A higher-order perturbation analysis of the nonlinear Schrödinger equation

Research

paper

A

higher-order

perturbation

analysis

of

the

nonlinear

Schrödinger

equation

J. Bonetti

a,c

_,

_S.M.

_Hernandez

a

_,

_P.I.

_Fierens

b,c,∗

_,

_D.F.

_Grosz

a,c

a_{GrupodeComunicacionesÓpticas,InstitutoBalseiro,Argentina} b_{InstitutoTecnológicodeBuenosAires,Argentina}

c_{ConsejoNacionaldeInvestigacionesCientíficasyTécnicas(CONICET),Argentina}

Keywords: Opticalfibers Noise

Modulationinstability

a

b

s

t

r

a

c

t

A well-known and thoroughly studied phenomenon in nonlinear wave propagation is that of modulation instability (MI). MI is usually approached as a perturbation to a pump, and its analysis is based on preserving only terms which are linear on the perturbation, dis- carding those of higher order. In this sense, the linear MI analysis is relevant to the un- derstanding of the onset of many other nonlinear phenomena, such as supercontinuum generation, but it has limitations as it can only be applied to the propagation of the per- turbation over short distances.

In this work, we propose approximations to the propagation of a perturbation, con- sisting of additive white noise, that go beyond the linear modulation instability analysis, and show them to be in excellent agreement with numerical simulations and experimental measurements.

1. Introduction

Pulsepropagationinsingle-modelosslessnonlinearfibersismodeledbytheNonlinearSchrödingerEquation(NLSE)[1]

∂

A

∂

z−i

β

ˆA = i

γ

ˆA

|

A

|

2_{. (1)}

A(z,T)isthepulseenvelope,zisthedirectionofpropagationandTisthetime referredtoaco-movingframewithgroup velocity

v

g =

β

₁−1 (i.e.,T = t −z

β

1). Lineardispersion ismodeled bythe operator

β

ˆ,while

γ

ˆ isrelatedtothethird-order susceptibility:

ˆ

β

=

k≥2 ik

β

k

k!

∂

k

∂

Tk,

γ

ˆ =

k≥0 ik

γ

k

k!

∂

k

∂

Tk. (2)

Wemustnote that,forthesake ofsimplicity,we haveomittedthecontributionofthe stimulatedRamanresponseofthe medium.Furthermore,wehavenotincludedanynoisesourcesuchasspontaneousRamanemission.

AnalyticalsolutionsofEq.(1)areknownina varietyofsimplifiedcases.Forinstance,solitonicsolutionscan befound by means of the inverse-scattering methodoriginally proposed by Zakharov andShabat [2] (see also,e.g., [3]), butonly

∗ _{Correspondingauthorat:MathematicsandPhysics,Av.EduardoMadero399,CiudadAutónomadeBuenosAiresC1106ACD,Argentina.}

undersomesimplificationssuchasnohigher-orderdispersion(

β

_{k =}0fork_≥3).Animportantfamilyofperiodicsolutions, knownasAkhmedievbreathers[4],hasattractedattentioninrelationtosupercontinuumgenerationandroguewaves[5,6]. AlthoughAkhmedievbreathers wereoriginally found forlow-dispersion cases,Eq.(1)hasbeenfound tobe integrablein morecomplexcases(see,forexample,[7–11]andreferencestherein).However,thenumberofexactlyintegrablevariations oftheNLSEisstillverylimited.

AlthoughexactsolutionsofsimplifiedversionsofEq.(1)provideimportantinsightintomanyfeaturesofthepropagation ofpulsesinnonlinearfibers,theydonotprovideaprecisedescriptioningeneral.Forthisreason,theNLSEisusuallystudied bymeansofsimulations basedonefficientalgorithmssuchassplit-stepFourier(SSF)[1]orafourth-orderRunge-Kuttain theinteractionpicture(RK4IP)[12].

Inthiswork,we putforth aperturbation analysisoftheEq.(1)when acontinuous-wave (CW) laserpumpsthe non-linearfiber.TheCWpumpisalwaysaccompaniedbytechnicalandquantumnoiseandwe focusonthenoisepropagation alongthefiber.Ourgoalisnot toproposeanefficientmethodologythat cansubstitutenumericalsimulations ofthe non-linearSchrödingerequation,buttointroduceapproximateexpressionsthatcanprovideamoreintuitiveandcomprehensive understandingofthemainprocessesinvolvedinhigher-ordermodulationinstability.

