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Coherence and incoherence in extended broad band triplet interaction∗ G.I. de Oliveira Departamento de F´ısica, Centro de Ciências Exatas e Tecnologia Universidade Federal do Mato Grosso do Sul, Caixa Postal 549, 79070-900 Campo Grande, MS, Brasil, F.B. Rizzato Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970 Porto Alegre, RS, Brasil. 8 In the present analysis we study the transition from coherent to incoherent dynamics in a non- 0 linear triplet of broad band combs of waves. Expanding the analysis of previous works, this paper 0 investigates what happenswhen thebandof available modesis muchlarger thanthat of theinitial 2 narrower combs within which the nonlinear interaction is not subjected to selection rules involving n wavemomenta. Hereselectionrulesarepresentandactive,andweexaminehowandwhencoherence a can be defined. J 1 3 I. INTRODUCTION in a very short time interval the initial modes scatter off theinhomogeneities,creatinggroupsofmanymodes,the ] initial combs. Another but equivalent way to see how h Wavetripletinteractionsmodelavastnumberofcases p combsarerelatedto inhomogeneitiesis to realizethatin where nonlinear wave dynamics of physical systems can - the interaction of tightly packed group of modes, neigh- m be described in terms of three dominant modes. The in- boringwavevectorscannotbe properlyresolvedin finite teraction is seen in a variety of situations, ranging from s size spatial scales, a typical occurrence in experimental a three wave interactions in laser-plasma and optical sys- settings[10,11]. Inthiscasewholegroupsofmodeswith l tems to pulsar emission of electromagnetic radiation,in- p similar wave vectors are altogether excited forming the cluding wave interaction in fluids and in several other . combs. The interaction acquires the aspect of a mean s settings [1, 2, 3, 4]. c field theory, where modes of one comb interact with av- si The conservative interaction, which will be our focus erages taken over modes of the remaining combs [10]. y here,ismoreeasilyhandledwhentheinteractioninvolves h onlythethreepuremodesofthetriplet. However,amore In the past, models for wave combs were based on p realistic view should allow for a microscopic description, combswithfixednumberofmodes. Oncethecombswere [ where each of the pure modes is replaced with a comb formed, submodes could evolve in time, but always pre- 1 with many submodes. This has been done in a number servingaprefixedtotalnumberwithineachofthecombs. v of papers [5, 6, 7, 8] where several results have been de- A recent analysis [12] shows that combs with prefixed 0 rived along recent years. The main lesson one learns is number of modes cannot actually maintain this number 5 that the dynamics canbe correctlydescribed interms of if the interaction takes place in a homogeneous environ- 8 three single or central modes, as long as nonlinearities ment. As one may conclude from the comments above, 4 are strong enough to lock all submodes into a single co- this happens in virtue of the fact that homogeneity is . 1 herentmode. For practicalpurposes the coherentmodes unable to create a naturalwavevector scale which could 0 canthenbeviewedasthepuremodesofthetripletinter- accommodate a given finite number of modes. Ref. [12] 8 action. Ontheotherhand,iflockingisnoteffective,each indeed shows that as the wave dynamics develops, more 0 of the submodes follows its own linear dynamics and co- and more modes are gradually excited and included in : v herence is lost. Random phase approximations can then the interaction. i X be invoked to analyze the problem [9], but the concept This leads us to