Prolific interaction structures and shape-changing collisions in an erbium-doped fiber system

1.4 MB·

by  Liang Duan

Prolific interaction structures and shape-changing collisions in an erbium-doped fiber
system
Liang Duan1, Zhan-Ying Yang1,∗ Chong Liu1, and Wen-Li Yang2
1School of Physics, Northwest University, Xi’an 710069, China and
2Institute of Modern Physics, Northwest University, Xi’an 710069, China
(Dated: November 27, 2015)
Prolific interactions of nonlinear waves on a plane-wave background in an erbium-doped fiber
system are unveiled, based on explicit coexistence conditions extracting from the general higher-
order solution of a coupled nonlinear Schro¨dinger and the Maxwell-Bloch equations. In contrast
to previous results, it shows that rich different types of nonlinear waves can coexist and interact
with each other. In particular, we reveal an interesting kind of shape-changing collision, in which
the multi-peak solitons exhibit intensity redistribution characteristics. This interaction involves
the mutual collision between multi-peak solitons as well as the interplay of multi-peak soliton and
6 other types of localized nonlinear waves (breather and antidark soliton). We find that the shape-
1 changing collision preserves the power of each multi-peak soliton, which is verified via optimal
0 numerical integration for the energy of light pulse against the plane-wave background. Our results
2 demonstrate that when the interactions occur, each multi-peak soliton exhibits the coexistence of
the shape change and energy conservation.
n
Keywords: Erbium-doped fiber system; Nonlinear wave interaction; Shape-changing collision.
a
J
PACSnumbers: 05.45.Yv,02.30.Ik,42.81.Dp
0
2

I. INTRODUCTION NLS systems [25–34], such as attractive interactions be-
]
S tween solitons and rogue waves [25–30], inelastic colli-
P sions between breathers and other-type waves (solitons
Nonlinear wave interactions are an integral part of
. and rogue waves) [31], and others [32–34]. As a result,
n nonlinear wave theory, which often exhibit highly non-
CMsystemopensavenueintorichinteractiondynamics
li trivial features [1–4]. The intraspecific interactions, of nonlinear waves, which needs more exploration.
n
including the well-known soliton interaction [5–7] and
[ breather interaction [8–14] with characteristic particle In the present paper we extend nonlinear wave in-
2 collision properties, commonly appearing in scalar non- teractions to a coupled nonlinear Schr¨odinger and the
v linear wave evolution equations, has been extensively Maxwell-Bloch(NLS-MB)model,whichdescribesoptical
5 studiedinmanyfieldsboththeoreticallyandexperimen- pulse propagation in a resonant erbium-doped fiber [35–
4 tally. Anotherinterestingcaseistheinterspecificinterac- 37]. Recently, the well-known localized nonlinear waves,
1
tion,i.e.,interactionbetweendifferenttypesofnonlinear i.e., soliton, breather and rogue wave and their interac-
3
waves [15–21]. In fact, research on interspecies interac- tions with shape-unchanging structures were studied in
0
. tions is in an emerging state. Recently, interactions be- [38–42]; different types of localized and periodic waves
1 tweenlocalizednonlinearwaves(soliton,roguewave,and onaplane-wavebackgroundwerealsorevealed[43]. The
0
breather) and periodic waves (standard cnoidal waves) next step of interest and significance is to investigate
6
1 havebeendemonstratedinthestandardscalarnonlinear whether these nonlinear modes can coexist and interact
: Schr¨odinger(NLS)systems[19–21]. Essentially,thesein- with each other. However, to our knowledge, the impor-
v teractions are shape-unchanging. tant interplays of these nonlinear modes have not been
i
X studied, not even for the possibility of the coexistence
In contrast to the standard scalar NLS case, interac-
andinteraction,butalsopropertyofinteractionshasnot
r tions, especially interspecific interactions exhibit struc-
a been analyzed at all.
turaldiversityincoupledmulticomponent(CM)systems.
