randomperturbationsinthenonlocalnonlinear schr¨odingerequationskupin/article/npcs_15_312.pdf ·...

Nonlinear Phenomena in Complex Systems, vol. 15, no. 4 (2012), pp. 312 - 325

Random Perturbations in the Nonlocal NonlinearSchrodinger Equation

F. MaucherMax Planck Institute for the Physics of Complex Systems, 01187 Dresden, GERMANY and

Laser Physics Centre, Research School of Physics and Engineering,Australian National University, Canberra, ACT 0200, AUSTRALIA

W. KrolikowskiLaser Physics Centre, Research School of Physics and Engineering,Australian National University, Canberra, ACT 0200, AUSTRALIA

S. SkupinMax Planck Institute for the Physics of Complex Systems, 01187 Dresden, GERMANY and

Friedrich Schiller University, Institute of Condensed Matter Theory and Optics, 07743 Jena, GERMANY(Received 01 December, 2012)

We discuss the nonlinear nonlocal Schrodinger equation affected by random perturba-tions both in propagational as well as transverse directions. Firstly, we revise the stabilityproperties of fundamental bright solitons in such systems. Both numerical simulations andanalytical estimates show that the stability of fundamental bright solitons in the presence ofrandom perturbations increases dramatically with the nonlocality-induced finite correlationlength of the noise in the transverse plane. In fact, solitons are practically insensitive tonoise when the correlation length of the noise becomes comparable to the extent of the wavepacket. Fundamental soliton stability can be characterized by two different criteria based onthe evolution of the Hamiltonian of the soliton and its power. Moreover, a simplified meanfield approach is used to calculate the power loss analytically in the physically relevant caseof weakly correlated noise. Secondly, we discuss how these criteria and concepts carry overto higher-order soliton solutions. It turns out that while basic results hold, noise inducedrandom phase shifts may trigger additional instability mechanisms which are not relevant inthe case of fundamental solitons.

PACS numbers: 42.65.Tg, 05.45.Yv, 03.75.Lm, 42.70.DfKeywords: nonlinear Schrodinger equation, higher-order soliton, random noise

1. Introduction

The nonlinear Schrodinger equation(NLS) [1] appears in diverse physical settings,among them, for instance, nonlinear optics [2],Bose-Einstein condensates (BEC) [3–5], andwater waves [6]. A particular class of solutions tothe NLS equation are solitons, particle-like wavepackets that do not change their shape upontemporal evolution or spatial propagation [7].Solitons appear due to the compensation ofdiffraction or dispersion, respectively, whichnaturally tends to spread the wave, by the

nonlinearity. They are ubiquitous in nature andcan be found in many nonlinear systems rangingfrom optics [2], physics of cold matter [8, 9]and plasma [10, 11] to biology [12]. In realisticsettings nonlinear systems supporting solitons areoften subject to random perturbations [13]. Suchperturbations may arise from the fluctuation ofthe external linear potential confining the wave,as in the case of BEC’s in spatially and tem-porally fluctuating trapping potentials [14, 15],optical beams in nonlinear dielectric waveg-uides [16], or waveguide arrays [17] with randomvariation of refractive index, size, or waveguide

312

Random Perturbations in the Nonlocal Nonlinear Schrodinger Equation 313

spacing. Furthermore, the optical nonlinearityof nematic liquid crystals can exhibit stochasticvariations due to fluctuations of the crystaltemperature or conditions of surface anchoring(e.g., roughness) affecting the orientation of thecrystal’s molecules [18, 19]. Similarly, fluctua-tions of temperature will introduce randomnessin colloidal suspensions [20, 21], while noise inthe magnetic field employed to control scatteringlength of the BEC via Feshbach resonance [22]will result in the stochasticity of its nonlinearinteraction potential [23]. Also, stochasticityin the Gross-Pitaevskii equation must be oftentaken into account to describe quantum effects indilute ultra-cold Bose-gases (e.g., [24]). Finally,fabrication-induced spatial fluctuations in peri-odic ferroelectric domain patterns in quadraticmedia act as a source of spatial disorder in thenonlinearity of the quasi-phase matched para-metric wave interaction affecting the propagationof quadratic solitons [25–27].

