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Soliton streaming: Transient energy growth of ac-driven damped breathers
Volume 143, number 9
PHYSICS LETTERS A
5 February 1990
SOLITON STREAMING: TRANSIENT ENERGY GROWTH OF ac-DRIVEN DAMPED BREATHERS A.L. G E R A S I M O V Institute of Nuclear Physics, 630090 Novosibirsk, USSR Received 28 April 1989; accepted for publication 19 June 1989 Communicated by A.R. Bishop
The dynamics of a breather in an ac-driven damped sine-Gordon (sG) system is considered. Under certain conditions, the transient phase-locked evolution, in the course of which the velocity damps to zero while the energy and the amplitude of the breather can increase, is demonstrated. Presumed implications for the statistical mechanics of an ac-driven sG are discussed.
Over the last decade, the influence o f small perturbations on solitonic solutions of exactly integrable pde's has been extensively studied (see, e.g., refs. [ 1-4] ). Among such, the perturbed sine-Gordon (sG) equation
02U
02U
Ot 2
Ox 2 + s i n v = e R ( v )
(1)
was one of the most popular, motivated by applications in long Josephson junction [4,5], quasi-one-dimensional ferromagnets [6], plane dislocations in crystals [7], to name but a few. The particular case o f ( a c - d r i v e n + d a m p i n g ) perturbation Ov e R ( v ) = e cos o J t + a - Ot
(2)
is also rather well documented [ 8 - 1 0 ] . The effect of perturbation is mainly studied for two types of (unperturbed) solitonic excitations in the sG system. The first one is the kink Vk(X, t) = 4 a arctg exp[ ( x - - Vt) ( 1 - V 2) -1/2] ,
(3)
where a characterizes the kink polarity a = _+ 1 (kink, antikink), and V< 1 is the kink velocity. The second type is the breather Vb (x, t)
( 1 -s-~n V?2 )~] ~ -~-2~ t _ lV--5-/~(-~_ 2(/ 1 - 2 V 2~) -1/2 sin y x])_ , 4 arctg(tg_ y cos [cos 7c_h_[
(4)
where the parameter y characterizes the internal oscillation frequency o f the breather -Q= cos ? ( 1 -- V 2 ) --1/2
(5)
and V< 1 is the velocity of the breather motion as a whole. The energy o f the breather is E b = s i n ~ (1-- V 2 ) - 1 / 2 .
(6)
In accordance with the already conventional marginal stability of solitons under the influence of perturbations, allowing one to consider them as particles and oscillators [ 1 ], the dynamics o f the perturbed breather is described [4] as the slow evolution o f the parameters V and y. The set of equations describing this evolution 448
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includes also the phase 0 "conjugate" to 7- On the whole, the perturbed dynamics of a breather is very similar to the two-dimensional ac-driven damped nonlinear oscillator. Usually the transient behaviour is not of interest, so that the velocity V is initially set to zero, and in the resulting perturbed dynamics of 7 and 0 the most important is the effect of the formation, at not very strong damping tx < ~, of the phase-locked solution at 7¢ 0 [8]. Such a solution appears e.g. in the asymptotic space-time structures of the spatially periodic version of the system ( 1 ), (2) [ I 1 ]. The same type of phase-locking for zero V was described in ref. [ 1 ] for the soliton of the nonlinear Schr6dinger equation. In the present paper we consider the ac-driven damped sG breather for initial velocity V¢ 0 and demonstrate the existence of a specific (transient) solution, staying phase-locked in the process of the velocity V being damped to zero. The breather can evolve in this case from the rather low and fast one to a high and resting one, and its energy increases in the process. In terms of the coordinates 7, 0 and V the process looks like a damping-induced motion along the nonlinear resonance - the well-known phenomenon of resonance "streaming" [ 12] in near-integrable damped anharmonic many-dimensional oscillators. This led us to call the described breather evolution "soliton streaming". In considering the effect of perturbation (2) on the breather dynamics we will make use of the most fundamental result of ref. [ 1 ], already ubiquitous in perturbed soliton literature - the possibility of neglecting the effect of the continuous spectrum component of the inverse scattering transform on the times of the order l/e. At most, the continuous component can manifest itself only in the formation of the tail of a certain "resonant" harmonic e ik~ of small ~ e amplitude behind the moving soliton (sometimes k is zero and the "tail" is a "'shelf"). The energy, accumulated in the tail on the times ~ 1/e is obviously of the order e, so that the solitonic discrete degrees of freedom dominate the dynamics. Thus, the perturbation in the first order may be considered to act only on the discrete solitonic degrees of freedom and the response of the system is the same as that of particles and oscillators [ 1 ]. For our particular system and perturbation the equations of motion of the breather coordinates, derived through the inverse scattering transform technique, can be taken directly from ref. [4], where the dc-driven damped sG breather was considered. The only modification needed is the substitution of ~ cos cot instead of E in the amplitude of the drive. Thus, we introduce, following ref. [4 ], the parameters 7 and V defining the breather solution (4) as (see expressions ( 1 . 3 ) - ( 1 . 6 ) of ref. [4] with 7=arctg(v/q))
v(7, V,O,xo)=-4arctg
ch[sinT(l_VZ)_l/Z(x_xo)]
tg7
].
(7)
The unperturbed breather (4) corresponds to constant 7 and V while 0 and Xo evolve as 0=cosT(1-V2)l/2t+0o,
xo=Vt+xoo.
