low-dimensional chaos in a driven damped nonlinear schrödinger equation

Physica 21D (1986) 381-393 North-Holland, Amsterdam

LOW-DIMENSIONAL CHAOS IN A DRIVEN DAMPED NONLINEAR SCHRODINGER EQUATION

K. NOZAKI and N. BEKKI Department of Physics. Nagoya Un&ersit),. Nagoya 464, Japan

Received 15May1985

We study bifurcations of attractors in a driven damped nonlinear Schr~dinger equation, which models nonlinear responses of a plasma driven by a rf field and a sequence of period-doubling bifurcations is shown to lead to a chaotic attractor in a low-dimensional subspace spanned by a soliton and long-wavelength radiation.

1. Introduction

Chaotic states in infinite-dimensional dynamical systems with dissipation have been increasingly interesting subjects, for example, in connection with turbulence in a continuous media. One of the remarkable findings about such chaotic states is the so-called temporal chaos with spatial coherence [1], which suggests that a low-dimensional chaotic attractor exists in an infinite-dimensional system. However, it is very difficult to find a low-dimensional subspace confining a chaotic attractor for a general infinite-dimensional dynamical system. In the case of a perturbed soliton system, the reduction to a low-dimensional system may be achieved by means of the perturbation method, in which solitons and radiations are taken as base functions. In fact, we succeeded in describing chaotic states of some perturbed soliton systems in terms of soliton parameters [2]. In these cases, the perturbations were assumed to be so small that the soliton does not couple with radiation and direct interactions between the soliton and the external force cause chaos. However, no specific physical examples were presented for perturbed soliton systems.

In this paper, we consider a plasma driven by a capacitor rf field and derive a driven damped nonlinear SchriSdinger equation from the fluid equations of plasmas. Then, we search for attractors of the driven damped nonlinear Schr~Sdinger equation and a sequence of bifurcations of attractors is studied by means of perturbational and numerical methods. In the present case, direct interactions between a soliton and a small pump field bring about a phase-locked soliton as an attractor but no chaos. As the strength of the pump becomes not so small, it is shown that there occurs a sequence of bifurcations due to couplings of the soliton and the long-wavelength radiation and a low-dimensional chaotic attractor appears. This chaotic attractor is confined in a low-dimensional subspace spanned by the soliton and the long-wavelength radiation. When the pump field increases further, we observe in numerical experiments a variety of interesting phenomena such as a crisis of the chaotic attractor and more complicated chaotic states.

2. Physical model

We consider a collisional plasma driven by an external rf field Eoexp(-i~%t), where o~ o is assumed to be slightly less than the plasma frequency a~p. Then linear plasma waves, of which the frequency is greater

0167-2789/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

382 K. Nozaki and N. Bekki / Low-dimensional chaos in a driven damped NLS equation

than O~p, do not resonate with the rf field. However, nonlinear plasma waves can exhibit resonance with the pump since a density depression caused by the ponderomotive force decreases the plasma frequency. A well-known phenomenon caused by a similar effect is the oscillating two-stream instability (O.T.S.I.) [3]. Here, we study nonlinear responses of the plasma to a pump below the threshold of the O.T.S.I.. For nonuniform plasma, formations of solitary waves (cavitons) were studied both experimentally [4] and theoretically [5]. Our results in this paper will also be checked experimentally when a nearly uniform plasma is produced.

Let us start with the fluid model for a plasma driven by an external rf field,

On Onv ot--g + = o,

Ov Ov

a t- o + . . . . .

3 Ot ° E = - no,

ogn az v~v- ( E°exp( - i °~° t ° ) + C'C'} '

(1)

(2)

(3)

where all variables and constants are normalized: n, v, E(Eo) and ~'e are the electron density, the electron fluid velocity, the electric field and the collision frequency normalized by n o, ( T i m ) 1/2, r n % ( T J m ) l / 2 / e and %----(4~re2n o/m)t/2, respectively (n 0: an equilibrium electron density, T~: the electron temperature, m: the electron mass); space and time variables z and to are normalized by the Debye length and 0~p. In order to consider slow responses (~ I% - ~01/% << 1) of the plasma to a small pump, we assume that E o - 0(~3), r e - tP(82) and ~ is a small parameter given by 8 2 = I% - ~001/%- By means of the reductive perturbation method [6], that is, introducing the slow variables t = 8 2t 0 and ~ = 8z and expanding E (and n, v) as E = 8E~1) exp ( - i~%t 0) + 82E (2) + . . . + C.C., we have the driven damped nonlinear Schr~dinger equation

iqt + q,,,, + 2[qi2q = - iTq - ieexp ( i~t) , (4)

where

q= - i E O ) / ( 2 8 ) , , / = 2 - 1 y j 6 2, e=4- tEo /83 ,

o~ = ( % - ~ % ) / % / 6 2 ( > 0) and x = ~ .