the central question of the present of a pure triplet has to be abandoned. r analysis, namely can the wave interaction in inhomoge- a A recent paper [8] shows how combs of modes can be neoussettingsbewelldescribedwithcombsoffinitenum- very naturally formed in a wave system: the essential ber of modes? We shall see that the answer depends on requirement, as we shall review,is that the nonlinear in- the time scales and the wave vector scales one is inter- teractiontakesplaceunderspatiallyinhomogeneouscon- ested in. ditions. Whenthe inhomogeneityispresent,wavevector matching among the interacting modes needs not to be The plan of the paper is the following. In 2 we first § exact since it includes the reciprocal vectors of the in- define a convenient interaction model allowing for an inhomogeneities. What happens then is that even if the homogeneous environment and explore how the model initial conditions involve a small number of wave modes, can be used to create the picture of interacting combs with fixed number of modes, simultaneously analyzing its inherent limitations. In 3 we examine what hap- § pens when the constraint of a constant prefixed number ∗Accepted inPhys. Rev. E(2008). of modes is relaxed. In 4 we summarize our results. § 2 II. THE MODEL set. We first need a structure for the function s(x). We define it as an even function centered at x = 0 and a The investigationstarts as we consider the set of fully characteristic half width ls, as follows: dimensionless space time equations governing the decay of mode “1” into modes “2” and “3”: s=s(x/ls)=s(x/ls); s(x/ls 1) 0, s(0)=1, | | | | ≫ → (4) where for mathematical convenience,andwith no loss of i∂ta1(x,t)+ivg1∂xa1(x,t)=s(x)a2(x,t)a3(x,t), (1) generality, we assumed a scaling that renders s(0) = 1. i∂ a (x,t)+iv ∂ a (x,t)=s(x)a (x,t)a (x,t) , (2) This kind of function restricts the effective interaction t 2 g2 x 2 1 3 ∗ regionascommentedintheIntroductionandcanbeused i∂ a (x,t)+iv ∂ a (x,t)=s(x)a (x,t)a (x,t) . (3) t 3 g3 x 3 1 2 ∗ to introduce the basic wave vector associated with the inhomogeneities of the system in the form k 1/l . Set (1) - (3) actually describes the slow modulational s s ∼ Nowwewriteeachofthewavesa (x,t)ascombsofmany dynamics for the complex wave amplitudes a (x,t) (p= p p modes 1,2,3) of corresponding carrier waves whose frequencies and wave vectors are matched. The combs are thus the multitude of sideband modes forming around each of three high frequency carriers. i2 = 1, and the real a (x,t)= aˆ (κ ,t)eiκpxdκ , (5) − p p p p function s(x) is the spatially dependent form factor in- Z troducing inhomogeneity in the problem. Function s(x) couldbetypicallyassociatedwithinhomogeneousdensity where κp denotes the wave vectors of submodes within distributions in plasma systems for instance. each comb. Letusfirstofallseehowtheclassicalpictureofcombs Spatial Fourier analysis of set (1) - (3) produces the with given number of modes can arise from the basic following group of equations for the various submodes: iaˆ˙ (κ )=v κ aˆ (κ )+ sˆ(κ κ κ )aˆ (κ )aˆ (κ )dκ dκ , (6) 1 1 g1 1 1 1 1 2 3 2 2 3 3 2 3 Z − − κ2,κ3 iaˆ˙ (κ )=v κ aˆ (κ )+ sˆ(κ κ κ )aˆ (κ )aˆ (κ ) dκ dκ , (7) p 2 g2 2 2 2 1 2 3 1 1 3 3 ∗ 1 3 Z − − κ1,κ3 iaˆ˙ (κ )=v κ aˆ (κ )+ sˆ(κ κ κ )aˆ (κ )aˆ (κ ) dκ dκ , (8) 3 3 g3 3 3 3 1 2 3 1 1 2 2 ∗ 1 2 Z − − κ1,κ2 with s(x/ls)= ∞ sˆ(κs)eiκsxdκs, sˆ(κs)= 1 ∞ s(x/ls)e−iκsxdx, (9) Z 2π Z −∞ −∞ sˆ(κ )alsoeven. OnethusseesfromthesecondofEqs. (9) one recovers the matched selection rule κ = κ +κ , s 1 2 3 that in general, wave vector mismatches of magnitudes but for finite l ’s any triple of modes within the bands s upto κ κ κ π/l