Many-particle effects in nearly integrable systems

Physica 24D (1987) 125-154 North-Holland, Amsterdam

M A N Y - P A R T I C L E E F F E C T S I N NEARLY I N T E G R A B L E S Y S T E M S Yuri S. K I V S H A R * Physico-Technical Institute of Low Temperatures, Kharkov, Lenin's prospect 47, 310164, USSR

and Boris A. M A L O M E D Institute for Biological Tests of Chemical Compounds, Staraya Kupavna, 142450, Moscow district, USSR

Received 14 February 1985

For the sine-Gordon and nonlinear Schr~Sdingerequations with conservativeperturbations three-particle effectsin collision of a fast soliton with two more slow ones are considered in the first order of the perturbation theory. The momentum and energy exchanged between the fast soliton and slow ones are calculated. For the latter system the three-particle contribution to the perturbation-induced phase shift of the fast soliton is calculated too. The effects vanish when the overlapping between the slow solitons is absent at the collision moment. In the former system inelastic interaction between the fast kink and a weakly bound breather is considered. The threshold value of the breather's binding energy admitting breaking the breather into a kink-antikink pair is calculated. Analogously, the inelastic collision of two weakly bound breathers is considered. For the collision of two "relativistic" sine-Gordon kinks the energy radiated under the action of the conservativeperturbation is found and is demonstrated to be asymptoticallyindependent on the relative polarity of the kinks.

1. Introduction The study of systems exactly integrable by means of the inverse scattering transform (IST) is far advanced at the present time [1]. However, m a n y physically interesting effects (for example, inelastic or many-particle interactions) in principle are absent in these systems. Therefore, it is of great interest to study such effects in nearly integrable systems, i.e. those differing from exactly integrable ones by small perturbation terms which break exact integrability. The principal ideas of the perturbation theory (PT) based on IST were formulated in [2-4]. It should be emphasized that the methods of this theory are relevant just for studying the mentioned effects, while more simple problems (concerning affects of a perturbation upon one-soliton dynamics or interaction of slightly overlapping solitons) can be effectively solved by means of the "direct" perturbation theory [5-8]. Inelastic interactions of solitons in the nonlinear Schr/Sdinger equation (NSE) and sine-Gordon equation (SGE) with small dissipative or conservative (Hamiltonian) perturbations were considered by one of the authors in [9] (the important numerical results in this field were recently obtained in [10]). In this connection we may also cite the papers [11, 12] where the perturbation-induced decay of a SGE breather (bound state of two solitons) into a pair of free solitons and the inverse process were treated. The present p a p e r is devoted to the investigation of many-particle effects in the same two equations with conservative perturbations. These effects are of principle interest for, e.g., field theory problems [13] as well * The author to whom correspondence should be forwarded.

0167-2789/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

as for elucidating the very notion of a nearly integrable system. In the second section we consider the collision of a fast kink (soliton) with a pair of more slow kinks in SGE with the perturbation proportional to the sine of the double argument [9, 14]. As is well known [1], in absence of perturbation the interaction of solitons results only in their phase shifts, the shift due to the collision with a pair of solitons being equal to the sum of partial shifts resulting from the separate collisions with each soliton of this pair what is commonly referred to as absence of many (three)-particle effects. Besides, it is known [9] that the collision of two solitons in presence of a conservative perturbation does not cause energy and momentum exchange between the solitons in the first order of PT. In the second section we demonstrate that in the first order the exchange is possible in the three-particle situation: dealing with the collision of a fast kink with a pair of two more slow ones we explicitly calculate the changes of velocities of the three "particles", i.e. the changes of their energies and momenta, the energy and momentum conservation laws being met. The three-particle nature of this effect manifests itself in the dependence of the exchanged energy and momentum on the distance between the slow solitons at the moment of their interaction with the fast one. The effect vanishes when the distance tends to infinity. In the third section we investigate three-particle effects (TPE) for NSE with polynomial Hamiltonian perturbation. Firstly, we calculate the energy and momentum exchange for the same process as treated above. As the NSE solitons have two independent parameters, amplitude */ and velocity - 4 ~ [1], on contrary to one-parameter SGE kinks, we accomplish this calculation for the two cases when the result may be obtained in an explicit form: when the amplitudes */1 and */2 of the two slow solitons are equal, and when */x << */2- Conservatively perturbed NSE conserves one more elementary integral of motion, namely the charge (or "number of particles" [1]), along with energy and momentum. We show the charge exchange induced by the three-particle interaction to be absent in the first approximation. Though the energy and momentum lost by the fast soliton are equal to those acquired by the slow ones, the change - 4 A ~ of the fast soliton's velocity - 4 ~ proves to be much smaller than the changes -4A~l, --4A~2 of the slow solitons' velocities - 4~1,2, f~zif~-f2af2-fa~