The nonlinear term comes from the ponderomotive force - 3 (v2/2)/Oz. Eq. (4) may be applicable to the other physical objects such as a one-dimensional condensate in the presence of an applied ac field [7].

3. Stability of homogeneous oscillations

For the homogeneous case (q~x = 0), eq. (4) has a bistability and the two attractors are given by

q = c j e x p ( i w t ) ( j = 1 , 2 ) , (5)

where cl and c2 are stable roots of (21cj 2 - w)c = - i v c - ie and shown as functions of e in fig. 1. When e and ~ are smaU, the two attractors represent a forced oscillation and a resonant oscillation, that is,

K. Nozaki and IV. Bekki / Low-dimensional chaos in a driven damped NLS equation 383

c t = i e / ~ and c z = - i x / ~ . Let us study the stabilities of these oscillations for irdaomogeneous perturbations (qxx* 0). Substituting q = cjexp(itot) + a l ( t ) exp ( i kx + i~0mt ) + a 2 ( t ) e x p ( - i k x + ito2t ), where ]aj I << Icj.t, and ~1 + *02 = 200, into eq. (4), we have the following linear|zeal equations:

ial,t + ( 4 1 c 1 2 - t o , - k 2 + i y ) a l + 2c2a~ = 0 ,

-ia'~., + (4lcl ~ - ~02 - k z - i r)a~' + 2 c 7 2 a l = 0.

Then we have the dispersion relation

( ot + 3' + |(to t - (o2)/2} 2= -12(Icj] 2 - ga/2) (Icjl 2 - t5/6),

where aj oc e ~t and ~ = o~ + k 2. A necessary condition for instabilities of homogeneous oscillations is given by

(~ + k 2 ) / 6 < Icjl 2 < + k2 ) /2 . (6)

For "y << 1, the resonant oscillation c 2 is always unstable (the modulational instability) while the forced oscillation becomes unstable or disappears when I c l ] : ( - le12/~, 2) > ~o/6. This instability of the forced oscillation may be called a stimulated O.T.S.I., in which the threshold field (Ig012- (%- ,,,0)3/00~) is much smaller than the threshold of O.T.S.I. (IE0] z - (~0p - %)/o~p). It will be shown later by numerical experiments that sol|tons are produced as a result of the modulational instability of the resonant oscillation (fig. 5). Therefore, in the following section, we investigated the responses of a sol|ton to a pump field, which is smaller than the threshold of a stimulated O.T.S.I.

1.0

Jql

0.5

0 27I

arg(cj)

7X"

CR

0 | t l

0 0.2 O. 4 .--,- E

Fig. 1. Complex amplitudes q and c 2 of homogeneous attractors defined in eq. (5) for y = 0.1 and to = 1.


4. Reduction to low-dimensional system

Since any localized nonlinear solution of eq. (4) must be connected to the stable forced oscillation q = c a exp(it~t) = c ( t ) at Ixl --' oo, a new field variable r ( x , t ) = q ( x , t ) - c ( t ) is introduced and eq. (4) is rewritten as

~H i t , = 6-77- - iTr. (7)

The Hamiltonian is given by H = H 0 +/q , in which

io= _ dx(IrA 2 - Jrl4), (8)

4 = - dx (zt t=(r.c + re*) + ( c r ' ) z + ( c ' r ) z + 4[rc[ z }. (9) oo

By introducing the new variable r, we can avoid the difficulty appearing in obtaining the first order correction to a soliton [8]. Assuming that leJ, T << 1, i.e., Ic[ - ]el/c0 - 0 ( 0 - ~9(,/), we expand a solution of eq. (7) around a sofiton as

r = r s + rl, (10)

in which rs= 2.i s e c h ( 2 ~ x ) e x p ( - 2 i o - i~r/2) and r 1 is the first order correction given in terms of the "Fourier" amplitudes p(~, t) of radiations [8, 11]

/ ~b 1 = ~/e -2i°+itx sech 2, /x/(~ + i~), (11)

qJ2 = (~ + i~ tanh 2,ix) ei~x/(~ + i-i).