It is because that CM systems possess some additional Ourtreatmentbelowgoesasfollows. InSectionII,ex-
coupling parameters and allow for interaction between actnonlinearwavesolutionsonaplane-wavebackground
different components, which potentially yield rich and areconstructed; inparticular, possibilitiesofcoexistence
significant interaction structures. One of the most well- and interaction between different-type nonlinear waves
known examples is the shape-changing collision for the are analyzed in detail, based on the explicit coexistence
standardsolitonsinthecoupledNLSsystems[22–24],re- conditions. In Section III, some striking coexistence and
vealed by the analysis of intensity redistribution among interaction, including the interspecies interactions be-
the components. More remarkably, rich interspecific in- tweenbreathersandothertypesofnonlinearwaves(peri-
teractions have been recently identified in the coupled odic wave, antidark soliton, and multi-peak soliton), the
mutualcollisionbetweenmulti-peaksolitons,andthein-
terplay of multi-peak soliton and antidark soliton, are
presented. Especially,akindofshape-changingcollision,
∗Electronicaddress: [email protected] whichisidentifiedastheresultoftheself-shapingeffects
2
of multi-peak solitons, is revealed analytically and nu- Let us then reveal the potential possibility of the in-
merically. Moreover, it is demonstrated that the shape- terplays of nonlinear waves in the system. We find that
changing interactions have no analogues in the standard not all of these nonlinear waves can interact with each
scalar NLS systems. The final section presents our con- other. Three interspecific interactions (periodic wave,
clusions. anti-dark soliton, and W-shaped soliton) do not exist in
the system due to their existence conditions. Addition-
ally,withoutlosinggenerality,weomitthecasesforrogue
waves and W-shaped solitons, since they are merely the
II. NONLINEAR WAVES IN NLS-MB SYSTEM
limiting cases of breathers and periodic waves, respec-
AND INTERACTION ANALYSIS
tively. Intheremainingcases,ourinterestismainlycon-
fined to providing some intriguing interaction patterns
We consider a resonant erbium-droped fiber system
of the rich different-type nonlinear structures in the sys-
governed by a coupled system of the NLS-MB equations
tem, which have not been reported before. Specifically,
[35–37]
we present two types of interaction structures, including
(cid:18) (cid:19) shape-unchanging and shape-changing interactions. Re-
1
E =i E +|E|2E +2P, markably, we find that the multi-peak solitons exhibit
z 2 tt
shape-changing properties before and after interactions.
(1)
P =2iωP +2Eη, It is noteworthy that, if k =0, implying P =0, η =0,
t
η =−(EP∗+PE∗), the NLS-MB model reduces to the standard NLS equa-
t
tion. Inthiscasethemulti-peaksoliton,antidarksoliton,
W-shapedsoliton,andperiodicwavereportedinthispa-
where E(z,t) is the slowly varying envelope field; P(z,t)
per cannot exist in the standard NLS system, since the
is the measure of the polarization of the resonant
medium, which is defined by P = v v∗; η(z,t) denotes existence condition k (cid:54)= 0 [see Table I]. As a result, the
1 2
interactions, which these waves are involved, are specific
the extent of the population inversion, which is given by
η =|v |2−|v |2, v and v are the wave functions of the to the NLS-MB system, and could exhibit unique and
1 2 1 2
significant dynamics properties.
two energy levels of the resonant atoms; ω is the carrier
frequency, and the index ∗ denotes complex conjugate.
Nonlinear waves type Existence condition
Tostudytheprolificinteractionstructuresbetweendif-
Breather and rogue wave b2+A2 (cid:54)=k (A =a +ω)
ferent types of nonlinear waves in an erbium-doped fiber j j j j
Multi-peak soliton b2+A2 =k, q(cid:54)=−2a
system,weshallconstructtheexactgeneralhigher-order j j j
Periodic wave b2+A2 =k, q=−2a , a2 >b2
nonlinear wave solutions on a plane-wave background. j j j j
Antidark soliton b2+A2 =k, q=−2a , a2 <b2
Explicit exact first-order and second-order solutions are j j j j
W-shaped soliton b2+A2 =k, q=−2a , a2 =b2
given in the Appendix. j j j j
The expressions of the solutions depend on the back-
TABLEI:TypesofnonlinearwavesinNLS-MBsystemwith
ground wave amplitudes a, k, the background wave fre- corresponding explicit condition.
