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1

Weak Langmuir turbulence in disordered multimode optical fibers

Kilian Baudin1, Josselin Garnier2, Adrien Fusaro1,3, Nicolas Berti1, Guy Millot1, Antonio Picozzi11 Laboratoire Interdisciplinaire Carnot de Bourgogne,

CNRS, Universite Bourgogne Franche-Comte, Dijon, France2 CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France and

3 CEA, DAM, DIF, F-91297 Arpajon Cedex, France

We consider the propagation of temporally incoherent waves in multimode optical fibers (MMFs)in the framework of the multimode nonlinear Schrodinger (NLS) equation accounting for the im-pact of the natural structural disorder that affects light propagation in standard MMFs (randommode coupling and polarization fluctuations). By averaging the dynamics over the fast disorderedfluctuations, we derive a Manakov equation from the multimode NLS equation, which reveals thatthe Raman effect introduces a previously unrecognized nonlinear coupling among the modes. Ap-plying the wave turbulence theory on the Manakov equation, we derive a very simple scalar kineticequation describing the evolution of the multimode incoherent waves. The structure of the kineticequation is analogous to that developed in plasma physics to describe weak Langmuir turbulence.The extreme simplicity of the derived kinetic equation provides physical insight into the multimodeincoherent wave dynamics. It reveals the existence of different collective behaviors where all modesself-consistently form a multimode spectral incoherent soliton state. Such an incoherent soliton canexhibit a discrete behavior characterized by collective synchronized spectral oscillations in frequencyspace. The theory is validated by accurate numerical simulations: The simulations of the general-ized multimode NLS equation are found in quantitative agreement with those of the derived scalarkinetic equation without using adjustable parameters.

I. INTRODUCTION

Multimode optical fibers (MMF) constitute ideal test-beds for the study of complex spatio-temporal nonlinearoptical phenomena. The phenomena that can be testedinclude multi-octave spanning supercontinuum genera-tion involving intense visible frequency combs, multiplefilamentation processes, or multimode solitons [1–6]. Ac-tually, light dynamics in MMFs involves a variety of non-linear effects whose complexity requires a deep under-standing of spatiotemporal nonlinear propagation, witha multitude of applications such as the improvement ofoptical signal processing techniques for spatial divisionmultiplexing [7], or the development of novel high-energyversatile fibre sources [8].

Aside from potential applications, MMFs also providea natural platform for the study of the interplay of nonlin-earity and disorder [9–11], which is a fundamental prob-lem of general interest [12–16]. As a matter of fact, lightpropagation in a conventional MMF is known to be af-fected by a structural disorder of the material due toinherent imperfections and external perturbations (e.g.,bending, twisting, tensions, or core-size variations in thefabrication process) [7], a feature which is relevant toendoscopic imaging for instance [17]. When such a nat-ural disorder of the fiber dominates over nonlinear ef-fects, the nonlinear Schrodinger (NLS) equation describ-ing the propagation of light can be reduced to an effectiveequation through the so-called Manakov approximation,a procedure originally developed for single-mode fibers[18] and more recently extended to MMFs [19–24].

From a different perspective, a fundamental phe-nomenon of spatial beam self-organization, termed“beam self-cleaning”, has been recently discovered in

(graded-index) MMFs [25, 26]. At variance with an ap-parently similar phenomenon driven by the dissipativeRaman effect in MMFs, known as Raman beam cleanup[27], this self-organization process is due to a purely con-servative Kerr nonlinearity [26]. While the detailed un-derstanding of spatial beam cleaning is still debated, dif-ferent works indicate that certain regimes of beam self-cleaning can be described as a natural process of opti-cal wave thermalization to thermal equilibrium [28–32],a feature that has been recently demonstrated experi-mentally [33–35]. This has motivated the developmentof a wave turbulence formalism that takes into accountthe structural disorder inherent to light propagation inMMFs. Following this approach, a wave turbulence ki-netic equation has been derived, which revealed that thestructural disorder leads to a significant acceleration ofthe process of thermalization and condensation, a fea-ture that can help to understand some regimes of spatialbeam-cleaning in MMFs [29].

Our aim in this paper is to study the interplay ofdisorder and nonlinearity in the framework of a differ-ent regime of light propagation in MMFs. At variancewith the previous theoretical works describing the purelyspatial dynamics[28, 29], here we consider the spatio-temporal multimode dynamics where temporally incoher-ent waves propagate through the MMF. On the basis ofthe wave turbulence theory [36–44], we show that thetemporal multimode turbulent dynamics is dominated bythe Raman effect. More precisely, under the assumptionthat the structural disorder dominates over nonlinear ef-fects, we derive a multimode Manakov equation from themultimode NLS equation. The new Raman term in thisManakov equation unveils a previously unrecognized non-trivial coupling among the modes, which is responsible for

2

a collective behavior of the multimode incoherent field.Indeed, applying the wave turbulence theory to the mul-timode Manakov equation, we derive a simple scalar ki-netic equation that governs the evolution of the temporalaveraged spectrum of the multimode optical field. Thekinetic equation has a form analogous to that developedin plasma physics to describe weak Langmuir turbulencein plasma [44–47]. The derived kinetic equation thengreatly simplifies the multimode NLS equation and pro-vides physical insight into the incoherent wave dynamics.It reveals the existence of several multimode collective be-haviors of the incoherent waves that propagate throughthe MMF. As a general rule, the multimode turbulentsystem exhibits a self-organization process, in which allmodes self-consistently form a vector spectral incoher-ent soliton (VSIS). This provides a generalization of thescalar (or bimodal) spectral incoherent solitons that werepreviously investigated by always ignoring the structuraldisorder of the fiber [6, 41, 48–50]. The reported VSIScan also exhibit a discrete behavior, which is character-ized by collective synchronized spectral oscillations of thediscrete soliton in frequency space. The numerical sim-ulations of the generalized multimode NLS equation arefound in remarkable quantitative agreement with the de-rived kinetic equation without using adjustable parame-ters.From a broader perspective, we remark that Lang-

muir turbulence in the strongly nonlinear regime hasbeen widely studied both theoretically and experimen-tally [51, 52], in particular in hydrodynamics [52–55],or in laboratory [56, 57] and space plasma experiments[58, 59], while cavitating Langmuir turbulence has beenevidenced in natural Earth’s aurora driven by solar wind[60]. However, aside from preliminary experiments in[50], experimental evidence of weak Langmuir turbulencehas not been reported in the context of nonlinear op-tics [41]. In this work we show that random mode cou-pling in optical fibers has a stabilizing role on the dy-namics of spectral incoherent solitons, which makes dis-ordered MMFs promising for the experimental study ofweak Langmuir turbulence in optics.

II. MULTIMODE NLS EQUATION

We consider the generalized NLS equation describingthe propagation of the optical field in a multimode fiberwith N modes (i.e., 2N modes accounting for polariza-tion effects) [61]. Following the notations of Ref.[61], thevector electric field can be expanded into a superpositionof the individual modes E(r, z, t) =

∑p Fp(r)Ap(z, t),

where Fp(r) denotes the normalized transverse vectormode profile and Ap(z, t) the modal envelope with z thelongitudinal propagation variable, r = (x, y) the vectorin the transverse plane and t the time variable. Themodal vector can be written A(z, t) = (Ap(z, t))

2Np=1,

where the components A2j−1, A2j refer to the orthog-onal linear polarization components of the j−th mode.