Substituting eq. (10) into eq. (9) and neglecting the third or higher order terms, we have

/ I = / t ( r r s ) - f_~ dx {2(r2rl * + 2[rsl2r])c * + c . c . ) . (12) oo

When action-angle variables of the unperturbed system are introduced, H o is given in terms of action variables only and we obtain, in principle, an infinite number of canonical equations up to the second order. Here, we make the following crucial assumption which will be justified by numerical simulations. Since radiations with finite wavenumbers disperse away, only long-wavelength radiation (4 = 0) is assumed to interact with the soliton significantly and we truncate finite wavenumber radiations. Then, the Hamiltonian part of eq. (7) can be reduced to a low-dimensional system for the parameters of the soliton and the long-wavelength radiation (see appendix A). Damping effects are calculated by means of the perturbation method based on the inverse scattering method (appendix B). Thus, we have a reduced system of eq. (7) or eq. (4),

n, = -2~rlclnZ cos X - 2r~ + 4~rlcl2n sin2× + (8 /~r) f r l lcP[ sin ( 2 X - ,~) + (T/2)fI~] s i n ( x - ~) ,

(13)

X, =- ~0 - 4,12 + 4wlcl*i sin X + 2{cl 2 cos2x - 4[cl 2 + (4/7r)flc~l cos (2X - ~)

+ ((2 -~r)v/2~rn}flPl cos (x - ~), 04)

/5, = iw~ + 8i~llcl e 2ix + i~rve ix - "t~, (15)

K, Nozaki and N. Bekki / Low-dimensional chaos in a driven damped NLS equation 385

where X = 20 + arg(c), /5 = t~exp ( ia rg(c)} , g = p(~ = 0) and ~ = arg(/5). The first order radiation is approximately given by eq. (A.6) in appendix A, that is,

r t = - ( f / T r ) { ~ e - 4io sech 2 (2 ~x ) + ~* tanh 2 (2 +/x ) ) , (16)

where f is the constant weight-factor ( f < 1). Although the value of f is chosen in accordance with the results of numerical simulations, the qualitative properties of eqs. (13)-(15) are not so sensitive to the choice of f ( < 1).

5. Attractors of the reduced system

Before searching for attractors of the reduced system, it may be instructive to study a simple version of the system. When l el (or I c I) and "r are sufficiently small, the second order terms in the reduced system may be neglected and the soliton and the radiation are decoupled. Then, from eqs. (13) and (14), we have

*P,- -2 ' / ,1 - 2~rlcl*1Zcos x, (17)

X+ - - to - 4r/2 + 4 'n ' lcl~ / s in X" (18)

Eqs. (17) and (18) have two attractors given as

, / = 0 and X=c0t, (19)

*1= ¢ ~ / 2 and c o s x = -V/(rr*11cl) = - E C ~ ' r / ( ~ r l e l ) , (20)

where pc I -- lel/o~ and sin X > 0 in eq. (20). An attractor (20) representing a phase-locked soliton exists only if

2 ~ y (21) ~rlt I < I ,

which is the same condition as obtained in [8]. If condition (21) is not satisfied, the system has only one attractor, (19), which corresponds to the homogeneous forced oscillation in eq. (4). Although the attractor (19) is trivial, the corresponding equations derived in [8] do not have this trivial attractor and so they have a difficulty near , /= 0, that is, , / can become negative.