Onepossibilityistostudynoisepropagationasaperturbationto theCW.Thefirst-orderperturbationorlinear stabil-ityanalysisis relatedto the studyofthe modulation instability (MI)phenomenon [4,5,13–23,23–29] (see also Chapter 5 ofRef. [1]andreferencestherein.) ExactsolutionsofMIaccountingforafull modeloftheNLSE,includingtheRaman re-sponseandthe dependenceofthenonlinear parameterwithfrequency, havebeendeveloped[30,31].The particularcase ofthepropagationofadditive noisehasbeendealtwithintheliterature (see,e.g.,[32,33]).Note,however,thatthewave propagationanalysisofanoisyCWpumpina MIsettinghasseverallimitations.The continuous-wavepumpisassumed undepletedand,hence,theresultsareonlyvalidovershortpropagationdistances.Furthermore,asitisafirst-order pertur-bationanalysis,itdisregardsthe’cascadingeffect’offour-wavemixing,inthesensethatperturbationstothepumpcanas wellactaspumpsthemselvesoncethey haveattainedenoughpower.Onealternativetoincorporatesuchcascadingeffect istosolvetheNLSEthroughPicard’siterations. Resultingexpressionsare,nevertheless, noteasilytractable andeventheir numericalevaluationmayturnoutto bean expensivecomputational effortascomparedto purenumerical solutions ob-tainedfromtheusual SSForRK4IPalgorithms.Forthisreason,we putforthseveralsimplificationsthatallow ananalysis ofhigher-orderperturbations.Thevalidityofthesesimplificationsistestedthroughnumericalsimulationandexperimental measurements.

Wemustnotethattherearealternativeapproacheswhicharerelatedtoideaspresentedinthiswork.Inparticular,many toolshavebeendevelopedforthestatisticalanalysisofopticalwaveturbulence(see,e.g.,[34–38]).

The remaining of thispaper is organizedas follows.In Section 2 we develop a higher-orderperturbation analysis of thenonlinearSchrödingerequationandmotivatethesimplificationsthatallowtractability.Wevalidateourapproachwith experimentalresultsandnumericalsimulationsinSection3.Finally,wepresentsomeconclusionsandlinesoffuturework inSection4.

2. Perturbationanalysis

LetusagainconsiderthenonlinearSchrödingerequation.Itisusefultonormalizethepropagationdistanceas

ζ

₌

γ

0P0z. Westudythe propagationofasmall perturbationa(

ζ

,T) to thestationarysolutionof Eq.(1), i.e.,we considerA

(ζ

,T

)

=

P0[1+a

(ζ

,T

)

]eiζ.Fouriertransformation(withrespecttotimeT)leadstothefollowingcoupleddifferentialequations

i

∂

_ζa˜

(

ζ

,

)

+ B

()

a˜

(

ζ

,

)

+

γ

˜

()

a˜

(

ζ

,−

)

= −

γ

()

N˜

(

a˜

(

ζ

,

))

, (3)

−i

∂

_ζa˜

(

ζ

,−

)

+ B

(

−

)

a˜

(

ζ

,−

)

+

γ

˜

(

−

)

a˜

(

ζ

,

)

= −

γ

(

−

)

N˜

(

a˜

(

ζ

,−

))

, (4)

wherea˜

(ζ

,

)

istheFouriertransformofa(

ζ

,T),B

()

=

_β

˜

₍₎

₊₂

_γ

_˜

₍₎

_−1,

˜

β

()

= 1

γ

0P0

M

m=2

(

−1

)

m

m!

β

m

m_,

_γ

_˜

₍₎

₌ 1

γ

0

N

n=0

(

−1

)

n

n!

γ

n

n_{, (5)}

˜

N

(

a˜

)

= a˜

(

ζ

,

)

∗a˜

(

ζ

,−

)

+ a˜

(

ζ

,

)

∗

a˜

(

ζ

,

)

+ a˜

(

ζ

,−

)

+ a˜

(

ζ

,

)

∗a˜

(

ζ

,

)

∗a˜

(

ζ

,−

)

. (6)

2.1. Linearstabilityanalysis

Analysisof modulationinstability (MI)proceedsby neglecting thenonlinear termsinEqs. (3)–(4). Letusassumethat ˜

a

(

0,

)

isanoisyperturbationsuchthat

˜

forsomepositiveconstants.Itcanbeshownthatthefirst-orderMIapproximationisgivenby(see[33])

|

a˜

(

ζ

,

)

|

2

_≈s·M2

()

+ G21

()

+

γ

˜2

()

4G2

1

()

·e2G1()ζ, ₍₈₎

where

M

()

=

_β

˜_e

₍₎

_{+ 2}

_γ

_˜e

₍₎

_{−1, (9)}

1

()

=

˜

γ

()

γ

˜

(

−

)

−M2

()

_{, (10)}

G1

()

=

1

()

if 1

()

∈ R,

0 otherwise, (11)

and

β

˜e and

γ

˜e containeventermsof

β

˜and

γ

˜,respectively.

Eq.(8)describeshowwhitenoisewithmeanspectraldensitysisamplifiedbyanMIgainG1(

).Modulationinstability analysis,however,suffersfromseveralshortcomings.Sincehigher-ordernonlinearinteractionsareneglected,expressionsso farcannotcapturethecascadingeffectoffour-wavemixing.