quency q, the carrier frequency ω, and real parameters
a , b (j = 1,2). When a = 0, the first-order solu-
j j 1
tion will reduce to the case in Ref. [43]; different abun-
dant types of nonlinear structures including multi-peak
III. CHARACTERISTICS OF INTERPLAYS
soliton, periodic wave, antidark soliton, and W-shaped
soliton (as well as the known bright soliton, breather,
and rogue wave) were revealed by the analysis of veloc- A. Shape-unchanging interactions between
itydifferencebetweenhyperbolicandtrigonometricfunc- different types of nonlinear waves
tions. However,theobviousdrawbackofthisapproachis
thatitdoesnotallowustoinvestigatesomenewspecific Wefirstconsidertheinterplaysofbreathersandother-
interspecies interplays. For instance, we cannot obtain type nonlinear waves. We present some interesting coex-
theinteractionbetweenmulti-peaksolitonsandantidark istence and interaction structures, including the interac-
solitons by nonlinear superposition of the first-order so- tion between breathers and periodic waves and the colli-
lutions with one selected value of q. sion between breathers and anti-dark solitons. The cor-
(cid:112)
We note that the additional degrees of freedom pro- responding typical amplitude [I(z,t) = |E(z,t)|2] dis-
videdbya ,a allowformoreabundantinterplaysofdif- tributions are depicted well in Figs. 1 and 2.
1 2
ferent nonlinear waves without affecting the wave types, Figure 1 exhibits the coexistences and interactions be-
which will be displayed in the following. In this case, tween breathers and periodic waves. The corresponding
we present existence conditions of nonlinear waves with coexistence conditions are chosen as b2 +A2 = k,q =
1 1
a general and concise form, which is shown in Table I. −2a , a2 > b2, and b2 + A2 (cid:54)= k, based on the ex-
1 1 2 2
Theinterestingfindingisthattheexistenceconditionsof act classification in Table I. To better understand the
types of nonlinear waves are the same for an arbitrary- coexistences and interactions, we set the periodic wave
order solution. stationary and select two well-know breathers, i.e., the
3
form a large amplitude peak. They then separate after
the collision, to restore the original shape; there is no
energy exchange during the collision.
(a) (b)
(a) (b)
FIG.2: Shape-unchanginginteractionbetweenbreathersand
anti-dark solitons. The setup is a = 1, a = 0, b = 1.5,
1 1
a =0.8, b =1.6, ω=1, and q=−2a .
2 2 1
(c) (d)
FIG. 1: Shape-unchanging interactions between breathers
and periodic waves, (a) the Kuznetsov-Ma breather and a
stationery periodic wave with b =1.5; ; (b) the Akhmediev
2
breatherandastationeryperiodicwavewithb =0.5. Others
2
are a=1, b =0.9, a =a =−1/3, ω=1, and q=−2a .
1 1 2 1
Akhmediev breather (AB) [44] and the Kuznetsov-Ma
breather (KMB) [45] propagating on the periodic wave (a) (b)
background.