Now, we study bifurcation phenomena in the reduced system (13)-(15) by numerical integrations, in which t is taken as a bifurcation parameter (y = 0.1, to = 1 and f = 0.268). When condition (21) is not satisfied, i.e., e _< 0.6, a soliton-like attractor does not exist and the only attractor is given by ~ = 0 (the homogeneous forced oscillation). For 0.6 _< e < 0.105, there is a stable fixed point representing the phase-locked soliton and the phase-locked radiation. This fact indicates that the phase of the total field q is also locked, which is confirmed by numerical simulations of eq. (4) (fig. 3a). At e = 0.105, the fixed point becomes unstable due to interactions betwen the sollton and the radiation and a limit cycle appears (the Hopf bifurcation). For 0.105 _< t < 0.14, a period of the limit cycle is approximately given by (2¢r/~) = 2~r (a period-1 limit cycle). As t increase further, a sequence of period-doubling bifurcations occurs and a chaotic attractor appears at t = 0.148. For 0.148 _< e < 0.152, we have a relatively small chaotic attractor (type 1), while a larger chaotic attractor (type 2) exists for 0.152 _< t < 0.165. Fig. 2 shows a sequence of typical attractors, where a trajectory on each attractor is projected to the plane of "soliton coordinates"


( - X, ~/) and of "radiation coordinates" (0r -= Re b, 0i = I m ~). When e exceeds 0.165, a chaotic attractor suddenly disappears (a so-called crisis) and a limit cycle with period - 2 r r reappears. However, the numerical simulations of eq. (4) in the next section show that a crisis of a chaotic attractor occurs at e = 0.157 and a remaining attractor is the forced oscillation. Therefore, the reduced system apparently becomes invalid for e >_ 0.157.

6. Numerical simulations

In order to obtain the maximum E, up to which the reduced system is valid and to see what happens when e exceeds the maximum value, numerical simulation of eq. (4) is performed by means of pseudospectral method [9]. A time step (At = 0.05) is advanced by the Crank-Nicolson scheme for linear terms. We take 256 modes and the length of a periodic region is 50, which is much greater than the width of a soliton. In our numerical experiments, we trace the trajectories of a field value q(x m, t) at a location xm, where Iql is maximum, so that the results of the numerical experiments are easily compared to those obtained from the reduced system. Corresponding trajectories in the reduced system are

qr(0, t ) = - 2i~/exp ( - 2io) - ( / / , r ) ~ exp ( - 4 i o - i a rg (c ) ) + c(t).

Fig. 3 shows these trajectories on the attractors in a X - Y plane, in which

X= arg { q(xm, t)} -~ot (mod 2¢r),

Y= Iq(xm, t)I/2,

(22)

(23)

for trajectories obtained from eq. (4) and

X=arg(qr(O,t)}-o~t (rood 2,r),

Y= Iqr(0, l)I/2,

(24)

(25)

for results of the reduced system. A fairly good agreement between the two sequences of bifurcations justifies the reduction from eq. (4) to the four-dimensional reduced system for e _< 0.155. For 0.148 < e < 0.152, we have a chaotic attractor of type 1, which is characterized by a quadratic return map shown in fig. 4, while such a one-dimensional map is not found in the chaotic attractor of type 2 for ~ >_ 0.152. It should be emphasized that soliton-like attractors discussed here can be reached even from a modulationaUy unstable uniform state with small random perturbations (fig. 5).

When e increases over 0.155, discrepancies between the results obtained from eq. (4) and the reduced system are increasingly significant and the reduced system becomes invalid. According to the numerical simulation of eq. (4), a period-3 limit cycle exists at e = 0.155985 (fig. 6) and the chaotic attractor of type 2 suddenly disappears at e ~ 0.157. For 0.157 _< e < 0.176, the uniform forced oscillation given in eq. (5) is the only attractor of eq. (4). Near e-= 0.176, there appears another chaotic state, in which irregular productions and destructions of the solitons occur (fig. 7). As e increases further, a chaotic state becomes more and more complicated, while the forced oscillation is still stable until e ~ 0.26. For e >__ 0.26, the forced oscillation becomes unstable or disappears and a very complicated chaotic state is developed (the onset of a stimulated O.T.S.I. discussed in the section 3). These result are summarized in table I.

K. Nozaki and N. Bekki / Low-dimensional chaos in a driven damped NLS equation 387

"7

0.~

o -OC 2~: -o.5

©

©

0

© !

p,. 0.5

Fig. 2. A sequence of bifurcations for y=0 .1 , co=1 and f = 0.268 in the reduced system (13)-(15). From the top figure, e = 0.13 (a perlod-1 limit cycle), e = 0.144 (a period-2), e = 0.1465 (a period-4), r = 0.149 (a chaotic attractor of type 1) and e = 0.153 (a chaotic attractor of type 2). A fixed point is also obtained but not shown in this figure.

