The cubic complex Ginzburg-Landau equation for a backward bifurcation

i

PHYSICA ELSEVIER

Physica D 114 (1998) 81-107

The cubic complex Ginzburg-Landau equation for a backward bifurcation Stefan Popp a, Olaf Stiller b'*, Evgenii Kuznetsov c, Lorenz Kramer a a Physikalisches lnstitut der Universitiit Bayreuth, D-95440, Bayreuth, Germany b Department ofMeteorolog3,; Universi~ of Reading, Early Gate, PO Box 243, Reading RG6 6BB, UK c Landau Institute for Theoretical Physics, 117334 Moscow, Russian Federation

Received 5 November 1996; received in revised form 1 May 1997; accepted 23 June 1997 Communicated by A.C. Newell

Abstract The one-dimensional complex Ginzburg-Landau equation (CGLE) with a destabilizing cubic nonlinearity and no saturating higher-order terms has stable bounded solutions. We consider a simple pedagogical model exhibiting qualitatively the mechanism which may suppress the divergence of the solutions. Then we investigate the functional form of the blow-up (collapse) solutions immediately before the divergence. From this analysis we find analytic boundaries for the existence of collapse solutions in the parameter space of the CGLE. A comparison with numerical simulations demonstrates that for parameters without collapse solutions the solutions of the CGLE remain bounded for all times. Finally we discuss the implications of our results for the solutions of the CGLE when saturating higher-order terms are included. Copyright © 1998 Elsevier Science B.V. PACS: 47.20.-k; 05.45.+b Keywords: Hydrodynamic stability; Bifurcation theory; Theory and models of chaotic systems

1. Introduction Spatially extended oscillatory media in the weakly nonlinear regime, often near the threshold of an instability, can be described by the complex Ginzburg-Landau equation (CGLE) [ 1] which may be derived from an amplitude and slow-variation expansion [2-6]. In I D it reads OtA = ~ A + (bl + ib2)O2A - (Cl + i c 2 ) l A I 2 A .

(1)

Here the small parameter ~ characterizes the distance from the instability threshold. All coefficients are real and bl > 0. After the usual rescaling of length, time and the complex order parameter A t ~

t/l~l,

x ~

bl~-~lx,

a ~

V/-~-/ClIA,

* Corresponding author. 0167-2789/98/$19.00 Copyright © 1998 Published by Elsevier Science B.V. All rights reserved PII S01 67-2789(97)00 170-X

(2)

s. Popp et al./Physica D 114 (1998) 81-107

82 the CGLE

OtA=#A+(l+ib)O2A-(g+ic)JAlZA

with/~=±l,

g=-+-I

(3a)

contains only two real parameters b = bz/bj and c = c2/Ictl and the signs/z = sgn(~) and g = sgn(cj). The sign of the nonlinear coefficient g is of utmost importance. If g is positive, the nonlinear term saturates any linear exponential growth and, as a result, A remains bounded for all times t > 0. In accordance with the rescaling (2) this means that in dimensional variables above the instability threshold (# ----÷ I ) A ~ vq~-T, i.e. we have a supercritical bifurcation. In the following we focus on the opposite situation /z=+l

and

g=-l,

(3b)

which occurs in particular slightly above a bifurcation that is commonly considered to be subcritical. In this form the CGLE appears, e.g., in the description of an unstable plane Poiseuille flow [2], in the context of binary mixtures [7-10], or in the description of the pulse generation by Kerr lens mode-locking laser systems [11,12]. At first glance one would expect that Eq. (3) without stabilizing higher-order terms does not admit any stable bounded solutions. Indeed, when looking at space-independent solutions a simple integration of Eq. (3) yields

I

A = V/exp(2(t0 _ t)) - 1 -- ~

exp

exP(21n(e2(to_,,l)+ict+iqSo

In(t0 - t) + iq~

for t < to.

)

(4a) (4b)

Thus an initially nonvanishing solution diverges at a time to, i.e., after afinite period of time. This phenomenon is called explosion, blow up or collapse l of the solution. Similar behavior is found for spatially periodic solutions exp(iQx) with wave number Q2 < 1. From Eqs. (4) it is seen that for c -¢ 0 the blow up is accompanied by an accelerating rotation of the phase. (The asymptotic expression (4b) valid in the collapse regime can be derived directly by integrating Eq. (3) without the linear growth term/zA.) It was already noted very early by Hocking and Stewartson [13] that, unlike the solution (4a) as well as other homogeneous solutions with Q2 < 1, the imaginary nonlinear term ~ c in Eq. (3) can suppress blow up of spatially inhomogeneous solutions of pulse type if Icl is not too small 2 and b is within some range around the axis b -- 0 bounded by bmin and bmax (see Fig. 1). In some sense the existence of such a region of noncollapsing solutions can be considered to be a consequence of a specific symmetry breaking of homogeneous solutions. This process is nonlinear and cannot be understood in the framework of linear stability analysis. Outside that range there exist collapse solutions of functional form more general than (4) and these solutions can be considered to be the relevant attractors there [13,14]. We will construct these different kinds of collapse solutions analytically in Section 3 and give a detailed account on their range of appearance in Section 4.1. Our approach differs somewhat from that introduced in [ 13], but it leads to similar conclusions. Between the lines bmin and bmax one finds from computer simulations that the suppression of blow up is very robust, i.e. with generic (random) initial conditions the solutions appear to remain bounded for all times. The conclusion is that, although the bifurcation described by Eq. (3) with g = - 1 looks like a subcritical one, where saturation of explosive growth can be provided only by higher-order nonlinear (e.g. quintic) the relevant attractor(s) I Although the term collapse for a blow-up phenomenon seems to be a misuse of the English language, we will continue to apply it here since it has become common. 2 Without loss of generality we will often choose c > 0.

S. Poppet al./Physica D 114 (1998) 81-107

83

20

/

(=bmin)

C

0 5"-0 / /

10 4.0

2.5

S.-~

0.2

-6

0.8

f

~ -

0

b

6

Fig. 1. Curves in the b-c plane that may separate different qualitative behavior of solutions of the CGLE (3): between the lines bmin and bmax the CGLE (3) possesses stable bounded solutions (except in a small area left to the line bmax below the point SI ). At the right-hand side of the line bmax (Eq. (36)) one has ot3 > 0 and a collapse via broad pulses is possible, t~5 is negative between the two dashed lines (or5 = 0) and positive otherwise; see Eq. (44). Below the line bmin(C0 = c0(b), Eq. (49)) left to the critical point (bcr, co(bcr)(cross, Eq. (61)) self-similar collapse solutions exist. A more detailed explanation of this figure is give in Section 4.1. in fact bifurcate supercritically and exhibit the usual supercritical scaling b e h a v i o r (in dimensional variables A

~). The b o u n d e d solutions are disordered e x c e p t sufficiently far inside the focusing 3 quadrant bc < 0. The influence o f b on the character o f the l o n g - t i m e solutions as e v o l v e d f r o m small-amplitude white noise is shown in Fig. 2 in the f o r m o f c o n t o u r plots o f IA(x, t)[. ( S o m e aspects o f these changes were already noted s o m e time ago in the investigation o f the case c --* ~

[15].) T h e simulations were done for c =

15. A t b = - 3 (Fig. 2(a)) a rather

regular grid o f stable stationary pulses develops after s o m e transient. For b = - 1.5 (Fig. 2(b)) the amplitude of the pulses oscillates w h i c h indicates the o c c u r r e n c e o f an oscillatory instability b e t w e e n b = - 3 and b ----- - 1.5. The amplitude o f the oscillation increases with increasing b until for b = 0 (Fig. 2(c)) one observes a permanent, spatially uncorrelated f o r m a t i o n o f transient pulse-like structures. For b = 2 (Fig. 2(d)) one finds extended areas with almost constant amplitude. On this b a c k g r o u n d there arise fairly long-lived localized structures in the form o f rather deep and sharp depressions traveling through the system. These objects are reminiscent o f the dark solitions o f the d e f o c u s i n g nonlinear S c h r t d i n g e r equation ( N L S E ) and the N o z a k i - B e k k i hole solutions o f the C G L E [ 16--18]. 4 Besides these sharp depressions one observes the formation of several very short-lived pulse-like 3 The terms "'focusing" and "defocusing" are taken over from the limiting cases - b . c --+ to and b, c --~ to where the CGLE (3) becomes the focusing or the defocusing NLSE. 4 In the CGLE (3) the Nozaki-Bekki solutions exist if c > ½(7b + 9b 3) or if c < l b (for c > 0). The parameters b = 2, c = 15 are somewhat outside that existence range. Nevertheless the observed hole-like objects should be related to the Nozaki-Bekki holes,

S. Poppet al./Physica D 114 (1998) 81-107

84

(a)

o

,4-0

tO 0

(b)