amongtheinteractingsub- are connected. The approximate dynamics of bands can 1 2 3 max s | − − | ≈ modes are allowed. If one defines a band width ∆ in the be obtained if one assumes sˆ(κ κ κ ) sˆ(0) for 1 2 3 − − ∼ form ∆/2 < κ < ∆/2, one concludes that all modes κ ∆/2, discarding all modes outside the combs; we p p − | | ≤ initially placed within the bands will interact simultane- note that under this approximation,and considering the ously,withnoconstraintsduetoselectionrules,provided normalization choice s(0) = 1, the first of Eqs. (9) in- π/l 3∆/2. We shallrefer to this regimeas the regime forms us that sˆ(0) 1/(3∆). In this case, and mov- s ∼ ∼ of democratic interaction because selection rules are not ing into the discrete version of our continuum equations operativehere;underthisregime,anythreemodeswithin with κ κ =2πm/L mκ (“m” is an integer p=1,2,3 m L → ≡ thebandsarecoupledwiththesamestrength. Ifl denoting the modal number), dκ aˆ (κ ) = κ aˆ (κ ) = s p p p L p p →∞ 3 (2π/L)aˆ (κ ) aˆ , and L as the system length, one the broad band triplet interaction, where selection rules p p pm → arrives at the set already explored by various authors amongthe wavevectorsareabsentinvirtueoffinite size [5, 6, 7, 10, 13] ofthe interactionregion. Severalinterestingresultshave been obtained, the most prominent of which concerning the competition between the linear and nonlinear terms. 1 iaˆ˙ =v κ aˆ + aˆ aˆ , (10) Ifthe linearbandwidthterms associatedwiththe group 1q g1 q 1q 2m 3n 3N ∆ Xm,n velocities are absent, one shows that in steady state the 1 wavesystemsoscillateswithasinglenonlinearfrequency iaˆ˙2m =vg2κmaˆ2m+ 3N∆ Xq,n aˆ1qaˆ∗3n, (11) (Ωw.hIefn|Ωu|ninselacergssearrtyh,amnotdhaellaarngdesctolminbeasrufbrienqdueexnecsyavrge∆o/c2- 1 casionallysuppressedtosimplifynotation),aphaselock- iaˆ˙ =v κ aˆ + aˆ aˆ . (12) 3n g3 n 3n 3N 1q ∗2m ingmechanismispresent,preventinganinitiallycoherent ∆ Xq,m comb to decohere. In general, when a linear band width is present a time propagator g(t) can be constructed for To obtain set (10) - (12) the prefactor κ /(3∆) = L the total amplitude, or macroscopic field of each comb (1/3)(κ /∆) of the nonlinear terms in the discrete ver- L sion is written as (1/3)/(1/N ), N ∆/κ being ∆ ∆ L ≡ therefore a measure of the number of modes composing Ap aˆpj, (13) ≡ the combs in the Fourier reciprocal space; the factor of Xj 3 can be absorbed into convenient rescalings. As mentioned, set (10) - (12) comprises the classical form of in the form [8] g(t)= 1 ∆/2 ieivgκtdκ= isin vg2t∆ (time domain) g →gˆ(ω)=∆ln[R(ω−−∆v/g2∆−/2)2]−ln[(ω+vg∆−/2)2] (cid:0)vg2itπ∆Si(cid:1)gn(vg∆/2−ω)+iπSign(vg∆/2+ω) (frequency domain). (14) 2vg∆ − 2vg∆  The factor i exp(iv κt) in the time domain expres- which corresponds to one third of the total interaction g − sion is essentially the propagator for the microscopic rangedefinedbytheformfactorsˆ(κ ). Thewidth∆con- s mode with wave vector κ, and the total propagator is tains a number N =∆/κ of modes which in the past ∆ L obtained through an integration over the whole comb. were supposed to be the only active modes of the wave If in the second of Eqs. (14) one identifies the Fourier system. However,the traditionalmodel set (10) - (12) is frequency ω with the dominant nonlinear frequency Ω, only an approximation to the full nonlinear system (6) - theconclusionisthatadissipative-liketermariseswhen- (8),whereonedeliberatelydiscardsallmodesoutsidethe ever Ω < v ∆/2. In extreme nonlinear cases with comb of the given width ∆. The assumption looks right g | | Ω > v ∆/2, coherence is preserved. In fact, a rela- because, as mentioned, modes within the comb are ex- g | | tively straightforward