Spatiotemporal chaos in nonintegrable three-wave interactions

PHYSICA ELSEVIER

Physica D 81 (1995) 237-270

Spatiotemporal chaos in nonintegrable three-wave interactions C a r s o n C. C h o w 1 Department of Astrophysical, Planetary and Atmospheric Sciences, University of Colorado, Boulder, CO 80309, USA

Received 29 March 1994; revised 14 September 1994; accepted 14 September 1994 Communicated by J.D. Meiss

Abstract

The nonlinear three-wave interaction is ubiquitous, having many physical manifestations. The conservative equations are integrable by inverse scattering transforms and can possess soliton solutions. Two nonintegrable three-wave models were numerically simulated. Spatiotemporal chaos, where coherent structures evolve and interact chaotically, was observed in the long time, large system limit. Correlation functions were measured. The behaviour was analysed by using perturbation theory around the inverse scattering transform solutions of the nonperturbed integrable equations. Saturation amplitudes, correlation times and lengths were estimated.

1. Introduction

T h e nonlinear three-wave interaction (3WI) has applications to plasma physics, water waves, nonlinear optics and many other areas [1-3]. It is a paradigmatic equation much like the nonlinear Schr6dinger, K o r t e w e g - d e Vries and sine-Gordon equations. Like those systems the conservative form of the 3WI is fully integrable by the inverse scattering transform (IST) and has soliton solutions [1,3-5]. This paper examines the dynamics of nonconservative, nonintegrable forms of the 3WI in one spatial dimension. It will be shown that the nonconservative 3WI with growth and dissipation exhibits spatiotemporal chaos (STC) in the large system and long time 1Present address: NeuroMuscular Research Center, Boston University, 44 Cummington St., Boston, MA 02215, USA. E-mail: [email protected].

limit. The two models examined consist of "a linearly unstable wave that is nonlinearly saturated by coupling to two damped waves [2,6,7]. In the first model the wave with the highest frequency has the middle group velocity of the three waves; in the second model the highest frequency wave has the greatest group velocity of the three waves. The term STC is employed here to refer to the chaotic dynamics of coherent structures or spatial patterns with a specific length scale in a spatially extended system [10-14]. This is contrasted to fully developed turbulence and to low dimensional chaos. The former is notable for its extremely complicated behaviour involving many scale lengths, energy cascades and intermittency. The latter concerns the unpredictable dynamics of systems that have a few degrees of freedom. Low dimensional chaos has been demonstrated in time-only versions of the 3WI [15-17]. O t h e r

0167-2789/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0167-2789(94)00198-7

238

C.C. Chow / Physica D 81 (1995) 2 3 7 - 2 7 0

examples of systems with STC include large aspect ratio Rayleigh-Bernard convection and the Kuramoto-Sivashinsky equation, which models several phenomena including chemical reactions and flame fronts. Ref. [10] contains an extensive review of these concepts. The advantage of the 3WI in understanding STC is that it has an integrable limit so perturbation theory around integrable solutions can be employed. A significant body of work exists for the study of nearly integrable soliton systems in periodic boundary conditions [18-22]. Ref. [18] contains a thorough review. The emphasis there is on the chaotic behaviour of perturbed systems containing a small number of solitons. The machinery of the inverse spectral transform is used to probe the actual mechanics of the chaos arising from homoclinic structures in the integrable solutions. In this paper, the statistical long time, large system properties are studied. The resulting STC arising from the interaction of many coherent structures is the main concern. However, a study in the line of Ref. [18] would be very useful and remains to be done. The nearly integrable limit of the 3WI is studied with numerical simulations and perturbation theory about the IST solutions. The 3WI can arise whenever [1,2,9] (1) a weakly nonlinear medium supports a set of discrete waves ~o = o~(k), (2) the nonlinearity is manifested as a slow variation of the field amplitudes, (3) the lowest order nonlinearity involves the quadratic coupling of the amplitudes and (4) the waves satisfy the resonance condition (Di q- O)j q- O.)k : O ,

k i + ky + k k : O .

(1)

If these conditions are satisfied then a straightforward perturbation expansion will yield the amplitude equations [2,9] OtOi ~- V i "VO i

= 0j* Oh* ,

(2)

where (i, j, k) are cyclically permuted to yield three equations. The left side of Eq. (2) involves simple advection along characteristics with group velocities v i and the right side involves a quad-

ratic coupling with the other two amplitudes. Eq. (2) is integrable in 2 + 1 dimensions [23]. However, the 3WI in a sense is integrable for any number of dimensions because the equation can always be reduced to three characteristic coordinates, two of which can be called space and the other time. In 1 + 1 dimensions solitons can exist

[1,3-5]. For a medium with a weak growth and dissipation, Eq. (2) can be modified [2,6]. The specific model to be studied involves a linearly unstable high frequency wave coupling to two damped lower frequency waves in one spatial dimension. After some rescaling the equations take the form [2,71 Ota i + Ui Oxa i -- D Oxxa i - Tiai :- - a j a x , 3ta j +

V] 3 x a j

+

"yjaj =

aia * k,

Ota k + v k Oxa k + y k a k = a i a ~

(3a) (3b)

,

(3c)

where the a's are complex wave envelopes, the 3,'s are growth or damping coefficients, and D is a diffusion coefficient. The diffusion term is usually not included in the 3WI. This term arises if the growth of the linear wave has a slow spatial variation. It is the lowest order reflection invariant term that provides a cutoff in wave number of the growth. Without this term the problem is not well posed. There would be unbounded growth as the wave number increases. This term is essential for nonlinear saturation and is very important in determining the long time behaviour. The high frequency unstable wavepacket is indexed by i. For convenience the amplitudes will be referred to as waves although they are actually wavepacket envelopes. The unstable wave is called the parent and the other two waves are referred to as the daughter waves. The order of the group velocities determines the behaviour. If the high frequency wave has the middle group velocity of the three waves then soliton solutions exist for the integrable equations. The solitons can be transferred between the parent and daughter waves. This case

C.C. Chow / Physica D 81 (1995) 237-270

is known as the soliton decay interaction (SDI). This case also corresponds to the nonlinear saturation of an absolute instability. If the parent wave has the highest or lowest group velocity then solitons are no longer possible and the collisions between wavepackets become important. This will be called the stimulated backscatter interaction (SBS). This case is probably more c o m m o n than the SDI because the resonance conditions Eq. (1) are more easily satisfied. It is also the nonlinear saturation of a convective instability. The SDI and the SBS are the two models considered in this paper.

2. Soliton decay interaction 2.1.

The model

A n example of a physical application of SDI is the decay of an unstable lower hybrid wave to two daughter lower hybrid waves in a magnetized plasma [8]. However, the model applies to any generic situation that satisfies the given conditions. The equations are a special case of the general 3WI nonconservative equations (3). Numerical simulations are combined with linear and perturbation analyses in order to understand the STC manifested in SDI. The IST solutions for the conservative SDI are described in detail in Ref. [1]. The solutions involve the exchange of solitons between the three waves. The IST solutions are essential to understanding the behaviour of the nonintegrable SDI, and provide nonlinear solutions to perturb around. For SDI the group velocities satisfy the condition that the highest frequency parent wave has the middle group velocity. Eqs. (3) can be simplified by transforming to the frame of the parent wave. F u r t h e r m o r e , it is assumed that the daughter waves have group velocities with the same magnitudes and have the same damping coefficients (v k = - v j , yj = Yk). The length and time are rescaled so that the damping coefficient

239

and magnitude of the daughter group velocities is unity yielding c3ta i - D Oxxa i - "yiai = - a j a k ,

(4a)

Otaj - Oxa j + aj = aia ~ ,

(4b)

Ota k + Oxa k + a k = a i a I •

(4c)

In the normalized form given by Eqs. (4), there are two free parameters: the normalized growth rate ~ and the normalized diffusion coefficient D. Different regimes yield different behaviour. Three different regimes in the parameter plane (~, D ) exhibiting STC are considered. Results of numerical simulations are then presented and analysed. The first case is nearly integrable with weak growth and weak diffusion. This allows the nonconservative terms to be treated as perturbations about the integrable 3WI, and much can be understood about the dynamics. The second case has strong growth and the third has strong diffusion. Both of these cases are not as easily tackled by perturbation theory, so analytic results are not as complete as in the nearly integrable case. 2.2.

The IST solution

The conservative 3WI is integrable by the inverse scattering transform [1,3-5]. In 1 + 1 dimensions the SDI has soliton solutions while the SBS, as will be discussed later, does not. Solitons in the SDI are a little different from the ones in the K o r t e w e g - d e Vries or nonlinear Schr6dinger equations. In those systems solitons arise out of the balance between dispersion and nonlinearity and they preserve their form after collisions with other solitons. The SDI is nondispersive, all wave forms travel at the group velocity of the wave. Solitons are structures incorporating all three waves. The IST solution uses the Z a k h a r o v - M a n a k o v third order scattering problem [1,3,4] (see Appendix C). Soliton solutions correspond to bound states of the Z M problem. However, if the wave envelopes are well separated, the scattering problem can be

C.C. Chow / Physica D 81 (1995) 237-270

240

reduced to three second order Z a k h a r o v - S h a b a t (ZS) scattering equations [1]. Solitons can then be thought of as bound states of the individual ZS problems. T h e r e is some ambiguity to the term soliton in the SDI, but here a soliton will refer to a ZS soliton. In this description solitons can be thought of as existing in a particular wave and can be exchanged between the three envelopes. The IST analysis shows that the parent wave can decay to solitons in the daughters when an area threshold is attained. This threshold corresponds to the existence of a bound state with an eigenvalue given by the Bohr quantization condition y

f

(la] a - n 2 ) 1/2 d x = ½~r,

(5)

x

where x and y are turning points of a local pulse, a is the ZS potential (proportional to the wave envelope) and ~7 is the ZS eigenvalue. The simplest soliton solution shows that a soliton of the form lai[ = 2,/sech(2~x) in the parent wave will decay into solitons of the form lall = V2 ~1sech[~?(x + v~t)] in each of the two daughter waves. Colliding daughter solitons can also reform a soliton in the parent.

2.3. The nearly integrable regime 2.3.1. Simulation results Eqs. (4) were numerically simulated with periodic boundary conditions on the domain x E [0, L]. The 3WI is a hyperbolic P D E . It lends itself fairly easily to numerical integration because the characteristic curves are straight lines. Each envelope is first transformed to its characteristic moving frame. A fixed spatial grid is then laid out for each envelope and the P D E is reduced to coupled ODEs. The spatial grid was chosen to be able to resolve the smallest structures that may result. The specific details of the numerical m e t h o d are in Appendix D. T h e long time, large system dynamical behaviour was of interest. In each simulation, the

complete spatiotemporal history of each envelope was recorded. The integrated energy Ul(t ) = foc lat(x, t)l 2 dx was monitored. The saturated state was considered to be reached when the integrated energy began to fluctuate about some average value. The correlation function Sl(x, t) = (al(x - x ' , t - t') a~(x', t')), where the angled brackets denote spacetime averages, was then constructed. Actual spatiotemporal profiles proved to be of great use in comprehending the dynamics. In all the simulations, r a n d o m real small amplitude initial conditions were chosen. The correlation function for different runs and runs of varying length were compared to ensure the results were consistent. Conservation laws for the integrated energy are derived in Section 2.3. Compliance with these laws was a measure of the veracity of the simulations. The parameters for the nearly integrable case were D =0.001, ~ = 0.1 and L = 2 0 . This case exhibited STC and fell into the regime where perturbation theory could be used to explain the behaviour [6,7]. Fig. 1 shows the spatiotemporal evolution profile of the parent. The length shown is one half the simulation system size and t -- 0 is an arbitrary time well after the transients have decayed. The profile of the parent is irregular yet spatial, and temporal scales can be observed. There are coherent structures of a definite length scale that are seen to grow, deplete and collide with one another. The corresponding daughter

Fig. 1. S p a t i o t e m p o r a l profile of the p a r e n t w a v e a v

C.C. Chow / Physica D 81 (1995) 2 3 7 - 2 7 0

241

~d o

i° o

10 z 10-~ ~

6

O~~Z

o

i 0 -~

e

I 0 -~ I

[ I1[11

I

I

I

I [111[

10° q

Fig. 2. Spatiotemporal profile of the daughter wave aj.

profile is shown in Fig. 2. A sea of structures is seen convecting to the left. The structures are of a definite size and are constantly being created. T h e y then damp as they propagate. Only daughter aj(x, t) is shown. The other will be similar but with structures propagating to the right because of the parity symmetry of Eqs. (4b) and (4c). The correlation function for the parent is given in Fig. 3. It clearly shows decorrelation for long times and lengths. The spatial direction shows a definite length scale that was observed in Fig. 1. Correlations fall off gradually in the time direction. This is a clear signature of STC. There is a well-defined correlation length, and beyond this length the dynamics are decorrelated. Further information is gained by Fourier transforming S l ( X , t ). The spectrum of static fluctuations Si( q, t = 0) for the parent wave is shown in Fig. 4. T h e r e is a cutoff near q = 10, and a range of

I

10~

Fig. 4. Spectrum of static fluctuations Si(q, t = 0) of the parent wave q~.

modes show up as a prominent hump. The cutoff reflects the length scale seen in the spacetime profile and in t h e correlation function. The hump in the spectrum indicates weak periodicity. The spectrum becomes flat for wave numbers below the hump indicating decorrelation. The local power spectrum S i ( x = 0 , w ) given in Fig. 5 shows two time scales. The spectrum bends over near w = 0.02 which gives a long time scale, and a shoulder at o ) = 0 . 3 gives a short time scale. Longer runs with the same parameters hint that there may be a slow power law rise of undetermined exponent for frequencies below the low o) bend similar to that observed in the Kuramoto-Sivashinsky equation [24]. The short

10~

U3

~

~

w

~

~'~' "~

I

[ IIIIIII

I

i iiIrlll

i0 ~

i

i 0 -~

i IIIIII I0 °

co

10 Fig. 3. Correlation function Si(x , t) of the parent wave a~.

Fig. 5. Local power spectrum S~(x = O, w) of the parent wave ai.