0

10

20

30

Fig. 2. Time evolution of the modulus IA (x, t )l in simulations of the CGLE (3) with periodic boundary conditions. The initial condition was small amplitude white noise. Everywhere c = 15. ( a ) b = - 3 ; maxr.t IAI -~ 3.6, (IAI) := [Aldx d t / f dx dt ~- 0.66; ( b ) b = - 1 . 5 ; maxx,t IAI -----2.7, (IAI) "" 0.49; (c) b = 0; max~.,r IAI - 2.1, (IAI) -~ 0.44; (d) b = 2; maxx.t IAI --~ 7.3, ([AI) -~ 0.99.

f

S. Poppet al./Physica D 114 (1998) 81-107

(c)

90

85

101

.-..%

-5

X

(,4 0 14..%

(d)

~'1

t-

Fig. 2. Continued

-5

86

S. Poppet aL /Physica D 114 (1998) 81-107

structures similar to those seen in Fig. 2(c). The pulses become higher with increasing b until for b > bmax blow up occurs. We note that in a substantial parameter range with b < bmax the peaks, though remaining finite, may become extremely large. In this range direct simulation of an extended system, as presented in [ 19], would normally lead to the conclusion that there is blow up. Only a detailed analysis and simulations with adaptive discretization and system size, as will be presented in Section 3.5, disclose the boundedness and allow to capture the true onset of blow up. In real systems higher-order terms will usually provide the relevant saturation in this region. The periodic pulse solutions from Fig. 2(a) can be treated analytically in the limit - b , c ~ ~ [19,20]. Eq. (3) then reduces to the focusing NLSE which processes a two-parameter family of (nonmoving) periodic solutions expressible in terms of elliptic functions IA(x)l = C . dn(kxlm).

(5)

Here the prefactor C is a function of the parameters k and m (where k > 0 and 0 < m _5< 1). Ginzburg-Landau-type perturbations of the NLSE lead to a destruction of this solution manifold. However, a one-parameter subfamily of solutions depending on the ratio b/c survives the perturbation. Thus in Eq. (5) the parameters C and k both become functions o f m and b/c, i.e. k = k(m, b/c) and C = C(m, b/c). The (left) boundary of existence for these periodic solutions is found to be bmin(C) ---+ -41-c for c ~ ~ [19]. In the limit m ~ I the solutions merge with the familiar pulse solutions of the CGLE [ 13,21,22] see Appendix A. C A- - exp(iy In cosh(Kx) -- i,Qt). cosh(Kx)

(6)

These solutions exist for (assuming c _> 0) c > - b + 3~/1 + b 2

(7)

which in the limit - b ~ e~ reduces to b > bmin(C) ~ --41-C. Coming back to our remark that the bifurcation described by Eq. (3) is in some sense supercritical in the (b, c)parameter range where solutions remain bounded ("supercritical parameter range", between the lines brain and bmax in Fig. 1), one may ask if in addition the finite-amplitude solutions typical for a subcritical scenario exist in an interval around the linear threshold, provided that saturating higher-order terms are added. The answer is that in a subrange there indeed exist stable constant (and periodic) solutions. However, the pulse solutions, which are also known to exist in the quintic CGLE for appropriate parameter intervals [23-29], do not presumably occur in the "supercritical parameter range". This will be discussed at the end of this paper (Section 4.2). The paper is organized as follows: in Section 2 we consider a simple "quasiclassical approximation" exhibiting qualitatively the mechanism which may suppress the divergence of the solutions of Eq. (3). The material on the collapse solutions is contained in the various subsections of Section 3. We start from numerical observations in Section 3.1, which show that there are indeed two types of exploding solutions of the CGLE existing only in certain regions of the parameters b and c. These solutions are analyzed in detail in Sections 3.2-3.4. The comparison with numerical simulations of the CGLE in Section 3.5 demonstrates that for parameters without collapse solutions the solutions of the CGLE remain bounded for all times. A detailed discussion and outlook is given in Section 4.

especially since the objects become better visible for large values of c (at b = 2 fixed).

S. Poppet al./Physica D 114 (1998) 81-107

87

2. A "quasiclassical approximation" In the following we discuss a simple model illustrating qualitatively how a collapse of the solutions of the CGLE (3) may be suppressed for suitable parameters [30-32]. Whereas space-independent solutions of the CGLE (3) diverge in finite time the dynamics of pulse-like solutions may proceed quite differently. The imaginary cubic term in the CGLE provides a stabilization mechanism which can eventually suppress the collapse. For the understanding of this mechanism it is useful to write the CGLE in terms of modulus and phase (A = IAle i4', k := 0x40 OtIAI = IAI(1 -+- IAI2 - k 2) - 2bkOxlal - b[alOxk + o~lal,

(8)

021A] + "Ox (0x(lAl2k)~kAi2 i/. Otk = -c'O,:lAI 2 - 2bkOrk + bOx ~--~-

(9)

Since the bounded solutions occur for not too small values of c near the line b = 0 we first consider the case b = 0, c >> 1. Then the last term on the right-hand side of Eq. (9) is assumed to be negligibly small compared to the first one. Furthermore in Eq. (8) we omit the smoothing term OZIAI. Thus we are lead to the simplified model system ~tIA[ 2 = 21A[2( 1 + [AI2 - k 2 ) ,

(10)

Otk = -cOxlAI 2.

(I 1)

This approximation corresponds essentially to the "quasiclassical limit" when the amplitude changes slowly over a local wavelength ,~ = 2zr/k. Evidently, this model breaks down in the vicinity of points where the amplitude ]AI vanishes, which is a typical problem for the quasiclassic approximation. To resolve it one would need to match in these regions a quasiclassical solution to a solution of the linear CGLE. In order to understand the dynamics of pulse-like solutions of the model we consider an initial condition for [A[ made up of an extended small-amplitude plateau with rapid decay to zero on both sides. Let k ----0 initially. At the linear stage of the instability, and then in the following blow-up regime, the amplitude will remain approximately constant inside the plateau region. The gradients of IA[ in the boundary regions will generate phase gradients k which will, via Eq. (10), lead to saturation of the amplitude in these regions. As a result of this phase gradient mechanism (PGM) the two plateau boundaries will move towards the center of the pulse. Due to the formation of sharper gradients of the amplitude the propagation speed will grow steadily during the blow up which results in a narrowing of the pulse and (eventually) a suppression of the blow up. Thus the PGM provides the comparison of the pulse up to its disappearance. It is important to note that this mechanism has a nonlinear origin, representing a specific nonlinear interference of the Hamiltonian and non-Hamiltonian terms. It cannot be understood from the linear analysis. This scenario is confirmed qualitatively in numerical simulations of the model equations (10) and (1 1). In Figs. 3(a) and (c) we show the time evolution of the amplitude and the phase gradient for an initially wide and flat pulse. The narrowing of the pulse caused by the action of the PGM is obvious. A comparison with simulations of the full CGLE (3) indicates that the model equations (10) and (I 1) preserve the main aspects of the full dynamics, see Figs. 3(b) and (d). However, there are some differences: e.g. the pulse shape becomes sharper in the model system - an effect which can probably be traced back to the omission of the term b)~lA1. Furthermore in the CGLE the compression of a pulse is always accompanied by phase slips. Here our approximation breaks down and it is necessary to account for the term 0x (0x (IA 12k)/IA 12) neglected in Eq. (11). Instead of phase slips one observes a "freezing" of the phase gradient: when the amplitude at the pulse boundary (after the initial growth) is sufficiently faded out one has at k --+ 0 and OtlA[ ~ [AI(I - k2). Thus the phase gradient k becomes constant (freezes) and the

88

S. Popp et al./Physica D 114 (1998) 81-107

4.0j

6.0

(a)

IAI

! L

IAI

(b)

3.0

4.0 2.0

! 2.0 i

0.0

L__

12.5

1.0

17.5

22.5

x

27.5

0.0 40.0

45.0

5d.0

5, .0

X

60.0

20.0

50.0

k

(c)

k

25.0

10.0

0.0

0.0

-25.0

-10.0

(d)

I

-50.0 12.5

. . 17.5

.

.