(8)
Thus, Xo is the coordinate of the breather centre of mass, moving with constant velocity V, while 0 is the phase of the internal oscillations of the breather with frequency g2 (5). The parameter 7 defines, if the breather is interpreted as a bound state o f a kink-antikink pair [7 ], the degree of boundedness - for 7<< ~/2 the breather is strongly bound and has a small amplitude (flat profile), while for 7 close to ~/2 the breather is weakly bound and has a large amplitude (close to ~). Under the influence of perturbations the parameters 7 and 0 are slowly evolving, governed by the equations d7
dt--(l--V2)l/Z(4cosT)-lll'
dV
dO--COS 7 ( 1 - -
1/2
dt
V 2)
l/z_ (1 - -
V 2)
d--[=--(1--V2)3/2(4COST)-lI2'
dx° - V+ e ( 1 - V 2 ) I3 - V tg 7 I4 dt (2 sin 7) 2 '
VctgTI3+c°sZ7 ( l - VE)I,-I5 4 sin 7 COS27
'
(9) 449
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where the quantities
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5 February 1990
Ii(7, V, O) are
GO
I1= f
chzsin[O-(V/tgT)z]
R(z) dz
c h 2 z + tg27 cos z [ 0 - ( V/tg 7)z]
i I2 =
shzcos[O-(V/tgy)z]
ch2z+tg27 cos2[0 - ( V/tg 7)z]
R(z) dz
--oo
i
13 =
z ch z sin[ O- ( V/tg y)z]
R(z) dz
ch 2z+tg27 cos 2 [ 0 - ( V/tg 7)z]
i 14---
zshzcos[O-(V/tgy)z] R(z) dz ch2z+tg2ycos2[O_(V/tgy)z]
--oo
15= ~
chzcos[O-(V/tgT)z] R(z) dz, V/tg ?)z]
ch 2 z + tg2~ cos 2 [ 0 - (
(10)
and R = e cos tnt + OtOVb/Ot. Eqs. (9), (10) are the same as eqs. ( 3 . 6 ) - (3.15 ) of ref. [4 ], taken for the varied amplitude of the drive e--,~ cos o~t. The quantity OVb/Otin (10) should be obtained from the derivative with respect to time o f the breather solution (7) for the unperturbed evolution of 0 and Xo with the further substitution of V(1- VZ) -t/2 cos7 (X-Xo) by z [4]. A further simplification of the system (9) can be achieved by the identification o f fast and slow time scales o f the dynamics and averaging over the fast oscillations in the spirit of the Bogolyubov-Krylov method. To start with, consider the system (9) for zero damping a = 0 and nonzero drive t ~ 0. Then it takes the form d7 - ~ cos dt dV
dt
--
oat ( 1 - V 2) 1/2(4 cos 7 ) - l J l
,
~(l-VZ)3/2(4cosy)-lJ2,
dO d t = c o s 7 ( I - V 2) 1/2_ E cos cot ( 1 - V 2) l/z[ Vctg 7 J3 "[- COS27 ( 1 -- Vz)J4 - J s ] (4 sin 7 COS2) )) --1 , dxo
dt
- V - e cos
oJt Vtg
7 (2 sin ~ ) - 2 J 4 ,
(1 la)
(llb) (llc) ( 1ld)
where the quantities Ji-J5 are the same integrals 11-15 (10) with R = 1. In analyzing the system ( 1 1 ) note first that the system of ode's ( 1 1 ) is not Hamiltonian. Secondly, the quantity Xo does not enter eqs. ( l l a ) - ( 1 lc), so that the last one (l l d ) can be omitted. In considering the system (1 l a ) - ( 1 lc) let us introduce the local coordinates 7'-7o = 71, V - Vo= VI. Since we expect to obtain the behaviour of trajectories of ( 1 1 ), similar to that o f the nonlinear Hamiltonian oscillator in the vicinity o f an isolated nonlinear resonance [ 13,14 ], let us single out the resonance harmonic Jt cos ~, ~ = lO-cot (l integer) from the Fourier decomposition of the functions J~, ,/2 by the periodic variable 0 and omit all the nonresonance ones. The quantities Jl, J2 can be easily expressed through one another by differentiating the expression arctg(tg 7 cos [ 0 - ( V/tg 7 ) z ] ~ ch z / with respect to z and integrating over z from - o o to ~ . This yields the relation Jz = ( V/tg 7)J~. The terms of order ~ in dO/dt ( 1 lc) can be omitted, since they lead to higher order corrections (in powers of e) to the trajectory behaviour. Thus, we arrive at d7 = e ( 1 - Vo2)1/2(4 cos 7o)- IJlz(7o, Vo) cos T dt
450
(12a)
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dV--¢(l-V2°)3/2(4c°sy°)-dtiVoj ~ It(7o, Vo) cos ~ ,
(12b)
d~ dt - 5o9(7, V ) ,
(12c)
where the detuning is 8o9 = l cos y ( 1 - V 2 ) 1/2_ 09. Note that the amplitudes before the resonance harmonic cos were taken in the zero approximation ? = Yo, V= Vo in the same fashion as is done in the "universal" description of nonlinear resonance [ 13,14 ]. The resonance condition 8o9= 0 as it stems from the last expression (12c), is (13)
l c o s 70 ( 1 - V02)1/2--o9 •
Thus, the resonance o f the harmonic l occurs only for o9< l. Note that the resonance frequency turned out to be not the breather frequency (5) in the laboratory frame, but contains an additional factor 1 - V 2 due to the Lorentz time contraction. For to smaller than unity all the resonances l = 1, 2... are present in the (~, V) plane. The first two resonance lines are shown schematically in fig. 1. The resonances with higher I approach the lines 7 = ~ / 2 and V= 1. Eqs. (12) can be simplified by introducing the new variable
p=(V-Vo)+ I I ° ( 1 - I I ° ) 2 ( 7 - ~ ' o ) ,
(14)
tg 7o
which stays constant in the process o f the dynamics (12). Then, the system (12) reduces to d7 d t =EKI(7o, Vo) cos ~ ,
(15a)
~d P - _o,
(15b)
~ =,~(ro,Vo)(7-yo),
(15c)
where K~ = ( 1 - V~ ) 1/2 ( 4 cos 7o ) - ~J~t( 70, Vo). In eq. ( 15c ), the r.h.s, o f ( 12 ) was expanded in deviations 7 - Yo up to quadratic terms from 7 = 70 for p = 0 (similarly to the nonlinear resonance description [13,14], p can always be considered zero, since a nonzero p corresponds just to a shift o f the point 7o, Vo). Thus, 2 is
0(8o9) a (8o9) V o ( l - v 2) 0 (_8o9) 2_ ~ p=o- ~7 v:co,,st+ -tg- -70 OV ~=~o.st"
(16)
The pair o f equations (15a) and (15c) is a Hamiltonian system, describing the nonlinear resonance in the
V o
Fig. 1. Nonlinear resonances l = 1, 2, defined by the condition ( 13 ). Arrows indicate oscillations of 7, V on the separatrix, with the amplitude ( 17 ). Dashed line 7= n/4 corresponds to the maximal energy on the resonance line. The dotted line t g T = F demarcates growing and decreasing 7 regimes.