C.C. Chow / Physica D 81 (1995) 237-270

242

time scale appears as the growth and depletion cycle observed in the spatiotemporal profile in Fig. 1. The daughter correlation function is calculated along the characteristic curve x = - t . There is an abrupt drop in the time direction (direction along characteristic) followed by a very slow and long decay. In space there is a definite length scale where the correlation function drops to zero, but then along the x axis a very small but nonzero correlation is observed over the entire length. The temporal decay rate is very quick (of order t - - O ( 1 ) ) . The spectrum of static fluctuations Sj(q, t = 0) is shown in Fig. 6. It shows a softer cutoff than the parent wave around q -~ 6 giving the correlation length. For wave numbers lower than the cutoff the spectrum flattens out. The local power spectrum Sj(x = 0, o9) is shown in Fig. 7. The spectrum shows two peaks at high o9. One is where the shoulder of the parent wave power spectrum is, and the other is at twice the frequency. For low frequencies the spectrum begins to bend over at ~ o - 0.007. This bend is m o r e p r o n o u n c e d in longer runs, It is not known whether the spectrum becomes fiat below this bend or has a power law rise like that in the parent wave. The time series for the integrated energy Uz is shown for all of the waves in Fig. 8. The upper curve is the parent wave; the two

ld

Q

U3

I

I IIllll

i

I I[11111

10 =

I

10 -~

I IIIIII

10 °

co

Fig. 7. Local power spectrum Sj(x = 0, ~o) of the daughter wave a].

daughter waves are perfectly synchronized in the lower curve. The dynamics clearly fall into the realm of STC. There are coherent structures of a definite length scale that interact chaotically. Correlation functions are well defined in space and time and provide a good description of STC. The parent wave is composed of coherent structures that grow and deplete on a short time scale and drift and diffuse on a longer time scale. The daughter waves are composed of damped drifting structures. They are created at intervals of the short time scale seen in the parent wave and have very

150.0

ld lo o

"~

100.0

10 -~

10 -3

50,0 1G 4

10-s I

llllll

I

0.0

I

10°

0,0

ld

q Fig. 6. Spectrum of static fluctuations Sj(q,t= 0) of the daughter wave aj.

200.0

400.0

600.0

800.0 1000.0

t Fig. 8. Energy time series U(t). The upper curve is the parent energy; both daughter energies overlap on the lower curve.

C.C. Chow / Physica D 81 (1995) 237-270

long correlation times. An analysis of the observed dynamics is presented in the ensuing sections.

2.3.2. Qualitative descriptions of the dynamics The main features of the dynamics can be understood in terms of the linearized equations and by considering the growth and dissipation terms as perturbations around the integrable 3WI. The numerical simulations show that the spatiotemporal dynamics exhibits STC and can be described in terms of a correlation function which has a given amplitude (average energy density) and well-defined temporal and spatial scales. By combining linear analysis with the IST solutions, the correlation lengths for the parent and daughter are estimated. The trivial fixed state of Eqs. (4) found by setting O/Ot to zero is 2

Oxxai +qoa i = O ,

aj=a k=0,

(6)

where q0 = ~/y~/D. Hence, the parent wave fixed state is sinusoidal. The linearized dispersion relation for growth rate s and Fourier mode q for the parent wave obtained by linearizing Eq. (4a) is

(q)

s--3~ 1 - - - T . q0

(7)

Thus modes q > q0 are damped and those with q < q0 grow. The fixed state is unstable to long wavelength perturbations. Energy is injected into the parent at long length scales. The trivial fixed state is always ,nstable to long wavelength fluctuations. The nonlinear interaction with the two daughter waves saturates the long wavelength instability. The parametric interaction analysis for a spatially varying pump shows that the daughter waves will be unstable and grow [25] whenever an area threshold is satisfied by the parent. Using WKB analysis the threshold is [25] b f a

243

Condition (8) for the case of no damping 7j = 0, is identical to the Bohr quantization condition (5) for zero eigenvalue. The damping and growth will perturb the decay process, but for weak perturbations the essential elements of the interaction remain intact. Instead of decaying into solitons the parent wave will decay into quasi-solitons when the WKB threshold condition is met. A perturbative analysis around the IST soliton decay solution is detailed in Section 2.3.4. The depletion process spatially decimates the pump and saturates the growth of the low q. From any random initial condition, small scales are smoothed out by the diffusion process to length scales on the order of 27r/qo. The long scales grow until a local area exceeds the WKB depletion threshold condition Eq. (8). The parent wave then depletes to quasi-solitons in the daughters and generates smaller scales or higher q in the process. The steady state is attained when a balance between the nonlinear conversion of long length scales to small length scales and the linear elimination of the small scales is attained. Spatially the parent wave will be composed of long wavelength fluctuations about the principal wavelength 2,rr/qo. The correlation length for the parent is then

~p = 27r/q o •

(9)

The daughter waves are composed of quasisolitons in a constant state of creation and decay. The daughter correlation length will be given by the average quasi-soliton width. In Ref. [1] it was shown the soliton width in the integrable equations was related to the bound state eigenvalue of the parent pulse. This relation is assumed to carry over into the nearly integrable regime. If the average eigenvalue is denoted by r/ then the daughter correlation length is ~d = 2/~/.

(10)

2.3.3. Energy equations [ai2 - Yj211/2 I dx > ½7r.

(8)

Energy equations can be formed for Eqs. (4) by first forming the complex conjugate equa-

244

C.C. Chow / Physiea D 81 (1995) 237-270

tions. Multiplying the original equations by a~ and the conjugate equations by a t and adding yields o,lail +

Oxla

l 2 -

2 la, I=

{Uj) = ~

L

(11)

= " K a * a j a k - K *a i a ,j a k•, o,lajl 2 + oj oxlajl = + 2Tjlaj] 2

(12)

= Ka*aja k + K *aiajak., , • o la l 2 + 23,klak[ 2

(13)

= Ka*aja k + K*a,a~a*~.

For a periodic domain x E [0, L], the equations are integrated over x. Eq. (12) is added to Eq. (11) and Eq. (13) is subtracted from Eq. (12) to yield the integrated conservation or energy equations

0

(14) (15)

where Ul = f~ lall 2 dx. For the conservative 3WI (D =Yt = 0), Eqs. (14) and (15) are known as the Manley-Rowe relations, usually rewritten in the form O,(Ui + Ui) = O, O,(Ui + Uk) = 0. The Manley-Rowe relations are two of the infinite conserved quantities associated with integrable partial differential equations [26]. Eq. (15) shows that if the damping coefficients of the daughters are equal (yj = y~) then the energies of the daughters will approach one another exponentially. In the saturated state the energy U~ is composed of an average part plus a fluctuating part, Ul = (Ul) +. ~Uz, where ( I / L ) ( U I ) = S,(0, 0). Inserting this into Eqs. (14) and (15) and taking the time average yields L 2

"~J

(U~) ---~ J (lOxail ) d x = - ~ ( g ) , 0

where the convention f(x) = (1/2"rr) .f eiqXF(q) dq is used. Note the right hand side of Eq. (18) is really a sum rather than an integral. However, for L>>~p, where ~p is the correlation length, this approximation is fairly good. For the STC regime (]ai(q) l2) is approximately constant up to the highest unstable mode q o = V ~ i i / D , then decays rapidly. For this case the integral is evaluated to yield L

(19)

0

½OtU~ + ½0tUj = y,U~ - 'Yig - D J IO.ael 2 dx ,

~

(18)

0

f (lOxail2} d x = ( x1q 3 (I a i(q)l 2 })0q0

L

½ 0 t U ] -- l o t U k = - ' y j U ] -1- ")lkUk ,

c~

J (10xail2) dx = j q2(lai(q, 012} dq , 0

D

(17)

Applying Parseval's theorem to the integral in Eq. (16) gives

- O ( a * O.~a i + a i O~:xa* )

o,la l z +

{U~).

(16)

1"~ -5~ (U,),

(20)

since U i=qo[ai(qo) 2. Reinserting into Eq. (16) gives the relation 47j (U~) --- g-~-i ( Uj).

(21)

2.3.4. Short time behaviour The short time scale observed in the simulation was due to the constant growth and depletion of the parent wave into quasi-solitons. This time scale is contingent on several factors. The IST soliton decay solution shows that the soliton content of the parent is completely transferred to the daughters. The perturbing terms will convert some of this soliton content into radiation. The result is that the depleted parent will have some remaining area after depletion. This remaining area will then grow until it attains the threshold for depletion and repeat the process. The threshold area depends on the initial amplitude of the daughters. If the daughter amplitudes are zero, Eq. (4a) shows that the parent pulse cannot

c . c . Chow / Physica D 81 (1995) 237-270

deplete. Some nonzero amplitude is required to seed the depletion. Colliding daughter waves provide the stimulus for decay. This leads to a cycle. Depleting parent pulses generate quasisolitons which collide with other parent structures. These then decay and the process repeates indefinitely. This decay instability is the root of the chaotic behaviour. The precise decay times of individual parent pulses are extremely sensitive to the colliding daughter quasi-solitons that trigger the decay, and the amplitudes and phases of the generated quasi-solitons are very sensitive to the parent pulses that generate them. However, some general statements about the average cycling time can be made. Perturbation theory is used to answer two questions involved in this process: (a) Given that the parent depletes from some initial area (ZS eigenvalue), what is the remaining area after depletion (leftover radiation)? (b) What threshold area (threshold eigenvalue) is required for the parent to deplete? The first question is addressed with a multipletime scale analysis around the IST solution for soliton decay given in Appendix A. The IST one-soliton decay solution shows that a soliton in the parent wave decays to solitons in the daughters with a characteristic decay time. The growth and dissipation are relevant on a slower time scale. The second question is answered with a perturbation expansion in the Z M scattering space. A threshold condition on the parent wave bound state eigenvalue for the emission of a daughter quasi-soliton is derived in Appendix B. Many of the IST concepts dealt with in this section are outlined in Ref. [1] and Appendix C. The main results are summarized here. Again let 77 be the average threshold bound state eigenvalue of a localized parent structure. The remaining area after depletion can be represented by an 'effective eigenvalue' ~7'. This is the amount of soliton content converted to radiation and not transferred to the daughters. If the effect of diffusion is considered small and 77 has a growth rate if y~. then the time required to grow back to threshold after a depletion is

tg~--~ln , .

245

(22)

The cycling time t c would then be the decay time plus tg. The IST solutions show that the decay time is on the order of 1/2~/ (see Appendix A). However, for weak growth, 3~ << 2r/, the decay time can be neglected with respect to the growth time and t c = tg. In Appendix A a multiple scale perturbation expansion about the IST decay solutions is used to estimate ~7'. The calculation relies on the ordering 3~ << 27/ and the result obtained is ~/' = (2 + 3~) ln(3)/2.

(23)

A comparison of the Bohr quantization condition (5) with the WKB condition for decay (8) shows that ~/> yj. is necessary for a parent pulse to decay. Once ~/exceeds this critical value it will decay if a quasi-soliton collides with it. A more thorough analysis is given in Appendix B where a perturbation expansion in scattering space is developed. The depleting parent structures generate quasi-solitons on average at intervals of t~--t c. These quasi-solitons then collide with other parent structures triggering further depletions. The complex dynamics is a result of this feedback loop. Assuming that depletions are triggered by two colliding quasi-solitons generated two correlation lengths away give an estimate of ~/= 2yj + 4~03~ .