. 22.5

-20.0 i 27.5 40.0

45.0

50.0

55.0

X 60.0

x Fig. 3. Simulations of the model equations (10) and ( 11) ((a) and (c)) and the CGLE (3) ((b) and (d)) for b = 0, c = 15; we show the modulus IA [ and the local phase gradient k := 0x4~for some successive time steps. The initial conditions were A = 0.01 (tanh(x - 16) - tanh(x - 24)) ((a) and (c)) and A = 0.1 (tanh(x - 44) - tanh(x - 56))) ((b) and (d)).

amplitude IAI decays exponentially. Actually in the case of a strictly asymmetric pulse the blow up is not prevented right at the pulse center (the pulse presumably approaches a g-distribution). However, a nonsymmetric perturbation of the pulse will probably in general eliminate this behaviour. Because of the limited importance of the model we have not treated this question (which could probably be answered analytically). The influence of the parameter b on the action of the P G M can also be assessed qualitatively. We remind that this coefficient in the C G L E (3a) is responsible for the dispersive broadening of pulses. Thus with increasing Ib[ the dispersive effect becomes stronger, which opposes the P G M and tends to bring the solution "closer" to the homogeneous collapsing solutions. Therefore one expects the existence of bounding values of the coefficient b(bmin (c) and bmax (c), respectively) separating collapsing and noncollapsing regions. Note that the regions b > 0 and b < 0 are not expected to be equivalent, because of the defocusing (bc > 0) and focusing (bc < 0) effects of the nonlinearity, which reinforce or oppose the dispersive spreading of the pulse, respectively. This is born out of the rigorous treatment, although the situation turns out to be somewhat involved.

S. Polypet al./Physica D 114 (1998) 81-107

89

We conclude that study of the model system (I 0) and (11) contributes to the understanding of the dynamics of the CGLE by singling out the stabilization mechanism, the PGM, which may give bounded solutions. However, it is not possible to draw quantitative conclusions from this discussion.

3. Collapse solutions The aim of this section is to construct the collapse solutions of the CGLE analytically. Numerical simulations of the CGLE (Section 3.1 ) indicate that one has at least two qualitatively different types of collapse solutions. These will be considered after some introductory remarks (Section 3.2) in Sections 3.3 and 3.4. A quantitative comparison with simulations will be given in Section 3.5. 3.1. Numerical observations As already mentioned in Section 1 one finds pulse-like collapse solutions in a wide range of parameters b, c. Since during the blow up the width (at half maximum) of these pulses continuously decreases simulations of Eq. (3) represent a delicate numerical problem. We used two integration schemes both based on a pseudospectral code with a predictor corrector time step. For the investigations of extended systems - i.e. in Figs. 2, 9(b), and 10 - we implemented fixed space and variable time discretization, the later depending on the actual maximum amplitude in the system. Reliable simulations up to peak amplutides of about [AI ~ 102 . . . . . 104 could be achieved in this way. When looking at the behavior of single pulses - i.e. in Figs. 4, 5, 7, 8, 9(a) - we chose adaptive space and time discretization depending on the actual width and height of the pulse. (A similar approach was used for the investigation of the wave collapse for systems of the NLSE type, see e.g. the review [34] and references therein, as well as the paper [35].) With increasing space resolution we reduced the system length by shortening the system symmetrically around the actual amplitude maximum. In this way we were able to simulate collapse solutions up to almost arbitrary amplitudes (our calculations extended up to values of IAI ~ 108°). As a result of the numerical investigations one finds two qualitatively different types of collapse solutions appearing roughly in the focusing (bc < 0) or defocusing (be > 0) parameter range of the CGLE. The exact existence boundaries of these solutions will be determined in Sections 3.3 and 3.4. Figs. 4 and 5 show examples for the two types of blow up. In (a) and (b) amplitude IA(x)l and the local wave number Ocp/~x(x) of the solutions are plotted for some successive times. Furthermore, in (c), (d) and (e) we show the dynamics of the maximal amplitude in the pulse center ]A0l : = max~ IA(x)l, the width of the pulse at half maximum dl/2 and the frequency in the pulse center dq~/dr, respectively. Instead of t we use the adapted time variable r :=-In(t0-t),

(12)

which is motivated by the behavior of the space-independent collapse solution (4). Here t0 is the (numerically determined 5) time of divergence. Figs. 4 and 5 indicate that the amplitude IA (x) I and the wave number 0, ~P(x) converge to certain limiting functions. The amplitude IA0l in both cases behaves like d ln(A0l) dr

1 ~, 2

or, more precisely,

qA0l --~ const, e r/2 -

const. x/~ - t

(13)

5 The simulations were actually continued much closer towards the collapse than il is shown in Figs. 4 and 5. Thc time at which a simulation was stopped could then be taken as an approximation for the time of divergence to.

S. Poppet al./Physica D 114 (1998) 81-107

90

1.o IAI/IAol 0.8

20

"~

0.20

r

d ' ~ * ( d ~ d x ) ~ l l ~

0.15

0.6 0.4 0.2 0.0

-5

o

5

lO x/d1/2 15

-20

23

-5

0

5

5

10

-0.42

o.51

15 x/d~ 2

d(In(IAol)/d'~

~

(d/d17)(In(d~,2))

-0.44

0.50 -0.46 0.49 -0.48 0.48

o

50

100

1;

150

-0.50 0

,

i

50

100

17

50

-7.5

de d17

-8.0

-8.5

f

i

50

100

T

150

Fig. 4. Simulations of the CGLE (3) for b = 6, c = 15. The initial condition was A = e x p ( - 0 , lx2). (a) Normalized amplitude IA(x)l for some succeeding time steps between r --~ 1 and r "'~ 20. The analytic limiting function IAI/IAol = (1 + 3(x/dl/2)2) -1/2 from Eq. (31) is drawn as a dashed line. (b) Normalized local wave number O(b/Sx(x) together with the analytic limiting function from Eq. (32) (dashed line). (c) Instantaneous growth exponent d(ln(IAol))/dr of the m a x i m u m amplitude IAol; here r := - In(t0 - t) was calculated directly from the numerically determined instant of collapse to. The roughness of the curve is caused by the numerics, especially the rescaling procedure. (d) Time evolution of the width at half m a x i m u m dl/2. (e) Frequency d~b/dr(r) in the pulse middle (x = 0).

S. Poppet al./Physica D 114 (1998) 81-107

91

2

1.0 IAI/IAol

dl/2*(d~/dx)

le-07

0.8 0.6 0e+00 . . . . . . 10.0 15,0 20.0 25.0 30.0

0.4 0.2 0.0 --" 0

5 \

-1

0

10

x/d1/2

20

-10

0

10

x/d1/2

20

1°0

IAol*exp(-z/2 ) 2.0

dlta *exp('d2)

0.5

0.0 1.0 d(In(IAol)/d~

0.0 ~ 0

10

(d/dl:)(ln(dla))

-0.5

-1.0

20

30

0

10

i

i

0

10

20

20

30

0 d._~_~ d'~ -10

-20

-30

-'40

30

Fig. 5. Simulations of the CGLE similar to Fig. 4, but with parameters b = - 4 , c = 15. Initial condition was A = e x p ( - 0 . Ix 2) again. (a) Normalized amplitude IA(x)d for some shortly succeding time steps at about r -~ 22. The dashed line shows the analytic limiting function from Eq. (53). (b) Normalized local wave number "dga/Ox(x). The strong variations at x/dl/2 ~-- 14 indicate the occurrence of phase slips. The dashed line comes from the analytic formula (53). (c) Time evolution of the m a x i m u m amplitude [A01. (d) Width at half m a x i m u m dl/2 = d l / 2 ( r ) . (e) Frequency d~b/dr(r) in the pulse middle (x = 0).

S. Poppet aL /Physica D 114 (1998) 81-107

92

for r --~ oo. However, Figs. 4 and 5 show differences in the dynamics of dl/2 and dqS/dr. In the focusing case (b = - 4 , c = 15; Fig. 5) one has dl/2e r/2 --+ const, so that

dl/2 --~ const, e -~/2

d49/dr --~ const.

and

(14)

In the defocusing case (b = 6, c ---- 15; Fig. 4) one finds dl/ze r/2 --+ oo. In fact the quantity dl/2 behaves as

du2 = e-r/2/.f(r)

for .f(r) sl°wly ~ 0

(15)

with a function ./~(r) which converges to zero more slowly than exponentially. Similarly, the frequency d4~/dr shows a slow convergence to the value - ½ c ; i.e. dq~/dr slowly - ½ c .

(16)

In summary this means that the exploding pulses tend to get "broader" in the defocusing case when compared with the focusine one. This behavior is reminiscent of the space-independent collapse solution (4) where one also has dfb/dr --+ -½c (and of course [A0[--~ const, e~/2).

3.2. General remarks on the analysis of collapse solutions The numerically observed asymptotic behavior of the exploding pulses as described by Eqs. (13)-(16) motivates the introduction of the following self-similar variables, in addition to r from Eq. (12) x Y:= x/~-t

-- xe r/2,

p A(x, t) --. ~ e x- t) , ~ exp

17)

i~ln(to-t) ic)r

B ( "v , r )

B(v, r).

18)

In these variables the CGLE (3) becomes

OrB = ½(1 - i e ) ( - 1

+ IBIR)B - ½YOyB + (1 + ib)O2B + e - r B .