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pendulum approximation. The resonance width ATr (the amplitude of oscillations of 7 on the separatrix) is given by AYr=x/ZelK11/I21 •
(17)
The first two resonance lines (13) l = 1, 2 (for the case ¢o< 1 ) with the resonance oscillations in its points in the direction p = const are shown schematically in fig. 1. At each line there exists a characteristic point, in which the direction o f resonance oscillations is tangent to the resonance line. This point is the point o f intersection of the line V = t g 7 (shown by the dotted curve in fig. 1 ) with the resonance line (13). For the motion with damping in the vicinity of a certain isolated resonance, we can average the r.h.s, of (9) over the "fast" phase 0, holding the "slow" phase 7j constant and omitting nonresonant harmonics. Since the same averaging was performed in ref. [4] for a dc-driving, and because the averaged damping terms in the r.h.s, of (9) will be the same, we can make use ofeqs. (3.32) o f ref. [4]. Finally, for the averaged slow quantities 7, V, 7j we obtain d7 d t = eKl cos ~ - - OtT, dV dt - E V ( I - V
(18a)
2)ctgTK~cos~P-aV(l-V
d ~ = 809. dt
2)(1-7ctg7),
(18b) (18c)
The trajectories o f the system (18) in the (7, V) plane in the absence of drive e = 0 are shown schematically in fig. 2. It is interesting to note [ 4 ] that while the parameter 7, defining the amplitude of the breather, damps t o zero, the velocity V does not. The dynamics under the influence of damping of the nonlinear oscillator in the vicinity of an isolated nonlinear resonance was considered in ref. [ 12 ] (see also ref. [ 15 ] ). It was demonstrated that for small enough damping the particle with the initial conditions inside the separatrix of resonance will move along the resonance line - a p h e n o m e n o n called "resonance streaming". In other words, the trajectory, being phase-locked to the driver initially, stays phase-locked in the course o f evolution under the influence of damping, though large-scale deviations in 7, V at times t ~ 1/or occur. The system ( 18 ) can be presented in a form more convenient for a description of resonance streaming [ 12,15 ] by introducing other variables, yielding d j = ~Kj - a T , dt
(19a)
V
452
Fig. 2. Damped undriven breather evolution, governed by ( 18 ) with ~= 0.
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@
dt - - ° i V ( l -
V 2) ,
(19b)
d~U =2p dt
(19c)
The direction of motion of a damping-pulled particle along the resonance is defined [ 12,15 ] by the direction of the component Ft of the decomposition of the damping "force" vector F on the components in the direction of resonance oscillations Fr and along the resonance line Ft. Thus, we can easily describe the motion of particles with the initial conditions inside the resonance separatrices under the influence of vanishingly small damping. Below the line tg 7= V the particles move along the resonance lines in the direction of decreasing y - for vanishing a up to ?=0. Above the same line the particles move upwards - increase their y and for vanishing ot reach the point V= 0. Thus, above the line tg y = V the breather evolves from the moving and lower one to the immobile and higher one. Below the line in the process of the transient phase-locking the breather accelerates, but decreases its height (for vanishing o~ to zero). We do not discuss the behaviour of the small inside-separatrix oscillations in the course of such motion, since we suppose the resonance width (17) to be small relative to n/2, rendering these oscillations irrelevant. Consider now the variation of the breather energy in the process of the phase-locked transient motion described above. For this, we observe that from expression (6) for the breather energy and formula (13) for the resonance line it may be easily shown that the breather energy along each resonance line is maximal at 7= n/4 (shown by the dashed line in fig. 1 ) and monotonically decreases in both directions from this point. Thus, we see that for initial conditions inside the resonance separatrix above the line tg 7= V the breather energy first grows (until 7 reaches the value ~,=n/4) and afterwards decreases, while below this line the energy monotonically decreases. The energy of the breather at the line tg 7 = V is E~, = (1/2~o)x/1 -~o 4/l 4 , at the point 7= n/ 4 it is Ebmax= l/2tO, and the energy of the stationary phase-locked breather is Ebs = x/1 --to2/l 2. These energies are ranged as Ebs < E~, < Eb . . . . Note also that in the process of the breather energy increase from E~, to Ebmax (initial condition at tg 7= V) the energy is drawn from the driver, but the role of dissipation is essential - in the absence of dissipation only a minor ( ~ x/~) amount of energy will be periodically exchanged between the driver and the breather. Let us consider now the limitations on e, a for which the effect takes place, since it is quite clear that for large enough damping the resonance will be destroyed - the particle will "fall out" of resonance. To find the critical value of damping, we substitute in (19a) the value of 7 at the resonance line 7 ~ 7r (V), which is defined by the resonance condition (13) (approximate equality means to the order of the resonance width ~ x/~). Making use of the relation d T J d t = ( d T J d p ) d p / d t and expressing the first factor from (13) to be dT~
