Apr 25, 1983 - points of view of turbulence of nonlinear disper- sive waves and statistics of solitons in nonequi- librium systems. Many works have been done ...
VOLUME 50, NUMBER
PHYSICAL REVIEW LETTERS
]7
25 APRIL 1983
Chaos in a Perturbed Nonlinear Schrodinger Equation DePaxtrnent
Kazuhiro Nozaki and Naoaki Bekki of Physics, Nagoya University, Qagoya 464, Japan (Received 21 January 1983)
It is shown that a nonlinear Schrodinger soliton behaves stochastically with random phases in both time and space in the presence of small external oscillating fields and emits small-amplitude plane waves with random phases. Statistical properties of random phases give the energy spectra of the soliton and plane waves. PACS numbers:
02. 50. +s, 03.40. Kf
Although soliton systems are completely integrable, the integrability could easily be broken which generalin the presence of perturbations, ly exist in real physical situations, and systems will turn into chaotic states in some cases. Such
chaotic states are of great interest from the points of view of turbulence of nonlinear dispersive waves and statistics of solitons in nonequilibrium systems. Many works have been done on chaos in dynamical systems of a few degrees of freedom but studies of chaos in perturbed soliton systems, which have infinite degrees of freedom, have just begun recently for the sine-Gordon (SG) equation. ' In this Letter we study chaotic states governed by a perturbed nonlinear Schro'dinger (NS) equation which has a generalized form of Landau's nonlinear saturation model for an unstable plane wave. ' The perturbed NS equation describes a much broader range of physical phenomena than the SG system does because almost all strongly dispersive waves with small but finite amplitudes are governed by the NS equation. ' In addition there is an important difference between a finite-amplitude breather of the SG equation and a NS soliton; that is, the oscillation of a breather is unharmonic while a NS soliton oscillates harmonically. This difference plays a crucial role in the response of solitons to external oscillating fields. In fact a breather behaves stochastically in the presence of an oscillating field having a single frequency component' but a NS soliton does not become chaotic under the same situation and phase locks occur. When the external field has many frequency components, it is shown that the soliton phase locks are broken and a NS soliton behaves stochastically because of Chirikov's resonance overlapping mechanism. This stochastic soliton emits small-amplitude plane waves with random phases. Our method of approach to chaotic states of the NS system is based on the inverse scattering method which makes it possible to introduce the concept of a stochastic soliton when the parameters specifying a soliton become
'
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stochastic. The previous method, which studies the time evolution of Fourier coefficients, ' failed to elucidate such an important concept as a stochastic soliton. Our method has another advantage, that approximate equations describing the evolution of soliton parameters are reduced to simple mapping equations which enable us to obtain statistical properties of the chaotic soliton and the energy spectrum of the system semianalytic ally. With small growth and damping terms, the nonlinear Schrodinger equation becomes'
.
—iq. —»Iql'q
qs
=(&, —&, Iql')q+
&,
q„„-(e, /T) &exp(in~, t), (1) n=-~
j
where v, =2&/T, e; ( =0, 1, 2, 3) are small positive constants, and the last term represents idealized contributions from the other stable modes. When derivatives with respect to x are neglected, Eq. (1) is reduced to the model equation by which Zaslavsky studied the stochasticity of an unstable plane wave. ' Let one soliton be assemed to exist initially, then various perturbation methods' yield the following equations for soliton parameters up to the first order of perturbation (nonsoliton parts are also excited and are discussed later): ~
=-»/'&+2n(
-be. ~'
~, (&~
+p'/&')}}, (2a)
g, =
p,
se/ari,
(-'b)
= —&tI!BX+ 2 p I e,
—(&e, rl'
+
e, (4 ri'+ p'/rt'))
],
(2c)
X,
= &&/BP,
where H= —re'+p'/ri
-zrrf(t)sech-
f (t) = —(e, /T)g„exp(in~,
1983 The American Physical Society
(- zrrp/ri')cosg,
t) = —~~„O(t —nT ).