19)

In this new notation the collapse solution (4) simply reduces to B = 1. The expression e - ~ B comes from the destabilizing linear term A of the original CGLE (3). Since this term looses importance during the blow up it will be neglected in most of the following discussion. Thus there is no explicit time dependence and we are left with

OrB = 1(1 - i c ) ( - I + ]BI2)B - ½YOyB + (1 + lb)OvB.

(20)

The CGLE (3) without the linear term A admits a similarity transformation: if A(x, t) is a solution then also XA(kx, xzt) satisfies the CGLE for any real X. Solutions of the form B(y, r) = /~(y)e - i ~ r of Eq. (20) correspond to solutions Aro(x, t) = (l/x/2)/~(y)e ¢l-ic)r/2-iy)r of the original CGLE which essentially map onto themselves under the similarity transformation. More precisely one has

)~Ato()~x, )~2t) = Ato/x2 (x, t)e i(''+2s~) In(X)

(21 )

Such solutions Ato will be called self-similar. We point out that the collapse solutions represented by Eqs. (13) and (14) (focusing case, Fig. 5) are of self-similar type• The second class of collapse solutions described by Eqs. (13), (15) and (16) (defocusing case, Fig. 4) is more

S. Prq~p et al./Physiua D 114 (199~') ,k'l 107

93

complex. Nevertheless, also here Eq. (20) may serve as a starting point for a systematic perturbation expansion of the solutions. Finally we note that the attractive collapse solutions A of Eq. (3) in general correspond to unstable solutions B of Eq. (20). Perturbations of B (and the corresponding perturbations of A ) generally change the collapse lime tn. If the actual collapse time is earlier than the value t0 assumed in Eq. (19) B will explode, if it is later. B will vanish for

l

~

7x:.

3.3. Broad i)ul.ves In this section the blow-up solutions of the defocusing paralneter range (Fig. 4) will be considered. From relations (13). (15) and (16) one sees that for these exploding pulses B (see Eq. (17)) gets continuously broader and thus presumably approximates the space-independent collapse solution B = 1 for r --, ~ . Accordingly these pulses should become asymptotically slowly varying functions in space and time. This behavior is exploited in the followin!~ expansion where time derivative and the Laplacian in Eq. (20) are treated perturbatively. To this purpose we rescale the spatial variable again r := f ( r ) y =

. t i r ) x e r/-~,

(22t

D(r. r) : = B(y. r).

1231

with a yet unknown function .f(r). Now Eq. (20) becomes ([" := i~r [') 2iJ, D

(I

iu)(-I +

IDle)D

(1 +~.1 /.l)rlJi. D + _ f - ( I

+lh)i)7:l).

(241

We seek a function f ( T ) such that in the limit ~- ----> ~ the solution D(r. r) of (24). becomes tin]edndependent tip to a possible global rotation. Thus. up to corrections of higher order, f corresponds to .t: flom Eq. (15). So f describes the slow divergence of the pulse width within the self-similar coordinates (r. 3'. B). Inserting D = Re i~, with R. ~ real, into Eq. (24) yields ,l

2i)~R = I R 2 - 1)R - rR, - 2 f '/"r R , .

+ 2f21R,.,. - R~7: - 2hR, qh - hRv),.,.1.

2iL¢=,'(I-Re)-r"°"-2f'r~"+2f2Ih~--h~r-'~R"q;'t"

' -

R

+¢J"'l"

(25~ i2(~)

Here R, := it,. R . . . . denote partial derivatives. One solves these equations with the ansatz R(#, r) = R0(r) + Rl(r.-c) ÷ R2(r. r) . . . .

(27

~ ( r . r ) = ~//(7) + ~()(r) + ~i 0 , T) + . . -

(2S

(The systematics will become clear without introducing a formal expansion parameter.) Because of Eq. (I 6) one slov, l\

assumes i~r ~/~ ~

0. Since also f vanishes s l o w l y in the l i m i t r --* ~c, Isee Eq, (15)) one anticipates this b e h a v i o r

for the other corrections RI, Pi . . . . in the ansatz (27) and (28) too. Consequently it seems natural to assume . [ " / I : ~ " O. ( i ) r R l ) / R I ~ 0 . . . . etc. Thus Eqs. 125) and (26) yield to leading order (R(] -

1)Ro -- rR(i,- = 0,

c(I -- R ~ ) - r ~ o j .

=0,

(20) (30!

94

S. Popp et al./Physica D 114 (1998) 81-107

from which follows 1

R02 = 1 -t- y r 2"

(31)

Without loss of generality the constant y can be chosen equal to one. (V can be absorbed in the definition of f . ) For qg0 one finds cF ~00r = 1 + r 2

or

c ~ In(1 + r2),

qg0

(32)

where the choice of the integration constant plays no role. At next order one finds from Eq. (25) /~RI := (rOr h- 1 - 3R2)R1 ~

(

rOr

1 -+ r 2

t

R1

= -2frRor + 2fZ[Rorr - Rogo2r - 2bRor~OOr - bRo~oOrr] f r2 2f2 f t = 2 f (1 + r2)3/2 h- (1 -k- r2) 5 / 2 ( - ( 1 q'- bc) q- r2(2 + 3bc - c2)).

(33)

The linear operator 12 has the neutral mode h (r) := r 2 / ( l -1--r2) 3/2 connected with scale invariance of Eq. (29) (or, equivalently, the existence of the one-parameter family of solutions (31)). Eq. (33) is solved by RI = P1 ( r ) f ' ( r ) / f ( r )

q- P2(r)fZ('r)

(34)

with Pl(r)=21n(r)h(r),

P2(r)-- (1+r2)3/2 +2(3+4bc-c2)

(

ln(r)-

)

l n ( l + r 2) h(r).

An additional contribution to R1 proportional to the neutral mode h(r) can be absorbed in the definition of f and is therefore neglected here. The correction RI and its spatial derivatives should be bounded at least in some inner region around the pulse center. However, from Eq. (34) one sees that OrrR1 (r, r) in general diverges at r = 0 and R1 is ill defined for negative r. Thus the function f has to be chosen such that the contributions ~ ln(r)h(r) are suppressed in the solution R1. From this condition one finds f'+ot3f 3=0,

where

~3:=3+4bc-c

z.

(35)

For ~3 > 0 the function f vanishes asymptotically in the way envisaged, i.e. more slowly than exponentially (in fact f = l/~/2ot3(r - r0)). Then the above expansion is consistent. For or3 < 0, on the other hand, an explosion via broad pulses is impossible. The boundary of existence for this type of collapse solutions is given by the relation or3 = 0, or equivalently bmax (c) =

c2 - 3 4c

(36)

This line is shown in Fig. 1. One sees that a divergence in the form of broad pulses is possible in a wide parameter range of the defocusing regime (bc > 0) but also in a small strip of the focusing regime (bc < 0). From Eq. (35) one can assess the influence of the parameters b and c on the dynamics of these pulses: the contribution - c 2 in or3 essentially describes the action of the PGM (see Section 2), whereas the term 4bc can either strengthen (bc < 0) or weaken (bc > 0) it. The critical line bmax(¢) is essentially determined by the balance of these two effects.

95

S. Popp et al./Physica D 114 (1998) 8 1 - 1 0 7

It is instructive to calculate the next correction to the dynamics of f , which will lead to a term ~ .f5 in Eq. (35). For this purpose we d e f i n e OrrRi(r

= O, r) = 0

V't', i ---- 1,2 . . . .

(37)

These conditions fix the function f unambiguously. 6 Using Eqs. (35), (37) one gets from Eq. (34)

Rl(r,

r) -- (1 f+Zr (Zr)) 3/2 ( l + b c + r Z (

3 +bc) ~(1

(3 + 4 b c - c Z )

-

ln(l

+ 1 "2) ) ) .

(38)

The corrections ~Or and ~0h- to the phase gradient ~00r are found by going in Eq. (26) to the next order, which gives

2 O r ~ + rgOlr = - Z c R o R I

-

2-yr~P0r + 2.fZ

b

r

_

b~6r + 2

Ro

+ ~o0,-r

(39)

2f 2 = -ZcRoRI + " ~ [c - b + rZ(Zb + 3 b e 2 - c 3) + r4(3c + 4be 2 - c3)]. (1 + rZ) 2 Evaluating Eq. (40) at r = 0 leads to Or ~ = - b ( l _

~Plr

r,fZ(r)

(40)

+ c 2) f2. From this follows

~,

(yT~rZ)2 [~b - 3c + 7 b c 2 - 2c 3 + r2(6c + 2b - 2c 3 + 10be z) + In(1 + r2)(6c q- 8 b c 2 - 2c3)].