(24)

The estimates for ~7 and ~7' can then be inserted into Eq. (22) to obtain the average cycling time. This estimate does not take into account radiation and diffusion effects which can delay the decay. It must be considered a lower bound of the average cycling time. 2.3.5. L o n g time b e h a v i o u r a n d S T C

The long time behaviour of SDI stems from the diffusion of the parent wave and the many collisions between the parent structures and the daughter quasi-solitons. The phases of the collid-

246

C.C. Chow / Physica D 81 (1995) 237-270

ing quasi-solitons are very important for the outcome of the result. Consider real envelopes. F r o m Eq. (4a) it is apparent that if the two quasi-solitons have opposite sign they reinforce ia positive pump and cause a negative one to deplete. Conversely two quasi-solitons of the same sign will deplete a positive pump but feed a negative one. In the STC situation a parent at a given location will be involved with collisions with quasi-solitons at random times with random phases. Certain collisions will cause the parent structure to grow and others will deplete it. Thus the threshold eigenvalue will have a spread around some average value depending on the phases of the quasi-solitons and the frequency of collisions. The parent structures also act as amplifiers for the quasi-solitons. When a quasisoliton collides with a parent structure of the right phase it can trigger a depletion and create new quasi-solitons. The quasi-soliton regenerates itself and continues to propagate. By this mechanism the effect of a quasi-soliton could extend over very long distances. During the process there will be a small phase shift due to the time required for decay. So the long correlations will be close to but exactly not along the characteristic curve. In the STC situation the parent structures are not truly in isolation but are in the close proximity to one another. A more appropriate (although more difficult) way to handle the problem may be to consider the ZS problem with a perturbed periodic potential. The lack of isolation can lead to interesting effects. For instance tunnelling between adjacent parent structures can occur. This could lead to bound states of double or even n-tuple wells. These effects were sometimes observed in the simulations. If the diffusion coefficient is zero then there will be no nonlinear saturation. The generated high q modes would not be damped. The parent structures would simply cascade to shorter and shorter length scales with higher and higher amplitudes. The diffusion is responsible for the long time behaviour. The parent structures tend

to grow and deplete in one location for very long times compared to the cycling time scale t c. However, the action of the diffusion combined with the growth and depletion will cause them to drift and shift position. A simplistic estimate of the long time scale for the parent rp is obtained by considering the diffusion time across a correlation length. Thus D

(2~) 2

(25)

The long time scale observed in the daughter wave dynamics will be related to the parent correlation time. As the quasi-solitons collide with the parent structures they will deplete creating new quasi-solitons. As the parent structures drift, so will the location for creation of the quasi-solitons. However, the quasi-solitons have a different width than the parent structures so the long time scale of the daughters r d is given by 4 r d -- r/2D

•

(26)

The average saturated energy density can be crudely estimated from the threshold eigenvalue ~/. Recall that the simulation results show that the saturated state of the parent wave is in the form of coherent structures with the average amplitude related to ~/. Locally, the shape of the coherent structures are roughly sinusoidal with wave n u m b e r %. Let the parent wave be composed of locally sinusoidal coherent structures of average amplitude %. Then the average energy density (Ui)/L ~Si(O, 0) is given by 2 Si(O, O) = ao/2.

(27)

The average amplitude can be estimated by using the Bohr quantization condition Eq. (5). Taking a square wave form with a width given by the correlation length ~p gives a condition relating the average amplitude to the average eigenvalue ( a 2 -[- 7 ] 2 ) 1 ' 2 ~ p = I ' ] T .

(28)

Solving for a 0 and plugging into Eq. (27) yields

C.C. Chow / Physica D 81 (i995) 237-270

Thus Si(0, 0) depends on the correlation length and the threshold eigenvalue previously calculated. Given this estimate for S~(O, 0) the average daughter energy density Sj(0, 0) follows from Zq. (21). 2.3.6. Discussion T h e analytical estimates can be compared to the simulation results. In the simulation 3~j~ E = 1, 3~ ~- o- = 0.1 and D = 0.001. From Eqs. (6) and (9), the principal mode is q0 = 10, which gives a correlation length of ~ p = 0 . 6 . The threshold eigenvalue equation (24) comes out to be r / = 2.2. This then gives a daughter correlation length of ~a = 0 . 9 from Eq. (10). The cycling time is given by t c -~ 8 from Eq. (B.48). From Eq. (25) the parent correlation time is rp -----400. From Eq. (26) the correlation time of the daughter is % = 800. T o compare with the results of the simulation the values can be expressed in terms of frequencies and wave numbers. For the parent wave the long time scale rp translates to a frequency of ~Op=0.016 which corroborates with what was observed in the parent power spectrum in Fig. 5. The predicted cycling time translates to wc = 0.78. This value is high by about a factor of two compared to the shoulder observed in the parent power spectrum. H o w e v e r , the spacetime profiles in Fig. 1 do show some of the parent structures cycling near the predicted time scale, and Fig. 2 shows quasisolitons being created at a rate close to the predicted value so the calculation does give a lower bound for the cycling time. The daughter power spectrum in Fig. 7 showed two peaks in the high frequency regime. The lower frequency one corresponds to the cycling time and the higher frequency one is at twice the frequency and is probably a harmonic of the first. The predicted long time scale translates to a frequency of o)d --~0.008 which again corroborates well with the simulation result.

247

The spectrum of static fluctuations in Fig. 4 for the parent wave shows a cutoff near q0 = 10 as expected. The daughter wave has a much softer cutoff as seen in Fig. 6. Since the correlation length is due to the quasi-soliton width this is expected. The quasi-soliton width is inversely proportional to the threshold eigenvalue ~/ and thus has a spread about an average value. The estimate for the width given by the correlation length £d corresponds to a wave n u m b e r of qd = 7 which is somewhat high for the same reason the cycling frequency estimate was too high. The ratio of the parent energy to the daughter energy was predicted to be U i / U j ~- 13 from Eq. (21). The energy time series in Fig. 8 corroborates this. The peak height of the correlation functions of the parent and daughter waves Si(O, O)/Si(O, 0) shows the same ratio. The two daughter wave energies are synchronized in Fig. 8 as expected from Eq. (15). The parent average energy density was calculated to be Si(0, 0) -~ 5.8 from Eq. (29) which is quite good considering the cavalier approximations made. The power spectrum of the energy showed a peak at around ---5 which is a little closer to the estimate for t c. Recall that the cycling time perturbation calculation in Section 2.3. was actually done for the eigenvalue or area cycling time. A word should be said about the system size. It is clear with the very long correlation times for the daughters that they cycle the box many times before correlations decay away. Thus for long times, the temporal correlation function along the characteristic or at a single spatial location would be the same. This was borne out in the simulation. It is unknown what the precise boundary effects are since it was impossible to numerically test a system large compared to this long time scale. However, with other runs of varying length, it was found that the above time scales seems to be unaffected by the system size as long as it is much larger than ~p. The power law rise for the parent power spectrum below 2~/rp, seems to decrease in exponent as the system size increases. It appears that the main

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C.C. Chow / Physica D 81 (1995) 237-270

features of STC in terms of correlation functions can be understood for this parameter regime.

The parameter space is extremely vast. The above parameters were chosen because they exhibited STC and fell into the regime where perturbation theory was possible to provide analytical estimates for the properties of the correlation functions. However, the system is very rich and other regimes are equally complex and interesting albeit less tenable to analytical analysis. Two different cases demonstrating other forms of STC in SDI are considered here. These are by no means an exhaustive representation of the behaviour of the system. In this section a case where the growth of the parent is comparable to the damping and a case where the diffusion is large will be studied. The parameters for the first case were 3~= 1, D = 0.01, L = 20. This is an example of strong growth. Again the system was numerically simulated and evolved until a saturated state was reached. Fig. 9 shows the spacetime profile of the parent wave. As in the nearly integrable case there is still evidence of structures growing and depleting. However, unlike that case the depletions are very violent, often destroying the structures. The complete decorrelation of a par-

ent structure seems to require a few depletions. The principal mode for these parameters is q0 = 10. The spacetime profiles show that the violent depletions tend to broaden the widths of the coherent parent pulses. The spacetime profile of the daughter wave is in Fig. 10. The most notable feature is that the amplitudes of the quasi-solitons are of the same order as the parent structures. This is because the growth rate and the damping rate are equal. Thus the energies will be roughly equal according to Eq. (21). The dynamics involve collisions between parent and daughter structures of comparable size so radiation effects are more important than in the nearly integrable case [1]. The spacetime profiles show that decorrelation is due mostly to the damping. The correlation length is around ~a 1. The energy ratio is Si(0, O)/Sj(O, 0 ) = 1.33 as expected from the energy condition Eq. (21). Applying the perturbation results of Section 2.3.4, realizing that these parameters are beyond its validity, yields the results ~ = 4 . 5 , ~'---1.6. This gives t c --1 and ~d ~ 0.4. The result for the cycling time does agree with the simulation. However, the daughter correlation length does not fare as well. The spacetime profile shows that well-defined quasi-solitons do not exist as they did in the nearly integrable case. In the strongly diffusive case the parameters were 3~=0.1, D = 0 . 0 5 , L = 8 0 . The parent

Fig. 9. Spacetime profile of the parent wave a i in the strong growth regime,

Fig. 10. Spacetime profile of the daughter wave aj in the strong growth regime.

2.4. Other parameter regimes

C.C. Chow / Physica D 81 (1995) 237-270

249

o

Fig. 11. Spacetime profile of the parent wave a i in the strongly diffusive regime.

Fig. 12. Spacetime profile of daughter wave aj in the strongly diffusive regime.

spacetime profile is shown in Fig. 11. As in the nearly integrable case, there are spatial structures that persist for very long times. The principal m o d e for these parameters is q0 = V2 corresponding to a length of ~p = 4.4. The spacetime profile shows structures of that size, but it is apparent that correlations exist well beyond that scale. T h e r e appear to be large compound structures where individual structures are seen t o grow and deplete yet remain part of a collective conglomerate. The correlation function showed long range correlations in both space and time. T h e spatial alternation between a single structure and a squarish collective structure seems to be a robust state. Different runs with different initial conditions and slight variations in the parameters p r o d u c e d similar looking profiles. The spacetime profile of daughter wave aj is shown in Fig. 12. Unlike the nearly integrable case there is not strong advection along the characteristic. In fact it is difficult to distinguish between the two daughter profiles. The damping length for the daughters is much smaller than the parent structure length scale. Thus the daughter structures only exist within the confines of a parent structure and do not interact very much with neighbouring structures as with weak diffusion. In this case the daughter waves are slaved to the parent wave. All the spatial scales are much larger than the damping lengths so for all effective purposes

the daughters are static and merely react to the parent. However, unlike the case of spatially uniform amplitudes the instability can be saturated by forming quasi-solitons. In the nearly integrable case, the daughter waves transferred information between the parent structures, constantly perturbing the parent structures, and instigated the diffusion. In the strongly diffusive case, information is transferred very slowly between the parent structures. H e n c e the extremely long correlation times. However, the mechanism for decorrelation is poorly understood. Somehow it must be related to the rate of information transfer between the different structures, but no quantitative estimates have been made.

3. Stimulated backscatter interaction

3.1. The model Numerical simulation and analysis is again used to understand the dynamics of the stimulated backscatter interaction (SBS).The IST solutions of the integrable equations will form the basis for understanding the nonintegrable behaviour. In SBS, soliton solutions do not exist. Instead it is the collision of the waves which drive the dynamics. Radiation effects dominate

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C.C. Chow / Physica D 81 (1995) 237-270

and as a result perturbation theory is not as readily employed as it was in SDI. However, much can still be gleaned from the IST solutions. Some of the analysis of SBS will overlap with that done for SDI. In normalized form the SBS equations are Otq i --

D Oxxa i - Na i = - a j a k ,

(30a)

Ota ] --

Oxaj + aj = a i a ~ ,

(30b)

Ota k - 2 0 x a k + a k = aia ~ .