x=-~p-
d7JdI~
=
OplOVIr=r,
V ctg 7 (1-- V2)(1 - V2 ctg27)
(20)
and the second factor from (19c), the solvability condition of the resulting equation yields (see ref. [ 15 ] for more details): ot
Kt
(~')cr ~ /~V(I_V2)_ 7 ,
(21)
where Kt, 7, x are taken at the resonance line 7=Yd V). Thus, the threshold (21) is different for different points on the resonance line (depending on V). The analysis shows, that most interesting from the point of view of observation of "soliton streaming" phenomena is the strongly underdamped regime a/E << 1, only the section of the resonance line, lying between zero and a certain (relatively small) y~ << 7t/2 will be overdamped - or~ E for this section will be smaller than the threshold (21 ), and no streaming at this part of the line occurs. The part of the resonance line above 7= y~ will be underdamped (above threshold), and the particle, starting with 453
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a p r o p e r initial phase ~ = n / 2 (see ref. [ 15 ] ) on the resonance line, will m o v e along it as described previously. The d a m p e d ac-driven sG system ( 1 ) is usually studied for c o m p a r a t i v e l y large p e r t u r b a t i o n c ~ 1 when the complex s p a c e - t i m e structure o f attractors proliferate [ 11 ]. We considered the from this point o f view trivial case ~ << 1, which for the u n d e r d a m p e d regime ot << 1 a n d a < E possesses a simple attractor o f phase-locked breather, but d e m o n s t r a t e d a nontrivial transient to this attractor breather evolution, in the course o f which the energy grows. It is supposed that this effect will strongly manifest itself in the statistical mechanics o f an ac-driven sG system for large relaxation t i m e and low temperatures. This conjecture is based on the recently found strong influence on the ac-drive o f the " t a i l s " o f the d i s t r i b u t i o n function a n d escape rate from the metastable potential well o f m a n y - d i m e n s i o n a l nonlinear oscillators subject to d a m p i n g a n d noise [ 15 ]. The statistical mechanics o f the sG system appears in a n u m b e r o f applications e n u m e r a t e d in the beginning. One o f the f u n d a m e n t a l processes in this k i n d o f p r o b l e m is the statistical birth o f k i n k - a n t i k i n k pairs, responsible for relaxation a n d n o n l i n e a r mass t r a n s p o r t in the system [ 7,16 ]. Since a k i n k - a n t i k i n k pair has nonzero rest energy Eo, the p r o b a b i l i t y W o f birth at each finite length a n d equilibrium density o f kinks and antikinks is exponentially small for low temperatures, kT<< Eo, i.e. W ~ exp ( - Eo/kT). The k i n k - a n t i k i n k p a i r is born [ 7 ] from the large-amplitude b r e a t h e r ~,=n/2 with zero velocity. We suppose, a n d this presents itself as a field o f further research, that in a certain leading a p p r o x i m a t i o n for the probability o f low t e m p e r a t u r e ac-driven k i n k a n t i k i n k birth, the d y n a m i c s o f d a m p e d , ac-driven a n d noisy breather can be uncoupled from the continuous degrees o f freedom. F o r such an oscillatory finite degree o f f r e e d o m system we can apply the theory [ 15 ], and expect therefore the b r e a t h e r to reach the state 7 = n / 2 , F = 0 , with the energy E o = 1, a n d to have to m o v e to the p o i n t on the resonance line tg 7 = I / w h e r e the streaming starts (the energy E ' there is smaller than Eo) and then m o v e along the resonance line up to the p o i n t y=n/4 (where the energy is larger than E ' ). This will exponentially strongly increase the probability o f k i n k - a n t i k i n k pair birth: W ~ exp ( - G~kT); G < Eo, Eo- G ~ Eo. Such a regime is expected for the p a r a m e t e r s ~<< 1, 09< 1, ot<~, kT<
References [ 1] D. Kaup and A. Newell, Proc. R. Soc. A 361 ( 1978 ) 413. [ 2 ] B. Malomed, Physica D 27 ( 1987 ) 113. [3] R. Bullough, P. Caudrey and H. Gibbs, in: Solitons, eds. R. Bullough and P. Caudrey (Springer, Berlin, 1980). [4] V. Karpman, E. Maslov and V. Soloviev, Zh. Eksp. Teor. Fiz. 84 (1983) 289. [ 5 ] A. Scott, in: Solitons and condensed matter physics, eds. A. Bishop and T. Schneider (Springer, Berlin, 1978 ). [6] F. de Leeuw et al., Rep. Prog. Phys. 43 (1980) 689. [7] A. Seeger, in: Lecture notes in physics, Vol. 249. Trends in applications of pure mathematics to mechanics, Bad-Honnef, 1985 (Springer, Berlin, 1985). [8] P. Lomdahl and M. Samuelsen, Phys. Lett. A 128 (1988) 427. [9] A. Mazor (Bin-Mizrachi) and A. Bishop, Physica D 27 (1987) 269. [ 10] M. Fordsmand, P. Christiansen and F. If, Phys. Lett. A 116 (1986) 71. [ 11 ] A. Bishop, M. Forest, D. McLaughlin and E. Overman II, Physica D 23 (1986) 293. [ 12] J. Tennyson, Physica D 5 (1982) 123. [ 13 ] B. Chirikov, Phys. Rep. 52 ( 1979 ) 263. [ 14 ] M. Lieberman and A. Lichtenberg, Regular and stochastic motion (Springer, Berlin, 1983 ). [ 15 ] A. Gerasimov, Phys. Lett. A 135 ( 1989 ) 92; Institute of Nuclear Physics Preprint 87-100, Novosibirsk ( 1987 ). [ 16 ] S. Trullinger, M. Miller, R. Guyer, A. Bishop, F. Palmer and J. Krumhansl, Phys. Rev. Lett. 40 ( 1978 ) 1603.