(41)

The next order in Eq. (27) gives £R2 = 3 R o R 1 + 0 ` 3 f Z ( Z r R l r + 2 f Z [ R I , . , . -- Rl~p2r -

+

4R1)

2R0~o0,-~Olr - 2b(Rlr~OOr + R0r99h-) --b(Ro~Ph.r + Rl~O0r,-)].

(42)

AS before the contribution R2 may not contain terms ~ ln(r)h (r). This yields the following condition for 3`: 3`" -t- 0`3.1.3 -t- 0`5f 5 = 0,

(43)

where ')

O'3 : = 3 + 4 b c - c-,

0`5 : = 2(c 2 + 5)(5b 2

10bc +

2c 2 - 3).

(44)

The expansion could be continued to arbitrary order. We note that the term 0`53`.5 is particularly important near the line 0`3 = 0 where one may have I0`33`'31 < or ~ I0`5f51 at least during a long transient. The curve 0`5 = 0 is shown in Fig. 1. We will discuss the implications of the above results in Section 3.5, where a comparison with numerical simulations of the CGLE is performed, and then in more detail in Section 4. The asymptotic pulse solutions from Eqs. (31) and (32) are shown in Figs. 4(a) and (b) (dashed lines). The convergence of the numerical pulses towards these analytical limiting functions is obvious.

6 The precise definition of f is not essential. An alternative definition would be, e.g.. R i ( r would imply that R0 describes the actual pulse exactly at a position r = a.

=

a, r )

=

O, Vr. i = 1.2 . . . .

This definition

S. Popp et al./Physica D 114 (1998) 81-107

96

3.4. Self~imilar pulses In the following section the blow-up solutions of self-similar type will be considered. Accordingly to Section 3.2 they correspond to solutions of Eq. (20) of the form B(y. r) = /~(y)e -i'or. The function /~ is a solution of the stationary equation •

9

( - 3 + iS2)B + !(1 - ic)lBl2B - ~vO~,B + (1 + ~b)O~,B = 0.

(45)

Here the notation was simplified by introducing the frequency $2 : = ~ + ½c and dropping the tilde ..... on the amplitude B. Eq. (45) together with the boundary conditions,

OyB=O

fory=0,

(46)

B ~ 0

for y ~ + 2 ,

(47)

and

constitutes a nonlinear eigenvalue problem for $2 with symmetric eigenfunction B. In general, i.e. for arbitrary values of the CGLE parameters b and c, the search for eigenfunctions B has to be done numerically, which is not an easy task. Fortunately in some important limiting cases the eigenvalue problem (45)-(47) can be treated (semi-) analytically: without the term ~ yOyB Eq. (45) has analytic pulse solutions of the following form (see Section 1 and Appendix A): C - exp(iy In cosh(Ky)), cosh0cy)

(48)

which exist in the range (see Eq. (A.6a) and Eq. (A.7)) 7

c < co(b) := - b + 3 , f l + b 2.

(49)

Near the existence boundary c = co(b) of these pulse solutions, i.e. for c < co(b) the parameters in Eq. (48) behave like if2

K2 ~ C 2 --+ ~ ,

y ~ const,

< co. for c --~

(50)

From this follows that in the limit c - +< c0 the terms iS2B, IBI2B and O~.B comprise the dominant contributions in Eq. (45) and therefore (48) should provide a consistent solution of Eq. (45) in this limit. These simple considerations become more precise in the following construction of self-similar collapse solutions near their (presumed) boundary of existence c = co(b). For that purpose we introduce new variables e : = xy,

E(~) : = B(y)/~c,

S2E := S2/x 2.

(51)

Eq. (45) then becomes

0 = Fb.c,se~:.l/~z(E) :=

(- 2 ~, 5 )+ i£2E

1

E + ~(1 - ic)]E]2E -

7 In the following we assume again c _>0.

O~E + (1 + ib)O~E.

(52)

S. Poppet al./Physica D 114 (1998) 81-107

97

At lowest order of our expansion we have c = co(b) and 1/x 2 = 0. Then Eq. (52) has a one-parameter family of pulse solutions with different widths (see Appendix A). Because of the scaling (51 one may select the solution E0(~) --

Co

cosh(~ )

exp(iy0 In cosh(~))

(53)

with V0 = b + v/-1 + b 2,

C o = 2y0x/I + b 2,

X?F.~, = 27/o(I + b2). 8

(54l

The linear corrections to the zeroth-order solution (53) can be found by differentiating Eq. (52) with respect to c at fixed b.

S. Poppet al./Physica D 114 (1998) 81-107

98

50 .

0

.

.

.

i

.

.

.

.

1/rll

25 -10

-5

.

.

.

. . 0 b

5

0

-25

*

*

*

|

I

i

0

-5

|

,

b

|

5

Fig. 6. Parameter r/i = ql (b) from Eq. (56) along the critical line c = c0(b) (see Eq. (49)) as determined from numerical shooting. At b = bcr -----0 . 6 2 5 9 . . . the parameter r/i changes sign via a pole of first order.

The linear boundary value problem (56), (59) and (60) can be solved using a numerical shooting method. In this way one finds the two real parameters r/l, r/2 along the line c = co(b). The result for r/l (b) is shown in Fig. 6. One sees that r/l : = Ocx -2 is negative for b < bcr = 0 . 6 2 5 9 . . .

(61)

and becomes positive for b > bcr. Since the above expansion assumes x to be real and thus x 2 positive, we conclude that for b < bcr the collapse solution E(~) can only exist below the line c < co(b). Correspondingly it can exist only above the line c > co(b) for b > bcr. We note that this interchange of the range of existence across the critical line c = co(b) could not be seen in the previous simple analysis (see Eq. (49)) because the term --- yOyB in Eq. (45) was neglected there. The boundary of existence c = co(b) of these self-similar collapse solutions together with the critical point bet (cross) are shown in Fig. 1. In the following the analytic results will be compared with numerical simulations of the CGLE. The possible existence of other self-similar collapse solutions will also be discussed. 3.5. Comparison with numerical simulations

In this section the analytic results from the previous sections are compared with numerical simulations of the CGLE. We start with the self-similar collapse solutions from Section 3.4. Near the critical line c = co(b) these solutions should be of the form A (x, t) ~--

er/2KCo

v/2 coshOcxer/2)

exp(iy0 In cosh(x x e r /2 ) - i$2E0X2r).

(62)

99

S. P o p p e t a l . / Physica D 114 (1998) 8 1 - 1 0 7

2

& dl/2e K Q A

~

•

•

J=A

•

•

•

•

0

0

3 A I~ IAI/I ~=

A

~-

-1

(d~/d~)/(2~)

-2

-

•

•

•

ml

--3

m

I

-2

*

I

-1

i

0

1

log(bo-b) Fig, 7. Self-similar collapse solutions: The theoretical limiting values of the quantities from Eq. (63) (lines) are compared with the corresponding values from numerical simulations (symbols). c = 15, b = - 3 . 6 7 . . . . . - 8 . Here the existence boundary of the selfsimilar pulses lies at b ---- bo(c) ---- - 3 . 6 4 9 0 9 . . . The numerical data were prepared like in Fig. 5; i.e. r was determined from the numerical instant of collapse to and x was calculated from the relation l( - 2 ~__ (b - bo)(OhK-2lc)bo = --(b - bo)Tll(ObCO)bo = - ( b - b0)ql ( - 1 + 3 b 0 / / 1

+ b 2) -~ - 0 . 3 5 7 8 5 ( b - b0).

Here e r/2 = 1/~/to - t and t¢ - 2 ~" (C - - CO)OcK-2 shown in Fig. 6. From Eq. (62) one finds [AI----~3(x= 0 ) -IAlxx iA01e_r/2 _ -

C~ _ 2

xCo ~/2'

~

(C -- c0)r/1. 90, Co

and 12E0 are given in Eq. (54) and rh is

v / ~ + b 2 ( b + x / 1 q_b2),

dl/2er/2

-

arcosh(2) x

-~r(x = 0) = -K2S-2e0

(63)

(Ao . A(x . 0))). The . relation . IA(x. 0, r 0)1 xCo c~cor cc implies that the divergence of the pulse solutions becomes progressively slower when approaching the line c = co(b). The four quantities listed in Eq. (63) are compared with data from numerical simulations in Fig. 7. The simulations were performed at c = 15 for some values of b in the range - 8 < b < -3.67. It is clearly seen that near the critical line lo b --~ bo(c) := ~ (cq: 3 ~ - 8) ~ -3.64909 . . . . the data from simulations (symbols) approach their theoretical limiting values (lines). Consistently one finds that the description of the pulse shapes by Eq. (62) improves near the line b = bo(c), see Figs. 5(a) and (b). Here we note that the numerically observed collapse solutions do not necessarily have the self-similar shape/~(y)e i~r in the whole range of y. Instead one often observes a rapid succession of phase slips at the boundary of the pulses (at least during a long transient). These phase slips replace the asymptotic decay B ~ y - l + 2 i t 2 expected from Eq. (47) for y >> x as can be seen, e.g., in Figs. 5(a) and (b). The applicability of the boundary condition (60) for c ~ co is, however, not touched by this fact. 10 This corresponds to relation (49) which is now written in the form b = bo(c).