2.2. Perturbationansatz

Eq.(8)motivatesaperturbativeapproximationtothesolutionoftheform

˜

a

(

ζ

,

)

≈ ∞

n=1

sn2

_n

()

eiφn(ζ,)_eGn()ζ, ₍₁₂₎

wherethefollowingrandom-phaseassumptionissatisfied

eiφn(x,μ)_e−iφm(y,ν)

=

δ

n,m

δ

(

x−y

)

δ

(

μ

−

ν

)

, (13)

Notethat,toafirstorder,Eq.(12)agreeswithEq.(8)withG1givenbyEq.(11)and,forG1(

)= 0,

|

1

()

|

2

= M2

()

+ G21

()

+

γ

˜2

()

4G2

1

()

. (14)

Ifwealsoassume that

nareindependentof

φ

mforall n,m,

n isindependentof

mform= n,andGnisdeterministic

andreal,themeansquaredvalueoftheperturbationmustevolveas

|

a˜

(

ζ

,

)

|

2

≈

∞

n=1

sn

|

n

()

|

2

e2Gn()ζ. ₍₁₅₎

Inordertofindexpressionsfor

|

n(

)|2 andGn(

),wesubstituteEq.(12)inEqs.(3)–(4)anduseEq.(15).However,

tomakecalculationstractable andfinalexpressions simpler,weproposeseveralsimplifyinghypotheseswhichare detailed inAppendixA.Althoughthetrueextentoftheireffectcanonlybecomprehendedinthecontextofthedetailedcalculations presentedintheappendix,someofthesesimplificationsareeasytounderstand:

1. WeassumethatthefunctionsGn(

)areeven.ThisassumptionismotivatedbythefactthatG1(

)(theMIgain)iseven. 2. Wealsoassumethat

|

n(

)|2 areevenfunctions.Again,thissimplificationismotivatedbythemodulationinstability

case:asitcanbeshown,fromEq.(14),

|

1(

)|2 iseven.

3. Weneglect the interactionofhigher-order MIterms: we onlykeep theinteraction ofn 1termsin Eq.(12)withthe modulationinstability(n=1)term.

4. Wealsoneglectthree-foldinteractionsoftermsinEq.(12).

5. SubstitutionofEq.(12)inEq.(6)leadstoanumberofconvolutionintegrals.Weconsiderthattheweightofthe corre-spondingintegrandsismaximizedwhentheexponents

(

G1

(

u

)

+Gn−1

(

u−

v

))

andG1

(

u

)

+Gn−1

(

v

−u

))

aremaximized. ThisapproximationisveryimportanttoobtainsimpleexpressionsforGn,asitisexplainedinAppendixA.

6. Finally,werepeatedly useEq.(13),weusethefactthat

∂

_ζ

φ

n

(ζ

,

)

=0andneglecthigher-ordermomentsof

∂

_ζ

φ

n(

ζ

,

±

).

Aftersomelengthymanipulations,wearriveatthefollowingexpressions:

Gn

()

= max

u G1

(

u

)

+ Gn−1

(

u−

)

, (16)

|

n

()

|

2

=

α

n−1

_γ

_˜2

₍₎

_·

_|

_B

₍

₋

₎

_−iG

n

()

|

2+

γ

˜2

(

−

)

|

(

B

()

+ iGn

()

)

(

B

()

−iGn

()

)

−

γ

˜

()

γ

˜

(

−

)

|

2

Fig.1. G1,G2andG3.Thecascadingfour-wavemixingprocessisreadilyobserved.

Thepositiveconstant

α

inEq.(17)isrelatedtotheMIgainbandwidth.

AlthoughEq.(15)correctlydescribestheevolutionoftheperturbation,itisnot accurateat

ζ

= 0.Indeed,asitcan be readilycalculated,

|

a˜

(

0,

)

|

2

_≈∞

n=1 sn

_|

n

()

|

2

. (18)

Thisequationdoesnot leadtothe knownvalue

|

a˜

(

0_,

)

|

2

₌s.Eq.(15)can bemadeaccurate evenat

ζ

₌0_,that is,for theinitialrandompertubation,bymakingaminorcorrectiontoEq.(12):

|

a˜

(

ζ

,

)

|

2

≈s+

∞

n=1 sn

_|

n

()

|

2 e2Gn()ζ−1

. (19)

2.3.Discussion

Eq.(16)is aresultofthecascadingeffectoffour-wavemixing. Fig.1showsanexampleofGi fori=1,2,3that helps

understandthe cascadingeffectwhen perturbations attainenough powerand themselvesact asnewpumps.The result-ing higher-order MI sidebands have already been discussed in the literature. Erkintalo et al. [22], for example, describe howan Akhmediev-breather evolvesandsplits intosubpulses usingtheDarboux transformationanddemonstratea good agreementwithexperimental results.While Ref.[22] developshigher-ordersolutions by iteratively applyingtheDarboux transformation,Zakharov etal.[29] present aclass of multisolitonicsolutions which maybe used to describe MI devel-opment.Kimmoun etal.[39] study a similar higher-ordercascadingprocess in surfacegravity waves indeep-water and Armarolietal.[40]alsoanalyzethesecond-ordersidebandsinthecaseoftheDystheequation.

Forthe sake of simplicity,we have omitted theinfluence of stimulated Raman scattering. However, simple modifica-tionstotheformulaspresentedhereallowtoincorporateinstraightforwardfashionthemolecularRamanresponseofthe medium.