(30c)

These equations are identical to the normalized SDI equations (4) except that the group velocity of envelope a k is v = - 2 instead of v = 1. The frame of reference of the unstable wave is again chosen. This small change makes an enormous difference in the behaviour. The terminology of stimulated Brillouin scattering in laser-plasma interactions is often convenient to use. The high frequency wave is referred to as the pump wave (PW), the middle group velocity wave is referred to as the ion-acoustic wave (AW) and the slow wave is referred to as the backscattered wave (BW). In order to satisfy the resonance conditions (1) it is much more c o m m o n for the high frequency wave to have the greatest group velocity. Thus SBS is applicable to many more physical situations than SDI. A n example is stimulated Raman or Brillouin scattering in l a s e r - p l a s m a interactions. The particular model of a growing wave coupling to two damped waves considered here could occur in a situation where the parent wave of SBS is the daughter wave of another 3WI. T h e parent wave grows because it is an unstable daughter wave of another 3WI. The STC induced in SBS may explain some recent ex-' perimental results in laser-plasma interactions [31]. H o w e v e r , the same equations are just as readily applied to other phenomena; for instance, the interaction of gravity-capillary waves where winds may excite and induce a particular m o d e to grow which then couples to two other modes [32].

3.2. I S T s o l u t i o n s

Without loss of generality the high frequency wave can be taken to have the highest group velocity. For this case the following applies in Appendix C: (i, j, k) = (3, 2, 1), ( % , Yz, 3'3) = ( - , - , +). From Eqs. ( C . 1 8 ) - ( C . 2 3 ) only the B W potential is not self-adjoint so soliton exchange effects do not play a role. This implies that the interesting effects are radiation dominated and due to the collisions between the envelopes. The interaction between the AW and the PW generates the B W hence the name stimulated backscatter. The waves also have a decimated structure after the interaction. Closed form analytic results cannot be obtained for this case as in the SDI case because the behaviour is radiation dominated. However, the IST solutions can deduce some of the properties of the envelopes. The ensuing follows from Ref. [1]. Define the density of radiation as C(a) = [ 1 - I p ( a ) l 2]

- 1,

(31)

where p(A) is the reflection coefficient and A is the eigenvalue of the ZS equation. The density of radiation F is analogous to a 'power spectrum' of the linear theory. When the initial pulses are square pulses, d o s e d form solutions for F can be found [1]. With the choice c 1 = - c 3 and c 2 0 in Appendix C we get =

F~3)(A) = A 2 G ( L Z A 2 - A 32),

(32)

F~02)(A) = A 22G(412A2 - A 2 ) ,

(33)

where A = A (1), L ( l ) i s the length of the L W ( A W ) pulse, A 3 (A2) is the area of the PW (AW) pulse and G ( x ) = ~ s i n h 2 ( - x ) l / 2 /( " x) [sin2xl/2 /x

ifx _ O .

(34)

Using Eqs. (C.27)-(C.29) for F(01) = 0 (BW initially zero) the final reflection coefficients can be derived as

C.C. Chow / Physica D 81 (1995) 237-270

F,(2) F,(3)

io

,12_ 1 + r(0 -0

io = l =_

'

(2)

(35) (3)

/"0 (1 +/"0

1 + F~2)(1 + F(03)) '

(36)

F(03)

(1 +/"(03))(1 + F(02)) "

(37)

T h e basic structure of the B W can be inferred from p~ 1). The Fourier transform of p~1) gives the wave envelope in the linear limit, but qualitative results should apply in the nonlinear regime. W h e n the areas of the AW and PW are small, Io~l~l --~--0r(Z)r(3)~0 and has a ( s i n x / x ) 2 behaviour when the initial conditions are square pulses. This would give a triangular shaped pulse of the BW. A large area AW pulse will not fundamentally alter the reflection coefficient because of the denominator in Eq. (35), but a large area PW pulse will. For example, from Eqs. (32) and (33), if A 3 increases from 1 to 10, then at A = 0,/"(03) rises from an order unity to an o r d e r of e 2°. For A > A 3 / L , /"(03) is bounded by A3, while for 0 < A < A3,/"(03) becomes of order e 2a3. Thus given A 3 > > 1, /"(03) a n d hence ]p~l)] 2 b e c o m e almost square shaped with a width A 3 / L -- Q3. The Fourier transform of a square wave is a sinc (sin(x)/x) function. Thus the wavelength of the oscillations of the B W structure are inversely proportional to the initial height of the PW. Causality will chop off the forward half of the behaviour. Also since O~1) is continuously differentiable, the corners of the square pulse are rounded and this implies the Fourier transforms will fall off faster than any power of x as X---). oo.

Similar analyses can be made with the AW and the PW. However, the estimates will not be as concrete as for the BW. For the AW initially small, the final reflection coefficient of the PW from Eq. (37) behaves approximately as ]p~3)[2 _ /"(o3)/(1 + r~3~). For A 3 >> 1, [p~3)l2 = 1/(1 + F(02)) in the region 0 < A < A 3. Outside this region goes to zero. The qualitative shape of the reflection coefficient is a symmetric func-

Io(03)1

251

tion that rises from zero to an amplitude of unity, then dips in the centre with a width of Q3The Fourier transform of such a function will be roughly some localized structure with a characteristic width given by 2"rr/Q 3. For a large PW, the reflection coefficient of the AW, Eq. (36), is also unity in the region 0 < A < A 3 and behaves as /"(2)/(1 +/"(2)) outside of this region. For a small initial AW the Fourier transform will behave somewhat like that of the BW. Collisions with varying initial amplitudes and between the B W pulse and the P W yield similar results. Ref. [1] presents a detailed discussion of these and other results.

3.3. Simulation results The SBS equations (30) were simulated on the domain x ~ [0, L] with periodic boundary conditions. The long time, large system limit was of interest. Simulations were started with r a n d o m real initial conditions. As in the SDI case it can be shown that the envelopes remain real for all time [1]. The numerical scheme is the same as that used for SDI. The spatial and temporal grids were chosen so as to resolve all the dynamics. The integrated energy Ul = f ]a~(t)] 2 dx was monitored for each run. When it reached a state where it fluctuated about an average value, the saturated state was considered to be attained. The spacetime history was recorded for all the envelopes. In the saturated regime the correlation functions St(x, t) = ( al(x - x', t - t' ) al(x', t' ) ), averaged over time, were taken. Several different parameters sets were used in the simulations. In the first example the parameters were "~ = 0 . 1 , D = 0 . 0 0 4 and L = 2 0 . The spatiotemporal profile of the P W is shown in Fig. 13. Again, furrowed ridgelike 'coherent' structures observed in the SDI are seen but with a definite drift towards the right. T h e r e appear to be length and time scales where things are correlated, but beyond which the dynamics become chaotic. The correlation function for the P W is shown in Fig. 14. The function approaches

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C.C. Chow / Physica D 81 (1995) 237-270

~5

101

I 1/111

10 -~

Fig. 13. Spacetime profile of the PW a~.

zero in space and time indicating STC, but a nonlinearity induced mode with a definite phase velocity is clearly observed. This effect was observed in the spacetime profiles as the drifting coherent structures. The correlation function shows that these structures are very long-lived. Along the x axis, a sin x/x behaviour similar to the SDI is observed. The spectrum of static fluctuations Si(q, t = 0) is shown in Fig. 15. A box-like function as expected is observed with a cutoff at approximately q = 5 , translating to a correlation length of ~p = 1.3. The local power spectrum Si(x = O, w) is shown in Fig. 16. A definite peak at w = 0 . 1 is observed, the spectrum then flattens out at around ~o = 0.007 defining a correlation time. The spacetime profile of the AW is shown in Fig. 17. Ridgelike coherent structures are seen to drift towards the left. For large scales the dynamics are chaotic. The correlation function

k

10 -~

10 0

co

Fig. 15. Spectrum of static fluctuations of the PW ai.

ld 1 0 -~ 10 -~ O3

10 -= 10 -4 1 0 -5 10 -6 _1

I IIIII

t

I

1o0

I

I IIILq

I

io ~

q Fig. 16. Power spectrum of the PW a r

measured along the characteristic x = - t is given in Fig. 18. There is strong decay in space and time confirming STC. However, there is a hump

~ 10

Fig. 14. Correlations function of the PW a r

oSw - ~

~

Fig. 17. Spacetime profile of the AW aj.

C.C. Chow / Physica D 81 (1995) 2 3 7 - 2 7 0

253

100

10-1

~

1 0 -~

10-~

z -= I

I IIIII

I

I

lo 0

Fig. 18. Correlation f u n c t i o n of the A W a r

I

I IFIfl

I

ld

q located at S ( x ~- 10, t---- 10), and another at S ( x ~1, t-~20). Note that the correlation function shown is over the entire length of the system, and the periodicity for t = 0 is seen. The power spectrum is shown in Fig. 19. The correlation time corresponds to a frequency of w = 0.3. The spectrum of static fluctuations is shown in Fig. 20. T h e r e is a cutoff at q = 9 corresponding to a correlation length of ~a--~ 0.7. The spacetime profile of the BW is shown in Fig. 21. Again irregular yet distinct structures are seen to drift towards the left. The correlation function measured along the characteristic x = - 2 t is shown in Fig. 22. Correlations approach zero in space and time indicating STC. A nonlinear mode similar to the parent is also observed. The propagating mode implies that the structures found in Fig. 21 are not aligned along the characteristic curve but are actually slightly

Fig. 20. Spectrum of static fluctuations of the A W a r

4

O ¸

Fig. 21. Spacetime profile of t h e B W a k.

skewed to the right. The phase velocity in the moving frame v - 0 . 1 indicates that the shift away from the characteristic velocity is not very great. Correlations in the direction of the coherent structures are fairly long compared to the damping times. The power spectrum along the

0L "5"

lo ~

Ct3

I

0 -2

I 1111111

I

i0 <

I I IIIIII

I

I ~1

i0 ° CO

Fig. 19. Power spectrum of the A W a/.

Fig. 22. Correlation function of the B W a k.

254

C.C. Chow / Physica D 81 (1995) 237-270 3.4. Analysis and discussion

ld

o~

lo o

U3

I

I II1[111

10 -~

I

I IIIIIII

10<

I

I II

10 ° CO

Fig. 23. Power spectrum of the BW a k.

characteristic in Fig. 23 shows a cutoff around t o - 0.4. The spectrum of static fluctuations in Fig. 24 shows a cutoff around q-----5 giving a correlation length of ~b = 1.3. Different p a r a m e t e r sets were explored and showed similar results. In the cases tested the spacetime profiles all showed drifting coherent structures evolving chaotically. The correlation functions showed qualitatively similar results to those shown here. H o w e v e r , these p a r a m e t e r sets were relatively close to the example case. A discussion of the different behaviour that could arise with different p a r a m e t e r sets is given in the next section.

101 100

10-1 10-2 .~

10-~ 10-' 10- 5 10- ~ 10-~ I

I Jill[

I

I

100

I

I Illl[

I

101 q

Fig. 24. Spectrum of static fluctuations of the BW ak.

The simulation results can be understood with the aid of linear analysis and the IST solutions. The SBS dynamics exhibit STC and are described in terms of correlation functions. T h e correlation functions have spatial and t e m p o r a l scales and a definite amplitude. T h e r e is also the additional feature of a nonlinear propagating d a m p e d m o d e in the P W and to a lesser extent in the BW. T h e linearized equation for the P W (30a) is exactly the same as that for the parent wave in SDI. The trivial fixed point Eq. (6) gives a principal m o d e for the P W at q0 = ~/y~/D. Higher m o d e s are d a m p e d and lower m o d e s are growing. As in SDI there is a competition b e t w e e n linear growth and nonlinear saturation. Instead of depletion to quasi-solitons seen in S D I , the saturation mechanism is due to the radiation transferred during collisions between the envelopes. The balance between the competing effects is also responsible for the propagating m o d e as will be shown. The IST results show that the interesting dynamics are due to collisional radiation effects. A collision between the AW and the P W generates the B W and decimates the waves. Similar behaviour occurs when the B W collides with the PW. The decimation of the parent wave is always on the side opposite to that of the collision. This is seen in the IST solutions and can be understood f r o m the nonlinear saturation of the corresponding parametric instability. W h e n the AW collides with the PW, the B W and AW grow f r o m the colliding edge as a convective instability. This is because both of their group velocities are in the same direction. W h e n the two envelopes attain a significant amplitude the P W begins to saturate. H o w e v e r , the two daughter waves will continue to grow and continue to take energy from the PW. T h e area of the P W will be reduced. T h e depleting P W cuts off the growth of the AW and BW; they saturate and begin to d a m p as well. If the

255

C.C. C h o w / Physica D 81 ( 1 9 9 5 ) 2 3 7 - 2 7 0

original amplitude of t h e PW was large enough or the growth rate 3~ high enough, the reduction in area continues until the PW becomes negative. This effect was seen in the integrable case. The negative part of the PW can again be a source for a convective instability and the same process ensures. In this way the envelopes are spatially decimated into the oscillatory structures seen in the simulation. The decimation is always on the side of the PW away from the colliding edge. The low q's are converted to high q's by this process. Modes higher than q0 get damped, so the P W will settle into structures of size ~p ~-- 2"rr/qo. The values q0 = 5 and ~p = 1.3, obtained for the simulation parameter set, agree well with the simulation. The P W equation (30a) has the form of a growing diffusion equation. Thus any localized pulse will spread and grow. The propagating PW m o d e is a result of the combination of this spreading effect and the decimation effect. The wavepackets decimate nonlinearly on one side and they spread and grow linearly on the other side. A pulse moves like a sandbar near an ocean shore, building on one side and receding on the other. A parabolic equation does not have a well-defined phase velocity, yet one was observed in the PW correlation function. However, a 'spreading' velocity can be defined by considering the trajectory of a point of constant amplitude on a localized pulse. The phase velocity of the sandbar mode, as it will be referred to, will then be given,by this spreading rate. F r o m the simulations of several different cases it was discovered that the phase velocity behaves as Up ~---g~/" This dependence can be demonstrated by considering the linearized P W equation Ota i = Tiai q- Oxxa i .