S. Poppet al. IPhysica D 114 ~1998) 81 107

I O0

In summary, from our numerical simulations we conclude that the self-similar collapse solutions near the line c = c0(b) are indeed described by the expansion in Section 3.4. Close to collapse the broad pulses from Section 3.3 can be written as e r/2

(

A(x.t)_~ ~-

I V/1 + f 2x2er

+R1

exp

7_ ln(l+ f2x2er)+iq21-7_

-I-i~ .

(64)

Here f is a solution of the differential Eq. (43) and RI is given in Eq. (38). Separation of variables yields 1

.)t'2

c~5 ln(O~_~3 +oes~ = 2 ~ 3 ( r - r o ) .

(Y3

\./`:

j

(65)

Neglecting in Eq. (43) the term ~ otsJ 5 one simply gets

1

./,'2 - - 2 ~ 3 ( r - ~ ) .

(66)

The integration constant r0 is determined by the initial amplitude. From Eq. (64) one finds the relations

f2 = _ v q e - 3 ~ / 2 1 A ( x = 0, t)l.,-.~, r = 2 In IAol + In 2 - 2f2( 1 + be) + O ( f 4 ) . I -_v IA01-~ -3(1 +bc)+O([2). .i`.2 - I A ( x -- 0. t)l.,-,

(67) (68) (69)

The validity of Eqs. (68) and (69) for the description of pulse solutions in the delocusing parameter range was checked in the following way: the quantities r and f were determined from the numerical amplitude A(x, t) from simulations according to Eqs. (68) and (69). (This method does not involve the moment of collapse to and can thus be applied independent of the fact whether the pulses finally explode or not.) Using these data the left-hand sides of Eqs. (65) and (66) were plotted against r and then differentiated numerically. If higher-order corrections are negligible, the slopes thus obtained should be 200. The results of such a numerical procedure are shown in Fig. 8. Here the numerical values of the slopes (symbols) are compared with 2cL~ from Eq. (44) (full lines) for c = 15 and 2 < b < 6, Since the slopes in fact depend through f on time, the instantaneous slopes at f 2 = 3000 and ,1:.-2 = 8000 were determined. One sees that for f - 2 = 8000 the slopes from Eq. (65) (rhombs) reproduce almost perfectly the theoretical value 2o~3 whereas there remain some deviations when using Eq. (66) (squares). At ./,.-2 = 3000 one has some differences in both cases. From this comparison it is apparent that Eqs. (65) and (66) in fact describe the asymptotic dynamics ( f -~ 0) of the pulses correctly. It is also seen, however, that the theory is only applicable to fairly broad pulses: i.e. f 2 > 3000 for the values of b and c considered. This restriction can also be seen in Fig. 9(a) where the time evolution of the initial pulse A = 1/x/2( l + x2/12 500) exP(½ic ln(l + .v2/12 500) in the CGLE (3a) for parameters ht = 0, g = 1, b = 2, c = 15 (nonexploding range) is shown. The characteristic quantities - 2 r A ] 3 / I A ].rx and 2 In ]A]+ln 2 corresponding to f - 2 and r, respectively, are plotted against each other (time goes from left to right). One finds that the initially large pulse widths -2]AJ3/IA Ix., indeed dacays with an almost constant slope of approximately 200 = - 2 0 4 , as predicted by Eq. (66). At smaller values of -21AI3/IAI ....... near the end of the pulse growth, i.e. near the maximum values of IAI. one, however, observes clear deviations (see inset).

3oot,

3oof,

,

S. Poppet al./Physica D 114 ¢ 1998) 6'1 107

200

200

100

~

2°t3

101

100 [

o -100

-100

-300'

""

2

' 3

.

.

4

.

.

5

.

. b

. 6

.

3OoT

.

.

.

2

.

3

.

.

4

5

b

6

Fig. 8. Broad pulses: Comparison of simulations (symbols) and theory (full lines) for c = 15 and 2 < b < 6. The numerically determined derivatives of the expressions on the left-hand sides of Eqs. (65) and (66) with respect to r at ¢.-2 = 3000 {a) and .[' 2 = 8000 (b) are shown as rhombs and squares, respectively. In the case of the rhombs f and r were determined from the numerical amplitude A(,r. t) according It) Eqs. (6g) and (69). For the squares the relations r ~_ 2 In IAol + In 2 and ./-2 ~_ _21Ai3/[Ai~ ~ were used. The theoretical collapse boundary lies at bmax (c = 15) = 37/10 corresponding to the zero of the full line. Dashed lines are drawn It) guide the eve.

12000

8bo K

"

400

"<~

400l~'\\\ t

-21AI 3 IAlx~

(a)

\

\

\ \

I/

;

\

\

\

200

6000

0

\

i .

-21AI 3 IAIxx

0

40

21n(IAI)+ln2

80

0

(b)

0

5

21n(IAI)+ln2

10

Fig. 9. (a) Time evolution of the pulse A = I/x/-2(I + x 2 / 1 2 5 0 0 ) exP(½icln(1 + . r 2 / 1 2 5 0 0 ) ) in the CGLE (3a) at the parameters ti = O. t~ = 1. h = 2. c = 15. The dashed line gives the asymptotic slope 2u 3 = 204 calculated from Eq. (66). The inset shows the breakdown o f the pulse in more detail, The dashed line is shifted to the right to allow a comparison of the asymptotic slope with that near the breakdown. Ib) Long time simulation of the CGLE (3) at b = 2, c' = 15 (as in Fig. 2(d)) using periodic boundar~ conditions with period P = 60. The quantities -21A 13/IA I~.v and 2 In[AI + In 2 were determined at l he global spatial maximum. The corresponding curve from the pulse m (a) is copied into the picture (shifted along the abscissa) as a dashed line.

4. Discussion and outlook In the following (,Section 4.1) we will discuss the significance of our results for the "'typical" solutions of the cubic CGLE (3), i.e. the long-time dynamics with generic initial conditions. Finally, in Section 4.2 we will comment on the question of having stable finite-amplitude solutions at (and around) the linear threshold when higher-order perturbations of the CGLE (3a) are included.

102

S. Poppet al./Physica D 114 (1998) 81-107

4.1. The cubic CGLE The regions in the b - c plane of different qualitative behavior of typical solutions of Eq. (3) can be read off Fig. I. According to Eq. (43) blow up should be possible if u3(b, c) > 0, i.e. to the right of the curve or3 = 0. Simulations indeed show that for essentially arbitrary initial conditions eventually a collapse takes place. We remark that in all cases investigated the collapse occurred via the broad pulses from Section 3.3 in spite of the fact that in a large parameter range our analysis would also allow a self-similar blow up. Presumably in this range the broad pulses form the only attractors, which is consistent with results from [14] that Gaussian initial conditions always evolve into broad blow-up pulses. The situation becomes more complicated for a3(b, c) < 0. Below the point S 1 at (b, c) = (½~T5, ~']5) where the curves ot3 ~---0 and u5 = 0 intersect one has to the left of the curve ot3 = 0 the relations a3 < 0 and a5 > O. Here Eq. (43) possesses stable fixed point solutions f 2 p ---~ __0t3/0~5"

(70)