Itmust benoted thatthe proposed approximation assumesthat theCWpumpactsasan unlimited sourceofoptical power.Asamatteroffact,Eq.(19)predictsacontinuousgrowthoftheperturbation.Sincethepoweroftheperturbation cannot exceed that of the pump atthe input endof the optical fiber, the proposed analytical model doesnot apply to anarbitrarylongpropagated distance

ζ

,ashortcomingalsopresentinthelinearmodulationinstabilityanalysis.However, first-orderMIanalysisdoesnotaccountforhigher-ordernonlinearinteractions,suchasthecascadingfour-wavemixingand, thus,itfailstogive anaccurate descriptionoftheevolutionoftheperturbationforevenshorterpropagationlengths. We verifythisassertioninthenextSection.

3. Experimentalandnumericalresults

Fig.2. First-order(dashedline)andfourth-order(dottedline)analyticalapproximationsvs.experimentalresults(solidline).ACW30-dBmpumplaserat 1590.4nmwaslaunchedattheinputendofthe770-mlongdispersion-stabilizedHNLF.

Fig.3. Analyticalapproximations(dashedlines)vs.simulationresults(solidlines).Numericalresultscorrespondtotheaverageof100noiserealizations. Resultscorrespondtodistances5LNL,6LNL,7LNL,8LNL.

k 0,

β

2 =−3.9198ps2/km,

β

3 =−0.1267ps3/km,

β

4 =1.7594×10−4 ps4/kmand

β

k =0fork 4.Asitisreadilyobserved,

experimental andanalyticalresultsare inexcellentagreement.Fig.2alsoshowsthefirst-orderperturbative solution,that is,thesolutionpredictedbytheclassicalmodulationinstabilityanalysis.MIcannotaccountformuchofthedetailobserved asitonlypredictstwogainsidebands.

Inordertofurtherexplore thevalidity oftheapproximations,we performedcomputersimulationsusingthealgorithm in[42].Fig.3showsresultsfortheaverageof100realizations.Thedistanceisnormalizedtotheso-callednonlinearlength

LNL =

(γ

0P0

)

−1, whereP0 isthe input power,givinga parameter-independent distancemetric.It is observedthat the ac-curacyoftheapproximation decreaseswiththepropagationdistance,althoughreasonablegoodresultsareobtainedeven after7LNL(≈800m).

Fig.4. Totalsignalpowerfortheanalyticalapproximation(dashedline)vs.simulation(solidline).Thecrossingatnearly7LNL (cf.Fig.3)marksthe

maximumpropagateddistanceatwhichtheanalyticalapproximationremainsvalid.

that enablescontinuous growthof theperturbation, asshownin Fig. 4and inaccordance withEq.(19). However, since total power (pumpplus perturbation) must remain constant, we can expect the analytical model to be valid aslong as thecalculatedpoweroftheperturbationremains lowerthanthat oftheinputpump. AsshowninFig.4,thisconditionis satisfiedupto ∼7LNL,entirelyconsistentwithresultsinFig.3.

4. Conclusions

Modulationinstabilityinnonlinearwave propagationiseitherapproachedbymeansofalinearperturbationanalysisto acontinuous-wavepump,orbythenumericalsolutionoftheNonlinearSchrödinger equation.Whiletheformerapproach givessome insightintotheinitialstagesofpropagation, itfailsatprovidinganaccuratepictureoverlongpropagated dis-tances;thelattercanprovideaccurateresultsoverlongerdistances,buthidestheunderlyingphysics.

Inthis work, we putforth a perturbation analysisthat goesbeyondthe linear modulationinstability, offering both a moreprecise analyticaldescriptionandmeaningfulphysicalinsights thatcapturehigher-ordercascadingfour-wavemixing effects.Weshowedthisanalysistobeaccuratebycomparingitspredictionstoactualexperimentalresults.Furthermore,we successfullyvalidatedtheapproximationsmadewithnumericalsimulationsforpropagateddistancesuptonearly7LNL.

Themathematicalderivationpresentediscomplexandinvolvesanumberofsimplifyingassumptionsbutleadstosimple andtractableformulas.Itisamatteroffutureworktolookforashorterpathandlessrestrictivesimplifications.

Inthis paper, we do not deal withthe nonlinear stage of modulation instability,that is,when the energyof the MI sidebandsiscomparabletothatofthepump.Thereisalsothequestionoftheeffectoftheparticularstatisticsoftheinitial perturbation onthis stage. Theseproblems are a subjectof futureresearch. We study thecase ofan homogeneous, un-doped,single-coreandsingle-modeopticalfiber.Amorecomplexsettingcanbefoundin,e.g.,dual-core[43]andresonant [44]opticalfibers.

Finally,webelieveourresultstobe ofvaluewhentacklingthestudyoftheearlystagesofsupercontinuumgeneration, andtocontributetoolsforthebetterunderstandingofnonlinearprocessessuchasrogue-waveformation.

Acknowledgments

WegratefullyacknowledgeS.RadicforhostingJ.B.’sresearchstayatthePhotonicSystemsGroup,UCSD,andE.Temprana forassistance with experiments. We also acknowledge financial support from project PIP 2015, CONICETgrant 1122015, 0100001CO,Argentina.