Figure 1(a) shows, the KMB propagation on a W-
shaped wave train background. One should note that
thenonzerovelocityoftheKMBisinducedbythestruc-
tural parameter ω, which is different from the case of
the standard NLS system. Specifically, when the KMB
passesthroughtheW-shapedwavetrain, itcollideswith
eachW-shapedwaveandsomelargeamplitudepeaksare
formed. The KMB and the W-shaped wave restore the
original shapes after each collision with a small phase
(c)
shift. Thisindicatesthatthewholeinteractionstructure
is elastic. Figure 1(b) illustrates the interaction between
FIG.3: Shape-changingcollisionbetweentwomulti-peaksoli-
the AB and a periodic wave which are well coexisting. tons (incident solitons S , S and outgoing solitons S(cid:48), S(cid:48))
1 2 1 2
The each peak of the AB is localized between two adja- with the conditions b2+A2 = k, q (cid:54)= −2a , j = 1,2. In (c),
j j j
centW-shapedwaves. Theperiodicwaveexhibitsaslight the dashed circles represent the main peaks with maximum
snakelike oscillation as it coexists and interacts with the intensities,andthesolidcirclesarethesubpeakswithsmaller
AB, while the AB remains its innate structure. intensities. The setup is a = 1, a = 0, b = 1.5, a = 0.5,
1 1 2
For the choice of parameters a2 > b2, and the rest b2 =1, q=5, ω=1.
1
remains the same, one obtains the interaction between
breathers and antidark solitons. Figure 2 depicts the
correspondinginteractionstructure. Althoughthewaves
aretwodistincttypesoflocalizedwaves,thecollisionbe- B. Shape-changing interactions
tweenthemexhibitsanelasticfeatureonthesameplane-
wave background. From z = −10, two waves approach Let us now pay our attentions to the collision between
each other, at z = 0 they collide with each other and multi-peak solitons and the interactions between multi-
4
peaksolitonsandother-typenonlinearwaves. Theinter-
esting finding is that, all these interactions in which the
multi-peak soliton is involved, exhibit self-shaping char-
acteristics. Namely, the features of the multi-peak soli-
ton, such as the multi-peak features, and the intensity
distribution, are changed significantly before and after
the interaction, while the properties of the other nonlin-
ear waves remain invariant.
Wefirststudythemutualcollisionbetweentwomulti-
peak solitons with the compatibility condition of the
multi-peaksoliton, i.e., b2+A2 =k, q (cid:54)=−2a (j =1,2).
j j j
As shown in Fig. 3, two incident multi-peak solitons S , (a) (b)
1
S withdifferentpeakdistributions[seeintensityprofiles
2
in Fig. 3(c)] move from z → −∞ and approach each
other; they undergo collision around (z,t) = (0,0) and
form higher peaks. They then separate after the colli-
sionandtheintensityprofilesoftheoutgoingmulti-peak
structures S(cid:48), S(cid:48) distinct from the ones of S , S . Re-
1 2 1 2
markably, it is shown, the numbers of the peaks stay es-
sentiallythesameeitherS (S(cid:48))orS (S(cid:48)),buttheinten-
1 1 2 2
sities are redistributed in a striking way. Specifically, by
comparison of the peak intensities of S , S(cid:48) (or S , S(cid:48)),
1 1 2 2
(c)
the intensities of main peaks (dashed circles) increase
while the intensities of subpeaks (solid circles) decrease
after the collision [see Fig 3(c)]. Namely, this fascinating FIG. 4: Shape-changing interaction between the Akhmediev
breatherandamulti-peaksoliton(theincidentsolitonS,and
shape-changingcollisionstemsfromtheintensitytransfer
outgoing soliton S(cid:48)). In (c), the dashed circles represent the
from subpeaks to main peaks of the multi-peak soliton
main peaks with maximum intensities, and the solid circles
itself. In contrast to the shape-changing collisions be-
arethesubpeakswithsmallerintensities. Thesetupisa=1,
tweenstandardsolitonsinthecoupledNLSsystemsthat
a = −1/3, b = 0.3, a = 1, b = −1.1, q = −2a , and
1 1 2 2 1
describeaprocessofenergytransferencebetweensolitons ω=1.
[22–24], we may regard this shape-changing property as
self-shaping effects of the multi-peak soliton.
In order to reveal further the collision characteristics solitonsbeforeandafterthecollisioncanbegivenclearly
betweentwomulti-peaksolitons,weintroducetheenergy via optimal numerical method at different z. The total
oflightpulseagainsttheplane-wavebackgroundwiththe energyofthetwosolitonsiscalculatednumericallywhen
form t = −∞, t = ∞. As shown in Table II, the total lo-
1 2
calizedenergyremainsinvariableatz =−15,−10,10,15.