They describe (another type of) self-similar collapse solutions of the functional form (64), which should exist in the limit u3 -~ 0. These solutions are in fact observed in numerical simulations of the CGLE (3). Thereby one finds a different behavior above and below the point $2 at b = ½ ~ / 9 v / ~ - 39 = 0 . 4 4 5 4 8 . . . where the lines c = co(b) and or3 = 0 meet. Below one appears to have self-similar collapse solutions in the whole range between the lines ot3 = 0 (below) and c = co(b) (above) practically independent of the initial conditions. For or3 ~ 0 the solutions < are of the form (64) with f = fFP whereas for c ~ co they are described by Eq. (62). From the simulations we could not decide clearly whether both branches are connected. Above the point $2 (but still below Sl), however, the self-similar collapse solutions seem to exist only in a very small strip to the left of the line a3 = 0. Indeed in simulations of the CGLE (3) at c = 3.2 we never found a collapse for values b < 0.55. Only at b = 0.565, i.e. directly besides bmax (c = 3.2) = 0.565625, a (self-similar) collapse occurred. Since one has j~p2 (b = 0.55) = 212 and f~p2(b = 0.565) "" 3795 this result is consistent with the calculations from Section 3.3 which were found to apply only at f - 2 > 3000. I I Apart from this small strip we only observed bounded solutions, as mentioned in Section 1, in the whole range between the lines c = co(b) and or3 = 0. Presumably the amplitude remains bounded there for all times. We emphasize, however, that in long-time simulations of extended systems the amplitude frequently takes on very large values without exploding finally. This behavior, which might be called an "almost collapse", is observed in particular in a wide range to the left of the line or3 = 0 as can be seen in Figs. 10(a)-(c). There the average spatiotemporal amplitude ([ A l) := f [A ] dx dt / f dx dt and an average (In Amax) 200 of the "highest peak" amplitudes are shown along three lines in the b~" plane. (In Amax)200 := ( l / N ) ~ N = I In Amax,n was calculated from the N highest local spatiotemporal amplitude maxima Amax.n with N chosen such that one maximum was included on the average from every 200 space-time units. The use of (In Amax)200 is of course quite arbitrary. But (In Amax)20o gives a reasonable impression of the height of typical pulses growing out of the chaotic small-amplitude background state, see e.g. Fig. 2(d). Furthermore (In Amax)200 can be estimated from a small amount of data (especially when compared with the number of data required for a full analysis of the distribution p(Amax)). In Figs. 10(a)-(c) both (IAI) and (In Amax)20o increase towards the analytic collapse boundary or3 = 0. The growth of (In Amax)2O0, however, is much more pronounced, in particular at c = 40 (Fig. 10(a)) and c = 15 (Fig. 10(b)) showing that in these cases the typical peak amplitudes are growing much faster than the average amplitude ([A I). We note that in all three figures the data terminate well before the critical line is reached. There remains a large gap of parameters where l I The range of f where the expansion in Section 3.3 applies may depend on the parameters b, c. However, f - 2 > 3000 gives a reasonable estimate also here.

S. Poppet al./Physica D 114 (1998) 81-107

103

4

4

2oo

/

m

3 2

0"01 1~'31~n(Amax)~200 _11~

2 1

~ , ~ e ~ _ _ 0 0 . 0 0 1 I~al2~ 0

'2o o /t

-2

0

J ~4..j_~_.~. ~ , o ~ J41,.-t \ In '

'

2

4

0

~--- I n < A > i

i

1

2

--1

b

0

6

b

3

0.01 II:z312oo

I

2

1

~oo In < A >

-1

10.0

12.5

C

15.0

Fig. 10. L o n g time s i m u l a t i o n s of the C G L E (3): In(IAI) : = In f IA[ dx d t / f dx dt and (In Amax)200 (see text) are shown along three lines in the p a r a m e t e r space. (a) c = 40: theoretical collapse boundary at bmax = 9.98125; (b) c = 15: bma x = 3.7; (c) b = 0: theoretical collapse boundary at c 0 = 3; ot3 = 0 at c = x/3. Periodic boundary conditions with period P = 60 were used. Typical durations of the s i m u l a t i o n s were T ~ 100 (c = 40), T ~ 600 (c = 15), T ~ 800 (b = 0). Lines are drawn as g u i d e s to the eye.

numerical difficulties (induced by the very short-lived high peaks) prevented reasonably long-time simulations in a large system. For the same reasons the largest (In Amax)200 values in Figs. 10(a)-(c) might exhibit a noticeable error. In the following we give a rough estimate of the behavior of the typical pulse height (In Amax)200 near the line a3 -----0 assuming that single pulses can be described by the dynamic equation (35) from Section 3.4. Taking an initial width 3'i- I and amplitude Ai at which a pulse enters the attractor Eq. (66) yields a characteristic period of time A r = 1/(2lot31fi2) > 0 after which the pulse should disappear ( f = z~). From this one finds A m a x "~

Ai ear~2 -~

Ai el/(41~31ji2),

i.e.

Iot3ll n A m a x

"~

1 4fi2 + l~3l

lnAi.

(71)

Assuming that near the critical line or3 = 0 the last term in Eq. (7 I) can be neglected and assuming further that the initial distributions of the pulse widths f / - l only slightly depend on the CGLE parameters b, c one concludes that near the line or3 = 0 1~3 I(ln Amax)200 should be independent of b, c.

(72)

104

S. Polyp et al./Physica D 114 (1998) 81-107

This prediction can be checked in Figs. 10(a)-(c) where the quantity lee31(ln Amax)200is shown. One sees that for b = 0 Eq. (72) is satisfied pretty well. At c ---- 15 and c = 40, however, 1~3](ln Amax)200grows significantly towards the collapse boundary. Thus the pulses become larger than expected from our naive approximation (72). Two reasons for these deviations are now discussed: (i) The description of the pulse attractor by Eq. (35) is not adequate for the simulations in Fig. 10. This can be seen from Fig. 9(b) where the behavior of -21AI 3/IA I.~x versus 2 In PAl + In 2, both evaluated at the actual global spatial maximum, is shown for a long-time simulation at b = 2, c = 15. Fig. 9(b) clearly illustrates that the dynamics of the spatiotemporally chaotic solutions becomes universal at large amplitudes. One finds a characteristic almost-linear relationship for large amplitudes immediately before the breakdown of the pulses. The typical pulse widths ( f 2 < 400) in this simulation are, however, not large enough to justify a description of the pulses by the dynamic Eq. (35) (or (43)) from Section 3.3, compare Fig. 9(a). In order to apply Eq. (35) one would need simulations much closer to the critical line ~3 -- 0. (ii) The change of the typical pulse initial conditions .fi 1 Ai with b, c is not negligible. In fact when comparing the simulation data from Figs. 10 in a representation similar to Fig. 9(b) one finds that the typical initial amplitudes Ai and widths .fi i increase considerably towards the collapse boundary. 12 This reflects a change of the chaotic background state which contributes to the growth of t~31(ln Am~)200 near the line 0~3 = 0, see especially Figs. 10(a) and (b). In summary, it is clearly understandable that the typical pulse height in the spatiotemporal chaotic attractor (characterized, e.g., by (In Amax)200tends to diverge when approaching the critical line (~3 ~ 0). A quantitative description of this divergence, however, requires the inclusion of the pulse-formation process out of the chaotic background state which goes beyond the simple asymptotic considerations leading to the simple prediction (72). Finally we remark that in all analytic considerations in Section 3 the linear growth term ~A from the CGLE (3) was neglected. One concludes that for parameters b, c where Eq. (3) has bounded solutions ('supercritical parameter range") in the CGLE (3a) with ~ = - 1, g = - 1 all (generic) initial conditions should converge to the zero solution.

4.2. The perturbed cubic CGLE In this section we discuss the question of the existence of stable finite-amplitude solutions below the linear threshold, i.e. Eq. (3a) with t~t = - 1 , g = - I , when "small" saturating higher-order terms are included. The simplest equation of this type is the quintic CGLE

'OtA = - A + (I + ib)O.2A + (I - ic)lAl2A + dlAI4A,

(73)

where d is a small complex constant d = d' + id'(Idl << I) with a saturating real part d' < 0. t3 Eq. (73) has a family of traveling-wave solutions

A

-~

Rqe iqx i~o(q)t

(74)

where Rq--

l d'-

I-

q2

and

o ) ( q ) = b q 2+cRq-d''l~q.n4

(75)

(The formulas in this section are given only to lowest order O(d).) In analogy to [33] the traveling waves are found to be stable against long-wavelength perturbations if l+b

c+~-7-/+2q

2 d'(l+

-bc/+

4c-b+-~7-/j

>0.

12 The initial amplitudes A i and widths .t}-I can be read off the coordinates of the upper turning points in Fig. 9(b). 13 In the scaled variables d becomes small sufficiently near to the linear threshold.

(76)

S. Poppet al./Phvsica D 114 (1998~ 81-107

105

For q = 0 this criterion reduces to l+b

c+

>0.