(38)

T h e general solution of this equation is given by

a~(x, t) = ~ )< e iqx

~ ( q , t = 0) exp[(3~ - Dq2)t]

dq ,

(39)

where fi/(q, t = 0) is the Fourier transform of the initial condition. Of interest is the spreading rate of a single parent structure. The simulation shows that the spectrum of static fluctuations is a box function of width q0 = V~-~JD. The inverse Fourier transform is sin(qox)/qoX which roughly describes the shape o f the P W structures. Therefore taking {A

if-q0
fii(q) = -

(40)

the solution of Eq. (39) is q0

Af

ai(x, t) = ~

exp[(y,- - Dq2)t] e iqx d q .

(41)

-q0 Completing the square and scaling out all the relevant factors yields

ai(x , t) = exp(3~t - o12) ~

A

erf(~/t - in), (42)

where a = x/4V~fft. In the limit a << 1, 3~t << 1, Eq. (42) can be expanded to yield

Aq° 2 ai(x , t) ~-----~--(1 + 3 " ~ t - OL2).

(43)

The 'spreading' velocity is defined by considering a point of constant amplitude. Imposing ai(x, t) = const, yields a 2 - ~7,-t = const. ,

(44)

giving a velocity of X up --- 7 =

(45)

if the constant is taken to be zero. This simple linear argument seems to capture the essence of the propagating mode in the PW. The peak in the PW power spectrum is given by the frequency of the sandbar mode. Using the relation to Opq o the frequency is found to be to = 3~ = 0.1. This is precisely what was observed in Fig. 16. As seen in the correlation function in Fig. 14 the structures remain coherent for very long times. The power spectrum in Fig. 16 was =

256

c.c. Chow / Physica D 81 (1995) 237-270

taken along the time axis. The long time scale observed was actually given by the transit time of the sandbar mode around the box ~-p L/Op; It is unknown what the decorrelation mechanism f o r the PW coherent structures actually is. They persist much longer than the diffusion time across a correlation length. T h e saturation energy of the PW can be understood as follows. The competition between the nonlinear and linear effects leads to coherent structures of size 2"rr/qo. The IST solutions show that the nonlinear interaction is radiation dominated so the ZS reflection coefficients for each envelope is the relevant quantity. The reflection coefficients in Eqs. (35), (36) show that structures of this size are generated in collisions when the PW has a height of a i-~q0. For taller structures, the collisions with the B W and AW will generate structures with smaller wavelengths. The simulations seemed to indicate that these results of the integrable case carry over the nonintegrable regime. As the PW grows, it gets depleted as it 'constantly collides with the other waves. If it grows higher than a / - - q 0 the generated structures damp away. Thus a i = q0 will be an upper bound to the height of the PW. For these parameters q0 = 5 and the tallest structures in the spacetime profile are of this order. Given the upper bound for the PW height, the saturated energy density can be estimated as in the SDI case in Eq. (27) by considering the PW to be composed of coherent structures locally resembling a sine wave with average amplitude of qo/2. This then gives an average energy density of Si(0, 0) -~ q~/4--~ 6. The simulation shows a value of Si(0, 0)~-5. Considering the assumptions used in the estimate this is remarkably good. T h e correlation lengths of the BW and AW structures are due to radiation effects and cross correlations. The final reflection coefficient of the B W in Eq. (37) shows that the wavelength of the decimated BW depends inversely on the height of the PW, and will be approximately the same as the wavelength of the PW. However, the --~

AW final width is not as strongly coupled to the widths of the other two waves. Simulations of pulse collisions confirm this fact. It is significant that the correlation length for the AW is one half the correlation length of the other two waves. This is due to the fact t h a t its group velocity is half of the BW. As discussed above, the PW settles into coherent structures of size ~p, and this fixes the size of the BW structures. The AW gets generated wherever the B E collides with the PW. In the time direction, along a PW coherent structure, the B W and AW will tend to have the same number of coherent~structures. This can be seen by comparing Fig. 17 with Fig. 21. However, since the AW has a group velocity half that of the BW, if it has the same n u m b e r of structures in the time direction, it must have twice as many in the spatial direction. In other words the coherent structures of the AW are half the size of the BW. This was observed in the simulation. In the saturated state, a lattice-like structure will become established. Only in special situations can a regular lattice be formed. In most cases the lattice will be frustrated. It would be very useful in the future to measure the cross correlation function between the waves to better understand these effects. The STC in SBS could be some frustrated phase turbulence akin to that seen in the Kuramoto-Sivashinsky equation [11,33]. The propagating mode of the B W seen in the correlation function can also be inferred from the IST solution. The correlation function showed that the propagation velocity of the coherent structures were slightly slower than the characteristic velocity. During a collision between the BW with the PW, the two waves will interact nonlinearly and this process retards the transmission of the BW, slowing the velocity. The AW spacetime profile in Fig. 17 shows a furrowed structure moving to the left like the BW, but the correlation function in Fig. 18 does not show the long correlations and evidence of a nonlinear mode like the B W and PW. Correlations are quickly damped out compared to the other waves. This is likely due in part to the fact

C.C. Chow / Physica D 81 (1995) 237-270

that since the group velocity is half that of the BW, it experiences twice as much damping between collisions. It may also be that the wave collisions affect the AW more han the other waves. The humps observed in the AW correlation function are due to collisions of the AW with the P W and B W waves. The one at (x ~ 1, t = 20), is due to repeated collisions of the AW with a particular P W structure. The correlation times of the PW structures are very long. Each time the AW circles the simulation box it will collide with the P W structure. The hump is slightly off from the characteristic. This is due to the fact that the P W structure is drifting. The hump at (x---10, t-----10) is due to collisions between a given B W structure and the P W structure. Whenever these two waves collide they generate the A W in the process. The B W has a group velocity twice that of the AW and so traverses the box in a time t = 10. In the frame of the AW the hump gets shifted in x as well. It would seem that the behaviour observed in Section 3.3 for SBS should persist as the PW growth rate increases or the diffusion decreases. T h e P W structures would reduce in width and this would lead to an increase in their amplitude. T h e ratio of the P W energy to the daughters would approach unity. However, if growth rate decreases or the diffusion increases the ratio of the P W energy to the daughter (AW, BW) energies would be large. The PW structures would b e c o m e wider and their amplitudes smaller. The daughter waves would damp more between collisions. The coherence times would likely b e c o m e longer as in the SDI case. The energies of the daughter waves would also get smaller in comparison to the PWs and the nonlinearity would become less important. Differences in the ratios of the velocity would change the ratio of the sizes of the AW and BW. Differences in the damping rates on the daughters would change the saturation energies. If the disparity were large then the wave with the lower damping would dominate the nonlinear collision processes. These effects were seen in preliminary

257

simulations. A detailed study remains to be done.

4. Conclusions

The 3WI has many properties that make it a useful and interesting paradigm for the study of spatiotemporal chaos. It represents the lowest order interaction of a generalized amplitude expansion. The nonlinear Schr6dinger and Ginzb u r g - L a n d a u equations both include higher order effects of n o n l i n e a r i t y and dispersion [9,34]. In this sense the 3WI will dominate as long as the resonance conditions can be satisfied. The 3WI has an integrable limit. The conservative equations are fully integrable by the IST. Low dimensional chaos has been observed in the time-only 3WI. Spatiotemporal chaos is exhibited when the highest frequency wave is linearly unstable and the other two waves are linearly damped. Two models, soliton decay and stimulated backscatter, were studied. The long time, large system limit was emphasized. The correlation function gives some statistical measure of the observed STC. It clearly shows that there are well-defined saturated amplitudes, length a n d time scales involved in the dynamics. W i t h the aid of the of the IST solutions of the 3WI, estimates for these quantities were possible. In both SBS and SDI, the competition between linear instability and nonlinear saturation led to complex behaviour and STC, although the nonlinear mechanisms responsible are different. However, in both cases, the STC involved the interaction of coherent structures of a particular length scale. A t long length and time scales the dynamics were stochastic; the correlation function showed uncorrelated dynamics. For a full statistical characterization of STC, one would require the distribution of fluctuations. It has been postulated that for STC the fluctuations are Gaussian [11,10]. Systems that exhibit fully developed turbulence exhibit non-

258

c.c.

C h o w / P h y s i c a D 81 ( 1 9 9 5 ) 2 3 7 - 2 7 0

Gaussian behaviour [35]. Measurements of higher moments would be necessary to resolve this issue for the 3WI. Knowledge of just the third and fourth moments would be very useful. The response function could be measured to check whether the dynamics obey a fluctuation-dissipation theorem. The contrast between STC and low dimensional chaos is quite clear. Specific routes to chaos and strange attractors- signatures of low dimensional c h a o s - may not be as relevant in the description of STC. If indeed the statistics of STC are relatively simple then the language of linear response theory [37] may be more appropriate to STC than the language of traditional chaos theory. For the large system limit, stochastically forced equations obeying the same symmetries may effectively describe the behaviour. This has been postulated for the Kuramoto-Sivashinsky equation [11,24,36]. Although the long time, large system behaviour of the 3WI lends itself to statistical analyses, the precise mechanisms that generate the chaotic behaviour are only understood at a rudimentary level. Studying smaller systems may shed some light onto that area. When the correlation length approaches the system size a transition to low dimensional dynamics would presumably occur. This needs to be investigated. There is a large body of work that has studied nearly integrable equations in periodic boundary conditions [1822]. There it was found that homoclinicity in the integrable IST solutions was at the root of the chaos. A study of this type is probably most suited to the SDI which has soliton solutions. The first step would be to formulate the inverse spectral theory for the 3WI, then carry out the prescription outlined in Ref. [18]. However, even without a complete theory, there are some hints of homoclinic behaviour in the SDI. For example, consider the integrable 3WI in a periodic domain x E [-~r, w] with a single parent soliton located at the origin. The IST solutions show that a single soliton in the parent wave will decay into two daughter solitons with the same phase. The two solitons will collide at x =-+w

and reform a negative parent soliton, which will decay again into two solitons but now with opposite phase. These two solitons will then collide at the origin and reform the original parent soliton. This type of behaviour was observed in simulations. From Section 2.3.1 the parent wave was seen to be approximately periodic at a wavelength of the correlation length, so the described possible homoclinic orbit may be at the source of the observed chaos. A study of the behaviour of the real line IST eigenvalues in the complex plane may be useful to check for this behaviour without a full periodic theory. Finally, given that the 3WI is the lowest order nonlinear, non-dispersive effect of a generalized amplitude expansion, the effect of the terms ignored in the equations studied comes into question. If the saturated amplitudes are fairly large, the self-modal interaction may become significant. On longer time scales the dispersion that was ignored could also play a role. The Ginzburg-Landau equation that ignores threewave coupling has a whole host of interesting behaviour [10]. A combination of the two systems awaits investigation.

Acknowledgments This work would not have been possible without Abraham Bers and Abhay Ram. I also wish to thank Terence Hwa and Dave Kaup for fruitful and interesting discussions. This work was supported in part by NSF Grant No. ECS88-2475, LLNL subcontract B160456, and DOE Grant No. DE-FG02-91ER-54109.