The field satisfies the generalized multimode NLS equa-tion [61]:

i∂zA+D(z)A+D0A+ iV∂tA−W∂ttA

+γ(1− fR)P (A) + γfRQ(A) = 0. (1)

Here, D0, V and W are deterministic 2N × 2N diagonalmatrices that model respectively the propagation con-stants, the modal inverse group velocity and the modaldispersion (relative to the fundamental fiber mode). Theterms P (A) and Q(A) are, respectively, the Kerr nonlin-earity and the Raman nonlinearity, which have the gen-eral forms

[P (A)

]p=

2N∑

l,m,n=1

SKplmnAlAmA∗

n, (2)

[Q(A)

]p=

2N∑

l,m,n=1

SRplmnAl[R ⋆ (AmA∗

n)], (3)

where R is the Raman response function, ∗ stands forcomplex conjugation and ⋆ denotes the convolution prod-uct. The Raman term contributes with a fraction fR tothe overall nonlinearity (fR = 0.18 for silica glass fibres)[6]. The nonlinear coefficient is γ = n2ω0/c = n2k0,where ω0 is the laser carrier frequency and λ = 2π/k0the corresponding wavelength. The tensors SK

plmn and

SRplmn are given in explicit form in Ref.[61].We consider the regime of strong random coupling

among the spatial modes and polarization states, which isrelevant for large propagation lengths in the MMF, typ-ically larger than a few hundred meters [62]. The mostgeneral form of random mode coupling that conserves the

total power P =∑2N

p=1 |Ap|2 is provided by a 2N × 2N

random matrix-valued process D(z) that is Hermitian.Note that the structural disorder of the MMF may alsoaffect the group-velocity and the group-velocity disper-sion of the propagating modes, which can be modelled byconsidering random matrices V(z) and W(z) in Eq. (9)as will be discussed later.

III. MANAKOV REDUCTION

We consider the so-called Manakov regime where theimpact of strong linear random coupling dominates overnonlinear effects [21], i.e., Lnl ≫ ℓc, 2π/σ, where Lnl =1/(γP) is the nonlinear length, while ℓc and σ denotethe correlation length and standard deviation of the ran-dom process D(z) that models the structural disorder.We recall that the Manakov reduction has already beenapplied to the multimode NLS equation without the Ra-man effect [19–21, 23]. On the other hand, the Manakovapproximation has been considered to study the Ramanamplification process in Ref.[22], and applied to the mul-timode NLS equation accounting for the first-order cor-rection of the Raman response [24]. Here, we generalizethe derivation of the Manakov equation, which reveals

3

that the Raman effect introduces a nontrivial couplingamong the modes that plays a key role for the incoherentpropagation regime.Let us introduce the unitary matrix U(z) solution

of i∂zU = DU, with U(0) = I, and define themode amplitudes in the local disordered axes B(z, t) =U

−1(z)A(z, t), with U−1(z) = U

†(z), where the super-script † denotes the conjugate transpose. We now fol-low the idea originally introduced for single mode fibersby Wai and Menyuk [18], in which birefringence fluctua-tions are assumed so strong that the probability densityof the polarization state uniformly covers the surface ofthe Poincare sphere, so that one can average the propaga-tion equation over all polarization states. By generalizingto MMFs, we assume that the linear coupling among themodes due to D(z) is the dominant effect, so that therandom matrix-valued process U(z) becomes uniformlydistributed in the set of unitary matrices. In this way, wederive in the Appendix the following homogenized Man-akov multimode NLS equation:

i∂zB + dB +i

v∂tB − β

2∂ttB

+γ(1− fR)P (B) + γfRQ(B) = 0, (4)

with

[P (B)

]p=(SK(1) + SK

(2)

)[ 2N∑

l=1

|Bl|2]Bp, (5)

[Q(B)

]p=SR

(1)

2N∑

l=1

Bl

[R ⋆ (BpB

∗l )]

+ SR(2)Bp

[R ⋆

( 2N∑

l=1

|Bl|2)]

, (6)

where we have for X ∈ {K,R}

SX(1) =

1

4N2 − 1

∑

p′,l′

SXp′l′p′l′ −

1

2N(4N2 − 1)

∑

p′,l′

SXp′p′l′l′ ,

SX(2) =

1

4N2 − 1

∑

p′,l′

SXp′p′l′l′ −

1

2N(4N2 − 1)

∑

p′,l′

SXp′l′p′l′ .

As already discussed in previous works [19–21], Eq.(5) shows that any instantaneous cubic nonlinearitygives rise to an effective (phase-insensitive) determinis-tic Manakov-type nonlinear term. Eq. (6) also shows thecontribution of the Raman effect, which gives rise to aneffective Raman-type nonlinear term that depends only

on two parameters, SR(1) and SR

(2).

Note that, as a result of the Manakov averaging pro-cedure, the propagation constant, the group velocity andthe dispersion coefficients have been homogenized in (4):

d =1

2NTr(D0),

1

v=

1

2NTr(V), β =

1

NTr(W).

Accordingly, all of the modes evolve with the same propa-gation constant, group velocity and dispersion coefficient.

Note however that this results from the assumption thatthe matrices V and W in (1) are deterministic and con-stant. If the matrix V(z) in (1) was randomly vary-ing, with random fluctuations that could be correlatedto those of D(z), then the transport term in (4) would

have the form V∂tB instead of 1v∂tB, with the effec-

tive matrix V =⟨U

†(z)V(z)U(z)⟩, where 〈·〉 stands for

the average with respect to the stationary distribution of(U(z),V(z)). The same argument holds for the disper-sion effects W, and the corresponding effective matrix

W =⟨U

†(z)W(z)U(z)⟩.

We finally remark that the nonlinear Raman-typeterms in Eq. (4) are different from those reported in [24],because in this latter work the authors made use of the as-sumption that the nonlinear terms are co-polarized withthe field, which is not justified in general. In particular,when one considers the propagation of incoherent waves,the terms that are not co-polarized (i.e., the ones asso-

ciated with SR(1) in Eq.(6)) are the only ones that give

rise to a coupling among the modes. These terms shouldnot be neglected in our framework, a feature that willbecome apparent from the weak Langmuir turbulence ki-netic equation discussed in the next section.

IV. WEAK LANGMUIR TURBULENCE

KINETIC EQUATION

In the following we derive the weak Langmuir turbu-lence kinetic equation governing the evolution of the av-eraged spectra of the incoherent waves that propagatein the MMF. For this purpose, we consider the weaklynonlinear regime where linear dispersion effects dominateover nonlinear effects Llin,j ≪ Lnl, where Llin,1 = tcv

and Llin,2 = 2t2c/β denote the characteristic propagationlengths associated to the first- and second-order disper-sion effects in the modal NLS Eq. (1), and tc is the cor-relation time of the incoherent waves.We are interested in the propagation of incoherent

waves, whereAp(z = 0, t) are random functions with fluc-tuations that are statistically stationary in time. Thenthe components Bp(z = 0, t) are also random functionswith statistically stationary fluctuations. By taking anaverage over the random initial conditions Bp(z = 0, t),we can derive the wave turbulence Langmuir kineticequation by following the procedure of Ref.[41]. Next,taking the Fourier transform, the spectra nBp

(ω, t, z) =∫ ⟨Bp(z, t+ τ/2)B∗

p(z, t− τ/2)⟩exp(−iωτ)dτ , satisfy

the multimode weak Langmuir turbulence kineticequations:

∂znBp(ω, z) = γ1nBp

(ω)

2N∑

j=1

∫g(ω − u)nBj

(u)du

+γ2nBp(ω)

∫g(ω − u)nBp

(u)du, (7)

where γj = γfRSR(j)/π for j = 1, 2, and g(ω) =

4

ℑ[R(ω)] denotes the Raman gain function, R(ω) =∫∞

0R(t) exp(−iωt)dt being the Fourier transform of the

response function. Note that the Raman gain g(ω) is anodd function (see the inset of Fig. 1(a)), reflecting the factthat the low-frequency components are amplified to thedetriment of the high-frequency components. We haveomitted to write the time dependence for the evolutionof the spectrum in Eq.(7), i.e., nBp

(ω, t, z) → nBp(ω, z).