It has been well appreciated that random-ness in the linear or nonlinear potential support-ing solitons may have dramatic consequences ontheir stability and dynamics depending on thestrength of disorder. The presence of randomnessleads to radiation being emitted by the self-guidedwave packet, the amount of which depends cru-cially on the typical length scale of the fluctu-ation or correlation of the noise. The emissionof radiation weakens the self-induced localizationand, ultimately leads to the decay of solitons. Infact, it has been shown that disorder is equivalentto the presence of an effective loss in the non-linear system [26–28]. On the other hand, it ap-pears that the interplay between nonlinearity andweak randomness can lead to diverse interestingphenomena, such as random walk of solitons inthe transverse plane [29–31] or Anderson local-ization [17, 23, 32–34]. Up to now, mostly localnonlinear interaction has been considered in stud-ies of solitons in nonlinear random systems. Thisamounts to Kerr-type nonlinear optical responseand contact boson interaction in BEC. Recentlyhowever, a few works appeared dealing with ran-

dom systems that exhibit spatially nonlocal non-linearity [18, 21, 31, 35].

Nonlocality of the nonlinear response ap-pears to be common to a great variety of non-linear systems. Physically speaking, nonlocalitymeans that the nonlinear response of the mediumin a specific location is determined by the waveamplitude in a certain neighborhood of this loca-tion. The extent of this neighborhood is often re-ferred to as the degree of nonlocality. Nonlocalityis common to media where certain transport pro-cesses such as heat or charge transfer [36], diffu-sion [37] and/or drift [38] of atoms are responsiblefor the nonlinearity. It also occurs in media withlong-range inter-particle interaction. This is thecase of nematic liquid crystals where nonlinearityinvolves the reorientation of induced dipoles [39]and in the context of BECs with non-contact long-range interatomic interaction [40–42]. Nonlocalityof nonlinearity and its impact on solitons has beenstudied extensively in the last decade. One of themost important features of nonlocality is its abil-ity to arrest catastrophic collapse of multidimen-sional waves [43–46] and stabilize complex soli-tonic structures [47–50]. These stabilizing prop-erties of nonlocality have also been identified inthe presence of randomness. For instance, in re-cent studies of many-soliton interaction in disor-dered nonlocal medium Conti et al. have demon-strated that nonlocality leads to the formation ofsoliton clusters and noise quenching [18, 21]. Batzet al. [35] reported nonlocality-mediated decreaseof the quantum phase diffusion and increased co-herence of quantum solitons while Folli et al. [31]have shown that the soliton random walk can beefficiently suppressed in highly nonlocal media.

In the present work we will firstly revise theeffect of nonlocality on the stability of fundamen-tal solitons in nonlinear random media [51]. Whilemany previous papers consider only “longitudinal”random perturbations [i.e., a situation where therandomness is only a function of the longitudinal(propagation) coordinate (e.g. [28])], we will dealhere with the general case when randomness is afunction of both propagation and transverse coor-

Nonlinear Phenomena in Complex Systems Vol. 15, no. 4, 2012

314 F. Maucher, W. Krolikowski, and S. Skupin

dinates. We will consider a random nonlocal sys-tem where the randomness contributes additivelytowards the nonlinear response of the medium.For sake of simplicity, we will restrict ourselvesto a prototypical Gaussian nonlocal model, whichis widely used in the literature [47, 49, 52, 53].As expected, nonlocality stabilizes fundamentalsolitons by effectively increasing the correlationlength of the random perturbation, and a simpli-fied mean field approach yields a lower bound forthe lifetime of solitary wave packets in the caseof weakly correlated noise. Secondly, we will in-vestigate the effect of random perturbations onnon-solitary input configurations or higher-ordersolitons. In particular the appearance of addi-tional decay mechanisms for higher-order solitonshighlight the crucial role of random noise inducedphase fluctuations, which are of minor importancein the case of fundamental solitons.