(f~,f2 <<~).

(1)

Retaining the quantities of the lowest order with respect to the small parameter ~-1 we may neglect A~ and calculate the phase shift of the fast soliton A¢. We demonstrate that there is a perturbation-induced three-particle contribution A¢123 to the phase shift, A~123- ~-1 (while A ~ - ~-2). This TPE, as well as others, vanishes when the slow solitons' overlapping is absent at the moment of interaction with the fast soliton. Note that the NSE solitons are equivalent to the small-amplitude SGE breathers in the sense pointed out in [4, 15]. Therefore comparing our results for SGE and NSE one concludes that TPE for the two types of "particles", i.e. kinks and (small-amplitude) breathers, in a conservatively perturbed SGE are essentially different. In the fourth section we consider the collision of a fast kink with a quiescent breather in SGE with the same perturbation as in section 2. The interaction stipulated by this collision may be regarded as three-particle since one may view the breather as a kink-antikink bound state. First we deal with the case when the breather has a small amplitude, i.e. the coupling between the kink and antikink inside the breather is strong. In this situation we investigate TPE analogous to that treated in the second section, i.e. the change of the fast kink's velocity. Besides, we calculate the changes of the breather's amplitude and velocity. These results are exponentially small in the breather's amplitude (cf. [14]). Then we proceed to the interaction of the fast kink with a weakly bound breather. Following the lines of [9] we investigate the possibility of breaking the breather into a kink-antikink pair. The threshold binding energy for the

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127

breaking process and the relative velocity of the breather's "splinters" are calculated, the threshold energy being much smaller than in the analogous problem for a dissipative perturbation [9]. The obtained results also enable us to consider the fusion of a kink-antikink pair with small relative velocity into a breather as a result of the collision with the fast kink. The fusion is as well possible on account of the energy emission in absence of the fast kink [9], but, on the contrary, the present process is nonradiative. The fifth section is devoted to the perturbation-induced interaction of two fast breathers. In the sense explained above this interaction may be regarded as four-particle. The break-up of a weakly bound breather due to the collision with small-amplitude fast one in the presence of the conservative perturbation was considered in [9]. In the present paper we deal with the collision of two weakly bound breathers. We demonstrate that, depending on the values of the internal oscillation phases of the breathers at the collision moment, the three channels of the interaction are feasible: the both breathers survive the collision; one breather survives, while another breaks into a kink-antikink pair; and, at last, the both breathers break. Finally, in the sixth section we consider the effect which is not a three-particle one, but which also is of interest from the viewpoint of investigating the perturbation-induced interaction of solitons. This is the energy emission accompanying the collision of two kinks in the conservatively perturbed SGE. In [9] this problem was considered for the case of small relative velocity V. In the present paper we deal with the opposite case of large relative velocity, v -- (1 - V 2) -x/= >> 1.

(2)

We calculate the emitted energy Eem and its spectral composition. It is interesting that the term of the lowest order in I,-x ( _ p-3) does not depend on the relative polarity (topological charge) of the two kinks, and the dependence arises only in the next term ( - ~,-5). This situation is opposite to that in the small relative velocity case where Eem crucially depends on the relative polarity [9].