procedure involving expansion of pectedtobemorestronglyandmorequicklyexcitedthan gˆ around Ω and a Fourier inversion from frequency to modesoutside. However,whenonelooksatthefullsetof time domain, allows to write a coupled set for the the equationsthereisalwaysanonlinearcouplingwhichmay macroscopicfields[8]whichgivesagoodqualitativeview eventually interlace and excite all modes, even those not of the dynamics in the democratic regime: initially placed inside the combs. In a real system with a band extension naturally much larger than the width 1 iA˙1 β1A1+ A2A3 (15) ∆,thepropagatorfortheentiremacroscopicfieldshould ≈ 3 be rewritten as in Eq. (14), but with ∆ replaced with 1 iA˙2 ≈β2A2+ 3A1A∗3 (16) ∆T,thelatterquantityrepresentingthetotalbandwidth available to the modes. Thus, even if Ω > v ∆/2, one iA˙3 ≈β3A3+ 31A1A∗2, (17) might still have |Ω|<vg∆T/2, a situa|tio|n whgere coherence decay might be present. Of course, if one takes ∆ T whereβ →(vg∆)2/(12Ω)ifvg∆≪Ω,andβ →−ivg∆if as the full band width, and ∆T > ∆, not all modes will vg∆ Ω. Oneseesthatgiventhe autonomousaspectof interactdemocraticallyandselectionrulesshallreappear. ≫ set (15) - (17) one predicts decay (shrinking of volumes In that case, previous results must be re-evaluated. In in the corresponding phase space) if ∆ becomes larger particular, from the stand point of macroscopic modes, than the nonlinear frequency. the systems ceases to be autonomous since the nonlin- We shall obtain Ω explicitly for some cases, but let us ear terms can no longer be written only in terms of A , 1 firstdwellontheroleofthewidth∆. Itisafixedquantity 4 A ,andA . Thereforeonecannotproveordisprovethat atleastinitially, we require1/l 3/l as explainedear- 2 3 s ρ ≥ volumesinthephasespaceofthemacroscopicmodesare lier - in all numerical work we actually take 1/l =3/l . s ρ shrinking,asithappenswiththeapproximateformgiven We shall also assume that ∆ ∆, and write for the T ≫ by Eqs. (15) - (17). Nevertheless a dissipative term is exponential distribution ∆=2π/l . ρ presentandthemacroscopicmodesarelikelytodecayin Independently of the choices we make for ρ (x) and 1 time - this is what really happens as we show next. s(x)wearealreadyinpositiontodefine coherenceinthe present case. We simply note that since III. A MORE ACCURATE VIEW: THE EXTENDED BROAD BAND INTERACTION a (x,t)= aˆ eiκjx, (19) p pj Xj As said, the full set (6) - (8) is equivalent to the its counterpart spatial set (1) - (3). The connection is relevant because if one discards space derivatives exact so- itistruethatthemacroscopicfieldsApintroducedearlier lutions can be obtained. These exact solutions form the in Eq. (13) obey simple expressions - we write down the basisforfurtherprogressasoneincludesthespacederiva- one obeyed by A1: tives. A = aˆ =ρ exp( 2iρ t). (20) 1 1j 0 0 − A. Neglecting space derivatives Xj Taking v ∂ 0 in the Eqs. (1) - (3) a station- From the equation above we see that the macroscopic gp x → ary solution can be obtained in the form A (x,t) = field oscillates harmonically with frequency Ω 2ρ p 0 ≡ − ρ (x)exp(iφ (x,t)), where φ = 2ρ (x)s(x)t, φ = and with constant amplitude ρ . This is what we shall p p 1 − 1 2,3 0 ρ (x)s(x)t, ρ = √2ρ (x), and where ρ (x) is an take as a coherent state: a non-decaying macroscopic 1 2,3 1 1 − arbitrary x-dependent function; we note that in the sta- mode oscillating with constant amplitude and constant tionary state phases depend on time, but amplitudes do frequency. Thequestionthatposesitselfhereistodeter- not. Once ρ (x) is defined, the complete solution is au- mine how many microscopic modes actually participate 1 tomatically found. And once the space time