This paper is organized as follows: In Sec. 2,we will introduce the aforementioned Gaussiannonlocal model with additive noise. This modelnaturally incorporates the interplay between ran-domness and nonlocality. Then, the propagationdynamics will be studied separately in the weakly(SEC. 3) and strongly (SEC. 4) nonlocal regime.Finally, we will conclude in Sec. 5.

2. Gaussian nonlocal model withadditive noise

We will consider the evolution of the wavefunction ψ(x, t) with x and t denoting generalizedtransverse and longitudinal (propagation) coordi-nates, respectively. The function ψ may representthe main electric field component of a linearlypolarized light beam or the wave function of aquantum object such as BEC. We assume thatψ is governed by the following model equations[47, 49, 52, 53]:

i∂tψ(x, t) + ∂xxψ(x, t) + ρ(x, t)ψ(x, t) = 0 (1a)

∞∫

−∞G(x− x′)

[|ψ|2(x′, t) + ε(x′, t)]dx′ = ρ(x, t).

(1b)

Here, the Gaussian nonlocal response function

G(x) =1√πσ

e−x2

σ2 (2)

was introduced, and ρ represents the resultingnonlinear response of the medium. In the contextof nonlinear optics, ρ is usually identified with anonlinear refractive index change, while it wouldaccount for the effective two-body interaction po-tential in the case of a BEC. The stochastic termε in Eq. (1b) is assumed to be a δ correlatedLangevin noise in both longitudinal and trans-verse coordinates. Then, the noise fulfills 〈ε〉 = 0and 〈ε(x, t)ε(x′, t′)〉 = n2δ(t− t′)δ(x− x′), where

〈f〉 = limN→∞

1N

N∑

j=1

fj

denotes ensemble averaging over different stochas-tic realizations fj , and n is the so-called couplingstrength.

Because the noise term ε(x, t) is additive inthe constituent equation Eq. (1b), it acts as asource term affecting the medium independent-ly of whether the actual signal (e.g., the opticalbeam) is present or not. Therefore, in the non-local NLS Eq. (1) noise plays the role of a ran-dom background potential and is not affected bythe nonlinearity itself. The parameter σ repre-sents the extent of the nonlocality of the nonlin-ear response and hence defines different nonlocalregimes. Without the noise term, Eq. (1) supportsstable nonlocal solitons [49].

In Eq. (1), the nonlocality parameter σ leadsto both nonlocal nonlinearity and finite correla-tion length. To separate the effect of each of thesetwo constituents, it is useful to discuss the effec-tive noise function

η(x, t) =∫

G(x− x′)ε(x′, t)dx′. (3)

Нелинейные явления в сложных системах Т. 15, № 4, 2012


Since the original noise term ε is δ correlated, onecan immediately find that

〈η(x, t)η(x′, t′)〉 = C(x− x′)δ(t− t′). (4)

For our Gaussian nonlocal model, the functionC(x) reads

C(x) =n2

√2πσ

e−x2

2σ2 , (5)

and describes the effect of nonlocality on thenoise. Namely, while the actual random source isrepresented by a white noise the nonlocal charac-ter of the nonlinearity transforms it into an ef-fective colored noise. The nonlocality acts as alow-pass filter eliminating the high frequenciesof the original noise source and thereby smooth-ing out the randomness. This is best appreciatedin the spatial Fourier domain (f = F [f ]) wherewhere the white-noise perturbation is modified bya bandpass filter defined by the Fourier spectrumof the nonlocal response function

η(k, t) = e−σ2k2

4 ε(k, t). (6)

As a result, the effective noise η exhibits a fi-nite correlation length determined by the degreeof nonlocality σ.