(77)

Thus Eq. (73) has stable finite-amplitude solutions J4 in a large parameter range, especially in the defocusmg quadrant. In the "supercritical parameter range" one has regions with and without attractive traveling-wave solutions. Besides these traveling waves the CGLE (72) is known to support stable pulse solutions for suitable parameters h, c, d 123-29,36,37]. We conjecture that these pulses do not exist as dynamically (and structurally) stable solutions in the (b, c)-parameter range where the corresponding CGLE (3) provides bounded solutions ("supercritical parameter range", between the lines brain and bmax in Fig. 1) since we expect stable pulses of Eq. (73) to merge with tile collapse solutions of the cubic CGLE (3) in the limit ]dl -+ 0. This scenario is obviously impossible in the "supercritical parameter range". Our conjecture is in fact supported by analytic investigations of the quintic CGLE (73) in the focusing NLS limit -I~, c -+ : v . Here Eq. (73) (with d" = 0) is known to possess two different pulse solutions which are selected ou! of the continuous one parameter family of nonmoving NLS soliton solutions bv the dissipative Gmzburg-kandau perturbations, The larger (stable) and the smaller (unstable) pulse both exist to the left of tile line - 4 h = c. At this line, which corresponds to the line brain = brain((') in Fig. 1, the two pulses merge and disappear 123,26,27]. We note furthermore that our conjecture appears to be consistent with all simulations explicitly mentioned in [ 2 3 - 2 9 , 3 6 , 3 7 ] . The stable pulses reported there were found outside the "'supercritical parameter range", in spite of the fact that Idl was not necessarily small in these simulations. The existence of stable pulse solutions in Eq. (73) is known to be correlated with properties of a so-called "nonlinear-front'" solution which connects the stable A = 0 state with a traveling-wave state. The velocity ~, of the front and the selected wave number q t can be both calculated as functions of the parameters h, c. d using a simple analytic ansatz. Stable pulses are allowed only for parameters where t ,+ < 0 (i.e. the A = 0 state invades the traveling wave) or where no nonlinear-front solution exists at all, In both cases, however, localized disturbances might also decay to zero [36,37]. According to the analytic results for t, v fi'om [37] (especially Fig. 8) stable pulses are in principle allowed in a large part of the "supercritical parameter range". Some numerical simulations of Eq. (73), which we performed in the relevant parameter range, however, always showed that localized initial conditions converged towards the zero solution in the long-time limit. In summary, our conjecture that stable localized finite-amplitude solutions should not exist in the "'supercritical parameter range", appears to be consistent with the literature 123-29,36,371 and there also exists some limited numerical conlirmation. Since extended finite-amplitude solutions are found stable only in a restricted part of the "'supercritical parameter range" (see Eqs. (76) and (77)) we expect the zero solution to be the only' attractor in the remaining parameter regime, It seems interesting to investigate these conjectures in more detail.

Acknowledgements One of us (L.K.) wishes to thank the Department E.C.M. of the University of Barcelona, where part of this work was performed, for its kind hospitality, and financial support through all Alexander yon Humboldt-J.C. Mutis Award lor the Scientific Cooperation between Spain and Germany. The work of E.K. has been supported in part by RFBR and INTAS. 14 Short-wavelength instabililies are expected to be of minor importance here.

S. Popp et al./Physica D 114 (1998) 81-107

106

Appendix A. Pulse solutions The C G L E (3a) with arbitrary real parameters b~, g, b, c possesses pulse solutions in the form [13,21,22] C A = - e x p ( i y In c o s h ( x x ) - iS2t). cosh0cx)

(A.I)

The constants C, ~c, y, ,(2 are functions of the parameters g , b, c, g

- ( g + bc) -

~'--2(bg--

tcZ=Iz(TZ-2b?'-l)-l=l C2:3K2VI

z

(g + bc) 2 + -~(bg - c) 2

l-y

,

3g + bc + 2b2g "] -J bg-c I

(A.2)

(A.3)

+b 2

c - bg'

(A.4)

= bK 2 -- yK 2 --~ cC 2.

(A.5)

Solution (A. 1) exists provided the right-hand side of Eq. (A.3) is positive; i.e. d2 < c < dl

for

c < d2 or dj < c c arbitrary

/_t < 0, g < 0 and b real, for

for

tz > 0, g < 0 and b real,

/x > 0, g > 0 and b real.

(A.6a) (A.6b) (A.6c)

Here

dl/2(b, g) : = gb 4- 3 l g l v / l + b 2.

(A.7)

I f / z = 0 and g < 0, solution (A.I) only exists if c = dl (b, g) or c = d2 (b, g). In these cases one has a family of pulse solutions with 7, C 2, I? from Eqs. (A.2), (A.4) and (A.5) and x as a free parameter. Here Eq. (A2) reduces to 7=b+v/l+b

2

for c = dl /2 (b, g).

(A.8)

References [1] A.C. Newell and J.A. Whitehead, Review of the finite bandwidth concept, in: Instability of Continuous Systems, Proc. IUTAM Symp. (Herrenalb, 1969), ed. H. Leipholz (Springer, Berlin, 1971). [2] K. Stewartson and J.T. Stuart, J. Fluid Mech. 48 (1971) 529-545. [3] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, Berlin, 1984). [4] A.C. Newell, The dynamics of patterns: A survey, in: Propagation in Systems Far from Equilibrium, eds. J.E. Wesfreid, H.R. Brand, P. Manneville, G. Albinet and N. Boccara (Springer, Berlin, 1988) p. 122, and references therein. [5] P. Manneville, Dissipative Structures and Weak Turbulence (Academic Press, San Diego, 1990). [6] M.C. Cross and P.C. Hohenberg, Rev. Mod. Phys. 65 (1993) 851. [7] E. Moses, J. Fineberg and V. Steinberg, Phys. Rev. A 35 (1987) 2757. [8] R. Heinrichs, G. Ahlers and D.S. Cannell, Phys. Rev. A 35 (1987) 2761. [9] P. Kolodner, D. Bensimon and C.M. Surko, Phys. Rev. Lett. 60 (1988) 1723, J.A. Glazier, P. Kolodner and H. Williams. J. Stat. Phys. 64 (1991) 945. [10] P. Kolodner, S. Slimani, N. Aubry and R. Lima, Physica D 85 (1995) 165-224. [11] E.P. Ippen, H.A. Haux and L.Y. Liu, J. Opt. Soc. Am. B 6 (1989) 1736; O.E. Martinez, R.L. Fork and J.P. Gordon, J. Opt. Soc. Am. B 2 (1985) 753; P.A. Belanger, J. Opt. Soc. Am. B 8 (1991) 2077. [ 121 H.A. Haus, J.P. Fujimoto and E.P. lppen, J. Opt. Soc. Am. B 8 ( 1991) 2068; IEEE J. Quantum Electron. 28 (1992) 2086; J.D. Moores, Opt. Comm. 96 (1993) 65.

S. Popp et al./Physica D 114 (1998) 81-107

107

[13] L.M. Hocking and K. Stewartson, Proc. Roy. Soc. London A 326 (1972) 289-313. [14] R. Haberman, SIAM J. Appl. Math. 32 (1977) 154. [15] C.S. Bretherton and E.A. Spiegel, Phys. Lett. A 96 (1983) 152. 116] N. Bekki and K. Nozaki, Phys. Lett. A 110 (1985) 133. I17] S. Popp, O. Stiller, I. Aranson and L. Kramer, Physica D 84 (1995) 398-423. 1181 O. Stiller, S. Popp and L. Kramer, Physica D 84 (1995) 424-436. 1191 W. Sch6pf and L. Kramer, Phys. Rev. Lett. 66 (1991) 2316. [20] J.A. Powell and P.K. Jakobsen, Physica D 64 (1993) 132-152. [211 N.R. Pereira and L. Stenflo, Phys. Fluids 20 (1977) 1733. ]22] K. Nozaki and N. Bekki, J. Phys. Soc. Jpn. 53 (1984) 1581. 1231 B.A. Malomed, Physica D 29 (1987) 155, [24] O. Thual and S. Fauve, J. Phys. France 49 (1988) 1829-1833. [25] B.A. Malomed and A.A. Nepomnyashchy, Phys. Rev. A 42 (1990) 6009. [261 S. Fauve and O. Thual, Phys. Rev. Lett. 64 (1990) 282. [27] V. Hakim, P. Jakobsen and Y. Pomeau, Europhys. Lett. 11 (1990) 19-24. [28] H.R. Brand and R.J. Deissler, Phys. Rev. Lett. 63 (1989) 2801. 129] R.J. Deissler and H.R. Brand, Phys. Rev. Lett. 72 (1994) 478. [30] E. Kaplan, E.A. Kuznetsov and V. Steinberg, Europhys. Lett. 28 (1994) 237. [31] E. Kaplan, E.A. Kuznetsov and V. Steinberg, Phys. Rev. E 50 (1994) 3712. [321 L. Kramer, E.A. Kuznetsov, S. Popp and S. Turitsyn, JETP Lett. 61 (1995) 904-910. [33] J.T. Stuart and R.C. DiPrima, Proc. Roy. Soc. London A 362 (1978) 27-41. [34] V.E. Zakharov, Collapse and self-focusing of Langmuire waves, in: Handbook of Plasma Physics, eds. A. Galeev and R. Sudan, Vol. 2 (Elsevier, Amsterdam, 1984)p. 81. [35] N.E. Kosmatov, V.E Shvets and V.E. Zakharov, Physica D 52 (1991) 16. [36] W. van Saarloos and P.C. Hohenberg, Phys. Rev. Lett. 64 (1990) 749. [371 W. van Saarloos and P.C. Hohenberg, Physica D 56 (1992) 303-367.