Appendix A. Multiple scale analysis For simplicity consider only real envelopes and write the 3WI as u t = -vw

+ eo'u + EDUxx,

v t = Vx + u w -

ev ,

(A,1) (A.2)

C.C. Chow / Physica D 81 (1995) 237-270 w,

=

-Wx

+ uv

-

Ew

,

(A.3)

w h e r e u = a~, v = a t and w = a k. It can be shown that if the amplitudes are initially real, they r e m a i n so for all time [1]. Numerically, it was found that the results for complex envelopes and real envelopes were similar, A small p a r a m e t e r e << 1 has b e e n scaled out. T h e p a r a m e t e r s ~r and D will also be considered small. A n ordering of D << cr << E << 1 is chosen. The effects of diffusion and growth are considered to be smaller than that of damping which is already conside r e d small. It is a simple task to transform back to the f o r m Eqs. (4) where the damping was scaled to unity. For • = 0 Eqs. (A.1), (A.2) are integrable by IST. T h e I S T solutions show that a single soliton in the p a r e n t wave will decay to solitons in the daughter waves. H o w e v e r , the same situation can be solved exactly in terms of the Z M p r o b l e m . This solution corresponds to two b o u n d states where a~l and a33 in the Z M scattering matrix Eq. (C.16) are zero in their respective u p p e r half planes. This was explicitly c o m p u t e d in Ref. [31 for the case where two daughter solitons at t = - % one at x = - ~ and one at x = w, collide. If the two daughter solitons are n o n r e s o n a n t (i.e. different eigenvalues) they pass through one another with only a phase shift. H o w e v e r , if the two are resonant then they form a soliton in the parent. The time reversed version of this solution is of i n t e r e s t - a parent soliton decaying to daughter solitons. T h e soliton solution for a degenerate Z M eigenvalue ~ = i7 is given by 47

u = e2nX + e_2n ~ + e2,t ,

(A.4)

v = w = 0,

u=0,

(A.10)

w = V 2 7 sech[7(x - t)].

(A.11)

The spectral p a r a m e t e r 7 will be estimated in A p p e n d i x B. Notice in Eqs. ( A . 4 ) - ( A . 6 ) that the characteristic time scale for the decay process is t ~ - 1 / 2 7 . The effect of the perturbations on this soliton solution is examined. With growth and dissipation, the parent soliton will not completely transfer its energy to the daughters. This remaining area is the desired result. Consider for the m o m e n t an isolated soliton governed by the linearized equation qt + cqx = - q •

(A.12)

For a soliton initial condition q ( x , t = O ) = q0 sech qo(x), Eq. (A.12) can be solved to yield the solution q = q0 e - ' sech qo(X - ct). Notice that only the amplitude is d a m p e d but the width of the soliton remains unaffected. With this in mind consider p e r t u r b e d solutions of the f o r m A ( x , t) u - e2,X + e_2n ~ + e2nX ,

(A.13)

B(x, t) e -n(x-0 v = e2,~ + e_2, x + e2nX ,

(A.14)

C(x, t) e n(x+O eZTt x -}- e _ Z n x + eZnX ,

W =

(A.15)

with the choice A = 47, B = C = 2X/27 at t = 0. Plugging this into Eqs. ( A . 1 ) - ( A . 3 ) yields the result --

2 7 A - B C e2,t S + e 2"t

q- eo'A + ED Ox~u,

(A.16)

(A.5) _

Bx _ A C _-_47__B e2nX _ eB S + ezm

(A.6) Ct - C . -

For t - - ~ - w the solution is u = 27 sech(27x),

(A.9)

v = V 2 7 sech[7(x + t ) ] ,

Bt 2~/27 en(X+o w -- eZnX + e_Zn x + e2nt .

(A.8)

and for t---> ~ the solution is

At 2V~ 7 e-n(~ - 0 v - e2nX + e_2n ~ + eZ, t ,

259

(A.7)

A_B _-_47___C S + e 2"t e - 2 " * - E C ,

where S = e 2"x +

e -2nx.

(A. 17)

(A.18)

Since D << e was chosen,

260

C.C. Chow / Physica D 81 (1995) 237-270

the effects of diffusion will be ignored. The diffusion term is only relevant for long time scales. The problem has two time scales, the soliton decay time given by t ~ - l / 2 r l and the damping time given by t = 1/E. The growth time scale will always be smaller than the damping time scale. For a separation of time scales 27/>> E, a multiple-time scale perturbation analysis can be considered. By a simple rescaling of time and distance the eigenvalue r/ can be adjusted to be of order unity. The validity of the perturbation theory is then for E << 1. When comparing with numerical results the parameters will be rescaled so that E will be unity. This just implies that ~7>> 1. For bookkeeping purposes ~7 will be carried throughout the calculation and is considered an C(1) quantity. Fast and slow time variables are assigned 7"0 = t ,

7"1=

et,

(A.19)

so that d / d t = d / d r 0 + Ed/d7.1 .

(A.20)

Consider perturbation expansions to first order, A = A o ( x , t) + e l x ,

(A.21)

B = B o ( x , t) + e v ,

(A.22)

C = Co(x, t) + e w .

(A;23)

Plug everything into Eqs. ( A . 1 6 ) - ( A . 1 8 ) and take order by order in e:

0/,6 e 2w% 0~-~° -- (4r/~ -- Cou - B o w ) S + e 2n~° OAo

1

0r a t-

e end°

(4~7A° - B ° C ° ) S

+ e 2n'°

+ o-Ao, (A.27)

0l,'

0p

O7"0

Ox OB o 07.~

e 2~x (Co/x - 4~7~ + Aoo)) S + e z~°

1 t- - - ( A ° C °

-

e 2nx 4~B°) S + e 2~r°

B 0

(A.28) Ow

Ow

e

-2~x

070 + - ~ - x - (B°/x + A ° v - 4~7w) S + e zn¢° -

OC o 1 Or1 ~---e

(A°B°

-

e -2nx e 2n'°

4~7C°) S +

Co " (A.29)

Notice that the following ordering is imposed: 2•A o - B o C o - e ( e ) ,

(A.30)

A o C o - 4~7Bo - 0 ( e ) ,

(A.31)

A o B o - 4~C ~ if(e).

(A.32)

Recall that for the unperturbed situation the left hand sides of the above equations are identically zero. The G(1) equations reveal that A o , B o and C o are independent of the fast time scale %. The ~(E) equations form an inhomogeneous system of equations with time dependent coefficients for the first order amplitudes. The if(e) equations can be written in the simplified form

6(a): OA o

0%

- O,

(A.24)

OB o

07.o = 0 , OCo

0%

e(,):

- 0.

(A.25)

(A.26)

where ~b is a column vector of the first order amplitudes, F is the vector formed by the right hand side of the ~(e) equations and L is the spatial differential operator of the system. Ideally one would like to solve for the eigenstates of L, expand F in those states and then remove the secularities as was done by Kaup [27] for the nonlinear Schr6dinger equation. In this m a n n e r the parts of F orthogonal to L which do not contribute secularities are identified. However,

C.C. Chow / Physica D 81 (1995) 237-270

the complexity and time dependence of L makes this proposition difficult. With this caveat in mind, the secularities of the G(E) equations were eliminated equation by equation. The justification comes a posteriori by comparing with numerical results. Removing the secular terms yields the set of slow modulation equations, 0A 0

e2nrl/e

Or1 -

(2hA° - B ° C ° ) e ( S + e 2nrl/" + °~A° ' (A.34)

Ao(x, t)

261

S -1- e ant

= @ e -2"' - 2e(2 + o-)

x [ln(1 + S e ant) e-a,, _ ln(1 + S) e'~'], (A.41)

where the initial condition A o ( x been applied. Recall that the solution for the parent is given Plugging in A 0 from Eq. (A.41) yields bt

47 e -2"t S + e aT'

B o,

(A.35)

OCo 0%

CO.

(A.36)

t = 0 ) = 4,1 has full perturbed by Eq. (A.13). into Eq. (A.13)

,

2 e ( 2 + o') S

x [In(1 + S e -ant) e -aEt

OB o Or s

S

ln(1 + S) eel']. (A.42)

Eqs. (A.35), (A.36) are readily solved to yield

The daughters can be considered separated from the parent for times t > - 1 / 2 7 . Thus, at separation the expression for the parent solution to first order in e is

B 0 = 2 V 2 7 e -'~ ,

(A.37)

U --

C O= 2 V ~

(A.38)

This is the remaining part of the parent after depletion to daughter solitons. Since ln(1 + S) is slowly varying compared to the denominator, simply evaluate it at x = 0, which gives s = 2. Notice that Eq. (A.43) has the form

e -rl ,

using B 0 = C o = 2V2 7 at t = 0. These results are then plugged into the modulation equation for Ao, Eq. (A.34), to obtain OA o Or1

= (2~7A° - 872 e-2"1) e(S + e 2n'1/') q- o-A0 . (A.39)

This is a first order linear differential equation. It can be immediately solved to yield e - ~°-t e ant -

-872

f eant e-e(2+o-)t 0 d t (S + e a n t ) 2 -k

einX + e_an x

u ~ Q sech(27x),

e 2nrl/e

A° S +

2e(2 + o-) ln(1 + S)

R

,

(A.40) where the substitution r~ = e t was made and R is an integration constant. The integral on the right hand side of Eq. (A.40) can be integrated by parts twice to obtain an expansion to first order in e. The expression for A 0 is then

(A.43)

(A.44)

where Q = e(2 + o-) In 3. The asymptotic solution for u has soliton with reduced amplitude[ The cycling time for a growth cycle is given by the time it takes Q to reach 27, its initial value. gives the expression

1 2,1 t c --~-~-~In e(2 + o') In 3 "

(A.45) the form of a and depletion the amplitude Imposing this

(A.46)

The total cycling time will actually be the depletion time plus this time (A.46). If the depletion time is much smaller than tc it can be ignored. The calculation was tested numerically. An

262

C.C. Chow / Physica D 81 (1995) 237-270

initial condition of a parent soliton with eigenvalue ~/= 2 and small daughter pulses was taken to represent the ZM soliton solution Eqs. (A.4)(A.6). The remnant of the parent wave after the daughters damped away had a reduced soliton shape as predicted by Eq. (A.44). The amplitude was a little less than the predicted value of Q = 2 from Eq. (A.45). Other simulations with different damping rates consistently had amplitudes that were close to but a little less than that predicted by the calculation. This may be due to the order e perturbations in Eqs. (A.21)-(A.23) that were not included. However, as the growth rate was increased the calculation began to fail. The calculation is only valid for very weak growth.

Appendix B. Scattering space perturbation theory In order to estimate the threshold eigenvalue, a perturbation expansion in scattering space, first developed by Kaup [28] and expanded by Kaup and Newell [29], is employed. This subject has recently been reviewed by Kivshar and Malomed [30]. The following calculation is a time dependent version o f the one given in Ref. [1], to estimate the threshold for noise induced decay of a soliton in the conservative 3WI. Those ideas have been used to construct a perturbation expansion to estimate the threshold for decay of the 3W! with growth and damping induced by collisions with daughter quasi-solitons. The parametric interaction [2,25] has shown that a parent pulse is unstable to decay when it exceeds a critical threshold in Eq. (8). However, a nonzero daughter amplitude (however small) is still required to induce the decay. The question of what threshold area or eigenvalue of the parent is required for the decay to occur remains. Many of the concepts and notation in this section are from Ref. [1]. The specific question addressed is the decay of the parent into the daughter. For this the sepa-

ration property of the ZM problem into three separate ZS problems will be used (see Appendix C). Since the pump decays into the two daughters symmetrically it is necessary to consider only one of them, q(3), where q(n) are the corresponding ZS potentials for the nth envelope detailed in Appendix C. In accordance with the notation of Ref. [1], the envelopes are indexed by (1, 2, 3) with index 1 corresponding to the lowest group velocity, index 2 corresponding to the middle group velocity and index 3 corresponding to the highest group velocity. As t--+ c~ the envelopes will be separated and a ZS reflection coefficient for q(3) (right going daughter wave) is defined as p ( 3 ) _ b(3) a (3)

b21

(B.1)

bll "

Of interest is the soliton produced in q(3) by the decay of the parent q(2). The one-soiiton solution for the ZS equation with eigenvalue A(3) = i~ has the form -

q(3)(x ) = 2D(k3)

2~x

e 1 + (D(k3)2/4~:2) e -4ex'

(B.2)

where D(k3)

• b (3) . b21 =--1 a-~P k~--l~l k"

(n.3)

The magnitude of the soliton solution Eq. (B.2) can be written as [q(3)(x)[ = 2~ sech[2(~ - r ) ] ,

(B.4)

where ]D (3) ] = 2~ e r .

(B.5)

The average position of the soliton is given by F X(3) = 2--~"

(B.6)

This gives the asymptotic position of the soliton after it has separated from the parent. An estimate of when the soliton was emitted can be found by extrapolating the position of the soliton to the position of the parent soliton q(2), where it was initially emitted. For a parent soliton with

C.C. Chow / Physica D 81 (1995) 237-270

263

zero group velocity (c 2 = 0) located at the origin, this would imply

bq,, = i £ c l c 2 c 3 ( 1 - ~ ) b q .