3. The weakly nonlocal regime

Let us first focus on the weakly nonlocalregime, where the degree of nonlocality σ is smallcompared to typical transverse length scales ofthe wavefunction ψ. For the fundamental brightsoliton ψS this means that its width σS is largecompared to σ. Then, it is actually possible toestimate analytically the noise induced radiativelosses to the soliton in the limit σ ¿ σS. Follow-ing [51], let us assume that the noise term ηψ inEq. (1) acts perturbatively (of the order δ ¿ 1)on the soliton ψS, and the total wavefunction ψ

can be written as the sum

ψ = (ψS + χ) exp(iλt), (7)where χ(x, t) is of the order δ ¿ 1 as well, and λ

is the propagation constant or soliton parameter.Then, in order δ1 we find

i∂tχ− λχ + ∂xxχ + χ

∫G(x− x′)|ψS(x′)|2dx′ + ψS

∫G(x− x′)

[ψ∗S(x

′)χ(x′, t) + c.c.]dx′ + ηψS = 0.

At initial time t = 0 we have the pure soliton ψS

without any perturbation, and thus χ(x, t = 0) =0. For small times ∆t we can therefore write downa formal solution for the perturbation

χ(x,∆t) ≈ i

∆t∫

0

η(x, t)ψS(x)dt. (8)

When the correlation length σ of the noise ismuch smaller than the width of the soliton σS,

the perturbation χ is spectrally much broaderthan the soliton ψS. Therefore, we can assumethat χ will essentially describe radiation, or, inother words, the part of the total wave functionwhich is completely alien to the soliton. To pro-ceed, we will compute the ensemble average ofP∆t

χ =∫ |χ(x,∆t)|2dx, that is, the power of the

radiation produced in the time interval [0, ∆t]:



⟨P∆t

χ

⟩=

⟨∫ ∆t∫

0

∆t∫

0

η(x, t)ψS(x)η(x, t′)ψ∗S(x)dt′dtdx

⟩= C(0)

∫|ψS(x)|2dx∆t = C(0)PS∆t

where we used Eq. (4) and introduced the solitonpower PS. On the other hand, Eq. (1) is conser-vative which dictates that the fraction of powerconverted to radiation P∆t

χ is lost to the soliton.It is known that the Schrodinger soliton can adaptadiabatically to losses by moving along the familybranch towards lower powers [54], i.e., the solitonparameter λ(t) and power PS(t) become slowlydecreasing in time. If we assume that the radia-tion, once produced, does not interact anymorewith the soliton and disperses quickly, Eq. (8) be-comes valid for any interval [t, t+∆t], and we canconclude that

〈PS(t + ∆t)〉 = 〈PS(t)〉 [1− C(0)∆t] . (9)

Here, PS(t) denotes the power in the solitonic partof the total wave function ψ(x, t). With ∆t → 0we therefore get

〈PS(t)〉 = PS(t = 0)e−C(0)t. (10)

Interestingly, we find that noise induced radiationlosses of the soliton are simply described by thespatial correlation function C(x) at x = 0.

We note that using a similar reasoning onecan estimate radiation losses for an arbitrary ini-tial condition ψ(x, t = 0), provided that it is spec-trally narrow compared to the noise. Moreover, itis possible to write down an evolution equationfor the so-called mean field ψMF, which fulfills

i∂tψMF(x, t) + ∂xxψMF(x, t) + iC(0)

2ψMF(x, t)

+∫

G(x− x′)|ψMF(x′, t)|2dx′ψMF(x, t) = 0,

(11)

and can be connected to the wave function ψ via|ψMF(x, t)|2 ≈ 〈|ψ(x, t)|2〉. We will see below that

predictions of the mean field equation (11) areparticularly useful as long as both random phaseshifts and random walk play no role, i.e., whendifferences between ensemble average and singlerealization are small and therefore 〈|ψ(x, t)|2〉 ≈|ψ(x, t)|2.

For technical convenience, in the followingexamples we will go to the local limit, i.e., σ → 0.In this case the spatial noise term in Eq. (1) hasto be interpreted appropriately, depending on theunderlying physics. According to Eq. (5), in thelimit σ → 0 and fixed coupling strength n ra-diation losses in Eq. (10) formally go to infinity,because C(0) diverges. From a physical point ofview this is not a problem because σ may be smallbut never actually zero. However, in the case thatwe do not resolve σ, e.g., in a macroscopic ap-proach, the noise can be considered to be formal-ly δ correlated, R(x) = δ(x) and C(x) = n2δ(x).In our numerical scheme δ-correlation means thatC(0) = n2/∆x. Provided that the wave functionψ is sufficiently resolved on the mesh with stepsize ∆x À σ, numerical simulations using δ corre-lated noise are supposed to mimic the physical ef-fect of interest. Thus, as far as radiation losses areconcerned, to mimic a small but finite σ, we haveto choose the same value for the auto-correlationC(0). This essentially means that we have to usean effective coupling strength neff ∝