PHYSICAL REVIEW LETTERS
VOLUME 50, +UMBER y7
Once
g) and (p, X) are determined,
71,
25 APRIL 1983
' the solutionn corres correspon d'ing tto a soliton is given by
q, (x, t) = —2i71 sech(2r(x —X)jexp! i((p/g)(x —X) —g) j. If perturbations
vanish (e/ =0), Eqs. (2a)-(2d)
(3)
after a single step for IEI » 1}, th t
'
!
give
.
0(f) =
~. = 4n'+
~—&+ 0(0),
(p/n)',
~„t
X(t) = (2P/q) t+ X(O),
are constant. Therefore, the soliton moves at a velocity 2p/q and oscillates at its own frequency u, . Without the external oscillating term f(f}, an equilibrium between growth and p, g
and damping
is reached'
when
q' = 3e, /4(2e,
+
e, )
arbitrary constant for e, =0. The latter case is more general in the sense that the velocity of the soliton is finite and is discussed hereafter. Using a velocity parameter v=-p/2q instead of p and expanding (q, v} around the equilibrium s'tate ( f/ p vc ) we integrate Eqs. (2a )-(2d) between two consecutive 5 pulses. Then we have approximate difference equations
q„„-e
(77„+ e, sing„ ),
g„„=p„- p~T —%sing„- BT v„~~ =
v„- ( vo/'go)
(4a)
I + vov„ (4c)
X„+, X„-4v,T+4(v, / 1,}7Teinsg„-4pT, ——
SC=8q, t, T((1-e
»
+g, q, =(3e, /Se, )'/',
(4d) p
«
l
f
{(~X)2)1/2
EOslng„',
where I'=4@,T, r/=g,
where g = g+ and angular brackets denote the ensemble average for initial values of g or the time average (both are equal because of the ergodicity of g). On the other hand, diffusion occurs in v and it is characterized by a timeeseae seal TD =v , /'a-TT/e, T T, where a diffusion coefficient is given by B=(vo/g, )'2,' /~p. Since lpl !vol, our simplified mapping (4) is valid for T & t «T~. When I' & 1 in addition to IEI &1, a strange attractor of Zaslavsky's type appears in the q-g plane (Fig. 1) and the distribution of soliton amplitudes becomes stationary and close to its equilibrium value (lg —g, &e r 8,). In accordance with the diffusion in the velocity parameter v, the location of the soliton center {X) randomly deviates from the path X= —4v 0 t and its average deviation is estimated as 8 Dl/2P /2
{(4~+)2)1/2di
Thus, the soliton behaves stochastically when both IKI and I'are greater than 1, in the sense that its phase g is random on the time scale r~ -T; its amplitude is attracted to the strange attractor close to the equilibrium value and the ranodm deviation of its center position from the unperturbed trajectory associates with the diffusion in its velocity parameter v.
')/r+(v, /q, )'), x!O ~-
and we assumed that lg «go Ipl «lvol ~ and T 1, which are justified later. The Jacobian of the map (4) is exp{- I ) and the map contracts areas since I' & 0. Equations (4a) and (4b) are reduced to Zaslavsky's map of a strange attractor' for v, =0, and Eqs. (4b) and (4c) are almost identical to a basic model of stoehasticity in Hamiltonian systems extensively studied by Chirikov. Therefore, the map (4) has the following properties, which are confirmed by nu-
——
I
»
IO
merical calculations.
N 0 ~
~
I««
~
0
.