Ie =(cid:90) t2(cid:8)|E(t,z)|2−a2(cid:9)dt, (2) Theenergyofthesinglemulti-peaksoliton[Ie(s1),Ie(s2)]
t1 is presented by the appropriate choice for t1, t2. At
a selected initial propagation distance, i.e., z < 0, we
resulting in that the localized energy of the multi-peak calculate I (s ) and I (s ) by (cid:82)t0 (cid:8)|E|2−a2(cid:9)dt and
e 1 e 2 −∞
(cid:82)∞(cid:8)|E|2−a2(cid:9)dt, respectively. Note that the trans-
Position Total I Multi-peak S I Multi-peak S I t0
e 1 e 2 e verse position t is located between S , S , leading to
0 1 2
z=−15 9.47891 5.66666 3.81225
|E(t ,z)|=a, and vice versa for positive z. One can see
0
z=−10 9.47891 5.66661 3.8123
from Table II that each multi-peak soliton preserves the
z=10 9.47891 5.66665 3.81225
localized energy conservation before and after the colli-
z=15 9.47891 5.66666 3.81225
sion. Namely,whenmulti-peaksolitonscollide,theshape
change and energy conservation coexist.
TABLE II: Numerical integration verification of energy I in
e
Fig. (3) at different z=−15,−10,10,15. For a better understanding of the interesting self-
shapingeffectsofmulti-peaksolitonsintheNLS-MBsys-
Position Multi-peak soliton I tem, we explore next interspecific interactions, i.e., the
e
z=−15 3.75045 interactions between multi-peak solitons and other-type
z=−10 3.75045 localized nonlinear waves. Our aim is to demonstrate
z=10 3.75045 that when the interspecific interaction occurs, the multi-
z=15 3.75045 peak solitons exhibit self-shaping intensity distribution
but the characteristics of other-type nonlinear waves re-
TABLEIII:NumericalintegrationverificationofenergyIe in main invariant.
Fig. (4) at different z=−15,−10,10,15.
We first consider the interaction between multi-peak
5
integrationofEq. (2). AsshowninTableIII,theenergy
of the multi-peak soliton remains unchanged at different
z, beforeandafterthemulti-peaksolitonpassesthrough
the AB.
Another interesting case is the interplay of the multi-
peak solition and antidark solitons. The correspond-
ing initial parameter condition is extracted exactly as
b2 + A2 = k, q = −2a , a2 < b2 and b2 + A2 = k,
1 1 1 1 2 2
q (cid:54)= −2a . Figure 5 shows the structure distribution of
2
the collision between a multi-peak soliton and an anti-
darksoliton. Itisevidentthattwosolitonsarewellsepa-
(a) (b)
ratedbeforeandafterthecollision,whichissimilartothe
elastic collision between the standard solitons reported
before. However, the interesting finding emerges when
we look in some detail into properties of the collision.
As shown in Fig. 5(c), multi-peak soliton shows differ-
ent intensity distributions before and after the collision
due to its self-shaping effects, while the anti-dark soliton
remains unchanged. This indicates that there is no en-
ergy exchange between these two different-type solitons
during the collision. Also, the corresponding numerical
(c) integration verification is displayed in Table IV.
It is noteworthy that, the anti-dark soliton can be re-
FIG. 5: Shape-changing interaction between a multi-peak gardedstrictlyasthelimitingcaseofmulti-peaksolitons
soliton and an anti-dark soliton. In (c), the dashed circles without periodic modulation along the propagation di-
represent the main peaks with maximum intensities, and the rection, as reported before in our previous work. Thus
solid circles are the subpeaks with smaller intensities√. The the unique self-shaping effect is closely associated with
setup is a = 1, a1 = 0, b1 = 1.5, a2 = 2/3, b2 = 17/6, the periodicity of multi-peak solitons, since it does not
ω=1, and q=−2a .
1 exist in anti-dark solitons.