Indeed, since the initial condition exhibits a stationarystatistics and the kinetic Eq.(7) does not explicitly in-volve the time variable t, then the stationary statistics ispreserved during the propagation and the averaged spec-trum does not depend on time.It becomes apparent in the kinetic Eq.(7) that only the

first term proportional to γ1 = γfRSR(1)/π gives rise to

a nonlinear coupling among the modes. In other terms,only the new terms in the derived Manakov equation (i.e.,

the ones associated with SR(1) in Eq.(6)) give rise to a

mode coupling in the kinetic Eq.(7).It is important to note that the initial conditions in

the basis Bp can be homogenized. Indeed, let the fieldpropagates over few correlation lengths ℓc, in such away that the matrix U becomes uniformly distributedwhile other linear and nonlinear effects are still negli-gible. Under such circumstances, the initial conditionsfor Bp(z, t) are such that 〈Bp(z = 0+, t)B∗

l (z = 0+, t′)〉 =δpl

12N

∑2Nj=1

⟨Aj(z = 0, t)A∗

j (z = 0, t′)⟩, where δpl = 1

if p = l and 0 otherwise. Taking the Fourier trans-form, we obtain a homogeneous initial condition for thespectra nBp

(ω, z = 0+) = 12N

∑j nAj

(ω, z = 0) forp = 1, . . . , 2N . An other important point to stress isthat this homogeneous initial condition is preserved dur-ing the propagation: As a consequence of the averag-ing Manakov procedure, the modal coupling coefficientsγ1 and γ2 in the kinetic Eqs.(7) are identical for all themodes, so that the spectra nBp

(z, ω) verify

nBp(ω, z) = nB(ω, z) for p = 1, . . . , 2N.

The multimode kinetic Eqs. (7) then reduce to a singlescalar kinetic equation:

∂znB(ω, z) = (2Nγ1 + γ2)nB(ω)

∫g(ω − u)nB(u)du,(8)

with nB(ω, z = 0+) = 12N

∑j nAj

(ω, z = 0).Following a similar argument, the spectra

nAp(z, ω) =

∫ ⟨Ap(z, t+ τ/2)A∗

p(z, t− τ/2)⟩e−iωτdτ

verify nAp(z, ω) = 1

2N

∑2Nj=1 nBj

(z, ω) = nB(ω, z) forany p and for z ≫ ℓc, even if the initial modal spectranAp

(z = 0, ω) are different from each other, see Ap-pendix. Then we arrive at the main conclusion that theaveraged spectra in the original basis A verify the scalarweak Langmuir turbulence kinetic equation:

∂znA(ω, z) = (2Nγ1 + γ2)nA(ω)

∫g(ω − u)nA(u)du.(9)

We stress the remarkable simplicity of the kinetic Eq. (9)as compared to the original multimode NLS Eq. (1).

First, as in the usual scalar case, both effects of lin-ear dispersion and instantaneous Kerr nonlinearity donot enter the kinetic equation, a property that has beenconfirmed by several previous works in different circum-stances [41]. Secondly, the structural disorder leads toan effective homogeneization that is characterized by anequipartition of the power among the modes, which fur-ther simplifies the vector kinetic Eq. (7) to the scalarkinetic Eq. (9). The kinetic Eq. (9) has two conservedquantities, the power P =

∫nA(z, ω)dω, and the ‘en-

tropy’ S =∫log[nA(z, ω)]dω [41].

Finally, we recall the formal analogy between the uni-versal form of the kinetic equation describing the weaklynonlinear regime of Langmuir turbulence [44] and thekinetic equations derived in this work. The formal math-ematical similarity mainly relies on the analogy betweenthe molecular vibrations mediated by the optical Ramaneffect in optical fibers and the excitations of ion-soundwaves mediated by the decay of plasma oscillations [41].

V. NUMERICAL SIMULATIONS

We have tested the validity of the theory by perform-ing numerical simulations of the original NLS Eq. (1)and of the derived scalar weak Langmuir turbulence ki-netic Eq. (9). We have considered a step-index bimodalfiber (core diameter 6µm, index difference ∆ = 0.005,wavelength λ = 1.55µm) in which the fundamental LP01mode is coupled to two degenerate modes LP01a andLP01b, which results in a total of 2N = 6 coupled equa-tions for the multimode NLS Eq. (1). As describedin the theory, we have considered the regime of strongrandom linear coupling among all modes, with vari-ance σ2 and correlation length ℓc. We have also con-sidered the standard form of the damped harmonic os-cillator Raman function in silica optical fibers, R(t) =H(t)(τ−2

1 + τ−22 )τ1 exp(−t/τ2) sin(t/τ1) with τ1 = 12.2fs

and τ2 = 32fs, H(t) being the Heaviside function (νR =1/(2πτ1) ≃ 13THz denoting the resonant Raman fre-quency) [6]. The corresponding Raman gain functionthen reads

g(ω) =1 + η2

2η

( 1

1 + (η + τ2ω)2− 1

1 + (η − τ2ω)2

), (10)

with the time ratio η = τ2/τ1.We consider a partially coherent optical field that is

injected in the MMF and populates different modes. Inthe following, for simplicity, we assume that the differ-ent modes are initially populated with partially coherentwaves with a Gaussian spectrum and random spectralphases, i.e., Ap(z = 0, t) has stationary Gaussian statis-tics with mean zero and Gaussian covariance functionand the modes p = 1, . . . , 2N are independent from eachother. Note that, because of the strong random couplingregime, this latter assumption is verified in practice aftera propagation length z ≫ ℓc that remains smaller thanthe nonlinear length z < Lnl. This latter property has

5

been verified by numerical simulations of the generalizedNLS Eq. (1).

A. Multimode discrete spectral incoherent soliton

In the following we illustrate different turbulentregimes of the system that depend on the spectral widthsof the launched optical field. Figure 1 reports a typicalexample of the evolution for a spectral width of the ini-tial condition of the order of ∆ν ≃ 15THz. Since thisspectral width is of the same order as the spectral widthof the Raman gain, the red-shift of the wave spectrumexhibits a discrete behavior, because the leading edge ofthe low-frequency tail of the spectrum exhibits a largergain as compared to the mean gain of the whole front ofthe spectrum (see the inset in Fig. 1(a)). As a result ofcascaded Raman scattering [6, 63], the spectrum exhibitsa discrete spectral shift that is determined by the Ramanfrequency νR. The remarkable result is that the globalspectral red-shift of the field is regular and exhibits a dis-crete soliton-like behavior. As already discussed in theliterature [50, 64], the discrete soliton propagates witha constant velocity in frequency space for arbitrary longdistances, without emitting apparent radiation. We re-call that the spectral incoherent soliton is ‘hidden’ in fre-quency space, in the sense that the soliton behavior can-not be identified in the temporal domain, where the fieldA(t, z) is a random wave featured by a stationary statis-tics [48]. In this respect, the VSISs are fundamentallydifferent in nature from optical solitons recently investi-gated in MMFs [65, 66]. Also note that a constant noisebackground has been added in the simulations. Such aspectral noise is important in order to sustain a steadyincoherent soliton propagation [41], otherwise the soli-ton would undergo a slow adiabatic reshaping so as toadapt its shape to the local value of the noise background.This noise background can also simulate the presence ofa quantum noise background.