From this, the threshold condition for soliton emission is found to be

Using Eqs. (C.10)-(C.12) with Eqs. (A.1)(A.3) the perturbed 3WI can be rewritten as an equation for the ZM potentials V. In generalized notation this takes the form

ID(fl)(t)l

V~ = Fo(V ) + eFI(V) ,

= 2~(t).

(B.8)

For the integrable case ~ is constant, and 3tD(k3)(t)=2~D(k3)(t). The perturbing terms evoke a time variation in ~. Scattering space perturbation theory is employed to determine the time dependence. In addition to the ZM eigenfunctions given in Appendix C, define the eigenfunctions qzo) with the following boundary conditions:

" ~o.J e -icjx qt n(J) ----

as x-+ oo .

(B.13)

(8.14)

where F 0 represents the unperturbed parts and F~ represents the first order perturbation (terms of 0(E)). Kaup's [28] strategy was to use this in Eq. (8.11) to construct an expansion in e for the time dependence of the scattering data. To first order Eq. (B.11) becomes

bij,t = -/(°)[t/~ (J), ap (/)] _ e/(1)[t/~ (J), 1/¢(i)] ,

(B.9)

(B.15)

A set of adjoint eigenfunctions for the associated adjoint problem can also be defined [1,28]:

where in 1 (1) the perturbed part of Eq. (B.14), /71, is used in combination with the unperturbed eigenfunctions. The emitted soliton is given by a zero in the upper half ~ plane of b n. Here ~ is the ZM eigenvalue. At a zero of b u designated by G the total derivative is also zero so

3 E i j k r"-l~n r~ ( k ) A ~- - - C n

E m,p=l

~; (~T)(i)(1)(J) ei~X(Cl+C2+C3) -nmp-- m --p --

(B.10) and correspondingly for ~ ) A , where %.~ is the totally antisymmetric tensor of rank three. Defining [bq] to be the inverse of [aq], the time evolution of the increase of the scattering data is

db11 dt

(3bn~ ~=gk \ 3t / ~e = 0.

[1] bq,t = _i[q~o), !/A0],

(B.11)

(Ob11~

d__~k (8.16)

Rearranging and using Eqs. (B.15) with (B.16) gives

where

I[U, W] = i f (]ac-q(V,) W dx ,

(B.12)

the ~ denotes matrix transpose, and the notation C ----diag[cl, c2, c3] , V=- [Vq] has been used. The ZM potentials Vq are defined in Eqs. (C.10)(C.12). Eq. (B.11) relates the time evolution of the scattering data directly to that of the potential. From Eq. (C.17) the time evolution of the scattering data for the integrable 3WI has the simple form

d~_ dt

1 (0bll ~

(B.17)

(b~l)k \ Ot /~k 1

(b.)k

i[q~(1),~O); ~'k]

(B.18)

where (b~l)~ = (0b11/3~)];~, and the third variable in I implies evaluation at K = ~k" Eq. (B.18) is the time dependence of the eigenvalue induced by the perturbing terms. To obtain the time evolution of the residue of p(3) it is necessary to account for he time variation of (k. Thus

264

C.C. Chow / Physica D 81 (1995) 237-270

0t(~,111 g=~k) = [

/b21"~

/'b21'~]

(B.19) Since D(~3) =-ibzl/b'11 the time evolution for D(k3) is D(3)

k,t = (

b'

ll)k

-2

- ½i(Cz

.

{--(bl,)kDk -

(3)

I[~

( 1 ) 1/t (1) ;

~k]

el)J[(/)(1), X; ~k]},

(B.20)

where X = b2a~ (1) - bll ap'(2) and co ~ A C - 1 (E)wII¢=#~. J[U,W; ¢A--i f o~[U

(B.21) This was found by using Eq. (B.11) in Eq. (8.19) 2. Assume at t = 0 that qO)= q(2) are small and q(2) is a soliton located at x = 0 with eigenvalue =g~ =iT. Let the group velocities be c 3 = - c 1 = 1 , c 2 = 0 . From Eq. (C.23) this gives = ~ / 2 . To lowest order the integrals in Eq. (B.20) can be evaluated with the unperturbed pure soliton states. For this initial condition [1,28] bll -

--

J =N

f

be so small that time scales. Eq. is then plugged to yield for the

eI O) = - e o - .

(B.26)

From Eqs. (A.1) and (A.2) the evolution equations for the two daughters have the form

q~l) = ,eqO) + . . . ,

(B.27)

q~3) = _eq(3) + . . . .

(B.28)

Let the initial conditions of the daughters have the form of amplitude reduced solitons, q(1) = B sech07x ) ,

B << ~ ,

(B.29)

q(3) = C sech(~Tx),

C <<,),

(B.30)

Plugging this into Eq. (B.24) yields for the •(e) term o~

Ej ( 1 ) = - - e2i~ f dx sech(2~/x) sech(~Tx) -oo

x (B e nx + C e-'X).

(B.31)

If B = - C

~-(1 ~ - ~1

(B.22)

o~ I ~

Again, consider the diffusion to it can be neglected on the short (B.25) with q(2)= 2rI sech(2~x) into Eq. (B.23) and integrated O(e) term

dx

e 2nx -~- e - 2,/x

q12)

(B.23)

e 2"~ + e-2"~ (enXq}1) + e-n~q}3)) •

then j(x)--0. This implies that two daughter pulses of opposite phase will not deplete the parent pulse. This fact is seen in the evolution Eqs. (4) where opposite phased daughters make a positive parent grow rather than deplete. Consider the case where C - - B . Then Eq. (B.31) is integrated to yield ej(x ) = -e-rrB

(B.32)

• 2

(B.24) From Eq. (A.1) the evolution equation of q(2) has the form _~(2) + . . . , q}2) = eorq(2) + e*"*tlxx

(B.25)

where only the perturbing terms are shown.

This situation gives the maximum depletion initial condition• Plugging this into Eq. (B.20), with the knowledge of the unperturbed part, yields D(3) 1,t = r/D~3) + 2Eo.D~3) _ e(~rB/2)

Plugging this into the (B.18) yields 2 There is a typographical error in Eq. F18 in Ref. [1]. The factor b ttn is missing.

bit = 2ieo-~7,

•

(B.33)

eigenvalue evolution (B.34)

265

C.C. Chow / Physica D 81 (1995) 237-270

which integrates to

e ~ = V 0 ezra'.

In the STC situation, the parent pulse decays and emits quasi-solitons which damp as they propagate. They can then collide with another parent pulse and trigger a second decay. With this in mind B is written in the form B =7 e-~,

(B.36)

where e -~ represents the damping experienced by the quasi-soliton before it collides with the parent pulse. Note that B differs from the daughter soliton amplitudes B o / 2 in Eq. (A.37) by a factor of V2. This is because for the group velocities chosen the wave amplitudes and the corresponding ZS potentials differ by a factor of In this calculation the parent pulse is a soliton. For very slow growth this should be a reasonable approximation because it is the soliton content of the pulse that is involved in the decay. The growth may cause some of the soliton content to be transformed into radiation but if it is slow the effect should not alter the results drastically. Plugging Eq. (B.34) into Eq. (B.33) yields

D(S)l,t='F]0 e2e°t~(3)/.31 + 2•O-D ~3) - •,rr(~0/2 ) e -6

e2e~t

(B.37)

This is a first order linear equation which can be integrated to yield D~ 3) exp( - r l ~ e 2"~t) e -2e(rt = - e~r-f e

)

(B.35)

( -"

exp - 2•o-

/ dt + R , (B.38)

where R is an integration constant. The integral on the right hand hand side of Eq. (B.38) can be expanded in powers of 2•o-/77o by integrating by parts twice. Performing this operation yields

+ R exp

2•o-

/ e

(B.39)

At this point the initial condition must be applied. The initial condition also splits into a zeroth order term plus a first order term. The zeroth order contribution is identically zero since the unperturbed state was chosen to be a single parent soliton. The first order contribution is due to the small but nonzero intial amplitudes of the daughters. Thus following the same calculation as above (see Ref. [1]), except for a variation in D~ 3) due to the perturbation, it can be shown that initially D~3)(t = 0) = I r B / 2 .

(B.40)

Using this in Eq. (B.39) and applying the threshold condition Eq. (B.8) yields 70 e

2Eo-t

•7r e-~ (1 _ 2 •o- e-2,~t ) = 2 ~70 /

+

('1_ 2 •°"] r/o/J]

2\

(

x exp - ~ ~/ e2~t

)

1 e 2~o-t . .

(B.41)

One observes immediately from the threshold equation (B.41) that for % < • the parent will not decay. The damping will be sufficiently strong to suppress the decay, This agrees with the conclusion deduced from the WKB threshold condition Eq. (8). The solution set (%, t) could be found numerically, although care must be taken to ensure that the correct branch is chosen. However, an analytic expression would be very useful. The eigenvalue ~7 grows at a rate e2~°tl If "/70< • then the decay will not occur and the eigenvalue will continue to grow until another collision occurs. On average nearest neighbor pulses will be depleting at roughly t¢ intervals generating quasi-solitons that may induce the decay of the pump. Suppose that % ~ 2• at a collision with a daughter pulse. Self-consistency can be checked

C.C. Chow / Physica D 81 (1995) 237-270

266

for a posteriori. Then a lowest order estimate of ~/ can be made. Plugging in ~70~ 2e into Eq. (B.41) yields 2e = ½e-rr e -6

e -2~'7t

Terms of if(o-) have been dropped; this approximation is valid in the weak growth limit. Furthermore consider the limit where 2e~rt << 1 (short times), so the exponentials in Eq. (B.42) can be expanded in a Taylor series. Truncating to first order and solving for t yields the result

1 lnk~/ 4 e - 6 ( 1 - ¼-rre - ~ ) ) . t---~e

(B.43)

This value for t can then be plugged into the expression for r / i n Eq. (B.35) to obtain ~ - - 2 e exp[~r l n ( 4 e - ~ ( 1 - ¼ ~ r e - ~ ) ) ] .

(B.44)

By taking ~b >> ln[(4/~)e-* - 1], Eq. (B.44) can be expanded to yield the simple form 77-- 2e(1 + o-~b).

(B.45)

In the STC regime as seen in Section 2.3.1, parent pulses are collided upon more or less randomly by quasi-solitons with random phases. Consider a situation where a collision is due to daughter pulses emitted two correlation lengths ~p away. The factor q5 is then the damping which occurred for propagation over this length, ~ 2~pE/C 3 : 26~p,

(B.46)

since c a has been scaled to unity. The expression for ~7 now takes the form -r/~ 2e(1

+ 2o'6~p) .

(B.47)

Plugging this result into the expression for cycling time Eq. (A.46) yields 1 46(1 + 2o-6¢p) t~ ~ - - In 6o6(2 + o-) In 3 "

(B.48)

This expression really should be considered to be a lower bound. It assumes that the daughters

collide with equal phases. In the STC situation this would not always hold true. Radiation and diffusion effects will also serve to suppress the decay of the parent pulse.

Appendix C. The Zakharov-Manakov scattering problem In this appendix, the IST equations for the 3WI will be stated to introduce the terminology and notation to be used in the text. All of what is presented in this section follows from Refs. [1,3]. The conservative 3WI has the form QI,t + ClQ1 ,x = T a Q 2*Q 3 ,*

(C.1)

Qz,t + c 2 e z , x =3'2-Q'Q*13,

(c.2)

Q3,t + c3Q3,x = T3Q*IQ~ ,

(C.3)

where Q t ( x , t) are the slowly varying amplitudes, c / a r e the corresponding group velocities satisfying c 1 < c2 < c3 ,

(C.4)

and ~ are the phases of the coupling coefficients given by ~/i = sign(w/x 0)i),

(C.5)

where w i is the energy of the ith wave, and w/ are the resonant frequencies, whose relative signs are determined from O)1 "q- 0-12 q- 0) 3 = 0 .

(C.6)

The appropriate scattering problem for the 3WI is the Z a k h a r o v - M a n a k o v (ZM) scattering problem given by [3,4] -iv1, x + V12u2 -]-g13u3 = - C l i o 1 ,

(C.7)

--i02, x + V21uI -}- Vz3u3 = - c 2 ~ u 2 ,

(C.8)

--iv3, x + V3av I "b V32/)2 = , C 3 ¢ V 3 ,

(C.9)

where Vii are the potentials and are given by

C.C. Chow / Physica D 81 (1995) 237-270 amn(~ , t) = amn(( , O)

-iOl 723 -- V(C2

267

-- C l ) ( C 3 -- C l )

'

V32 = - , ~ 3 ~ / 2 V 2 3 ,

×exp[i~tCxC2C3(-lm

ln-)].