454
PHYSICS LETTERS A
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SOLITON STREAMING: TRANSIENT ENERGY GROWTH OF ac-DRIVEN DAMPED BREATHERS A.L. G E R A S I M O V Institute of Nuclear Physics, 630090 Novosibirsk, USSR Received 28 April 1989; accepted for publication 19 June 1989 Communicated by A.R. Bishop
The dynamics of a breather in an ac-driven damped sine-Gordon (sG) system is considered. Under certain conditions, the transient phase-locked evolution, in the course of which the velocity damps to zero while the energy and the amplitude of the breather can increase, is demonstrated. Presumed implications for the statistical mechanics of an ac-driven sG are discussed.
Over the last decade, the influence o f small perturbations on solitonic solutions of exactly integrable pde's has been extensively studied (see, e.g., refs. [ 1-4] ). Among such, the perturbed sine-Gordon (sG) equation
02U
02U
Ot 2
Ox 2 + s i n v = e R ( v )
(1)
was one of the most popular, motivated by applications in long Josephson junction [4,5], quasi-one-dimensional ferromagnets [6], plane dislocations in crystals [7], to name but a few. The particular case o f ( a c - d r i v e n + d a m p i n g ) perturbation Ov e R ( v ) = e cos o J t + a - Ot
(2)
is also rather well documented [ 8 - 1 0 ] . The effect of perturbation is mainly studied for two types of (unperturbed) solitonic excitations in the sG system. The first one is the kink Vk(X, t) = 4 a arctg exp[ ( x - - Vt) ( 1 - V 2) -1/2] ,
(3)
where a characterizes the kink polarity a = _+ 1 (kink, antikink), and V< 1 is the kink velocity. The second type is the breather Vb (x, t)
( 1 -s-~n V?2 )~] ~ -~-2~ t _ lV--5-/~(-~_ 2(/ 1 - 2 V 2~) -1/2 sin y x])_ , 4 arctg(tg_ y cos [cos 7c_h_[
(4)
where the parameter y characterizes the internal oscillation frequency o f the breather -Q= cos ? ( 1 -- V 2 ) --1/2
(5)
and V< 1 is the velocity of the breather motion as a whole. The energy o f the breather is E b = s i n ~ (1-- V 2 ) - 1 / 2 .
(6)
In accordance with the already conventional marginal stability of solitons under the influence of perturbations, allowing one to consider them as particles and oscillators [ 1 ], the dynamics o f the perturbed breather is described [4] as the slow evolution o f the parameters V and y. The set of equations describing this evolution 448
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includes also the phase 0 "conjugate" to 7- On the whole, the perturbed dynamics of a breather is very similar to the two-dimensional ac-driven damped nonlinear oscillator. Usually the transient behaviour is not of interest, so that the velocity V is initially set to zero, and in the resulting perturbed dynamics of 7 and 0 the most important is the effect of the formation, at not very strong damping tx < ~, of the phase-locked solution at 7¢ 0 [8]. Such a solution appears e.g. in the asymptotic space-time structures of the spatially periodic version of the system ( 1 ), (2) [ I 1 ]. The same type of phase-locking for zero V was described in ref. [ 1 ] for the soliton of the nonlinear Schr6dinger equation. In the present paper we consider the ac-driven damped sG breather for initial velocity V¢ 0 and demonstrate the existence of a specific (transient) solution, staying phase-locked in the process of the velocity V being damped to zero. The breather can evolve in this case from the rather low and fast one to a high and resting one, and its energy increases in the process. In terms of the coordinates 7, 0 and V the process looks like a damping-induced motion along the nonlinear resonance - the well-known phenomenon of resonance "streaming" [ 12] in near-integrable damped anharmonic many-dimensional oscillators. This led us to call the described breather evolution "soliton streaming". In considering the effect of perturbation (2) on the breather dynamics we will make use of the most fundamental result of ref. [ 1 ], already ubiquitous in perturbed soliton literature - the possibility of neglecting the effect of the continuous spectrum component of the inverse scattering transform on the times of the order l/e. At most, the continuous component can manifest itself only in the formation of the tail of a certain "resonant" harmonic e ik~ of small ~ e amplitude behind the moving soliton (sometimes k is zero and the "tail" is a "'shelf"). The energy, accumulated in the tail on the times ~ 1/e is obviously of the order e, so that the solitonic discrete degrees of freedom dominate the dynamics. Thus, the perturbation in the first order may be considered to act only on the discrete solitonic degrees of freedom and the response of the system is the same as that of particles and oscillators [ 1 ]. For our particular system and perturbation the equations of motion of the breather coordinates, derived through the inverse scattering transform technique, can be taken directly from ref. [4], where the dc-driven damped sG breather was considered. The only modification needed is the substitution of ~ cos cot instead of E in the amplitude of the drive. Thus, we introduce, following ref. [4 ], the parameters 7 and V defining the breather solution (4) as (see expressions ( 1 . 3 ) - ( 1 . 6 ) of ref. [4] with 7=arctg(v/q))
v(7, V,O,xo)=-4arctg
ch[sinT(l_VZ)_l/Z(x_xo)]
tg7
].
(7)
The unperturbed breather (4) corresponds to constant 7 and V while 0 and Xo evolve as 0=cosT(1-V2)l/2t+0o,
xo=Vt+xoo.