solution is in the coherent state. In other words, would be true to found, Fourier transforms can be used to move into the assume that only the modes inside the initially defined reciprocal space. To further specify the system with ini- combsareactive? Atafirstglanceonemightsuspectthe tial conditions localized both in the real and reciprocal answer would be positive since those are the modes in- spaces, we shall make the following choice for the combs teracting more strongly in the system. However, we had and the form factor s in the spatial representation: already pointed out that due to the nonlinear cascading structure of the interaction, some energy may flow from ρ (x)=ρ exp x2/l2 , low to high wave vectors; and this is what actually hap- 1 0 − ρ s(x/l )=exp(cid:0) x2/l2(cid:1). (18) pens. This can be seen more formally with help of some s − s tools. Keeping focus on the first comb, one first chooses (cid:0) (cid:1) Thecombthusdefinedhaswidth 1/l inthereciprocal a range D defined by D/2< κ < D/2 and performs a ρ ∼ − spaceandinorderthatitsmodesinteractdemocratically partial summation over its internal modes, D/2 a (κ )dκ = 1 D/2 ∞ e iκ1xρ (x) exp[ 2iρ (x)s(x)t] dxdκ . (21) D 1 1 1 − 1 1 1 I ≡Z 2π Z Z − D/2 D/2 − − −∞ For a finite band D, the integral over κ , performed final result can be written in the form 1 firstly, yields a delta-like structure as a function of x, Erfi e3iπ/4π 2ρ ts (0)/2/D withheightD/2π andwidth2π/D. Ifonesupposes1/D 0 ′′ small,theremainingintegrationoverxcanbe donewith ID ∼e−2iρ0t√ρ0D h pts (0) i, ′′ help of a saddle approximation near x = 0 where the p (22) space derivative of fields and form factor vanish. The where s d2s/dx2, where Erfi(χ) denotes the imag- ′′ ≡ inary error function as a function of argument χ, and where we recall that s(x) varies faster than ρ (x). We 1 seethatalldepends onthe behaviorofthe imaginaryer- 5 ror function for large and small arguments. If χ 1, 2.0 (a) | | ≪ Erfi(χ) χ and if χ 1, Erfi(χ) i. One therefore 1.0 conclude∼s that | | ≫ ∼ I 0.0 D Constant when t<l2D2/2π2ρ -1.0 |ID|∼(cid:26)√1t when t>ls2D2∼/2sπ2ρ0. 0 (23) -2.0 (b)2.0 1.0 Inother words,givenarangeD there is anintrinsiclim- I 0.0 iting time for coherence, D -1.0 τ D2ls2 , (24) 0.02 (c)-2.0 D ≡ 2π2ρ0 0.01 where by intrinsic we understand the limiting time ob- ar 0.00 1d tainedintheabsenceofthelinearfrequencybandwidth, -0.01 i.e. by taking vg = 0. We know from our discussion re- -0.02 0 100 200 300 400 500 gardingEq. (9) that l π/(3∆/2),so, the intrinsic co- s ∼ time herence time for modes within the originalpacket would be given by τD=∆ ∼ 1/ρ0 which is relatively small since FIG.1: ID asafunctionoftimeforDls=2π×22 in(a)and this is essentially the period of the nonlinear wave. Our forDls=2π×26in(b). In(c)weshowthetimesseriesforthe conclusion is that the initial packet can be hardly called real part of the borderline mode with wave vector κ= D/2, acoherentstructureevenintheabsenceofthefrequency ls=29,lρ =3ls. Thegroupvelocityiszeroforallwavesand band width. The collection of modes that could be seen all quantities are dimensionless. as a coherent structure is anyone where D ∆. In that ≫ case it is still true that decay will be present, but for all practical purposes τ would be so much larger than the l = 3l , panel (a) displays the case D = 2π/L 28, D ρ s × period of the nonlinear wave that a physical setting or for which Dl = 2π 22, and τ 32, while in panel s D equipment resolving modes up to κ D would perceive (b) Dl = 2π 26 fo×r which τ ∼8200. Panel (a) re- ∼ s × D ∼ the wave system as coherent. veals a fast decay, but coherence is far more persistent Asecondimportanttimescalehastobedefinedforthe