AppendixA. Mathematicalderivation

Inorder tofind expressions for

|

n(

)|2 andGn(

),we substitute Eq.(12) inEqs. (3)–(4) anduseEq.(15).

Assumingthattheseriescanbederivedtermbyterm,substitutionofEq.(12)intoEqs.(3)–(4)

∞

n=1

sn2eGn()ζ·_B·

n

()

eiφn(ζ,)

∗

n

(

−

)

e−iφn(ζ,−)

= −

_γ

_˜

₍₎

_N˜

₍

_a˜

₍

_ζ

_,

₎

₎

˜

γ

(

−

)

_N˜

₍

a˜

(

ζ

,−

)

)

, (A.1)

where

B =

B

()

+ iGn

()

−

∂

ζ

φ

n

(

ζ

,

)

γ

˜

()

˜

γ

(

−

)

B

(

−

)

−iGn

()

+

∂

_ζ

φ

n

(

ζ

,−

)

. (A.2)

InthederivationofEqs.(A.1)–(A.2)wehavemadeuseoftheassumptionthatthatthefunctionsGn(

)areeven(simplifying

assumption#1inSection2.2). UsingEqs.(6)and(12),

˜

N

(

a˜

(

x,

μ

)

)

=

∞

n=1 sn2

n−1

m=1

+∞

−∞ 2

m

(

u

)

n−m

(

u−

μ

)

e

(Gm(u)+Gn−m(u−μ))x_ei(φm(x,u)−φn−m(x,u−μ))_du

+ +∞

−∞

m

(

u

)

n−m

(

μ

−u

)

e

(Gm(u)+Gn−m(μ−u))x_ei(φm(x,u)+φn−m(x,μ−u))_du

+

n−m−1

k=1

+∞

−∞ +∞

−∞

m

(

u

)

k

(

v

)

n−m−k

(

u+

v

−

μ

)

×e(Gm(u)+Gk(u)+Gn−m−k(u+v−μ))x_ei(φm(x,u)+φk(x,v)−φn−m−k(x,u+v−μ))_dud

v

. _(A.3)

In order to make these equations tractable, we resort to the approximations 3–5 spelled out in Section 2.2. First, ap-proximation #3 implies that we keep only the first term in the sum, that is, m= 1. Second, aproximation #4 means that we neglect the termswith doubleintegrals ashigher-orderperturbations. Finallyapproximation #5 is, perhaps,the mostrelevant:weconsiderthattheweightoftheintegrandsismaximizedwhentheexponents

(

G1

(

u

)

+Gn−1

(

u−

μ))

and

(

G1

(

u

)

+Gn−1

(μ

−u

))

aremaximized.SincewehavealreadyassumedthattheGn areeven,

max

u

(

G1

(

u

)

+ Gn−1

(

u−

μ

))

= maxu

(

G1

(

u

)

+ Gn−1

(

μ

−u

))

= max

u

(

G1

(

u

)

+ Gn−1

(

μ

+ u

))

= max

u

(

G1

(

u

)

+ Gn−1

(

−

μ

−u

))

. (A.4)

Withallthesesimplifications,weobtain

˜

N

(

a˜

(

x,

μ

)

)

≈

∞

n=1

sn2·emaxu (G1(u)+Gn−1(u−μ))x·

{

I

1

(

x,

μ

)

+I2

(

x,

μ

)

}

, (A.5)

where

I1

(

ζ

,

)

=

+∞

−∞

2

1

(

u

)

n−1

(

u−

)

ei(φ1(ζ,u)−φn−1(ζ,u−))du, (A.6)

I2

(

ζ

,

)

=

+∞

−∞

1

(

u

)

n−1

(

−u

)

ei(φ1(ζ,u)+φn−1(ζ,−u))du. (A.7)

UsingEqs.(A.4)–(A.7)inEq.(A.1),weget

∞

n=1

sn2·eGn()ζ·_B·

n

()

eiφn(ζ,)

∗

n

(

−

)

e−iφn(ζ,−)

= ∞

n=1

sn2·emaxu (G1(u)+Gn−1(u−))ζ·

₋

_γ

_˜

₍₎

₍

_I

1

(

ζ

,

)

+ I2

(

ζ

,

)

)

−

γ

˜

(

−

)

I1

(

ζ

,−

)

+ I2

(

ζ

,−

)

.

Thisequationleadsto

Gn

()

= max

withGn(

)anevenfunction,and

n

()

eiφn(ζ,)

∗

n

(

−

)

e−iφn(ζ,−)

= −B−1

˜

γ

()

(

I1

(

ζ

,

)

+ I2

(

ζ

,

)

)

˜

γ

(

−

)

I1

(

ζ

,−

)

+ I2

(

ζ

,−

)

. (A.9)

What remains is to take the mean squaredvalue of

n(

) which can be found fromEq.(A.9). In theprocess, a useful

simplificationistoassumethat

|

n(

)|2 areevenfunctions(asitcanbeshown,fromEq.(14),that

|

1(

)|2 iseven). Wealsoneglecthigher-ordermomentsof

∂

_ζ

φ

n(

ζ

, ±

)anduseEq.(13)(seesimplifyingassumption#6inSection2.2).