(8)
Thus, Xo is the coordinate of the breather centre of mass, moving with constant velocity V, while 0 is the phase of the internal oscillations of the breather with frequency g2 (5). The parameter 7 defines, if the breather is interpreted as a bound state o f a kink-antikink pair [7 ], the degree of boundedness - for 7<< ~/2 the breather is strongly bound and has a small amplitude (flat profile), while for 7 close to ~/2 the breather is weakly bound and has a large amplitude (close to ~). Under the influence of perturbations the parameters 7 and 0 are slowly evolving, governed by the equations d7
dt--(l--V2)l/Z(4cosT)-lll'
dV
dO--COS 7 ( 1 - -
1/2
dt
V 2)
l/z_ (1 - -
V 2)
d--[=--(1--V2)3/2(4COST)-lI2'
dx° - V+ e ( 1 - V 2 ) I3 - V tg 7 I4 dt (2 sin 7) 2 '
VctgTI3+c°sZ7 ( l - VE)I,-I5 4 sin 7 COS27
'
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where the quantities
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Ii(7, V, O) are
GO
I1= f
chzsin[O-(V/tgT)z]
R(z) dz
c h 2 z + tg27 cos z [ 0 - ( V/tg 7)z]
i I2 =
shzcos[O-(V/tgy)z]
ch2z+tg27 cos2[0 - ( V/tg 7)z]
R(z) dz
--oo
i
13 =
z ch z sin[ O- ( V/tg y)z]
R(z) dz
ch 2z+tg27 cos 2 [ 0 - ( V/tg 7)z]
i 14---
zshzcos[O-(V/tgy)z] R(z) dz ch2z+tg2ycos2[O_(V/tgy)z]
--oo
15= ~
chzcos[O-(V/tgT)z] R(z) dz, V/tg ?)z]
ch 2 z + tg2~ cos 2 [ 0 - (
(10)
and R = e cos tnt + OtOVb/Ot. Eqs. (9), (10) are the same as eqs. ( 3 . 6 ) - (3.15 ) of ref. [4 ], taken for the varied amplitude of the drive e--,~ cos o~t. The quantity OVb/Otin (10) should be obtained from the derivative with respect to time o f the breather solution (7) for the unperturbed evolution of 0 and Xo with the further substitution of V(1- VZ) -t/2 cos7 (X-Xo) by z [4]. A further simplification of the system (9) can be achieved by the identification o f fast and slow time scales o f the dynamics and averaging over the fast oscillations in the spirit of the Bogolyubov-Krylov method. To start with, consider the system (9) for zero damping a = 0 and nonzero drive t ~ 0. Then it takes the form d7 - ~ cos dt dV
dt
--
oat ( 1 - V 2) 1/2(4 cos 7 ) - l J l
,
~(l-VZ)3/2(4cosy)-lJ2,
dO d t = c o s 7 ( I - V 2) 1/2_ E cos cot ( 1 - V 2) l/z[ Vctg 7 J3 "[- COS27 ( 1 -- Vz)J4 - J s ] (4 sin 7 COS2) )) --1 , dxo
dt
- V - e cos
oJt Vtg
7 (2 sin ~ ) - 2 J 4 ,
(1 la)
(llb) (llc) ( 1ld)
where the quantities Ji-J5 are the same integrals 11-15 (10) with R = 1. In analyzing the system ( 1 1 ) note first that the system of ode's ( 1 1 ) is not Hamiltonian. Secondly, the quantity Xo does not enter eqs. ( l l a ) - ( 1 lc), so that the last one (l l d ) can be omitted. In considering the system (1 l a ) - ( 1 lc) let us introduce the local coordinates 7'-7o = 71, V - Vo= VI. Since we expect to obtain the behaviour of trajectories of ( 1 1 ), similar to that o f the nonlinear Hamiltonian oscillator in the vicinity o f an isolated nonlinear resonance [ 13,14 ], let us single out the resonance harmonic Jt cos ~, ~ = lO-cot (l integer) from the Fourier decomposition of the functions J~, ,/2 by the periodic variable 0 and omit all the nonresonance ones. The quantities Jl, J2 can be easily expressed through one another by differentiating the expression arctg(tg 7 cos [ 0 - ( V/tg 7 ) z ] ~ ch z / with respect to z and integrating over z from - o o to ~ . This yields the relation Jz = ( V/tg 7)J~. The terms of order ~ in dO/dt ( 1 lc) can be omitted, since they lead to higher order corrections (in powers of e) to the trajectory behaviour. Thus, we arrive at d7 = e ( 1 - Vo2)1/2(4 cos 7o)- IJlz(7o, Vo) cos T dt
450
(12a)
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dV--¢(l-V2°)3/2(4c°sy°)-dtiVoj ~ It(7o, Vo) cos ~ ,
(12b)
d~ dt - 5o9(7, V ) ,
(12c)
where the detuning is 8o9 = l cos y ( 1 - V 2 ) 1/2_ 09. Note that the amplitudes before the resonance harmonic cos were taken in the zero approximation ? = Yo, V= Vo in the same fashion as is done in the "universal" description of nonlinear resonance [ 13,14 ]. The resonance condition 8o9= 0 as it stems from the last expression (12c), is (13)
l c o s 70 ( 1 - V02)1/2--o9 •
Thus, the resonance o f the harmonic l occurs only for o9< l. Note that the resonance frequency turned out to be not the breather frequency (5) in the laboratory frame, but contains an additional factor 1 - V 2 due to the Lorentz time contraction. For to smaller than unity all the resonances l = 1, 2... are present in the (~, V) plane. The first two resonance lines are shown schematically in fig. 1. The resonances with higher I approach the lines 7 = ~ / 2 and V= 1. Eqs. (12) can be simplified by introducing the new variable
p=(V-Vo)+ I I ° ( 1 - I I ° ) 2 ( 7 - ~ ' o ) ,
(14)
tg 7o
which stays constant in the process o f the dynamics (12). Then, the system (12) reduces to d7 d t =EKI(7o, Vo) cos ~ ,
(15a)
~d P - _o,
(15b)
~ =,~(ro,Vo)(7-yo),
(15c)
where K~ = ( 1 - V~ ) 1/2 ( 4 cos 7o ) - ~J~t( 70, Vo). In eq. ( 15c ), the r.h.s, o f ( 12 ) was expanded in deviations 7 - Yo up to quadratic terms from 7 = 70 for p = 0 (similarly to the nonlinear resonance description [13,14], p can always be considered zero, since a nonzero p corresponds just to a shift o f the point 7o, Vo). Thus, 2 is
0(8o9) a (8o9) V o ( l - v 2) 0 (_8o9) 2_ ~ p=o- ~7 v:co,,st+ -tg- -70 OV ~=~o.st"
(16)
The pair o f equations (15a) and (15c) is a Hamiltonian system, describing the nonlinear resonance in the
V o
Fig. 1. Nonlinear resonances l = 1, 2, defined by the condition ( 13 ). Arrows indicate oscillations of 7, V on the separatrix, with the amplitude ( 17 ). Dashed line 7= n/4 corresponds to the maximal energy on the resonance line. The dotted line t g T = F demarcates growing and decreasing 7 regimes.
