (3.1) has the forms of
24)d2+2b2(5b3−7))z2)ω3b−4(9(b−d)2(b2+4d2)
rational functions. Note that the rational solutions of
(3b2+4d2)(2dt(dt+1)+1)(b+d)2−12(b−d)d(48
the modified NLS equation have been found that they
can describe various soliton states including a paired bd6+8(10b3+9)d4+b2(41b3+16)d2+6b4(b3−3))
bright-brightsoliton,singlesoliton,apairedbright-grey (2dt+1)z(b+d)+8d2(12bd4+(11b3+30)d2+2b2
soliton, a paired bright-black soliton, which is induced (b3−5))(12bd4+(17b3+6)d2+6b2(b3−1))z2)ωb+
by self-steepening and self phase modulation through
64(9(b−d)2(2dt(dt+1)+1)(b+d)2+8b2d2(2b2+3
tuning the values of the corresponding free parame-
ters [41,42]. Hereby, we display that the rational solu- d2)2z2−12d(2b5+d2b3−3d4b)(2dt+1)z)ω6+16(9
tionofSystem(1.1)characterizestheW-shapedsoliton (b−d)2(15b2+8d2)(2dt(dt+1)+1)(b+d)2−12(b−
which is plotted in Fig. 10. Different from the wave in d)d(24bd4+(61b3+6)d2+6b2(5b3−1))(2dt+1)z(b
Figs. 3 and 4, the soliton in Fig. 10 has only one (two),
+d)+8bd2(2b2+3d2)(24bd4+(61b3+12)d2+6b2(5
infinitely stretched, peak (valleys) without oscillatory
tails. It is worth noting that the W-shaped soliton can b3−2))z2)ω4+4(9(b−d)2(15b4+48d2b2+16d4)(2
be also converted by the periodic wave in Fig. 8 when dt(dt+1)+1)(b+d)2−12(b−d)d(48bd6+8(22b3
the period is infinite (hR → 0). In other words, there +3)d4+3b2(47b3+28)d2+6b4(5b3−8))(2dt+1)
are two orders to obtain the W-shaped soliton: one is
z(b+d)+8d2(144b2d8+48b(13b3+3)d6+(775b6+
thebreather→RW→W-shapedsoliton,andtheotheris
the breather→periodic wave→W-shaped soliton. 600b3+36)d4+12b2(31b6+4b3−6)d2+12b4(5b6
(cid:18) (cid:19) −16b3+3))z2)ω2,
Υ
EW[1] =d 1− Υ1 eΥ3 (3.1) Υ3 =i(z(−b2+((b4−12b2d2+6d4)((b−2ω)((b−2ω)2+
2
4d2)+4))/(6(b−d)(b+d)(b−2ω)((b−2ω)2+4d2))
with
−4/(b−2ω)+2d2)+bt).
Υ =4d(−8ω3(4b3dz+6bd3z−3(b−d)(b+d)(dt+1))
1
+12bω2(4b3dz+6bd3z−3(b−d)(b+d)(dt+1))
−3(b5+3b3d2−4bd4)(dt+1)+2bdz((11b3+30)
4 Conclusions
d2+2b2(b3−5)+12bd4)−2ω(2dz((17b3+6)d2+
6b2(b3−1)+12bd4)−3(3b4+b2d2−4d4)(dt+1)))
In this article, we have shown that the first-order
(4b6z−3b5(t+8ωz)+2b4(11d2z+3ω(3t+8ωz)) breathersolutionofSystem(1.1)canbeconvertedinto
−b3(d2(9t+68ωz)+4ω2(9t+8ωz)+20z)+6b2 several types of novel solitons including the W-shaped
soliton (multi-peak), M-shaped soliton, periodic wave
(4d4z+d2ω(t+12ωz)+4ω(tω2+z))+12bd2(d2
andantidarksoliton.Thetransitioncondition(2.3),de-
(t−4ωz)+3tω2−4ω3z+5z)−24d2ω(t(d2+ω2)
pending on the parameters m, n, ω, τ and b, has been
+z)), analytically presented. Additionally, we have exhibited
Υ =32d2z2b12−24dzb11−48d2tzb11+18d2t2b10+352 thatthefirst-orderRWsolutionofSystem(1.1)canbe
2
converted into the W-shaped soliton with single peak
d4z2b10+18dtb10+9b10−320d2z2b9−204d3zb9−
and two valleys when the ω, τ and b satisfy the condi-
408d4tzb9+54d2b8+108d4t2b8+1352d6z2b8+108
tion(2.3).Itwillbeinterestingtostudytheinteractions
d3tb8+120dzb8+240d2tzb8−800d4z2b7−444d5z among different kinds of nonlinear waves. We will give
b7−888d6tzb7+9d4b6+18d6t2b6+2112d8z2b6+ the details in another paper.