We stress the remarkable quantitative agreement thathas been obtained between the simulation of the multi-mode NLS Eq.(1) and of the scalar kinetic Eq.(9), with-out using any adjustable parameter. Such a good agree-ment is clearly visible in the normal and logarithmicplots reported in Figs. 1(c)-(d) at a particular propa-gation length. In this simulation, the different modes areinitially populated with different amount of powers, asillustrated in Fig. 1(c) (gray solid lines). As expectedfrom the theory, we can observe in Fig. 1(e) that randommode coupling leads to an equipartition of power amongthe modes, after a propagation length of the order of thecorrelation length z & ℓc = 10cm.

(e)

(d)(c)

(a) (b)

NLS KIN

z = 0z = 0

FIG. 1: Multimode discrete spectral incoherent soli-ton: Evolution of the spectrum of the field during the prop-agation obtained by simulation of the generalized multimodeNLS Eq. (1) (a), and of the scalar weak Langmuir turbulencekinetic Eq. (9) (b). Spectral profiles in normal scale (c), andlogarithmic scale (d), at the propagation length z = 16km:The blue line reports the result of the simulation of the NLSEq. (1) (averaged over the 6 fiber modes), the red line reportsthe result of the simulation of the weak Langmuir turbulencekinetic Eq. (9). The gray solid lines in (c)-(d) report the ini-tial conditions of the six modes, while the dashed black linesthe corresponding average. (e) Evolution during the propa-gation (z is in log-scale) of the relative amount of power ofthe 6 fiber modes from the simulation of the generalized NLSEq. (1): Random mode coupling induces an equipartition ofpower among the 2N = 6 modes after a propagation lengthz & ℓc (ℓc = 10cm, σ = 63m−1, P = 17W). The inset in(a) shows the Raman gain spectrum g(ν), with ν = ω/(2π).The quantitative agreement between the NLS (Eq. (1)) andkinetic (Eq. (9)) simulations is obtained without adjustableparameters.

B. Multimode continuous spectral incoherent

soliton

In this sub-section we illustrate a turbulent regimecharacterized by the emergence of a continuous spectralincoherent soliton. Indeed, when the spectral width ofthe initial field becomes larger than the resonant Ramanfrequency, then the low-frequency tail of the spectrumsees a gain comparable to the mean gain of the spectralfront as a whole. In the example of Fig. 2 we have con-sidered a spectral width ∆ν ≃ 50THz, which is muchlarger than in Fig. 1. As a consequence, we can see inFig. 2(a)-(b) that the red-shift of the wave spectrum isno longer discrete, but continuous, then giving rise toa continuous VSIS behavior. We remark that, for thebroad spectral widths considered in Fig. 2, higher-orderterms should be included in the NLS model to accuratelydescribe light propagation in the fiber [61] (also see [67]).

6

However, our purpose here is just to provide a qualitativeoverview of different possible incoherent dynamics, whilea more realistic regime of light propagation in MMFs willbe considered in the next sub-section.The continuous spectral incoherent soliton reported in

Fig. 2 can be described theoretically as a stationary soli-ton solution of the Langmuir kinetic Eq. (9) [68]:

nsolA (ω) = n0

A + (nmA − n0

A) exp[− log

(nmA

n0A

) ω2

ω20

], (11)

where ω = ω−V z, n0A refers to the constant background

noise, nmA (≫ n0

A) is the soliton spectral amplitude, andω0 denotes the typical spectral width of Raman gain de-fined by ω0 =

√2[−∂ωg(0)]

−1/2[−∫∞

0g(ω)dω]1/2. The

soliton (11) propagates in frequency space with a con-stant velocity given by

V = −(2Nγ1 + γ2)

∫nA(ω)− n0

Adω∫log(nA(ω)/n0

A)dω

∫ωg(ω)dω.(12)

We can observe a remarkable agreement between the an-alytical soliton solution given by Eq.(11) and the numer-ical simulations of both the generalized NLS Eq.(1) andthe kinetic Eq.(9), as illustrated in Fig. 2(c)-(d). Notehowever in the logarithmic plot in Fig. 2(d) a discrep-ancy between the solution (11) and the simulations inthe tails of the soliton, a feature that can be explainedby the fact that (11) is valid in the vicinity of the solitonpeak. As a matter of fact, the computation of the solitonvelocity is very sensitive to the tails of the soliton pro-file, as revealed by the expression of V in Eq.(12), whosedenominator involves the logarithm of the soliton profile.Consequently, the computation of V with the analyticalsolution (11) matches the numerics qualitatively but notquantitatively, while a very good agreement of the solitonvelocity (12) with the numerics is obtained by consideringthe soliton profile generated in the simulation, as illus-trated by the dashed white lines in Fig. 2(a)-(b) that areparallel to the soliton trajectory.

C. Synchronization of incoherent spectral

oscillations

To complete our study, we consider a more realis-tic numerical simulation in which the incoherent sourcelaunched into the MMF is characterized by a relativelynarrow frequency bandwidth, ∆ν ≃ 2THz, which can beaccessible from an amplified spontaneous emission (ASE)source, see e.g. [69]. In addition, we have included theimpact of the fiber losses in the numerical simulations,with a typical value of 0.2dB/km.The results of the numerical simulations of the gener-

alized NLS Eq.(1) and the kinetic Eq.(9) are reported in

(a) (b)

(c) (d)

V V

VSISVSIS

NLS KIN

FIG. 2: Multimode continuous spectral incoherentsoliton: Evolution of the spectrum of the field during thepropagation obtained by simulation of the generalized NLSEq. (1) (a), and of the derived scalar weak Langmuir turbu-lence kinetic Eq. (9) (b). Spectral profiles in normal scale(c), and logarithmic scale (d), at the propagation lengthz = 26km: The blue line reports the result of the NLS Eq. (1)simulation (averaged over the 6 modes), the red line reportsthe result of the weak Langmuir turbulence kinetic Eq. (9)simulation. The dashed black-line reports the initial con-dition. Parameters are the same as in Fig. 1 (ℓc = 10cm,σ = 63m−1, P = 17W). The spectral width is larger thanin Fig. 1, which induces a continuous motion of the VSIS.The dashed green lines in (c) and (d) report the analyticalsoliton solution from Eq.(11). The dashed white lines in (a)and (b) denote the soliton velocity V from Eq.(12) (see thetext for details). The quantitative agreement between theNLS (Eq. (1)) and kinetic (Eq. (9)) simulations is obtainedwithout adjustable parameters.

Fig. 3(a)-(b). As a consequence of the narrowness of theinitial spectrum, the discrete nature of the spectral shiftgets more apparent as compared to the discrete VSIS dis-cussed above through Fig. 1. This reinforces the idea ofsynchronization of the spectral oscillations of the fibermodes in the simulation of the NLS Eq.(1). Indeed, wehave reported in Fig. 3(a) the spectrum averaged overthe six fiber modes. If the spectral oscillations werenot synchronized, then the average among the six modeswould be characterized by a significant spectral broaden-ing. The good agreement between the NLS and kineticsimulations shown in Fig. 3(a) and Fig. 3(b) reflects theaccurate synchronisation among the spectral oscillationsof the fiber modes. Note that the fiber losses naturallyinduce a significant reduction of the nonlinear effects dur-ing the propagation, which thus limits the spectral red-shift of the field. Accordingly, we have removed the fiberlosses in the simulation reported in Fig. 3(c)-(d), whichconsiderably improves the visualization of the synchro-nization of the spectral oscillations of the fiber modes.