in panel (b). It is noticeable that in panel (b) function wavesystem. Itis the time scaleofexcitationofindivid- , although initially laminar, develops slight modula- D I ual modes in the reciprocal space. Looking again at the tionsafteraverysharpinstantalongthe timeaxis. This first comb - reasonings are similar for the other two, we verysharpinstantcorrespondstot=τ (D/2). Indeed, exc first recall the expression A (x,t) = ρ (x)exp(iφ (x,t)) the excitation time reads τ (D/2)= 100.5 in this par- 1 1 1 exc for the steady state field. ρ (x) is constant in time and ticular instance. This is confirmed in panel (c) where, 1 the phase φ (x) = 2ρ (x)s(x)t depends both on the for the same parameters of panel (b), we show the time 1 1 − spatial coordinate and time. If one evaluates the phase evolutionofthe realpartofmode withwavevectorD/2. gradient∂φ /∂xandlook at the maximum ofthis quan- Thesuperscript“r”means“realpart”andthesubmodal 1 tity as the largest wave vector involved in the dynamics, index d reads d (D/2)/(2π/L) = 211 in this case, as ≡ one derives the relation definedinthecontextofthediscreteequations,Eqs. (10) - (12). We emphasize that as mode κ = D/2 is excited, κ (2ρ /l )t, (25) max 0 s coherence of the packet D, although undergoing a mod- | |∼ ulational process, does not decay. which shows that the packet spreads over the recipro- Of course, the presence of a band width for the linear cal space at a rate 2ρ /l . The time for excitation of ± 0 s frequenciesmaychangetheentirepicture,andthisisthe any particular wave vector κ is thus τ (κ ) = max exc max subject of the next section. κ l /2ρ . If we take κ = D/2, we see that for max s 0 max the typical case Dl 1, τ τ (κ = D/2), s D exc max ≫ ≫ whichmeansthatintheabsenceoflinearfrequencyband widths, coherence time of a packet of range D is in gen- B. The effects of space derivatives and the associated linear frequency band width eral much longer than the time required to activate the modes at the borders of the packet. In Fig. 1 we display the contrasting behaviors for Sincethefullnonlinearsystemisnotautonomousfrom D I in the cases Dl 1 and Dl 1. In the simulations the perspective of macroscopic modes, one cannot make s s ∼ ≫ weintegrateset(1) -(3)withapseudo-spectralmethod, veryformalpredictionsaboutcoherencedecayduetothe using a grid of length L = N = 215, N denoting the frequencybandwidths,likewedidintheapproximations number of nodes which for scaling simplicity is equal to leadingtoEqs. (15)-(17). However,someestimatescan the length. In all numerical analysis we use ρ = 1; the still be made. 0 choice is not restrictive because field scales can always Let us consider our expression (25) for the maximal be absorbed in space and time. Considering l =29 and wave vector involved into the dynamics. When κ s max 6 reaches the value κ corresponding to the resonant fre- band width, i.e., for v = 0: τ = D2l2/(2π2ρ ). τ is r g D s 0 D quency, v κ 2ρ , coherence is expected to be lost, the largest coherence time of a collection of modes con- g1 r 0 ≡ but now due to resonant effects. Under this circum- tained within the limits D/2 < κ < D/2. Then, once stance,thelargestexcitedlinearfrequencywouldbecome 2.0 − comparableto the nonlineartripletfrequency Ω, andco- herent nonlinear effects would be no longer dominant. The time to attainresonance,letus callitτ τ (κ ), 1.0 r exc r ≡ can be obtained as one uses κ =κ in Eq. (25): max r τr ls/vg1. (26) I 0.0 ≡ D A given collection of modes of range D will remain coherent as long as t < τ min τ ,τ . To illustrate coh ≡ { D r} -1.0 this point, let us take the case analyzed in the panel (b) of Fig. 1. In that case τ is large and