Furthermore, a comparison of Figs. 3-5 implies that,
when the NLS-MB multi-peak soliton collides with an-
solitons and breathers for the choice of the parameters
other localized waves, the multi-peak pulse undergoes
b2+A2 (cid:54)= k and b2+A2 = k, q (cid:54)= −2a . To illuminate
1 1 2 2 2 self-shapingphenomenon; itsintensityisredistributedin
the interaction property in a more clear way, we illus-
a regular way; but the other-type waves recover original
trateatypicalcollisionbetweenamulti-peaksolitonand
shapes. We could not detect any energy exchange be-
aAB.AsshowninFig. 4, anincidentmulti-peaksoliton
tween the two localized nonlinear waves. We then could
S propagating along the distance passes through the AB
infer that the self-shaping effect is specific to the multi-
near z = 0; after that the outgoing soliton S(cid:48) exhibits a
peak solitons in the NLS-MB system.
different intensity distribution with a higher main peak
buttheABremainsunchanged. Ananalysisoftheinten-
sity profiles S and S(cid:48) in Fig. 4(c) shows that, the single
IV. CONCLUSION
multi-peak soliton allows the intensity to be transferred
from the subpeak to the main peak when it collides with
Insummary,wehaveinvestigatedabundantintriguing
the AB. It is noted that a complete reverse self-shaping
interactionstructuresofnonlinearwavesonaplane-wave
process occurs with the initial condition |b | = −b , in
2 2
backgroundinanerbium-dopedfibersystem. Incontrast
which a multi-peak soliton shifts the intensity from its
to the previous results that are mainly about intraspe-
main peak to sub peak.
cificinteractions(solitionandbreather),ourresultshave
Let us analyze the energy of the multi-peak soliton
shown that rich different types of nonlinear waves can
before and after passing through the AB by numerical
coexist and interact with each other. These interactions
involve the shape-unchanging collisions and the shape-
Position Total I Anti-dark I Multi-peak I changing collisions. In particular, a shape-changing col-
e e e
z=−20 6.17792 4.47212 1.70580 lision between multi-peak solitons has been revealed as
z=−15 6.17792 4.47212 1.70580 a result of the self-shaping effects of the multi-peak soli-
z=15 6.17792 4.47210 1.70582 tons. Itshowsthattheshape-changingcollisionpreserves
z=20 6.17792 4.47211 1.70582 thepowerofeachmulti-peaksoliton,whichisverifiedvia
optimalnumericalintegrationfortheenergyoflightpulse
TABLEIV:NumericalintegrationverificationofenergyIe in against the plane-wave background. Our results demon-
Fig. (5) at different z=−20,−15,15,20.
strate that when the interactions occur, the multi-peak
6
solitons exhibit the coexistence of the shape change and Substitute the eigenfunctions ψ (λ ) and ψ (λ ) into
1,1 1 2,1 1
energy conservation. It is expected that these results the Eq. (A1) with a tedious simplification, the general
willenrichourunderstandingoftheinterplayofdifferent and concise expression of the first-order nonlinear wave
types of nonlinear waves in a coupled multicomponent solution is presented as
system.
Note added. Recently, a mathematical derivation of
(cid:26) (cid:27)
analytical higher-order nonlinear wave solutions of the E =E 1− 8b1m1[sin(γ+µ1)−isinh(β+iµ1)] ,
NLS-MB system with some higher-order effects was ob- 1 0 m sin(γ+µ )−im sinh(β−iµ )
3 2 2 3
tained by Wang et al [46]. (6)
where
ACKNOWLEDGEMENTS
β =ζ(t+V z), γ =σ(t+V z),
1 2
This work has been supported by the National Nat- V =v +b σv /ζ, V =v −b ζv /σ,
1 1 1 2 2 1 1 2
ural Science Foundation of China (NSFC) (Grant Nos. v =k(a +ω)/[b2+(a +ω)2]+(a −q/2),
1 1 1 1 1
11475135, 11547302), andtheministryofeducationdoc-
v =1−k/[b2+(a +ω)2],
toral program funds (Grant No. 20126101110004). C. 2 1 1
LiuthanksT.KannaandL.-C.Zhaofortheirusefuldis- (cid:16)(cid:112) (cid:17)1/2 √
ζ = χ2+µ2+χ / 2,
cussions and valuable comments.