√∆x for the

δ correlated noise [51].Let us now test the applicability of Eq. (11)

to single realizations in the limit σ → 0. In thislimit, and if no perturbing noise is present, thefundamental bright soliton solution is given by

ψS(x, t) =

√2λ

cosh(√

λx)eiλt, (12)



where λ is the soliton parameter already intro-duced earlier in Eq. (7). In Fig. 1(a,b,c), the evolu-tion of the fundamental soliton Eq. (12) for λ = 1is shown for two different noise strengths, andcompared to the mean field prediction Eq. (11)in Fig. 1(d,e,f). Of course, the mean field equa-tion does not describe the stochastic fluctuationsof the peak intensity [see Fig. 1(a) vs. (d)]. More-over, the soliton in Fig. 1(c)] performs a randomwalk in the transverse plane [31], which is absentin Fig. 1(f) because Eq. (11) is an entirely deter-ministic equation. Besides these differences, boththe evolution of peak intensity as well as the soli-ton power are captured correctly. Note that therandom walk of the soliton plays no role for thesetwo quantities, because the peak intensity movesalong with the barry center of the soliton, andthe soliton power is the integrated over the wholenumerical box [a], so that the actual transverseposition of the soliton does not matter.

Next, we consider a particular non-solitaryinitial condition,

ψ(x, 0) =

√4√π

e−x2/2, (13)

again in the local limit (σ → 0) of Eq. (1). In thiscase, the mean field equation (11) has to be solvednumerically as well. When comparing the simpli-fied mean field prediction to the evolution of thefull Eq. (1) (see Fig. 2), it turns out that Eq. (11)still has good predictive power. The oscillationsof the peak intensity due to the non-solitary ini-tial condition are well-captured by the mean fieldmodel [compare Fig. 2(a) and (d)] as well as theloss of power [compare Fig. 2(b) and (e)].

Let us close our discussion of the weakly non-local regime with an example of complete failureof the mean field model, at least as far as the prop-agation dynamics are concerned. To this end, we

[a] Because we are operating in the local limit, the radiationcreated by the numerical white noise leaves our compu-tational box almost instantaneously. Therefore, we sim-ply plot P =

∫box

|ψ|2dx.

consider the initial condition

ψ2S(x, 0) = 2ψS(x, 0) =2√

2cosh(x)

, (14)

i.e., the famous two soliton solution. In this case,the unperturbed evolution can be written downanalytically as [2]

ψ2S(x, t) = 4√

2eit

× cosh(3x) + 3e8itcosh(x)cosh(4x) + 4cosh(2x) + 4cos(8t)

.(15)

Obviously, the two soliton solution is highly dy-namic, and |ψ2S|2 is periodic with period T =π/4. The dynamics predicted by the mean fieldmodel (see Fig. 3) resembles clearly the unper-turbed dynamics [see Fig. 3(d)], with the noteddifference, that the peak intensity decays slight-ly [see Fig. 3(a)] due to the loss of power [seeFig. 3(b)].

For the following discussion it will be essen-tial to consider the phase ϕ of the wavefunction,defined as ψ = |ψ|eiϕ and ψMF = |ψMF|eiϕMF ,respectively. As we can see in both Eq. (15) andFig. 3(c), the two soliton solution contains a fast(λ = 8) and a slow (λ = 1) ”propagation con-stant”. Clearly, in one period of the slow oscilla-tion, between e.g. two of the large blue bars, thereare eight fast oscillations, and upon propagation,this phase relation remains fixed.