we do not ex- D pect tosee coherencedecaysoonifthe linearfrequencies -2.0 are absent. But now let us add a frequency band width 0 50 100 150 time with v chosen such that a given mode of the spectrum g1 becomes resonant with τ <τ ; we achieve this require- r D ment with v = 1/(κ 29) which yields τ 50 < τ . FIG. 2: Coherence decay due to the resonant effect. Param- g1 l r D For completeness we take v = 0 which ≈corresponds eters are those of panel (b) of Fig. 1, with exception of vg1 to one wave moving relativge1ly,2 to the other two. The which here reads vg1 = 1/(29κL), defining a resonant time setting would be of relevance to Brillouin scattering, for τr ≈50. instance,wheretwoelectromagneticwaveswiththesame groupvelocity interactwith a slowerion wave;we would the range D is defined, one has to look at the excitation be examiningthe processinthe framewherethe electro- time τ ofthe resonantmode,whichdoesnotnecessarily magnetic wave is stationary. The resulting dynamics is r belongsto the rangeD; we found thatτ =l /v . Gath- then displayed in Fig. 2, where one clearly sees a fast r s g ering together both time scales, the final conclusion is decay whereas for v = 0 one sees persistent coherence g thatthecoherencetimeτ satisfiesτ =min τ ,τ . as previously shown in Fig. 1(b). coh coh { D r} We havealso observedandstressedthat coherencegains Expressions (24) and (26) therefore provide us with a some substantial meaning only when several nonlinear simple tool to make estimates on the circumstances al- oscillations occur prior to τ . Since in our normalized lowing coherence to be seen in the nonlinear triplet sys- D variablestheperiodofthenonlinearoscillationis 1/ρ , tem. Onceonehasdefinedanextendedcombdistributed ∼ 0 one concludes that the dynamics resembles a nonlinear over a range D/2 < κ < D/2 of wave vectors with D/∆ 1, and−once one knows the group velocity vg for phase locking processonly when D ≫∆ and vg∆≪ρ0. ≫ Letusconnectourresultswiththoseofpreviousworks. this particular class of wave, the coherence time can be Our macroscopic model does not look into fine micro- obtained. scopic scales of size, say l , where discrete effects be- mic come relevant. Therefore an upper limit D 1/l max mic ∼ does exist beyond which mode dynamics is naturally at- IV. FINAL REMARKS tenuated by microscopic effects. One can however imag- ine that modes with wave vectors κ > D /2 are In this paper we developed a technique to investigate | | max initially small and heavily damped; if this is true they coherence in nonlinear triplets, when the available band will be minimally excited during the dynamics. Under ofmodesismuchlargerthanthatoftheinitialcombs. If these circumstance the condition on τ for an inacces- modes remainrestrictedto their initial combs,the series D sible D > D ceases to exist (since τ in this of approximations outlined in 2 allows to describe the max D → ∞ § case)andweareleftonlywiththeconditiononthegroup system as an interaction of macroscopic modes. In the velocity and linear band width, which is similar to what presentlystudiedcase,onecannotresorttotheseapprox- is discussed in previous investigations. For D < D imationsbecauseinitiallylowamplitude,idlemodesout- max τ is finite and physically relevant. side the initial range will be gradually excited at a rate D 2ρ /l , whenever the whole available band is larger 0 s ∼ than ∆. Coherence in this, perhaps, more realistic case We acknowledgesupport by CNPq, Brasil,and by the is a little more involved subject to define. One first de- AFOSR, USA, under the grant No FA9550-06-1-0345. fines the range D of interest. 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