FromEq.(A.2),

−iJ

()

n

()

eiφn(ζ,)=

−B

(

−

)

+ iGn

()

+

∂φ

n

(

ζ

,

)

∂ζ

˜

γ

(

)

(

I1

(

ζ

,

)

+ I2

(

ζ

,

)

)

−

γ

˜

()

γ

˜

(

−

)

I1

(

ζ

,−

)

+ I2

(

ζ

,−

)

, (A.10)

whereJ

()

=det

(

B

)

.Multiplyingthisexpressionbyitsconjugate,

|

J

()

|

2

|

n

()

|

2=

γ

˜2

()

·

−B

(

−

)

+ iGn

()

+

∂φ

n

(

ζ

,−

)

∂ζ

(

I1

(

ζ

,

)

+ I2

(

ζ

,

)

)

−

γ

˜

(

−

)

I1

(

ζ

,−

)

+ I2

(

ζ

,−

)

·

−B

(

−

)

−iGn

()

+

∂φ

n

(

_∂ζ

ζ

,−

)

I1

(

ζ

,

)

+ I2

(

ζ

,

)

−

γ

˜

(

−

)

(

I1

(

ζ

,−

)

+ I2

(

ζ

,−

)

)

. (A.11)

Eq.(13)andEqs.(A.5)–(A.7)leadto

I1

(

ζ

,

)

I2

(

ζ

,

)

=

I1

(

ζ

,

)

I2

(

ζ

,−

)

= 0, (A.12)

I1

(

ζ

,

)

I1

(

ζ

,

)

=

I2

(

ζ

,

)

I2

(

ζ

,−

)

= 0, (A.13)

|

I1

(

ζ

,

)

|

2

= +∞

−∞ 4

|

1

(

u

)

|

2

|

n−1

(

u−

)

|

2

du, (A.14)

|

I1

(

ζ

,−

)

|

2

= +∞

−∞ 4

|

1

(

u

)

|

2

|

n−1

(

u+

)

|

2

du, (A.15)

|

I2

(

ζ

,

)

|

2

= +∞

−∞

|

1

(

u

)

|

2

|

n−1

(

−u

)

|

2

du, (A.16)

|

I2

(

ζ

,−

)

|

2

= +∞

−∞

|

1

(

u

)

|

2

|

n−1

(

−

−u

)

|

2

du. (A.17)

Usingsimplification#2inSection2.2(i.e,assumethat

|

n(

)|2 areevenfunctions),wemaywrite

|

I1

(

ζ

,

)

|

2

+

|

I2

(

ζ

,

)

|

2

=

|

I1

(

ζ

,−

)

|

2

+

|

I2

(

ζ

,−

)

|

2

= 5 +∞

−∞

|

1

(

u

)

|

2

|

n−1

(

u−

)

|

2

du. (A.18)

Finally,usingapproximation#6inSection2.2,

|

J

()

|

2

|

n

()

|

2

=

γ

˜2

₍₎

_·

_|

_B

₍

₋

₎

_−iG

n

()

|

2+

γ

˜2

(

−

)

·5 +∞

−∞

|

1

(

u

)

|

2

|

n−1

(

u−

)

|

2

du. (A.19)

Although we may incur in an error, we approximate

|J(

)|2_|

n(

)|2 ≈

|J(

)|2

|

n(

)|2 . Using the fact that

∂

ζ

φ

n

(ζ

,

)

=0 andneglecting higher-order moments of

∂

_ζ

φ

n(

ζ

, ±

) (i.e., simplifying assumption #6 in Section 2.2),

weobtain

|

J

()

|

2

=

|

(

B

()

+ iGn

()

)

(

B

()

−iGn

()

)

−

γ

˜

()

γ

˜

(

−

)

|

2. (A.20)

IntroducingEq.(A.20)inEq.(A.19),

|

n

()

|

2

≈

γ

˜

2

₍₎

_·

_|

_B

₍

₋

₎

_−iG

n

()

|

2+

γ

˜2

(

−

)

|

(

B

()

+ iGn

()

)

(

B

()

−iGn

()

)

−

γ

˜

()

γ

˜

(

−

)

|

2

·5 +∞

−∞

|

1

(

u

)

|

2

|

n−1

(

u−

)

|

2

Bythewaywedefined

1(seeEq.(14)),theintegralisactuallyadefiniteintegral.Sincetheintegrandsarenonnegative,by themean-valuetheoremwemaywrite

+∞

−∞

|

1

(

u

)

|

2

|

n−1

(

u−

)

|

2

du=

|

n−1

(

c

())

|

2

+∞

−∞

|

1

(

u

)

|

2

du. (A.22)

Thisequationmotivatesourlastsimplification.Letusdefine

n =5

+∞

−∞

|

1

(

u

)

|

2

|

n−1

(

u−

)

|

2

du. (A.23)

Weassumethat

n ≈

α

n−1 forn>1,

1 = 1. (A.24)

Usingthisapproximation,wehave

|

n

()

|

2

≈

α

n−1_·

γ

˜

2

()

_·

|

_B

(

₋

)

_−iG

n

()

|

2+

γ

˜2

(

−

)

|

(

B

()

+iGn

()

)

(

B

()

−iGn

()

)

−

γ

˜

()

γ

˜

(

−

)

|

2

. (A.25)

Inpractice,

α

mayhelp compensatesomeoftheapproximationsinthederivationofEq.(A.25)andneedstobe estimated foreachparticularscenario.