However, in presence of random perturba-tions, the evolution of the peak intensity looksentirely different [see Fig. 4(a) vs. Fig. 3(a)], eventhough the loss of power perfectly agrees with themean field model [see Fig. 4(b) vs. Fig. 3(b)].Apparently, a decay into two fundamental soli-tons, which separate upon further propagation,occurs [see Fig. 4(c)]. A zoom into this splittingprocess is shown in Fig. 4(d). To understand theunderlying decay mechanism, we have to considerthe evolution of the phase shown in Fig. 4(e), andin particular the zoom into the relevant regionin Fig. 4(f). Obviously, the random fluctuationsperturb the delicate phase relation between slowand fast “propagation constant”. This effect is notcaptured by the simple mean field model, since



FIG. 1. Soliton evolution under the influence of white noise (σ → 0) according to Eq. (1) in the upper row (a,b,c)and mean field prediction Eq. (11) in the lower row (d,e,f) for two different noise strengths C(0) = 6.4 × 10−4,C(0) = 1.024× 10−2 and initial soliton parameter λ = 1. (a) shows the evolution of the peak intensity, which iscompared to the mean field prediction in (d) given by |ψMF|2max = 2e−2C(0)t. (b) illustrates the loss of power dueto radiation, where the mean field prediction in (d) is given by PMF = 4e−C(0)t. (c) illustrates the dynamics of theintensity of the randomly perturbed soliton for the realization C(0) = 6.4×10−4, and (f) shows the correspondingmean field prediction. The fluctuations of the peak intensity as well as the random walk of the soliton accordingto Eq. (1) cannot be captured by the simplified Eq. (11).

it does not take local phase fluctuations into ac-count. After separation of the two fundamentalsolitons, one soliton carries the “fast propagationconstant” λ ≈ 8, and the other on the ”slow prop-agation constant” λ ≈ 1. Because the power ofan unperturbed fundamental soliton Eq. (12) isgiven by P = 4

√λ, we find a power ratio of the

emerging fundamental solitons of approximately4√

1 : 4√

8.After these three generic examples to illus-

trate the applicability of the mean field approachin the weakly nonlocal limit, let us now switch tothe strongly nonlocal regime.

4. The strongly nonlocal regime

In the strongly nonlocal regime, the degreeof nonlocality σ is comparable or larger than typi-cal transverse length scales of the wavefunction ψ.For the fundamental bright soliton ψS this meansthat σS ≤ σ. In this regime, noise induced ra-diative losses are of limited influence, and ba-sic assumptions needed for the mean field ap-proach Eq. (11) no longer hold. In this situation,an alternative criterion for soliton stability involv-ing the Hamiltonian of the unperturbed versionof Eq. (1) can be derived [51]. The Hamiltoni-an H is usually defined as the functional, thatyields Eq. (1) without noise (i.e., ε = 0) when ap-plying i∂tψ = δH/δψ∗. Hence, the Hamiltonian is



FIG. 2. Evolution of the non-solitary initial condition ψ(x, 0) =√

4/√

πe−x2/2 for two different noise strengthsC(0) = 6.4×10−4, C(0) = 1.024×10−2 (a,b,c) and corresponding (numerical) mean field results (d,e,f), presentedin the same manner as in Fig. 1. Apparently, the mean field equation (11) is capable to predict the dynamics forthis non-solitary initial condition as well, apart from fluctuations of the peak intensity and the random walk.

given by

H =∫|∂xψ(x)|2

−12|ψ(x)|2

∫G(x− x′)|ψ(x′)|2dx′dx.

(16)

The HamiltonianH is a conserved quantity for theunperturbed equation ε = 0, but it becomes timedependent for finite coupling strength. Several pa-pers [51, 55–57] already emphasized the fact that〈H〉 is a linear function of time t, 〈H(t)〉 = H0+γt,where H0 = H(t = 0) was introduced. In fact, itis possible to compute the ascent γ analytically(see App. 5). Then, the final result for the timeevolution of the averaged Hamiltonian reads [51]

〈H〉 = H0 − d2C(x)dx2

∣∣∣∣x=0

Pt = H0 +n2P√2πσ3

t.