(c]) (b)

Y 0 0 X 27{ 0 X 27~

Fig. 3. Sequences of bifurcations of attractors obtained from (a) eq. (4) and (b) the reduced system (13)-(15) for 3' = 0.1, ~o= 1 and / '=0.268. From the top figure, E=0.1, ~=0 . I3 , ~=0.144, e=0.1465, E=0.149 and e=0.153. Coordinates (X, Y) are defined in eq. (22)-(25).

388 K. Nozaki and N. Bekki / Low-dimensionat chaos in a driven damped NLS equation

3.6

Xn+l

JI

\ t .,'"

¢ °e

s , , - f

2.0 2.0 Xn 3.6

Fig. 4. A plot of successive minima of phase (X,, X,,+I) for = 0.149, where X is defined in eq. (22).

Y

0 X 2if_

Fig. 6. A period 3 limit cycle for e = 0.155985, to= 1 and 7 = 0 . 1 .

400

3 0 0

2 0 0 200 E

I00'

0 0 X -"" 5 0

Fig. 5. Contours of equal Iq(x, t) l versus x and t, in which e = 0.149, to = 1 and 7 = 0.1. An initial condition is taken as q(x, 0) = - i / v t 2 - + (small perturbations with random phases).

0 4 - v 0 - X 5O

Fig. 7. Contour of equal Iq(x , t ) l versus x and t, in which = 0.1767, o~ = 1 and 3, = 0.l. An initial condition is taken as

q(x , 0) == - i~ /2 + (small perturbations with random phases).

K. Nozaki and N. Bekki / Low-dimensional chaos in a driven damped NLS equation

Table I A sequence of bifurcations for o~ = 1 and y = 0.1

e Attractors

Reduced system 0.6 Forced oscillation (F.O.)

- 0.105 fixed point (phase-locked soliton) & F.O. 0.140 Period 1 & F.O. 0.145 Period 2 & F.O. 0.147 Period 4 & F.O. 0.148 Period 8 & F.O. 0.152 Chaotic attractor i (1-D map) & F.O.

- 0.156 Chaotic attractor 2 (no 1-D maps) & F.O. - 0.156 Period 3 & F.O.

0.157 Chaotic attractor 2 & F.O. - 0.177 F.O. (no chaotic attractors)

Irregular creations and annihilations of solitons & F.O. Onset of st imulated O.T.S.I. (complicated chaotic states)

- 0.26

389

7. Conclusions

We have investigated a sequence of bifurcations and resulting chaotic states in a rf driven plasma wave, using a driven damped nonlinear Schr/Sdinger equation. For a sufficiently smaller pump field than the threshold of a stimulated O.T.S.I., a driven damped nonlinear Schrrdinger equation is reduced to a four-dimensional system for a sohton and long-wavelength radiation by means of the second order perturbation method. In the four-dimensional subspace, a phase-locked soliton (a fixed point) is bifurcated to an amplitude oscillating soliton (a limit cycle) and a subsequent sequence of period-doubling bifurcations of a limit cycle leads to a low-dimensional chaotic attractor representing a chaotically oscillating soliton. When the pump amplitude becomes closer or exceed the threshold, there appear more complicated chaotic states, in which many degrees of freedom are excited and we have no reduced systems to explain them.

Acknowledgements

This paper is dedicated to Prof. Tosiya Taniuti in celebration of his sixtieth birthday. We thank Dr. Y. Kodama for valuable discussions.

Appendix A

Derioation of the Hamiltonian part of the reduced system Let us consider the following Hamiltonian equation:

8.0 irt = 8--r-~- + ~-~ ,


in which H 0 and a perturbed Hamiltonian /4 are defined in eqs. (8) and (12). Action-angle variables of the unperturbed nonlinear Schr~Sdinger equation for solutions including a single soliton are given by (a, 20) and { B(~), ~(~)}(- oo < ~ < oo) llO], in which

and

a=4~1, / ~ = - ( 2 / e r ) l n l a l, ~ = a r g ( b ) ,

a(~)=b(~)/a(~),

ln(a)=(2i)-'pf~8(~') in ( ~ - i n ] _ t - t ' d r + ~ , t + b l j - ( ~ r / 2 ) / ~ ( t ) ,

la[ 2+ lb l 2=1 .