We finally note a minor discrepancy between the sim-ulation of the generalized NLS Eq.(1) and the kineticEq.(9) in Figs. 3(e)-(f). This can be ascribed to the factthat the separation of scales between linear dispersioneffects and nonlinear effects is only partially satisfied be-

7

(c) (d)

(e) (f)

(a) (b)

NLS with

absorption

KIN with

absorption

NLS without

absorption

KIN without

absorption

FIG. 3: Synchronization of incoherent spectral oscil-lations: Evolution of the spectrum of the field during thepropagation obtained by simulation of the generalized NLSEq. (1) (a), and of the derived scalar weak Langmuir turbu-lence kinetic Eq. (9) (b). Fiber losses (0.2 dB/km) have beenincluded in (a)-(b). (c)-(d) report the simulation (a)-(b) butin the absence of the fiber losses, so as to improve the visu-alization of the multimode collective behavior of the spectraloscillations of the discrete soliton. Spectral profiles in normalscale (e), and logarithmic scale (f), at the propagation lengthz = 25km corresponding to (c)-(d): the blue line reports theresult of the NLS Eq. (1) simulation (avergaed over the 6 fibermodes), the red line reports the result of the weak Langmuirturbulence kinetic Eq. (9) simulation, the dashed black linethe initial condition. Parameters are ℓc = 2m, σ = 3.1m−1,P = 8.5W. The good agreement between the simulations ofthe NLS Eq. (1) and the kinetic Eq. (9) is obtained withoutadjustable parameters.

cause of the narrowness of the spectrum considered inFig. 3. Then at variance with the simulation reported inFig. 1 where Llin/Lnl ≃ 3.10−3, in the case of Fig. 3 weonly have Llin/Lnl ≃ 0.1, which can merely explain theslight discrepancy between the NLS and kinetic simula-tions observed in Figs. 3(e)-(f).

VI. CONCLUSION AND DISCUSSION

In summary, we have studied the propagation ofspatio-temporal incoherent waves in MMFs in the pres-ence of a random coupling among the modes. By aver-aging over the fast disordered fluctuations, we have de-rived the multimode Manakov Eq.(4-6). The new Ramanterm (6) in the multimode Manakov equation unveils acoupling among the modes, which is responsible for theemergence of a collective multimode behavior of the in-coherent waves. Indeed, applying the wave turbulence

theory to the multimode Manakov equation, we havederived a very simple scalar kinetic equation governingthe evolution of the averaged spectrum of the multimodefield. The theory has been validated by the numericalsimulations, which confirm the robustness of the processof modal attraction toward the dynamics described bythe scalar kinetic equation: A quantitative agreement be-tween the simulations of the NLS Eq. (1) and the kineticEq.(9) has been obtained, without using any adjustableparameters. The simulations reveal that the fields thatpropagate in different modes of the MMF self-organizeand self-trap to form a VSIS. In particular, the VSIScan exhibit a discrete behavior characterized by collec-tive synchronized spectral oscillations in frequency space.This work should stimulate the realization of optical ex-periments in MMFs. Aside from the discrete multimodespectral solitons, the reduction of the multimode NLSequation to the effective scalar kinetic Eq.(9) can be ex-ploited to study different turbulent regimes predicted inthe scalar case, such as the formation of incoherent spec-tral shock waves [70].

We recall that we have considered in this work thecase of strong random coupling among the modes. Asimilar analysis can be carried out by considering a weakrandom mode coupling, where only (quasi-)degeneratemodes are coupled to each other. Actually, weak randommode coupling is known to be relevant when relativelyshort propagation lengths in optical fibers are considered[62]. However, the validity of the kinetic approach re-quires a weak nonlinear regime, Llin ≪ Lnl, so that largepropagation lengths, typically larger than a few hundredmeters are required to observe the formation of multi-mode spectral incoherent solitons in optical fibers. Forsuch a large propagation length, it is commonly admit-ted that random coupling among non-degenerate modesshould not be neglected and must be taken into account[62], which legitimizes the consideration of strong modecoupling in our work.

We remark that the validity of the derived kineticEq.(9) becomes questionable when the optical spectrumfeels the presence of a zero-dispersion-frequency of theoptical fiber. Nearby a zero-dispersion-frequency, lineardispersive effects become perturbative. The dynamicsturns out to be dominated by nonlinear effects, whichinvalidates the weakly nonlinear assumption underlyingthe derivation of the kinetic equation. In this case oneneeds to include higher-order contributions in the clo-sure of the hierarchy of the moments equation in thewave turbulence theory. To next-order, the instantaneousKerr nonlinearity coupled to higher-order dispersion ef-fects leads to a collision term in the kinetic equation thatdescribes an incoherent (turbulent) regime of supercon-tinuum generation [41]. It would be interesting to de-velop a generalized kinetic formulation of spatio-temporaleffects in MMFs, which would unify the Langmuir formu-lation discussed here with the wave turbulence formula-tion accounting for random mode coupling discussed in[28, 29]. Such a generalized theory can also shed new

8

light on the recent experiments of supercontinuum gener-ation that can be characterized by spatial beam cleaningeffects [71–74]. From a broader perspective, this wouldcontribute to the development of a wave turbulence the-ory that accounts for the presence of a structural disorderof the nonlinear medium [28, 29, 75–77].

VII. ACKNOWLEDGEMENTS

We acknowledge financial support from the FrenchANR under Grant No. ANR-19-CE46-0007 (project

ICCI), iXcore research foundation, EIPHI Gradu-ate School (Contract No. ANR-17-EURE-0002),French program “Investissement d’Avenir,” Project No.ISITE-BFC-299 (ANR-15 IDEX-0003); H2020 MarieSklodowska-Curie Actions (MSCA-COFUND) (MULTI-PLY Project No. 713694). Calculations were performedusing HPC resources from DNUM CCUB (Centre de Cal-cul, Universite de Bourgogne).

Appendix A: Derivation of the Manakov equation

We derive the Manakov multimode NLS Eq. (4). Without approximations, the vector field B(z, t) is solution of

i∂zB +U†D

0UB + iU†

VU∂tB −U†WU∂2

tB + γ(1− fR)U†P (UB) + γfRU

†Q(UB) = 0, (A1)

where

[U

†D

0UB

]p

=∑

l

[ ∑

p′,n′

U∗p′pD

0p′n′Un′l

]Bl,

[U

†VU∂tB

]p

=∑

l

[ ∑

p′,n′

U∗p′pVp′n′Un′l

]∂tBl,

[U

†WU∂2

tB]p

=∑

l

[ ∑

p′,n′

U∗p′pWp′n′Un′l

]∂2tBl,

[U

†P (UB)]p

=∑

l,m,n

[ ∑

p′,l′,m′,n′

SKp′l′m′n′U∗

p′pUl′lUm′mU∗n′n

]BlBmB∗

n,

[U

†Q(UB)]p

=∑

l,m,n

[ ∑

p′,l′,m′,n′

SRp′l′m′n′U∗

p′pUl′lUm′mU∗n′n

]Bl[R ⋆ (BmB∗

n)].