References

[1] AgrawalG. Nonlinearfiberoptics.OpticsandPhotonics. 5thed. AcademicPress; 2012.

[2] ZakharovVE. Collapseoflangmuirwaves.SovietPhysJETP1972;35:908–14.

[3] AblowitzMA, ClarksonPA. Solitons,nonlinearevolutionequationsandinversescattering.CambridgeUniversityPress;1991.ISBN978-0521387309.

[4] AkhmedievN, KorneevV. ModulationinstabilityandperiodicsolutionsofthenonlinearSchrödingerequation.TheorMathPhys1986;69(2):1089– 93.

[5] DudleyJM,GentyG,DiasF,KiblerB,AkhmedievN.Modulationinstability,Akhmedievbreathersandcontinuouswavesupercontinuumgeneration. OptExpress2009;17(24):21497–508.doi:10.1364/OE.17.021497.

[6] AkhmedievN,Soto-CrespoJM,AnkiewiczA.Howtoexcitearoguewave.PhysRevA2009;80:043818.doi:10.1103/PhysRevA.80.043818.

[7] AkhmedievN,AnkiewiczA,TakiM.Wavesthatappearfromnowhereanddisappearwithoutatrace.PhysLettA2009;373(6):675–8.doi:10.1016/j. physleta.2008.12.036.

[8] AnkiewiczA,Soto-CrespoJM,ChowdhuryMA,AkhmedievN. Roguewavesinopticalfibersinpresenceofthird-orderdispersion, self-steepening,andself-frequencyshift.JOptSocAm B2013;30(1):87–94. doi:10.1364/JOSAB.30.000087.

[9] AnkiewiczA,AkhmedievN. Higher-orderintegrable evolutionequation and itssolitonsolutions. PhysLett A2014;378(4):358–61. doi:10.1016/j. physleta.2013.11.031.

[10]AnkiewiczA,WangY,WabnitzS,AkhmedievN.ExtendednonlinearSchrödingerequationwithhigher-orderoddandeventermsanditsroguewave solutions.PhysRevE2014;89:012907.doi:10.1103/PhysRevE.89.012907.

[11]AnkiewiczA,KedzioraDJ,ChowduryA,BandelowU,Akhmediev N.InfinitehierarchyofnonlinearSchrödingerequations andtheirsolutions. PhysRevE2016;93:012206. doi:10.1103/PhysRevE.93.012206.

[12]HultJ.Afourth-orderrunge-kuttaintheinteraction picturemethod forsimulatingsupercontinuumgenerationinopticalfibers.JLightwave Technol2007;25(12):3770–5.doi:10.1109/JLT.2007.909373.

[13]BenjaminTB,FeirJE.Thedisintegrationofwavetrainsondeepwaterpart1.theory.JFluidMech1967;27:417–30.doi:10.1017/S002211206700045X.

[14]HasegawaA.Observationofself-trappinginstabilityofaplasmacyclotronwaveinacomputerexperiment.PhysRevLett1970;24:1165–8.doi:10.1103/ PhysRevLett.24.1165.

[15]ZakharovV, ShabatA. Exacttheoryoftwo-dimensionalself-focusingandone-dimensionalself-modulationofwavesinnonlinearmedia.Soviet PhysJETP1972;34:62–9.

[16]HasegawaA,BrinkmanW.TunablecoherentIRandFIRsourcesutilizingmodulationalinstability.IEEEJQuantumElectron1980;16(7):694–7.doi:10. 1109/JQE.1980.1070554.

[17]JanssenPAEM.Modulationalinstabilityandthefermi-pasta-ulamrecurrence.PhysFluids1981;24(1):23–6.doi:10.1063/1.863242.

[18]AndersonD,LisakM.Modulationalinstability ofcoherentoptical-fibertransmissionsignals.Opt Lett1984;9(10):468–70.doi:10.1364/OL.9.00 0468.

[19]ShuklaPK,RasmussenJJ. Modulationalinstabilityofshortpulsesinlong opticalfibers.OptLett1986;11(3):171–3.doi: 10.1364/OL.11.000171.

[20]TaiK, HasegawaA,TomitaA.Observation ofmodulationalinstability inopticalfibers.PhysRevLett1986;56:135–8.doi:10.1103/ PhysRevLett.56.135.

[21]PotasekMJ.ModulationinstabilityinanextendednonlinearSchrödingerequation.OptLett1987;12(11):921–3.doi:10.1364/OL.12.000921.