2. Three-particle interaction of kinks in conservatively perturbed SGE In this paper we shall deal with the perturbed SGE [9, 14] u , - Uxx + sin u = c sin2u,

(3)

the subscripts " t " and " x " standing for the differentiation with respect to the corresponding variables, u being a real wave field, and c being a small real parameter. The kink solution to the unperturbed equation (3), i.e. that with c = 0, is [1] uf(x,

t) = 40 arctan (e-Z),

z = l'(x - Vt),

(4) (5)

where o is the kink's topological charge, V is its velocity, and v is determined in (2). We shall consider the interaction of the fast kink (4-5) with the pair of two more slow kinks described by the unperturbed

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Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

solution

Usl( x ~ t

)

= 4 arctan{ 0"' exp[-7,1 "~- D/21)d + 0"2e x p [ - z 2 - D/2u2] }

z. . v.(x .

.V.t),.

1). (1

V2) -1/2 ,

n = 1,2,

(6) (6')

612 =- [I)1(1 + Vl) -- 1)2(1 + V2)]2[p1(1 q- Vl) + 1)2(1 + 1"12)]-2, where

Vl= w, v 2 = - w

(w>o)

(7)

are the velocities of the two kinks, o 1 and 0 2 are their topological charges, and D (which may be sign-changing) is the distance between them determined as the difference of the coordinates of the first and second kinks. The evolution of a kink's velocity under the action of the perturbation in rhs of (3) is determined by the well-known equation [2-4] d

.

, c

T/ (lhn) = 1-4C.(t)L

oo

dx sin (2u(x, t))Q.(x, t),

(8)

where

['~¢n(1)(X, t)] 2 - [~/P(2)(X, l)l 2,

Qn(x,t)=

A n ~- 1(1

+

Vn)1/2(1-- Vn)-1/2.

(9) (10)

Here ~(l'2)(X, t) stand for the two components of the Jost function (related to the spectral problem employed for integrating unperturbed SGE [1]) taken at the complex spectral parameter value corresponding to the kink: ~ = iX., the subscript " n " designating the kink's number,

Cn(t ) = b(iXn)/a'(iX.) = - 2a.X.exp [(X. + 1/4h.)X(n °) + (X. -- 1 / 4 X . ) t ]

(11)

is the standard amplitude expressed in terms of the scattering data a(X) and b(M, x~.°) being the kink's coordinate at the moment t = 0 [1]. For the three-particle problem explicit results can be obtained in the case 1-V 2<
2.

(12)

Besides, in this section we shall assume the more slow kinks (6)-(7) to be relativistic too, i.e. 1 - W 2 << 1.

(13)

As is shown in [9], the condition (12) provides for the "splitting" of the full wave potential:

U = Uf "4- Usl -1- ~(1)-1)

(14)

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129

(p being determined in (2)). Analogously, the condition (13) provides for the "secondary" splitting of the wave potential (6) into the sum of two kink potentials: Usl = (u~l)l + (U~l)2.

(14')

As for the Jost functions, it (9) essential for (8) does split, "one-particle" approximation Using these simplifications [1],

g'(2)(x, t)

is demonstrated in [9] that they do not split, but their quadratic combination i.e. under the conditions (12) and (13) one may insert Q,(x, t) into (8) in the for each n. together with the standard expression for the Jost function of the fast kink

2cosh z

[ ioexp ( - z / 2 )

'

.

and similar expressions for the more slow kinks, we obtain from (8) the evolution equations for the fast kink,

dv -d-7

,

=-poj_ 4

£c~

dx ~o

sin[2u(x,t)] cosh z

(16)

and for the pair of more slow ones, ~t"

' f_co sin [2u(x, t)] = 4°"X" ~0dx cosh [z, + ( - 1 ) " D / 2 1 , , ] '

(17)

where ~ , ( t ) are expressed in terms of the velocities Vl(t ) and V2(t ) of the two kinks according to (10). The changes of these parameters AX 1 and AX 2 due to the collision can be found by direct integrating (17): coo d~,.