(17)

Note, that only the second derivative of the corre-lator C with respect to x contributes to the linearascent of the mean Hamiltonian with time. Fol-lowing [51], we assume that a fundamental soli-ton remains practically unchanged as long as themean Hamiltonian is negative, and we introducethe characteristic time tH via

〈H(tH)〉 = 0 ⇔ tH =H0

√2πσ3

n2P. (18)

The definition of tH is somewhat arbitrary, butit works reasonably well for fundamental solitons.Figure 5 shows the evolution of the soliton withinitial power P = 200 (σS ≈ 0.54), governedby Eq. (1) with degree of nonlocality σ = 2. Theunperturbed version of Eq. (1) cannot be integrat-ed analytically for finite σ, so that the initial soli-tary wave has to be computed numerically, whichwe did by using an iterative solver [49]. For a soli-



FIG. 3. Evolution of the two soliton initial condition Eq. (14) according to the mean field equation Eq. (11) forC(0) = 6.4×10−4. (a) shows the evolution of the peak intensity, (b) illustrates the loss of power due to radiation,and (c,d) display the dynamic evolution of phase ϕMF and intensity |ψMF|2, respectively.

ton power P = 200, one finds H0 ≈ −4928 for thefundamental soliton. For a coupling strength ofn = 4, Eq. (18) yields tH ≈ 30.9. The fundamen-tal soliton apparently survives the quite signifi-cant random perturbations, which lead a substan-tial random walk. The soliton power is practical-ly conserved, and drops by less than one percentover the propagation time tH. Hence, for t < tHthe soliton is well-localized and practically unaf-fected in spite of the large random fluctuations.

The situation is different when we consid-er the evolution of an initial higher-order soliton,here with three nodes (see Fig. 6), with samepower P = 200 and resulting initial Hamiltoni-an H0 ≈ −1926. The coupling strength as wellas the nonlocality is again chosen to be n = 4and σ = 2, respectively. If we simply apply our

stability criterion, we find tH ≈ 12.1, because theascent γ for time evolution of the mean Hamil-tonian is the same as in the previous example.However, contrary to the propagation of the fun-damental soliton, the intensity [see Fig. 6 (a-b)]changes significantly already after a short evolu-tion t with t < tH. This rapid departure fromits initial shape is can be well understood whenstudying the phase shown in Fig. 6 (c-d). Theloss of the initial phase structure happens alreadyaround t = 2 [see the zoom in Fig. 6 (d)]. Hence,we can conclude that our stability criterion is notapplicable for higher-order solitons. Similarly tothe situation in Fig. 4, it is again the change inthe local phase that leads to the break-up of thehigher-order soliton. Obviously, such phase effectsare not contained in our simple criterion Eq. (18).



FIG. 4. Randomly perturbed evolution of the non-solitary initial condition Eq. (14), again for C(0) = 6.4×10−4.Clearly, apart from the power P shown in (b), there is a dramatic difference to the mean field results shownin Fig. 3 in all other plots. Due to the random perturbations and resulting random phase fluctuations, a decayinto two fundamental solitons takes place (c-d). Each of the emerging fundamental solitons carries approximatelyone of the two ”propagation constants” contained in Eq. (15), which is also reflected in the power ratio of the twosolitons, as elaborated in the text. The oscillations of the peak intensity in (a) vanish after the splitting process,and the peak intensity is simply determined by the fundamental soliton with higher amplitude. Note that (d)and (f) show a zoom into the splitting region of (c) and (e), respectively.



FIG. 5. Evolution of the intensity (a) and phase (b) of a randomly perturbed initial nonlocal soliton with P = 200perturbed quite significantly (coupling strength n = 4, nonlocal parameter σ = 2). The strong perturbations leadto a pronounced random walk in the transverse plain. However, the soliton withstands the strong perturbationsover a distance t > tH.