~
~Or
OOI
«N
~
t«ANAt
Ot v«Art I ~ v«v
«
0
oo
Oo
W
~
~
,W N
.ort .»~» «
0
Aot
~
~
~
I
~ ~
~
t«t
Voto t«OV 0
I ~~
««I
« t 0 N too ot
~
0
~ IV«O
~
oo
w«t r
oot
~
—
~
0
It I ~ tA« ~ rto I N «A«A 0
rot
~
~ ~
Nt
t IIAN ~
0
» 0
t0 ~~
0 to«
AON ~ ~ ~ ~
0
o
0 t
r ««0
t« « ~ 0 It 0 0«0 0 ~ ~0 NON ~
The onset of stochastic
-I
'
~
It O
~
N»
«0 ~
~ ~
~
NO
~A
I ANWt
0 tovINor
O«t 00 «O«t
~
~
oN
Aot
N
O«O
I0
O ~
~ ~
NA« ~
0«N ~ OAOO ~ OOA 0 ~0Aoot 0 t«« ~ ~ 0 ~ I ~Aowo At ttANI O«A 0 ~ 0» O 0 ~ ~OA OA « I ~ AO ~
~0 ~ IO ~
«
At ~
~
~
Atl Ao
OIO ~ ~
0W
I
~
~
~
~ It
OWA
0 W
A
0
0 O«t ~ 0
IA« OWOt A«to« I OOOO« W tAO Ot«OIOA
AOA
w wo
t
.~ ~ «0
'
trajectories in the phase space occurs when I&l & 1, which gives T» 1. Then the distance of two neighboring trajectories grows exponentially with a rate It =in!~I/T and the sequences of g„become random in the range of 2n (mod2m) after a phase correlation time T~ = it ' = T/in!El (practically
~
Nt
~o
~
~ Ato« Ot V«
~
ttttA'0« A ~ Ir ~0 N 001
~
N OA
0I
Tl'
K
Trajectories m pgase spa ttractor appears for I~I =4y 9
p
~
(,
q)
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PHYSICAL REVIEW LETTERS
VOLUME 50, NUMBER $7
I
Here we compare the chaos condition of a soliton with that of a plane wave which is equivalent to a single oscillator. Zaslavsky's criterion' for chaos of a plane wave is obtained by neglecting vo and replacing eo and go with E'p and the amplitude of the plane wave, respectively, in the expression for K. Since 6'p c&Ep when v, g„ the spatial structure of soliton phase characterized by a wave number 2v, tends to prevent chaos. For convenience in later discussions, we list somewhat empirical relations which can be confirmed by numerical analyses:
»
"""')=exp(-
(e"~~'"e (e ikX&t + 7') e
- ik
~~~/~,
'),
L&t)
) = (2n/L) 6(k —k')e
(6b)
(e ia X&t')e ibg&t) ) 0
25 APRIL 1983
(6c) i
where Tp Tp X=X+4v, t, is the range of X, a and ft are nonzero constants, and g(k) can be numerically determined but details of its functional form are not necessary for later discussions. Since X is governed by Eq. (4d) which has a structure similar to the governing equation for g [Eq. (4b)], g(k) is of the order of ln~K~ for k -0(1) and vanishes as k approaches zero. Among various nonsoliton fields excited by perturbations, we are interested in those which remain finite at large t and far away from the soliton. The former condition excludes coherent wave packets which vanish at large t because of a strong dispersion and the latter one eliminates fields contributing to slight deformations of the soliton core. Then the remaining nonsoliton fields consist of small-amplitude plane waves with random phases which are estimated up to the lowest order of perturbations as
' d g {(g —v —i tl tanh2 8) /(g —v+ it) )j'p(g, t) exp (- i (&u + 2 8/7l + 2 q, (x, i) = —tt & &X)], t = — (~,/T) J, dt'(R($, v, tl)/($ —v- iq)')exp[2i((&& —2v)X-Q]p„exp(i(~&+2+ —nu, )i'), p($, i) where 6= rj(x+4vot —X),
f
it',
($,
v,
tl)
=-', tie
(7)
(8)
csch(tt($ —2v)/(2')],
&ut = 4)(( —2v), which is the frequency of a linear plane wave with wave number 2( in the reference frame moving with the soliton. Let us assume that 1 «T& / &c T~; then we can replace vand g by their equilibrium values and resonant contributions in Eq. (8), yielding
and
q (x, i) = —4~—
.Q=. p Q„,($„,)exp[i
n
i ((2$„,)'i+
where Jt,'(g,
v„q, ) '
($+ is@,)'(n~, —4v, —8tl, ')
' v, +(n&u, /4 —v, —2tl, ')'~' or co&+ 2«t„ —nuo=0 for g= s=1 and s= —1 in the sum indicate that plane waves with wave numbers 2$„, and are in the positive and negative regions away from the soliton, i.e. , tl(x-X)»1 and tl(x —X) « —1, respectively. Examining group velocities of plane waves in Eq. (9), we can see that plane waves in the positive and negative regions propagate in the positive and negative directions from the soliton, respectively. The phase of each