(C.17)

(c.10) --iO2 V31 -

V(c2

- Cl)(C 3 - c2)

'

V l 3 = ~/1,)/3V;1 ,

(C.11) -iQ3 7 1 2 ~--- V ( C 3 - - C l ) ( C 3 -- C 2 ) '

V21 = --')/1')/2712 •

(C.12) For Vii--+ 0 sufficiently rapidly as Ix[--+ m and for real ~, two sets of linearly independent eigenfunctions of Eqs. (C.7)-(C.9), ~(n) and g*(n), can be defined with the following boundary conditions: ~ n ) ~ 8] e-JCdx

as x---> - ~ ,

(C. 13)

~ n ) ~ 8j" e -icjcx

as x--> ~ ,

(C.14)

where n = 1, 2, 3 denotes the nth eigenfunction and j = 1, 2, 3 denotes the jth component. These two sets are then 'connected' through the relation

Given any initial condition going to zero sufficiently fast for ]x[ --> ~, the scattering data S(~', 0) are obtained, evolved in time with (C.17), then inverse transformed to obtain the time evolved potentials. The actual inverse scattering procedure for the Z M scattering problem is very involved and is detailed in Ref. [3]. However, for most purposes a simplification is possible. Whenever the three envelopes are separated, the scattering data for the Z M problem can be given in terms of the simpler second order Zakharov-Shabat scattering problem [1,26,38]. Consider Eqs. ( C . 7 ) (C.9) in a region of space where Q1 and Q2 are zero, but Q3 is nonzero. Then by Eqs. (C.10)(C. 12), the only nonzero potentials in this region are V12 and V21. The Z M system decouples into a trivial equation for v 3 and a second scattering problem for v I and v 2 which can be transformed into the ZS problem b y appropriate rescaling. Similarly this applies for the region where Q1 and Q2 are nonzero. For each envelope,

3

~(m) = ~' [am~(g)]

~.r(n) ,

(C.15)

--~/2"Y3

Q~

rO) =

1

qO) _ ~/(c2 _ Cl)(C3

n=l

,

(c.18)

defining the scattering matrix S(~) = [a~,,(~')],

T2T3q(1),

__ C l ) '

(C.16)

the elements of which are referred to as scattering data. The potentials Q1, Q2, Q3 have been mapped into the scattering data S. In the linear limit of Eqs. ( C . 1 ) - ( C . 3 ) for infinitesimal potentials the diagonal elements of S are unity and the offdiagonal elements are the Fourier transforms of these potentials. For the nonlinear situation such a simple relation no longer exists between the potentials and the scattering data. However, given S the potentials can be reconstructed uniquely. The time dependence of S is exactly the same as the linear limit and is given by

/~(1) = ¢(C3 - - C 2 ) / 2 ,

(C.19)

-Q2 = V(c

-

cl)(c3

r O ) = _Tl,y3q(2)* , -

c2)

'

(c.20) /~(2) ~- ~(C3 __ C l ) / 2 ,

q(3) =

-~/1~2Q3

(C.21) r(3) = TIT2q(3), ,

~v/(C 3 - - C l ) ( C 3 -- C2) '

(C.22) A (3) = ~'(c2 - c ~ ) / 2 ,

(C.23)

where q(n), r(n) and h (n) are the corresponding ZS potentials and eigenvalue for the nth en-

268

C.C. Chow / Physica D 81 (1995) 237-270

velope. If the three envelopes are separated then the scattering matrix S can be factored in terms of the scattering matrices of the three ZS eigenvalue problems. For instance if at some time the envelopes have negligible overlap and are arranged with the ith envelope to the right of the j t h which is to the right of the kth then S may be factored as (i, j, k = f, 2, 3).

S = S ( O S ( J ) S ! g>

(C.24)

For situations where the three envelopes are initially well separated and again at a later time after interacting, the ZM problem need never be used. The final ZS scattering data can be expressed directly in terms of the initial ZS scattering data. If at t = 0 the envelopes are ordered (3, 2, 1) from left to right then S -_- ~(3)~(2)~O) ~0 ~0 ~"0

(C.25)

where the zero subscript denotes the initial value. Since the time evolution of the scattering data is given by Eq. (C. 17) this determines S for all times. If the envelopes separate as t--~ oo then (C.26)

S : ¢(1) ~,(2)~(3) ~ f o f ~af .

Setting the expressions for S equal gives b}3)

a~3) =

~(1)/.(3) -4- .-7(3)h(2)/~ (1) UO UO - - ~ 0 t"O ~0

~(2)_(3)

,

(C.27)

~0 It0

b~2) a} 2)

b} 1) a~l)

a~3) ~(3) ~.(2) ~-(1) (1) (3) - rt(Z)rt(3) @% ° 0 u 0 - b 0 b 0 ) ~0

The numerical method is based on the one used by Reiman and Bers [39] to integrate the conservative 3WI. Each envelope is transformed to its characteristic frame by substituting hi(x, t) = ai(x + vit, t). The SDI and LDI simulations used the same code. Recall that the relevant equations were already transformed to the frame of the high frequency waves. Each wave was put on a grid that moved along the characteristic. The grid spacing Xp - x e_ ~ = h was fixed and chosen so as to resolve the smallest dynamical structures that may arise. The diffusion term in the high frequency wave Eq. (4a) was finite differenced by a five-point scheme. The resulting equations were then bi,~(x ,, t) = - bj(x m - vff, t) b k ( x . - vgt, t)

(C.28)

-- 30bi(xt, t) + 1 6 b i ( x t _ l , t) - bi(xt_2, t)],

(D.1) Z.(3) ~(2)/., (1)

--(3) (2)

_(1)~(2)("f u0 u0 + bf b 0 ) . u0

Appendix D. Numerical methods

D b + Yibi(xl, t ) + 1-i~h-2[ ( - i(xl+2,t) + 1 6 b i ( x l . ~ , t )

~0

a~2)

(C.29) zeros of a(~)0 and a~2) become zeros of a~1). Thus solitons are never lost from the slow and fast envelopes. The middle envelope always loses its solitons, giving solitons to both the slow and fast envelopes. In the stimulated backscatter case, the middle envelope can never contain solitons, so this exchange will never occur.

(C.29)

u0

The a's and b's are the usual ZS scattering data for a given envelope [26]. Eqs. (C.27)-(C.29) contain all the information about the exchange of solitons and radiation density. The boundstate eigenvalues are given by the poles of b(A)/ a(A) (zeros of a(h)) for A in the upper half plane. From Eq. (C.27) it is seen that zeros a(02) and a(03~ become zeros of a~3~. It then follows from Eq. (C.28) that a~2) has no zeros. Also from Eq.

bi,t(Xm, t) = bi(x , + vjt, t) b *k(x . + (vj - Vk)t, t) -- bj(xm, t) ,

(0.2)

bg,t(xn, t) = bi(x l + okt, t) b ~.(xm + (ok - vi)t, t) - bk(X n, t ) .

(D.3)

In the SDI case v~ = - o i = 1; in the L D I case v k = - 2 , vj = - 1 . The x coordinates of the nonlinear terms on the right hand sides often will not fall on the grid. A four-point polynomial interpolation formula was used to calculate the

C.C. Chow / Physica D 81 (1995) 237-270 amplitudes on the grid points. The time evolution was calculated using the IMSL O D E solver D V E R K , which employs the Runge-Kutta-Verner fifth and sixth order method. All calculations were performed on a CRAY 2 at the National Energy Research Supercomputing Center. This numerical method is very stable. At specified time intervals the entire grid would be recorded. The integrated energy was also calculated using Simpson's rule. They were then used with the energy equations in Section 2.3.3 to check for accuracy. The derivatives were evaluated with a simple two-point difference method. The grids were made periodic. To simplify the computations, only real envelopes were evolved. It can be shown that for real initial conditions the envelopes remain real for all time [1]. Comparisons between complex and real initial conditions did not show any noticeable behaviour in the dynamics. The correlation function was computed from the recorded spacetime data. The discrete correlation function was calculated and averaged over several time slices with

Sl(Xm, In ) --

1 M-1 N-, N M £ £ a(Xm+l' tn+j) a(xi' i=1 j=l

t/), (D.4)

where M is the number of grid points and N is the number of time slices for the correlation function. This double sum was evaluated for each grid point and for all of the time slices in the simulation. In a typical run, the simulation run was twenty to thirty times longer than the correlation function time corresponding to N. The correlation function was Fourier transformed along the x and t axes to obtain the respective spectra.

References [1] D.J. Kaup, A. Reiman and A. Bers, Rev. Mod. Phys. 51 (1979) 275, and references therein.

269

[2] A. Bets, in: Plasma Physics - Les Houches 1972, eds. C. De Witt and J. Peyraud (Gordon and Breach, New York, 1975). [3] D.J. Kaup, Stud. Appl. Math. 55 (1976) 9. [4] V.E. Zakharov and S.V. Manakov, Zh. Eksp. Teor. Fiz. Pis'ma Red. 18 (1973) 413. [Sov. Phys. JETP Lett. 18 (1973) 2431. [5] A. Bers, D.J. Kanp and A.H. Reiman, Phys. Rev. Lett. 37 (1976) 182. [6] C.C. Chow, A. Bers and A.K. Ram, Plasma Phys. Contr. Fusion 34 (1992) 1945, Special Issue: Invited papers for the 1992 ICPP, Innsbruck, Austria, 29 June-3 July 1992. [7] C.C. Chow, A. Bers and A.K. Ram, Phys. Rev. Lett. 68 (1992) 3379. [8] A. Reiman, Rev. Mod. Phys. 51 (1979) 311. [9] D.J. Benney and A.C. Newell, J. Math. Phys. 46 (1967) 133. [10] M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65 (1993) 65. [11] P.C. Hohenberg and B.I. Shraiman, Physica D 37 (1989) 109, and references therein. [12] P. Coullet, C. Elphick and D. Repaux, Phys. Rev. Lett. 58 (1987) 431. [13] F.T. Arrechi, G. Giacomelli, P.L. Ramazza and S. Residori, Phys. Rev. Lett. 65 (1990) 2531. [14] S. Ciliberto and M. Caponeri, Phys. Rev. Lett. 64 (1990) 2775. [15] S.Y. Vyshkind and M.I. Rabinovich, Zh. Eksp. Teor. Fiz. 71 (1976) 557 [Sov. Phys. JETP 44 (1976) 292]. [16] J. Wersinger, J. Finn and E. Ott, Phys. Fluids 23 (1980) 1142. [17] C. Meunier, M. Bussac and G. Laval, Physica D 4 (1982) 236. [18] D. McLaughlin and E. Overman II, Princeton preprint (1992). [19] N. Ercolani, G. Forest and D.W. McLaughlin, Physica D 43 (1990) 349. [20] A.R. Bishop, G. Forest, D.W. McLaughlin and E.A. Overman, Phys. Lett. A 144 (1990) 17. [21] N. Ercolani, G. Forest, D.W. McLaughlin and A. Sinha, J. Nonlin. Sci. 3 (1993) 393. [22] G. Forest, C. Goedde and A. Sinha, Physica D 67 (1993) 347. [23] D.J. Kaup, Physica D 1 (1980) 45. [24] S. Zalesky, Physica D 34 (1989) 427. [25] A. Bers, in: Handbook of Plasma Physics, Vol. 1, eds. M.N. Rosenbluth and R.Z. Sagdeev (North-Holland, Amsterdam, 1983) Ch. 3.2. [26] M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (SIAM, Philadelphia, 1981). [27] D.J. Kaup, Phys. Rev. A 42 (1990) 5689. [28] D.J. Kaup, SIAM J. Appl. Math. 31 (1976) 121. [29] D.J. Kaup and A.C. Newell, Proc. R. Soc. London Ser. A 361 (1978) 413. [30] Y.S. Kivshar and B.A. Malomed, Rev. Mod. Phys. 61 (1989) 763. [31] C.C. Chow, A.K. Ram and A. Bets, to be published.

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[32] L. McGoldrick, J. Fluid Mech. 21 (1965) 305. [33] B.I. Shraiman, Phys. Rev. Lett. 57 (1986) 325. [34] A.C. Newell, in: Propagation in Systems Far from Equilibrium, eds. J.E. Wesffeid, H.R. Brand, P. Manneville, G. Albinet and N. Boccara (Springer, Berlin, 1988) p. 122, and references therein. [35] R.H. Kraichman and S. Chert, Physica D 37 (1989) 160. [36] V. Yakhot, Phys. Rev. A 24 (1983) 642. [37] P.C. Martin, Measurements and Correlation Functions (Gordon and Breach, New York, 1968),

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