We assume here that the linear coupling between modes due to D(z) is strong enough so that this effect dominatesand the random matrix-valued process U(z) becomes uniformly distributed in the set of unitary matrices. We canthen replace the linear and nonlinear terms by the homogenized coefficients

[U

†D

0UB

]p

=∑

l

[ ∑

p′,n′

⟨U∗p′pUn′l

⟩D0

p′n′

]Bl,

[U

†VU∂tB

]p

=∑

l

[ ∑

p′,n′

⟨U∗p′pUn′l

⟩Vp′n′

]∂tBl,

[U

†WU∂2

tB]p

=∑

l

[ ∑

p′,n′

⟨U∗p′pUn′l

⟩Wp′n′

]∂2tBl,

[U

†P (UB)]p

=∑

l,m,n

[ ∑

p′,l′,m′,n′

SKp′l′m′n′

⟨U∗p′pUl′lUm′mU∗

n′n

⟩ ]BlBmB∗

n,

[U

†Q(UB)]p

=∑

l,m,n

[ ∑

p′,l′,m′,n′

SRp′l′m′n′

⟨U∗p′pUl′lUm′mU∗

n′n

⟩ ]Bl[R ⋆ (BmB∗

n)],

where the expectation is taken with respect to the stationary distribution of the random processU(z), that is the Haarmeasure on the unitary group in dimension 2N . Integration with respect to the Haar measure on the unitary grouphas been studied in the mathematical physics literature for a long time [78, 79]. A general formula for calculating

9

monomial integrals is given in [80]. In the case of monomials of rank 2 and 4, we have [81, Prop. 4.2.3]:

⟨UijU

∗i′j′

⟩=

1

2Nδii′δjj′ , (A2)

⟨Ui1j1Ui2j2U

∗i′1j′1

U∗i′2j′2

⟩=

1

4N2 − 1

(δi1i′1δi2i′2δj1j′1δj2j′2 + δi1i′2δi2i′1δj1j′2δj2j′1

)

− 1

2N(4N2 − 1)

(δi1i′1δi2i′2δj1j′2δj2j′1 + δi1i′2δi2i′1δj1j′1δj2j′2

). (A3)

Using these formulas we find

[U

†D

0UB

]p

= dBp,

[U

†VU∂tB

]p

=1

v∂tBp,

[U

†WU∂2

tB]p

=β

2∂2tBp,

[U

†P (UB)]p

=∑

l,m,n

SKplmnBlBmB∗

n,

[U

†Q(UB)]p

=∑

l,m,n

SRplmnBl[R ⋆ (BmB∗

n)],

with

d =1

2NTr(D0), (A4)

1

v=

1

2NTr(V), (A5)

β =1

NTr(W), (A6)

SXplmn = δlnδmp

{ 1

4N2 − 1

∑

p′,l′

SXp′l′p′l′ −

1

2N(4N2 − 1)

∑

p′,l′

SXp′p′l′l′

}

+δlpδmn

{ 1

4N2 − 1

∑

p′,l′

SXp′p′l′l′ −

1

2N(4N2 − 1)

∑

p′,l′

SXp′l′p′l′

}, X ∈ {K,R},

or equivalently

SXplmn = SX

(1)δlnδmp + SX(2)δlpδmn, X ∈ {K,R},

with

SX(1) =

1

4N2 − 1

∑

p′,l′

SXp′l′p′l′ −

1

2N(4N2 − 1)

∑

p′,l′

SXp′p′l′l′ , X ∈ {K,R}, (A7)

SX(2) =

1

4N2 − 1

∑

p′,l′

SXp′p′l′l′ −

1

2N(4N2 − 1)

∑

p′,l′

SXp′l′p′l′ , X ∈ {K,R}. (A8)

In other words the mode amplitudes B satisfy

i∂zB + dB +i

v∂tB − β

2∂ttB + γ(1− fR)P (B) + γfRQ(B) = 0, (A9)

with

[P (B)

]p

=(SK(1) + SK

(2)

)[ 2N∑

l=1

|Bl|2]Bp, (A10)

[Q(B)

]p

= SR(1)

2N∑

l=1

Bl

[R ⋆ (BpB

∗l )]+ SR

(2)Bp

[R ⋆

( 2N∑

l=1

|Bl|2)]

. (A11)

10

We remark that, if the initial conditions in Eq. (4) are deterministic, then the components |Bp|2(z, t) are de-terministic and the components Ap(z, t) of the field are random and given by A = UB, so that they verify|Ap|2(z, t) =

∑j,k Upj(z)U

∗pk(z)Bj(z, t)B

∗k(z, t) and, using (A2),

⟨|Ap|2(z, t)

⟩= 1

2N

∑j |Bj |2(z, t) for any t and z

much larger than the correlation length of D(z), say ℓc. In other words, we have equipartition in the A basis. Notethat this result is obtained for the expectation (with respect to the distribution of (D(z))z≥0) of the components|Ap|2(z, t).We also remark that, if the initial conditions are random (and independent of (D(z))z≥0), then the compo-

nents |Bp|2(z, t) are governed by the deterministic Eq. (4) and the components Ap(z, t) are given by A = UB, sothat they verify |Ap|2(z, t) =

∑j,k Upj(z)U

∗pk(z)Bj(z, t)B

∗k(z, t) and, using (A2),

⟨|Ap|2(z, t)

⟩= 1

2N

∑j

⟨|Bj |2(z, t)

⟩

for any t and z much larger than ℓc. Following the same remark, we have⟨Ap(z, t+ τ/2)A∗

p(z, t− τ/2)⟩

=12N

∑j

⟨Bj(z, t+ τ/2)B∗

j (z, t− τ/2)⟩for any p for z ≫ ℓc.

[1] L. Wright, D. N. Christodoulides, and F.W. Wise, Con-trollable spatiotemporal nonlinear effects in multimodefibres, Nature Photon. 9, 306 (2015).

[2] W. H. Renninger and F. W. Wise, Optical solitons ingradedindex multimode fibres, Nature Comm. 4, 1719(2013).

[3] K. Krupa, A. Tonello, A. Barthelemy, V. Couderc, B.M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz,Observation of geometric parametric instability inducedby the periodic spatial self-imaging of multimode waves,Phys. Rev. Lett. 116, 183901 (2016).

[4] M. Conforti, C. Mas Arabi, A. Mussot, and A. Kudlin-ski, Fast and accurate modeling of nonlinear pulse prop-agation in graded-index multimode fibers, Opt. Lett. 42,4004 (2017).

[5] K. Krupa et al., Multimode nonlinear fiber optics, a spa-tiotemporal avenue, APL Photonics 4, 110901 (2019).

[6] G. Agrawal, Nonlinear Fiber Optics, (Sixth Ed., Aca-demic Press, New York, 2019).

[7] I.P. Kaminow, T. Li, and A.F. Willner, Optical Fiber

Telecommunications, Systems and Networks (Sixth Ed.,Elsevier, 2013).

[8] L. G. Wright, D. N. Christodoulides, and F.W. Wise,Spatiotemporal mode-locking in multimode fiber lasers,Science 358, 94 (2017).

[9] M. Leonetti, S. Karbasi, A. Mafi, and C. Conti, Lightfocusing in the Anderson regime, Nature Comm. 5, 4534(2014).

[10] W. Schirmacher, B. Abaie, A. Mafi, G. Ruocco, and M.Leonetti, What is the right theory for Anderson local-ization of light? An experimental test, Phys. Rev. Lett.120, 067401 (2018).

[11] A. Mafi, J. Ballato, K.W. Koch, and A. Schulzgen, Dis-ordered Anderson Localization Optical Fibers for ImageTransport - A Review, J. Lightwave Techn. 37, 5652-5659(2019).

[12] Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti,D.N. Christodoulides, and Y. Silberberg, Anderson Lo-calization and nonlinearity in one-dimensional disorderedphotonic lattices, Phys. Rev. Lett. 100, 013906 (2008).