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pendulum approximation. The resonance width ATr (the amplitude of oscillations of 7 on the separatrix) is given by AYr=x/ZelK11/I21 •
(17)
The first two resonance lines (13) l = 1, 2 (for the case ¢o< 1 ) with the resonance oscillations in its points in the direction p = const are shown schematically in fig. 1. At each line there exists a characteristic point, in which the direction o f resonance oscillations is tangent to the resonance line. This point is the point o f intersection of the line V = t g 7 (shown by the dotted curve in fig. 1 ) with the resonance line (13). For the motion with damping in the vicinity of a certain isolated resonance, we can average the r.h.s, of (9) over the "fast" phase 0, holding the "slow" phase 7j constant and omitting nonresonant harmonics. Since the same averaging was performed in ref. [4] for a dc-driving, and because the averaged damping terms in the r.h.s, of (9) will be the same, we can make use ofeqs. (3.32) o f ref. [4]. Finally, for the averaged slow quantities 7, V, 7j we obtain d7 d t = eKl cos ~ - - OtT, dV dt - E V ( I - V
(18a)
2)ctgTK~cos~P-aV(l-V
d ~ = 809. dt
2)(1-7ctg7),
(18b) (18c)
The trajectories o f the system (18) in the (7, V) plane in the absence of drive e = 0 are shown schematically in fig. 2. It is interesting to note [ 4 ] that while the parameter 7, defining the amplitude of the breather, damps t o zero, the velocity V does not. The dynamics under the influence of damping of the nonlinear oscillator in the vicinity of an isolated nonlinear resonance was considered in ref. [ 12 ] (see also ref. [ 15 ] ). It was demonstrated that for small enough damping the particle with the initial conditions inside the separatrix of resonance will move along the resonance line - a p h e n o m e n o n called "resonance streaming". In other words, the trajectory, being phase-locked to the driver initially, stays phase-locked in the course o f evolution under the influence of damping, though large-scale deviations in 7, V at times t ~ 1/or occur. The system ( 18 ) can be presented in a form more convenient for a description of resonance streaming [ 12,15 ] by introducing other variables, yielding d j = ~Kj - a T , dt
(19a)
V
452
Fig. 2. Damped undriven breather evolution, governed by ( 18 ) with ~= 0.
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@
dt - - ° i V ( l -
V 2) ,
(19b)
d~U =2p dt
(19c)
The direction of motion of a damping-pulled particle along the resonance is defined [ 12,15 ] by the direction of the component Ft of the decomposition of the damping "force" vector F on the components in the direction of resonance oscillations Fr and along the resonance line Ft. Thus, we can easily describe the motion of particles with the initial conditions inside the resonance separatrices under the influence of vanishingly small damping. Below the line tg 7= V the particles move along the resonance lines in the direction of decreasing y - for vanishing a up to ?=0. Above the same line the particles move upwards - increase their y and for vanishing ot reach the point V= 0. Thus, above the line tg y = V the breather evolves from the moving and lower one to the immobile and higher one. Below the line in the process of the transient phase-locking the breather accelerates, but decreases its height (for vanishing o~ to zero). We do not discuss the behaviour of the small inside-separatrix oscillations in the course of such motion, since we suppose the resonance width (17) to be small relative to n/2, rendering these oscillations irrelevant. Consider now the variation of the breather energy in the process of the phase-locked transient motion described above. For this, we observe that from expression (6) for the breather energy and formula (13) for the resonance line it may be easily shown that the breather energy along each resonance line is maximal at 7= n/4 (shown by the dashed line in fig. 1 ) and monotonically decreases in both directions from this point. Thus, we see that for initial conditions inside the resonance separatrix above the line tg 7= V the breather energy first grows (until 7 reaches the value ~,=n/4) and afterwards decreases, while below this line the energy monotonically decreases. The energy of the breather at the line tg 7 = V is E~, = (1/2~o)x/1 -~o 4/l 4 , at the point 7= n/ 4 it is Ebmax= l/2tO, and the energy of the stationary phase-locked breather is Ebs = x/1 --to2/l 2. These energies are ranged as Ebs < E~, < Eb . . . . Note also that in the process of the breather energy increase from E~, to Ebmax (initial condition at tg 7= V) the energy is drawn from the driver, but the role of dissipation is essential - in the absence of dissipation only a minor ( ~ x/~) amount of energy will be periodically exchanged between the driver and the breather. Let us consider now the limitations on e, a for which the effect takes place, since it is quite clear that for large enough damping the resonance will be destroyed - the particle will "fall out" of resonance. To find the critical value of damping, we substitute in (19a) the value of 7 at the resonance line 7 ~ 7r (V), which is defined by the resonance condition (13) (approximate equality means to the order of the resonance width ~ x/~). Making use of the relation d T J d t = ( d T J d p ) d p / d t and expressing the first factor from (13) to be dT~