(cid:16)(cid:112) (cid:17)1/2 √
σ =± χ2+µ2−χ / 2,
APPENDIX χ=4b2−4a2−(2a +q)2, µ=−4b (2a +q),
1 1 1 1
m =(cid:8)(iζ−σ)2+[2b +i(2a +q)]2(cid:9)1/2,
For the erbium-doped fiber system governed by Eq. 1 1 1
(1), a general nonlinear wave solution in the E compo- m =(cid:8)(α +α )2−4[2b ζ+(2a +q)σ]2(cid:9)1/2,
2 1 2 1 1
nent for arbitrary order j is recursively constructed by
m =(cid:8)(α −α )2+4[2b σ−(2a +q)ζ]2(cid:9)1/2,
3 1 2 1 1
E =E −2i(λj −λ∗j)ψ1,jψ2,j, (3) α1 =4a2+4b21+(2a1+q)2, α2 =ζ2+σ2,
j j−1 |ψ1,j|2+|ψ2,j|2 tanµ = 2b1+i(2a1+q),
where ψ and ψ are eigenfunctions corresponding to 1 σ−iζ
1,j 2,j
j-fold Darboux transformation. In order to obtain rich tanµ = α1−α2 ,
nonlinear wave solutions on a background of Eq. (1), we 2 4b σ−2(2a +q)ζ
1 1
firstintroducethefollowingplane-wavebackgroundwith α +α
tanµ = 1 2 . (7)
a generalized form 3 4b ζ+2(2a +q)σ
1 1
E =aeiθ, P =ikE , η =ωk−qk/2,
0 0 0 0
where θ =qt+νz, ν =a2+2k−q2/2, a and q represent Here ‘±(cid:48) in σ depends on µ≤0 and µ>0, respectively.
theamplitudeandfrequencyofbackgroundelectricfield, Thehigher-ordernonlinearwavesolutionisgeneratedby
respectively,andk isarealparameterwhichisrelatedto theiterationofDarbouxtransformation. Herewepresent
the background amplitude of P component. By solving the second-order solution with the eigenfunctions ψ
1,2
the Lax pair in the Ref. [40], the eigenfunctions ψ = and ψ of two-fold Darboux transformation which are
(cid:0)ψ1,1(cid:1)correspondingtoone-foldDarbouxtransformation given 2,2
ψ2,1
with the eigenvalues λ =a +ib is given
1 1 1
ψ1,1(λ1)=a(φ1+φ2), ψ =(λ −λ∗−2ib M)ψ (λ )−2ib Nψ (λ ),
1,2 2 1 1 1,1 2 1 2,1 2
ψ2,1(λ1)=[iλ1(φ1+φ1)+(τ1φ1+τ2φ2)], (4) ψ =(λ −λ +2ib M)ψ (λ )−2ib N∗ψ (λ ).
2,2 2 1 1 2,1 2 1 1,1 2
where
φ1 =exp[τ1t+(iτ12+B+C)z], where
φ =exp[τ t+(iτ2+B+C)z],
2 2 2
(cid:18) (cid:113) (cid:19)
τ = iq+ −q2−4a2−4λ q−4λ2 /2, 4a2(coshβ+cosγ)
1 1 1 M = ,
m sin(γ+µ )−im sinh(β−iµ )
(cid:18) (cid:113) (cid:19) 3 2 2 3
τ2 = iq− −q2−4a2−4λ1q−4λ21 /2, N = 2am1[sin(γ+µ1)−isinh(β+iµ1)]eiθ. (8)
m sin(γ+µ )−im sinh(β−iµ )
3 2 2 3
B =q/2+λ +k/(λ +ω),
1 1
C =λ2+λ q/2+3a2/2−kq/(2λ +2ω), (5)
1 1 1
7
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