FIG. 6. Evolution of the intensity (a-b) and phase (c-d) of a randomly perturbed initial higher-order nonlocalsoliton with P = 200 (coupling strength n = 4, nonlocal parameter σ = 2). In this case, a significant change tothe initial soliton shape takes place after a short evolution, and its phase structure is already destroyed roughlyat t = 2. Clearly, the initial shape is entirely lost around t = tH



In a vivid picture, the individual humps of ahigher-order soliton start to perform individualrandom walks. Moreover, local phase fluctuationschange the strict phase relation between neigh-boring humps, which enables merging events andquickly destroys the characteristic structure of thesolution.

5. Conclusions

In this paper, we studied the interplay be-tween nonlocality and randomness and its impacton the evolution of solitary waves. First, we con-sidered the case where the nonlocality inducedcorrelation length can be considered to be smallcompared to the extent of the wavepacket. In thiscase, the random perturbations lead to a rapidloss of power of the soliton. We showed, thatindependently of the initial condition, the rateof the latter can be found by a simple formula,that is also valid for single realizations. The evo-

lution of the shape (apart from fluctuations) ofthe wavepacket can be determined by means of amean field approach, whenever local phase effectsplay no role. Second, we considered the case wherethe correlation length is no longer small comparedto the extent of the wavepacket. In this case, weused the mean Hamiltonian to obtain a criteri-on for the stability of the wavepacket. While thiscriterion works well for the fundamental nonlocalsoliton, it clearly fails for higher-order states. Thisfailure can be explained again as a result of localphase fluctuations.

Appendix A: The averaged Hamil-tonian

The ascent γ of the ensemble averagedHamiltonian 〈H〉 equals its time-derivative. Thus,we have to compute

∂t〈H〉 = ∂t

⟨∫|∂xψ|2 − 1

2|ψ|2

∫G(x− x′)|ψ(x′)|2dx′dx

⟩=

∫〈iη (ψ∗∂xxψ − ψ∂xxψ∗)〉dx. (A.1)

For the time-derivatives of the wave functionψ(x, t) we plugged in Eq. (1). For the nextstep, we have to introduce the Furutsu-Donsker-Novikov formula, which is given by [13, 58–60]:

〈ηψ〉 =

t∫

−∞

∞∫

−∞〈 δψ(x, t)δη(x′, t′)

〉〈η(x, t)η(x′, t′)〉dx′dt′

(A.2)Causality in time is reflected by the upper in-tegration limit. This formula involves the varia-tional derivative of the wave function ψ with re-spect to the noise term η. To compute this quan-tity we write Eq. (1) in integral form. Then, withδη(x,t)

δη(x′,t′) = δ(x−x′)δ(t− t′) (see, e.g., [13]) we find

in the limit t′ → t

δψ(x, t)δη(x′, t)

= iψ(x, t)δ(x− x′). (A.3)

This finally allows to compute the expression [b]

〈ηψ〉 = iC(0)

2〈ψ〉. (A.4)

Applied to Eq. (A.1) the Furutsu–Donsker–Novikov formula leads to

[b] Note that the factor 1/2 in Eq. (A.4) appears due to∞∫0

δ(t)dt = 1/2.



∂t〈H〉 = i

t∫

−∞

∞∫

−∞

∞∫

−∞

{⟨δ [ψ∗(x, t)∂xxψ(x, t)]

δη(x′, t′)

⟩−

⟨δ [ψ(x, t)∂xxψ∗(x, t)]

δη(x′, t′)

⟩}〈η(x, t)η(x′, t′)〉dxdx′dt′

=i

2

∞∫

−∞

∞∫

−∞〈δ[ψ

∗∂xxψ − ψ∂xxψ∗]δη(x′, t)

〉C(x− x′)dxdx′

In the second step, we performed the time-integration over t′. By evaluating the variationalderivatives Eq. (A.3) and integration by parts weread

∂t〈H〉 =12C(0)〈

∫ψ∗∂xxψ + c.c. dx〉

− 12〈∫∫

ψ∗δ(x− x′)∂xx

[C(x− x′)ψ

]+ c.c. dx′dx〉.

Most of the integrals above turn out to be zero,which can again be seen by integration by parts,and we find

∂t〈H〉 = −d2C(x)dx2

∣∣∣∣x=0

〈∫|ψ(x, t)|2dx〉. (A.5)

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