Then, the unperturbed Hamiltonian is written as

_ fo~ Ho = . 3 / 1 2 + _ 4~=/~(*)d~, O~

and the canonical equations are

1 a/-) "'= 2 a a ' (A.1)

2o, = - a 2 / 4 + 0I-) (A.2) a a '

#,=

g,,= 4~2+ (A.4)

where h is the perturbed Hamiltonian density (H = fdxh) and r 1 is the first order correction to the single soliton given in eq. (11). In eq. (A.4), we neglect the second or higher order terms such as (Sh/Srs)(Srs/Sfl), in which r s is the single soliton solution.

For convenience, the following notations are introduced:

b = b ( ~ = 0 ) , a = a ( ~ = 0 ) , ~ = p ( ~ = 0 ) , b = b * / a ,

r+=~r-l{p~b~+p*~bt2}, ? , = r l ( ~ = 0 ) , ?_+=r_+(~=0),

~=~(~=0), ~=¢(~=0), ~j=,/(~=0).

Then, we can show that

(ln a ) , = - (2 i ) -~ f / ~ , ( f ' ) / f ' d f ' - (¢r/2)~t,

ap a~ ap(~) a-7 = ip, g g = i~+, a~ p ( f ) / ( 2 i f ) (f ~= o),

08~ =rr0/(2lb]Z)' ot~°"- aB'9"- (2i)-'fd~,+/e.

K Nozaki and N. Bekki/Law-dimensional chaos in a drioen damped NLS equation 391

Using these relations, h, is calculated as

b, = ~ { (~r/2) } b l- 2/~, _ i~ t _ (ln a ) , )

(=i/2)afi_2f dx a T , _ ~_~l.(?.__r,

= -(i/a=) f dx { 8I~ -~T~14'1 - ~-~hrl,~} = - 8i~c* e-4i*/a 2.

Assuming that 1 >> ~ >> p(~) (~ ~ 0), i.e., 1 >> ~ >>/8(~), we have ~ = 1, ~* = h a / a * = a and

~' , - - a, = - 8 i ~ c * e - ~ ° . ( A . 5 )

This assumption also gives

r 1 = -- f { p e - 4 i ° sech 2 (2~lx) + ~* tanh 2 (2~x) }, (A.6)

where f is a constant weight-factor. A constant f may be explained as follows. In the leading order, #(~, t) -- ff(~)exp(4i~zt) and then

~,---~-~ f[¢($)~,~(~) exp (2i(~x + 2~zt) } - 0'*($) ~[2($) exp ( - 2i($x + 252t)}] d$

= (2 t f c ) - l / a [ la ' (~ )~ (~) - ia'*(~)d/~z(~)] (~= - x / 4 t )

= ~ { ~ , - ~*~ } (ix~it -. o),

where (rr /2t ) 1/2 is replaced by f . Substituting eq. (A.6) i n to /1 and using eqs. (A.1) and (A.2), we obtain

n,--- - 2rrlel 'lZcos X + 4¢rlc1='1 s in2x + (8/~r)fnlcFI sin (2X - ~) , (A.7)

2o¢ = -4~12 + 4~rlcln sin X + 21c[: cos2x - 41cl 2 + (4/~r)flc~l cos (2 x - ~) , (A.8)

where X = 2o + arg(c), ~ = arg(~) + arg(c). Eqs. (A.5), (A.7) and (A.8) constitute the Hamiltonian part of the reduced system (13)-(15).

Appendix B

Derivation of the non-hamiltonian part of the reduced system In this appendix, we derive the non-Hamiltonian part of the reduced system (13)-(15), following the

perturbation method developed by V.I. Karpman and E.M. Maslov [11]. In order to use their formulas, we consider

i u t + ½Uxx+ lulZu = --i(3'/2)U (0<y<<l),


which is reduced to eq. (7) with c = 0 when t is replaced by 2t. Karpman and Maslov showed that the parameters (~, o) of the soliton and the scattering coefficients are described by the following equations [121:

. , = - ~ Im (ct(bl , i~)/£~f*(x, i~)f(x, i~)dx}-- ~ ,m { A[u]} ,

- 2 o , = 2~/2+ ~ R e { B [ u ] +A/~},

iy b * ( ~ ) , = - 2 ~ b * ( ~ ) - T { ~(~)~*(~' ~) - t ,* (O ~(~, ~)),

ot(~k. )t) = f ? f* (x , h)R(u)f(x, )t)dx.