[13] M. Segev, Y. Silberberg, and D.N. Christodoulides, An-derson localization of light, Nature Photonics 7, 197(2013).

[14] D. Pierangeli, A. Tavani, F. Di Mei, A.J. Agranat, C.Conti, and E. DelRe, Observation of replica symmetry

breaking in disordered nonlinear wave propagation, Na-ture Comm. 8, 1501 (2017).

[15] C. Conti and L. Leuzzi, Complexity of waves in nonlineardisordered media, Phys. Rev. B 83, 134204 (2011).

[16] F. Antenucci, M. Ibanez Berganza, and L. Leuzzi, Sta-tistical physics of nonlinear wave interaction, Phys. Rev.B 92, 014204 (2015).

[17] D. Psaltis and C. Moser, Imaging with multimode fibers,Opt. Photonics News 27, 24 (2016).

[18] P.K.A. Wai and C.R. Menyuk, Polarization mode dis-persion, decorrelation, and diffusion in optical with ran-domly varying birefringence, J. Lightwave Tech. 14, 148-157 (1996).

[19] A. Mecozzi, C. Antonelli, and M. Shtaif, Nonlinearpropagation in multimode fibers in the strong couplingregime, Optics Express 20, 11673 (2012).

[20] A. Mecozzi, C. Antonelli, and M. Shtaif, Coupled Man-akov equations in multimode fibers with strongly coupledgroups of modes, Optics Express 20, 23436 (2012).

[21] S. Mumtaz, R.-J. Essiambre, and G. P. Agrawal, Nonlin-ear propagation in multimode and multicore fibers: Gen-eralization of the Manakov equations, J. Lightwave Tech.31, 398-406 (2013).

[22] C. Antonelli, A. Mecozzi, and M. Shtaif, Raman amplifi-cation in multimode fibers with random mode coupling,Opt. Lett. 38, 1188 (2013).

[23] S. Buch, S. Mumtaz, R.-J. Essiambre, A. M. Tulino, andG. P. Agrawal, Averaged nonlinear equations for multi-mode fibers valid in all regimes of random linear coupling,Opt. Fiber Technol. 48, 123 (2019).

[24] C. Antonelli, M. Shtaif, and A. Mecozzi, Modeling of non-linear propagation in space-division multiplexed fiber-optic transmission, J. Lightwave Tech. 34, 36-54 (2016).

[25] Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W.Wise, Kerr self-cleaning of femtosecond-pulsed beams ingradedindex multimode fiber, Opt. Lett. 41, 3675 (2016).

[26] K. Krupa, A. Tonello, B. M. Shalaby, M. Fabert, A.Barthelemy, G. Millot, S. Wabnitz, and V. Couderc, Spa-tial beam self-cleaning in multimode fibres, Nat. Photon-ics 11, 237 (2017).

[27] N.B. Terry, T.G. Alley, and T.H. Russell, An explana-tion of SRS beam cleanup in graded-index fibers and theabsence of SRS beam cleanup in step-index fibers, OpticsExpress 15, 17509 (2007).

[28] A. Fusaro, J. Garnier, K. Krupa, G. Millot, and A. Pi-

11

cozzi, Dramatic acceleration of wave condensation medi-ated by disorder in multimode fibers, Phys. Rev. Lett.122, 123902 (2019).

[29] J. Garnier, A. Fusaro, K. Baudin, C. Michel, K. Krupa,G. Millot, and A. Picozzi, Wave condensation with disor-der versus beam self-cleaning in multimode fibers, Phys.Rev. A 100, 053835 (2019).

[30] E. Podivilov, D. Kharenko, V. Gonta, K. Krupa, O. S.Sidelnikov, S. Turitsyn, M. P. Fedoruk, S. A. Babin, andS. Wabnitz, Hydrodynamic 2D Turbulence and SpatialBeam Condensation in Multimode Optical Fibers, Phys.Rev. Lett. 122, 103902 (2019).

[31] F. O. Wu, A. U. Hassan, and D. N. Christodoulides,Thermodynamic theory of highly multimoded nonlinearoptical systems, Nat. Photon. 13, 776 (2019).

[32] A. Ramos, L. Fernandez-Alcazar, T. Kottos, and B.Shapiro, Optical Phase Transitions in Photonic Net-works: A Spin-System Formulation, Phys. Rev. X 10,031024 (2020).

[33] K. Baudin, A. Fusaro, K. Krupa, J. Garnier, S. Rica, G.Millot, and A. Picozzi, Classical Rayleigh-Jeans conden-sation of light waves: Observation and thermodynamiccharacterization, Phys. Rev. Lett. 125, 244101 (2020).

[34] K. Baudin, A. Fusaro, J. Garnier, N. Berti, K. Krupa, I.Carusotto, S. Rica, G. Millot, and A. Picozzi, Energy andwave-action flows underlying Rayleigh-Jeans thermaliza-tion of optical waves propagating in a multimode fiber,EPL 134, 14001 (2021).

[35] H. Pourbeyram, P. Sidorenko, F. Wu, L. Wright, D.Christodoulides, and F. Wise, Direct measurement ofthermalization to Rayleigh-Jeans distribution in opticalbeam self cleaning, ArXiv 2012.12110

[36] V.E. Zakharov, V.S. L’vov, G. Falkovich, Kolmogorov

Spectra of Turbulence I (Springer, Berlin, 1992).[37] A.C. Newell, S. Nazarenko, L. Biven, Wave turbulence

and intermittency, Physica D 152, 520 (2001).[38] S. Nazarenko, Wave Turbulence (Springer, Lectures

Notes in Physics, 2011).[39] A.C. Newell, B. Rumpf, Wave Turbulence, Annu. Rev.

Fluid Mech. 43, 59 (2011).[40] Advances in Wave Turbulence, World Scientific Series on

Nonlinear Science Series A, Vol. 83, edited by V.I. Shriraand S. Nazarenko (World Scientific, Singapore, 2013).

[41] A. Picozzi, J. Garnier, T. Hansson, P. Suret, S. Randoux,G. Millot, and D. Christodoulides, Optical wave turbu-lence: Toward a unified nonequilibrium thermodynamicformulation of statistical nonlinear optics, Phys. Reports542, 1-132 (2014).

[42] E. Turitsyna, G. Falkovich, A. El-Taher, X. Shu, P.Harper, and S. Turitsyn, Optical turbulence and spec-tral condensate in long fibre lasers, Proc. R. Soc. LondonSer. A 468, 2145 (2012).

[43] D. Churkin, I. Kolokolov, E. Podivilov, I. Vatnik, S.Vergeles, I. Terekhov, V. Lebedev, G. Falkovich, M.Nikulin, S. Babin, and S. Turitsyn, Wave kinetics of arandom fibre laser, Nature Commun. 2, 6214 (2015).

[44] S. Musher, A. Rubenchik, and V. Zakharov, Weak Lang-muir turbulence, Phys. Rep. 252, 177 (1995).

[45] V. E. Zakharov, Collapse of Langmuir waves, Zh. Eksp.Teor. Fiz. 62, 1745 (1972) [Sov. Phys. JETP 35, 908(1972)].

[46] V.E. Zakharov, S.L. Musher, A.M. Rubenchik, Hamil-tonian approach to the description of non-linear plasmaphenomena, Phys. Reports 129, 285 (1985).

[47] C. Montes, Phys. Rev. A 20, 1081 (1979).[48] A. Picozzi, S. Pitois, and G. Millot, Spectral incoherent

solitons: A localized soliton behavior in frequency space,Phys. Rev. Lett. 101, 093901 (2008).