AX~= J_oo--~-dt.

(18)

The expressions (18)-(17) contain spatial and time integrations. To perform them explicitly we use (14) and, after simple transformations, insert the perturbation sin(2u) into (17) in the form sin (2u) = sin (2Ur) cos (2us1) + sin (2Usl) - 2 sin 2 uf sin (2us,).

(19)

The second term from rhs of (19) that does not depend on uf gives no contribution to (18). Actually this is equivalent to the above assertion that in the first order of PT the collision of two kinks in the presence of a conservative perturbation does not change their velocities [9]. As to the two remaining terms, their contribution takes a rather simple form since, in the first approximation, the spatial integration regards only the fast dependence on x related to u f, and one should set x --- t in all remaining functions of x (the same trick was employed in [9]). Using (4), it is easy to verify that

f?oo dx sin(2uf(x, t))=0,

(20)

f- ~ d x sin 2 uf(x, t) = 8/(3v).

(21)

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130

Thus we see from (20) and (21) that the integral over dx is dominated by the third term in rhs of (19). Using (14'), we can perform the further transformation of this term, sin (2u~,) = sin (2u~,)z cos (2u~1)1 + sin (2u~1)1 - 2 sin2 (Us1)2 sin (2Usl)l

(190

(cf. (19)). According to (6)-(6'), (us0 . is a function of z,, n = 1, 2. On the other hand, it is seen from (6), (7) and (10) that setting x = t after performing the spatial integration turns z, into t/(2X,). According to (10), (7) and (13), X2 << ½<< X1. Thus the fast dependence on t is related to (usl)2 while (us1)1 is a slow function of time. Therefore, by analogy with the above, and with regard to the form of the integrand in (17), it is clear that, if we calculate AXx, the time integration in (18) is dominated by the third term in rhs of (19'). So, substituting (19), (19') and (20)-(21) into (17)-(18), we arrive at the final expression 128 AX 1 = --~-u, f ( 8 ),

(22)

where the odd function f ( 8 ) = tanh 8 sech2 8 (1 - 2 sech2 ~ ),

(23)

and 8 = D/2X 1

(23')

characterizes the degree of overlapping between the more slow kinks at the moment of the collision with the fast one. For the second kink the analogous consideration results in

8 ,2~_4f(6). 9v

AX2 -

(24)

As, according to (13), ?~1 >> 1, we see from comparing (24) and (22) that lAX2] <<

IAXll.

(25)

This discrimination is connected with the fact that the first kink has the velocity coinciding in direction with the fast kink's velocity while the second kink's velocity is opposite to that of the fast kink, i.e. with regard to (13), the effective time of interaction with the fast kink is much larger for the first slow kink than for the second one. The change of the fast kink's velocity can be easily found with the aid of the energy and momentum conservation. Indeed, the energy AE and momentum Ap transferred by the fast kink to the more slow ones are 2

AE=8

Y'~ (1 - 1/4~2,,)AX,,~ 8A~.l,

(26)

n=l 2

A p = 8 Y'~ (1 + 1/47~2)AX,-- 8AX1 = A E

(27)

n=l

(we have used (22), (24) and (25) for neglecting the second terms in (26) and (27)). The change o f the fast

YgS. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrablesystems

kink's velocity can be found from (26): Av = - A E / 8 aj,_

= -AXx,

131

i.e.

128 , f ( 8 ) .

(28)

To verify the result (28) we have also derived it directly from (16), without using the conservation laws. As to the change of the velocity V itself, it can be easily expressed in terms of A~,,

A V = AI,/~, 3. Analogously, one can express the changes of the slow kinks' velocities AV1 and AV 2 in terms of AX 1 and AX 2. It is essential that the function f ( 8 ) determined in (23) vanishes when 8---, oo, i.e. the overlapping between the slow kinks is absent at the collision moment. This is the direct manifestation of the three-particle nature of the considered effect. It is also interesting to note that this function is odd, i.e. depending on the sign of parameter 8, the fast kink may both lose and acquire energy due to the collision with the pair of the overlapping slow kinks. At last, as is seen from (22)-(24) and (28), it is worth noting that the considered TPE does not depend on topological charges of the involved kinks.