[22]ErkintaloM,HammaniK, KiblerB,FinotC,Akhmediev N,DudleyJM, etal.Higher-ordermodulationinstability innonlinear fiberoptics.Phys RevLett2011;107:253901.doi:10.1103/PhysRevLett.107.253901.

[23]SolliD, HerinkG, JalaliB, RopersC. Fluctuationsandcorrelationsinmodulationinstability.NatPhotonics2012;6(7):463–8.

[24]GroszD,MazzaliC,CelaschiS,ParadisiA,FragnitoH.Modulationinstabilityinducedresonantfour-wavemixinginWDMsystems.IEEE PhotonicsTechnolLett1999;11(3):379–81.doi:10.1109/68.748242.

[25]Grosz D, Boggio JC, Fragnito H. Modulation instability effects on three-channel optically multiplexed communication systems. Opt Commun 1999;171(1–3):53–60.doi:10.1016/S0030-4018(99)00494-0.

[26]HammaniK,WetzelB,KiblerB,FatomeJ,FinotC,MillotG,etal.SpectraldynamicsofmodulationinstabilitydescribedusingAkhmedievbreather theory.OptLett2011;36(11):2140–2.doi:10.1364/OL.36.002140.

[27]SørensenST,LarsenC,MøllerU,MoselundPM,ThomsenCL,BangO.Influenceofpump powerandmodulation instabilitygainspectrumon seededsupercontinuum androguewavegeneration.JOptSocAm B2012;29(10):2875–85.doi: 10.1364/JOSAB.29.002875.

[28]Soto-CrespoJM,AnkiewiczA,DevineN,AkhmedievN.Modulationinstability,cherenkovradiation,andfermi-pasta-ulamrecurrence.JOptSocAm B2012;29(8):1930–6.doi:10.1364/JOSAB.29.001930.

[29]ZakharovVE,GelashAA.Nonlinearstageofmodulationinstability.PhysRevLett2013;111:054101.doi:10.1103/PhysRevLett.111.054101.

[30]BéjotP,KiblerB,HertzE,LavorelB,FaucherO.Generalapproachtospatiotemporalmodulationalinstabilityprocesses.PhysRevA2011;83:013830. doi:10.1103/PhysRevA.83.013830.

[31]HernandezSM,FierensPI,BonettiJ, SánchezAD,GroszDF.A geometricalview ofscalarmodulationinstability inopticalfibers.IEEE Photonics J2017;9(5):1–8.doi:10.1109/JPHOT.2017.2754984.

[33]BonettiJ,HernandezSM,FierensPI,GroszDF.Analyticalstudyofcoherenceinseededmodulationinstability.PhysRevA2016;94:033826.doi:10. 1103/PhysRevA.94.033826.

[34]ZakharovV,DiasF,PushkarevA.One-dimensionalwaveturbulence.PhysRep2004;398(1):1–65.doi:10.1016/j.physrep.2004.04.002.

[35]PicozziA,PitoisS,MillotG.Spectralincoherentsolitons:alocalizedsolitonbehaviorinthefrequencydomain.PhysRevLett2008;101:093901.doi:10. 1103/PhysRevLett.101.093901.

[36]PicozziA, RicaS. CondensationofclassicalopticalwavesbeyondthecubicnonlinearSchrödingerequation.OptCommun2012;285(24):5440–8.

[37]PicozziA,GarnierJ,HanssonT,Suret P,RandouxS,MillotG,etal.Optical waveturbulence:towardsaunified nonequilibriumthermodynamic formu-lation ofstatistical nonlinearoptics.PhysRep2014;542(1):1–132.doi:10.1016/j.physrep.2014.03.002.

[38]Soto-CrespoJM,DevineN,AkhmedievN.Integrableturbulenceandroguewaves:breathersorsolitons?PhysRevLett2016;116:103901.doi:10.1103/ PhysRevLett.116.103901.

[39]KimmounO,HsuHC,KiblerB,ChabchoubA.Nonconservativehigher-orderhydrodynamicmodulationinstability.PhysRevE2017;96(2):022219.doi:10. 1103/physreve.96.022219.

[40]Armaroli A, Brunetti M, Kasparian J. Recurrence in the high-order nonlinear Schrödinger equation: a low-dimensional analysis. Phys Rev E 2017;96:012222.doi:10.1103/PhysRevE.96.012222.

[41]KuoBP-P,Fini JM,Grüner-NielsenL,RadicS.Dispersion-stabilized highly-nonlinearfiberfor widebandparametric mixersynthesis.OptExpress 2012;20(17):18611–19. doi:10.1364/OE.20.018611.

[42]TraversJC, FroszMH, DudleyJM. Nonlinearfibreoptics overview.In:Dudley JM,TaylorJR,editors. Supercontinuumgeneration inoptical fibers.CambridgeUniversityPress;2010.p.32–51.

[43]GanapathyR,MalomedBA,PorsezianK. Modulationalinstabilityand generationofpulsetrains inasymmetric dual-corenonlinearopticalfibers. PhysLettA2006;354(5–6):366–72.doi:10.1016/j.physleta.2006.02.002.