g„„=
and
2$„,
S(k, k',
(a&)=
(„„;
2$„,Xj]exp[2i
((&&„, —2v
)X- g] ],
Iplane wave consists of a, coherent part (4g„,'t + x) and a random part [2($„,—2v, )X- 2f). Therefore, plane waves with random phases are emitted from the stochastic soliton, whose frequencies and wave numbers are determined by the resonance condition co& =neo, —2'~. The maximum amplitude of plane waves is attained when the wave number of the emitted plane wave is nearly equal to twice the characteristic wave number of the soliton (&t„, =2vo), where &j„, becomes maximum. Let us now calculate the energy spectrum of the stochastic soliton and emitted plane waves, using the properties of random variables g and X. The energy spectrum of the soliton is defined by
2)„,
J (q, *(x', f+ 7)q, (x, t))e '"" """"' dx'dxdr
(10) Eq. (3) into Eq. (10) and taking into account Eqs. (5), (6b), and (6c), we get for T & i «7D S(k, &u) = (n'/L)se h' (cn(k —2v, )/(4', ) ja~, /f~, '+ (a~, )']., (11) where S(k, k', &u) =2tt6(k —k')S(k, ~), &9, = ~+4v, k —~~, and A&a, =2'~, 1ln~K~+g(k)]. Although the width Substituting
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PHYSICAL REVIEW LETTERS
25 APRlL 1983
of the frequency spectrum is small (A&a, -&u, lniKi), spectrum broadening ton. Equations (6a)-(6c) and (9) give the spectrum of plane waves:
occurs even for a single soli-
VOLUME 50, NUMBER $7
(12)
~, = e+ 4$„,', b. ~, = 2s~, {lniR'i +g(2$„, —4v, )), and»I&'I =T/~. '=»I~I
where
Since we used the idea of the singular perturbation method (multiple time scales) in order to derive Eq. (2), the rough time scale for validity of Eq. (2) is 0(s ') where e-e, (j=0, 1, 2, 3) while the time scale for stochastic instability is T~ -T when chaos occurs (K-8q, e, T) 1). Therefore our results cease to be valid when T» e ', that is, 8g, sech(sv, /(2q, )f «1 which is not satisfied unless q, «1 or v, /q, 1. In conclusion, a NS soliton is shown to behave stochastically in the presence of external oscillating fields. The crucial statistical property of a stochastic soliton is the randomness of its spatial and time phases over the phase-correlation time 7~. This suggests that the randomphase approximation for solitons may become a powerful means to study the turbulence of the perturbed NS equation. Since a soliton covers the complementary frequency range to plane waves as shown in Eqs. (11) a, nd (12), the weakturbulence theory treating only random-phase waves cannot provide the frequency spectrum of
»
!
a soliton such as that given in Eq. (11). It should be emphasized that the energy of a soliton (cc 7I') is distributed on the strange attractor and does not obey the canonical distribution which is usually assumed in the theory for statistics of soliton. We wish to thank Professor T. Taniuti and Mr. H. Takayasu for many helpful discussions.
'J. C. Kilbeck, P. S. Lomdahl, and A. C. Newell, Phys. Rev. Lett. 87A, 1 (1981); K. Nozaki, Phys. Rev. Lett. 49, 1883 (1982). K. Stewartson and J. T. Stuart, J. Fluid Mech. 48, 529 (1971). 3T. Taniuti, Prog. Theor. Phys. , Suppl. 55, 1 (1974). 4D. J. Kaup and A. C. Newell, Phys. Rev. B 18, 5162
(1978).
5H. T. Moon, P. Huerre, and L. Q. Redekopp, Phys. Rev. Lett. 49, 458 (1982). 'G. M. Zaslavsky, Phys. Lett. 69A, 148 (1978). 'D. J. Kaup and A. C. Newell, Proc. Roy. Soc. London, Sec. A 361, 413 (1978); V. I. Karpman, Phys. Scr. 20, 462 (1972); J. P. Keener and D. W. McLaughlin, Phys. Rev. A 16, 777 (1977). N. R. Pereira and L. Stenflo, Phys. Fluids 20, 1733 (1977). ~B. V. Chirikov, Phys. Rep. 52, 265 (1979).
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