~(x, x) = f~S*(x,X)~(u)f(x,X)dx, ( 0 ) [ ] ( ia"(i~)}

R = u 0u* , e = ~ 2 aB(x,in)0)t a=i~-A(u) 'O-x + a'(i~) '

¢(x, x,) = X)R(u)f(x , X')dx,

(B.1)

(B.2)

(B.3)

f(x, X) and g(x, X) are two-component Jost functions corresponding to the eigenvalue X; ~ = (g~', g~') and f - - ( f * , - f l* ) - We expand u around the single soliton solution, i.e., u = 271 e-Zi°{sech z + w(z, t)}, where z = 2~x and w is the first order correction given by

if w = - ~ ( pg + (Oo - og) s~:h 2 z } , oo = ~ e - ~ ° ,

in our approximation (see eq. (A.6)). Then, f[ul, g[u], a'[u], a'[u], A[ul and B[u] are also expanded as

f = f [ u ~ l + f ( ' , g = g [ u ~ ] + g m ,

a ' = (2i~/) - l + a '~l~, a" = (2r/2) -1 + a ''~1),

A = A [ u ~ I + A (~), B = B [ u s l + B (~,

where u s is the single soliton solution. After some calculations, we can show that

A [ . s ] = 2in, B [ u d = 0, a '(1~ = a"") = 0,

A(" = (Oo - 0 ~ ) / 4 ,

Re { B (t) } = - (2 - ¢r)(0o + O ~ ) / ( 4 ~ r ) .

(B.4)

From eqs. (B.1), (B.2) and (B.4), we obtain, up to the second order,

~, = - "r n - / r ( po - o ~ ) / 8 i ,

- 2 o r = 2112 - f2 , (2 - tr)(p 0 + p ~ ) / ( 4 ~ r r ) .

K. Nozaki and N. Bekki / Low-dimensional chaos in a driven damped NLS equation 393

Eq. (B.3) gives the first o rde r te rms of Or---P(~5 = 0)t as

Or* = - - i ( T / 2 ) ~re-2i°- (B.5)

S u b s t i t u t i n g 2 t in to t, we have p e r t u r b a t i o n t e rms p r o p o r t i o n a l to ), in eqs. (13) - (15) . The las t t e rm in eq.

(15) is a d d e d in o rde r to take the s e l f -damping effect in to account .

References

[1] A.B. Bishop, K. Fisser, P.S. Lomdahl, W.C. Kerr, M.B. Williams and S.E. Trullinger, Phys. Rev. Lett. 50 (1983) 1095. [2] K. Nozaki, Phys. Rev. Lett. 49 (1982) 1883; K. Nozaki and N. Bekld, Phys. Rev. Lett. 50 (1983) 1226; Phys. Lett. A102 (1984)

383. [3] K. Nishikawa, J. Phys, Soc. Jpn. 24 (1968) 916. [4] HC. Kim, R.L Stenzel and A.Y. Wong, Phys. Key. Left. 33 (1974) 886. [5] G.J. Morales and Y.C. Lee, Phys. Rev. Lett. 33 (1974) 1016. [6] T. Taniuti, Prog. Theor. Phys. Suppl. No. 55 (1974) 1. [7] D.L Kaup and A.C. Newell, Phys. Rev. B 18 (1978) 5162. [8] D.J. Kaup and A.C. Newell, Proc. R. Soc. Lond. A 361 (1978) 413. [9] D.G. Fox and S.A. Orszag, J. Com. Phys. 11 (1973) 612.

[10] L.D. Fadeev, in: Solitons, Topics in Current Physics, vol. 17 (Springer, Berlin, 1980), pp. 339-354. [11] V.I. Karpman and E.M. Maslov, Soy. Phys. JETP 46 (1977) 281. [12] E.M. Maslov, Theor. Math. Phys. 42 (1980) 237.

low-dimensional chaos in a driven damped nonlinear schrödinger equation

Documents