5
Acknowledgements 17. Wang,Q.M.,Gao,Y.T.,Su,C.Q.,Zuo,D.W.:Solitons,br-
eathers and rogue waves for a higher-order nonlinear
Weexpressoursincerethankstoallthemembersof Schr¨odingerCMaxwellCBloch system in an erbium-doped
our discussion group for their valuable comments. This fibersystem.Phys.Scr.90,105202(2015).
18. Su,C.Q.,Gao,Y.T.,Xue,L.,Yu,X.:Solitons and Rogue
work has been supported by the National Natural Sci-
Waves for a Higher-Order Nonlinear Schr¨odinger-Maxwell-
ence Foundation of China under Grant No. 11305060, BlochSysteminanErbium-DopedFiber.Z.Naturforsch.A.
theFundamentalResearchFundsoftheCentralUniver- 70,935(2015).
19. Zuo,D.W.,Gao,Y.T.,Feng,Y.J.,Xue,L.:Rogue-wave in-
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Fig.1(a) Fig.1(b) Fig.1(c)
Fig. 1 First-order breather solution of the GNLS-MB system (E,p,η) with d = 1,b = 2,k1 = −k2 = 1,ω = 12,τ = 12 and
λ1=λ∗2 =−1.2+1i
Fig.2(a) Fig.2(b) Fig.2(c)
Fig.2(d) Fig.2(e) Fig.2(f)
Fig. 2 SolutionsofEq.(2.3)onacomplexplaneofλ1 =m+ni.d=1,b=2,ω= 21.(a):τ =6,(b):τ =3.8,(c):τ =3,(d):τ =1,
(e):τ =0.01,(f):τ =−1.
Fig.3(a) Fig.3(b) Fig.3(c)
Fig. 3 A breather transformed into a multi-peak soliton with d=1,b=2,k1 =−k2 =1,ω = 21 and λ1 =λ∗2 =9.6+i. (b) is the
contourplotof(a).(c)isthecross-sectionalviewof(a)atz=0.
8
Fig.4(a) Fig.4(b) Fig.4(c)
Fig. 4 A breather transformed into a multi-peak soliton with d = 1,b = 2,k1 = −k2 = 1,ω = 12 and λ1 = λ∗2 = 2.5+i. (b)is the
contourplotof(a).(c)isthecross-sectionalviewof(a)atz=0.
Fig.5(a) Fig.5(b) Fig.5(c)
Fig. 5 A breather transformed into a M-shape soliton with d=1,b=2,k1 =−k2 =1,ω= 21 and λ1 =λ∗2 =0.9+0.8i. (b) is the
contourplotof(a).(c)isthecross-sectionalviewof(a)atz=0.
Fig.6(a) Fig.6(b) Fig.6(c)
Fig. 6 A breather transformed into a multi-peak soliton (equal-amplitude) with d=1,b=2,k1 =−k2 =1,ω = 21 and λ1 =λ∗2 =
−1−0.1i.(b)isthecontourplotof(a).(c)isthecross-sectionalviewof(a)atz=0.
Fig.7(a) Fig.7(b) Fig.7(c)
Fig. 7 Abreathertransformedintoanantidarksolitonwithd=1,b=2,k1 =k2 =1,ω= 12 andλ1 =λ∗2 =−12b+1.5i.(b)isthe
contourplotof(a).(c)isthecross-sectionalviewof(a)atz=0.
9
Fig.8(a) Fig.8(b) Fig.8(c)
Fig. 8 A breather transformed into period wave with d = 1,b = 2,k1 = −k2 = 1,ω = 12 and λ1 = λ∗2 = −12b+0.8i. (b) is the
contourplotof(a).(c)isthecross-sectionalviewof(a)atz=0.
Fig.9(a) Fig.9(b)
Fig. 9 First-orderasymmetricRWsolutionwithd=1,b=2,k1=−k2=1,ω= 12,τ = 12 andλ1=λ∗2 =−12b+id.
Fig.10(a) Fig.10(b) Fig.10(c)
Fig. 10 AnasymmetricRWtransformedintoaW-shapedsolitonwithd=1,b=0,k1=−k2=1,ω= 12 andλ0=−21b+id.(b)is
thecontourplotof(a).(c)isthecross-sectionalviewof(a)atz=0.