[49] A. Fusaro, J. Garnier, C. Michel, G. Xu, J. Fatome, L. G.Wright, F. W. Wise, and A. Picozzi, Decoupled polariza-tion dynamics of incoherent waves and bimodal spectralincoherent solitons, Opt. Lett. 41, 3992 (2016).

[50] B. Kibler, C. Michel, A. Kudlinski, B. Barviau, G. Millot,and A. Picozzi, Emergence of spectral incoherent solitonsthrough supercontinuum generation in a photonic crystalfiber, Phys. Rev. E 84, 066605 (2011).

[51] M.V. Goldman, Strong turbulence of plasma waves, Rev.Mod. Phys. 56, 709 (1984).

[52] P. A. Robinson, Nonlinear wave collapse and strong tur-bulence, Rev. Mod. Phys. 69, 507 (1997).

[53] I. Langmuir, Surface motion of water induced by wind.Science 87, 119 (1938).

[54] A.D.D. Craik, S. Leibovich, A rational model for Lang-muir circulations, J. Fluid Mech. 73, 401 (1976).

[55] J. C. Mc Williams, P. P. Sullivan, C.-H. Moeng, Lang-muir turbulence in the ocean, J. Fluid Mech. 334, 1(1997).

[56] A. Y. Wong, P. Y. Cheung, Three-Dimensional Self-Collapse of Langmuir Waves, Phys. Rev. Lett. 52, 1222(1984).

[57] L. N. Vyacheslavov, V. S. Burmasov, I. V. Kandaurov, E.P. Kruglyakov, O. I. Meshkov, S. S. Popov, A. L. Sanin,Strong Langmuir turbulence with and without collapse:experimental study, Plasma Phys. Controlled Fusion 44,B279 (2002).

[58] M. P. Sulzer and J. A. Fejer, Radar spectral observationsof HF-induced ionospheric Langmuir turbulence with im-proved range and time resolution, J. Geophys. Res. 99,15035 (1994).

[59] B. Isham, C. La Hoz, M. T. Rietveld, T. Hagfors, andT. B. Leyser, Cavitating Langmuir Turbulence Observedduring High-Latitude Ionospheric Wave Interaction Ex-periments Phys. Rev. Lett. 108, 105003 (2012).

[60] B. Isham, M. T. Rietveld, P. Guio, F. R. E. Forme, T.Grydeland, E. Mjolhus, Cavitating Langmuir Turbulencein the Terrestrial Aurora, Phys. Rev. Lett. 108, 105003(2012).

[61] P. Horak and F. Poletti, Multimode nonlinear fibre op-tics: Theory and applications, In Recent Progress in Op-tical Fiber Research, M. Yasin, S.W. Harun, H. Arof,eds., Intech, 3-25 (2012).

[62] K.-P. Ho and J. M. Kahn, Linear propagation effects inmode-division multiplexing systems, J. Lightwave Tech.32, 4 (2014).

[63] H. Pourbeyram, G.P. Agrawal, and A. Mafi, StimulatedRaman scattering cascade spanning the wavelength rangeof 523 to 1750nm using a graded-index multimode opticalfiber, Appl. Phys. Lett. 102, 201107 (2013).

[64] C. Michel, B. Kibler, and A. Picozzi, Discrete spectralincoherent solitons in nonlinear media with noninstanta-neous response, Phys. Rev. A 83, 023806 (2011).

[65] L. Rishoj, B. Tai, P.Kristensen, and S. Ramachandran,Soliton self-mode conversion: revisiting Raman scatter-ing of ultrashort pulses, Optica 6, 304 (2019).

[66] A. Antikainen, L. Rishoj, B. Tai, S. Ramachandran, andG.P. Agrawal, Fate of a soliton in a high order spatialmode of a multimode fiber, Phys. Rev. Lett. 122 023901(2019).

12

[67] G. Xu, J. Garnier, M. Conforti, and A. Picozzi, Gen-eralized description of spectral incoherent solitons, Opt.Lett. 39, 4192 (2014).

[68] J. Garnier and A. Picozzi, Unified kinetic formulation ofincoherent waves propagating in nonlinear media withnoninstantaneous response, Phys. Rev. A 81, 033831(2010).

[69] M. Conforti, A. Mussot, J. Fatome, A. Picozzi, S. Pitois,C. Finot, M. Haelterman, B. Kibler, C. Michel, and G.Millot, Turbulent dynamics of an incoherently pumpedpassive optical fiber cavity: Quasisolitons, dispersivewaves, and extreme events, Phys. Rev. A 91, 023823(2015).

[70] J. Garnier, G. Xu, S. Trillo, and A. Picozzi, Incoher-ent dispersive shocks in the spectral evolution of randomwaves, Phys. Rev. Lett. 111, 113902 (2013).

[71] K. Krupa, C. Louot, V. Couderc,M. Fabert, R. Guenard,B. M. Shalaby, A. Tonello, D. Pagnoux, P. Leproux, A.Bendahmane, R. Dupiol, G. Millot, and S. Wabnitz, Spa-tiotemporal characterization of supercontinuum extend-ing from the visible to the mid-infrared in a multimodegraded-index optical fiber, Opt. Lett. 41, 5785 (2016).

[72] G. Lopez-Galmiche, Z. Sanjabi Eznaveh, M. A. Eftekhar,J. Antonio Lopez, L. G. Wright, F. Wise, D.Christodoulides, and R. Amezcua Correa, Visible super-continuum generation in a graded index multimode fiberpumped at 1064nm, Opt. Lett. 41, 2553-2556 (2016).

[73] A. Eftekhar, L. G. Wright, M. S. Mills, M. Kolesik, R.Amezcua Correa, F. W. Wise, and D. N. Christodoulides,Versatile supercontinuum generation in parabolic multi-mode optical fibers, Optics Express 25, 9078 (2017).

[74] A. Niang, T. Mansuryan, K. Krupa, A. Tonello, M.Fabert, P. Leproux, D. Modotto, O. N. Egorova, A. E.Levchenko, D. S. Lipatov, S. L. Semjonov, G. Millot,V. Couderc, and S. Wabnitz, Spatial beam self-cleaningand supercontinuum generation with Yb-doped multi-mode graded-index fiber taper based on accelerating self-imaging and dissipative landscape, Optics Express 27,24018 (2019).

[75] N. Cherroret, T. Karpiuk, B. Gremaud, and C.Miniatura, Thermalization of matter waves in speckle po-tentials, Phys. Rev. A 92, 063614 (2015).

[76] S. Nazarenko, A. Soffer, and M.-B. Tran, On the waveturbulence theory for the nonlinear Schrodinger equationwith random potentials, Entropy 21, 823 (2019).

[77] Z. Wang, W. Fu, Y. Zhang, and H. Zhao, Wave-turbulence origin of the instability of Anderson localiza-tion against many-body interactions, Phys. Rev. Lett.124, 186401 (2020).

[78] N. Ullah and C. Porter, Expectation value fluctuationsin the unitary ensemble, Physical Review 132, 948-950(1963).

[79] D. Weingarten, Asymptotic behavior of group integralsin the limit of infinite rank, Journal of MathematicalPhysics 19, 999-1001 (1978).

[80] B. Collins and P. Sniady, Integration with respect tothe Haar measure on unitary, orthogonal and symplec-tic group, Commun. Math. Phys. 264, 773-795 (2006).

[81] F. Hiai and D. Petz, The semicircle law, free randomvariables and entropy, no. 77, American MathematicalSociety, 2006.