x=-~p-
d7JdI~
=
OplOVIr=r,
V ctg 7 (1-- V2)(1 - V2 ctg27)
(20)
and the second factor from (19c), the solvability condition of the resulting equation yields (see ref. [ 15 ] for more details): ot
Kt
(~')cr ~ /~V(I_V2)_ 7 ,
(21)
where Kt, 7, x are taken at the resonance line 7=Yd V). Thus, the threshold (21) is different for different points on the resonance line (depending on V). The analysis shows, that most interesting from the point of view of observation of "soliton streaming" phenomena is the strongly underdamped regime a/E << 1, only the section of the resonance line, lying between zero and a certain (relatively small) y~ << 7t/2 will be overdamped - or~ E for this section will be smaller than the threshold (21 ), and no streaming at this part of the line occurs. The part of the resonance line above 7= y~ will be underdamped (above threshold), and the particle, starting with 453
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a p r o p e r initial phase ~ = n / 2 (see ref. [ 15 ] ) on the resonance line, will m o v e along it as described previously. The d a m p e d ac-driven sG system ( 1 ) is usually studied for c o m p a r a t i v e l y large p e r t u r b a t i o n c ~ 1 when the complex s p a c e - t i m e structure o f attractors proliferate [ 11 ]. We considered the from this point o f view trivial case ~ << 1, which for the u n d e r d a m p e d regime ot << 1 a n d a < E possesses a simple attractor o f phase-locked breather, but d e m o n s t r a t e d a nontrivial transient to this attractor breather evolution, in the course o f which the energy grows. It is supposed that this effect will strongly manifest itself in the statistical mechanics o f an ac-driven sG system for large relaxation t i m e and low temperatures. This conjecture is based on the recently found strong influence on the ac-drive o f the " t a i l s " o f the d i s t r i b u t i o n function a n d escape rate from the metastable potential well o f m a n y - d i m e n s i o n a l nonlinear oscillators subject to d a m p i n g a n d noise [ 15 ]. The statistical mechanics o f the sG system appears in a n u m b e r o f applications e n u m e r a t e d in the beginning. One o f the f u n d a m e n t a l processes in this k i n d o f p r o b l e m is the statistical birth o f k i n k - a n t i k i n k pairs, responsible for relaxation a n d n o n l i n e a r mass t r a n s p o r t in the system [ 7,16 ]. Since a k i n k - a n t i k i n k pair has nonzero rest energy Eo, the p r o b a b i l i t y W o f birth at each finite length a n d equilibrium density o f kinks and antikinks is exponentially small for low temperatures, kT<< Eo, i.e. W ~ exp ( - Eo/kT). The k i n k - a n t i k i n k p a i r is born [ 7 ] from the large-amplitude b r e a t h e r ~,=n/2 with zero velocity. We suppose, a n d this presents itself as a field o f further research, that in a certain leading a p p r o x i m a t i o n for the probability o f low t e m p e r a t u r e ac-driven k i n k a n t i k i n k birth, the d y n a m i c s o f d a m p e d , ac-driven a n d noisy breather can be uncoupled from the continuous degrees o f freedom. F o r such an oscillatory finite degree o f f r e e d o m system we can apply the theory [ 15 ], and expect therefore the b r e a t h e r to reach the state 7 = n / 2 , F = 0 , with the energy E o = 1, a n d to have to m o v e to the p o i n t on the resonance line tg 7 = I / w h e r e the streaming starts (the energy E ' there is smaller than Eo) and then m o v e along the resonance line up to the p o i n t y=n/4 (where the energy is larger than E ' ). This will exponentially strongly increase the probability o f k i n k - a n t i k i n k pair birth: W ~ exp ( - G~kT); G < Eo, Eo- G ~ Eo. Such a regime is expected for the p a r a m e t e r s ~<< 1, 09< 1, ot<~, kT<
References [ 1] D. Kaup and A. Newell, Proc. R. Soc. A 361 ( 1978 ) 413. [ 2 ] B. Malomed, Physica D 27 ( 1987 ) 113. [3] R. Bullough, P. Caudrey and H. Gibbs, in: Solitons, eds. R. Bullough and P. Caudrey (Springer, Berlin, 1980). [4] V. Karpman, E. Maslov and V. Soloviev, Zh. Eksp. Teor. Fiz. 84 (1983) 289. [ 5 ] A. Scott, in: Solitons and condensed matter physics, eds. A. Bishop and T. Schneider (Springer, Berlin, 1978 ). [6] F. de Leeuw et al., Rep. Prog. Phys. 43 (1980) 689. [7] A. Seeger, in: Lecture notes in physics, Vol. 249. Trends in applications of pure mathematics to mechanics, Bad-Honnef, 1985 (Springer, Berlin, 1985). [8] P. Lomdahl and M. Samuelsen, Phys. Lett. A 128 (1988) 427. [9] A. Mazor (Bin-Mizrachi) and A. Bishop, Physica D 27 (1987) 269. [ 10] M. Fordsmand, P. Christiansen and F. If, Phys. Lett. A 116 (1986) 71. [ 11 ] A. Bishop, M. Forest, D. McLaughlin and E. Overman II, Physica D 23 (1986) 293. [ 12] J. Tennyson, Physica D 5 (1982) 123. [ 13 ] B. Chirikov, Phys. Rep. 52 ( 1979 ) 263. [ 14 ] M. Lieberman and A. Lichtenberg, Regular and stochastic motion (Springer, Berlin, 1983 ). [ 15 ] A. Gerasimov, Phys. Lett. A 135 ( 1989 ) 92; Institute of Nuclear Physics Preprint 87-100, Novosibirsk ( 1987 ). [ 16 ] S. Trullinger, M. Miller, R. Guyer, A. Bishop, F. Palmer and J. Krumhansl, Phys. Rev. Lett. 40 ( 1978 ) 1603.
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