3. Three-particle interactions of solitons in conservatively perturbed NSE As is well known, there is one more type of localized stable "particles" in SGE besides the kinks, namely breathers [1]. Consideration of TPE concerning breathers encounters obvious technical difficulties in a general case. Therefore we shall resort to the case of small-amplitude breathers which, pursuant to [4, 15], can be reduced to NSE solitons. We take conservatively perturbed NSE of the form [9, 16] iu, + uxx + 21u12u = clul2Nu,

(29)

being a small real parameter, N being integer. The perturbation in rhs of (29) is nontrivial provided N > 2. We shall perform calculations primarily for N = 2. On the other hand, if we derive (29) from (3) as the equation for small-amplitude SGE breathers, we should substitute into (3)

u ( x , t) = e x p ( - i t ) U ( x ,

t) + exp ( i t ) U * ( x , t)

[4, 15], the asterisk standing for the complex conjugation. The perturbation term in (29) with N = 2 arises as the third nonvanishing term of the expansion of sin u and sin2u in powers of U. In the same time it is clear that the expansion of the basic term sin u and that of the perturbing term sin2u irremovably differ beginning from the coefficient before the third term, i.e. the perturbation in (29) adequately models the perturbation from (3). From the viewpoint of NSE proper, the physical sense of the perturbation with N = 2 also is obvious: it takes into account the second term of the expansion of dispersion in powers of the wave field amplitude a = ]u 12. The unperturbed soliton solution to (29) is [1] u(x,t)

[ - 2i~x - 4i(~ 2 - */2)t- iq,] 2i~ eXPcosh [2,/(x + 4 ~ t - x(°))]

(30)

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Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

Here 71 stands for the soliton's amplitude and - 4 ~ is its velocity; ¢ and x t°) are two arbitrary phase constants. We shall consider the collision between the fast soliton with the velocity - 4 ~ and amplitude ,/ and the two overlapping more slowly solitons possessing the velocities -4~1,2-- T-4× and amplitudes */1 and */2- We assume (cf. (12))

>> X,

~'~ >> X*/1,

X*/2

(31)

and (cf. (13)) x >> */1, */2.

(32)

The equation for evolution of the soliton's parameters ~ and ~/ under the action of the perturbation is analogous to (8) d~7(~. + i71.)=

-C.(t)f~°° dxlu(x,t

)14{ u( x , t )[ q'tn(1)(X, /)]2 _~_U * ( x , t )[ ~ 2 ) ( x , t)]2}.

(33)

Here X/'t(l'2)(X, t) are the Jost functions taken at X = ~. - ~. + i*/., and (cf. (11))

C.(t)

= 2i*/. exp (2,/.x~.°) + iq~. + 4i(~2 - ,/2)t - 8~/.~.t).

Besides (33), we shall need the evolution equation for the phase parameters q~. and x~°) defined in (30) [2-41,

dr.

dt - 2i'yn*/nC

oo

dxlu(x, t)14[u(x, t)G~l)(x, t) + u*(x, t)G~2)(x, t)],

(34)

where ~/. = exp (2*/.x.~°, + i¢.), and

Here ~(k)(x, t; X) and ~(k)(x, t; ~), k = 1, 2, are the components of the two Jost functions used in IST [1]. Proceeding to the calculation of the collision-induced changes of the solitons' parameters, it is necessary to take into account that the conditions (31) provide for the splitting of the field potential analogous to (14). The condition (32) provides for the "secondary" splitting (14') of Usl(X,t) into the sum of two solitonic potentials (30) with the velocities -4~1, 2 = -Y-4x, amplitudes */1, */2, and the phase constants q~,2, x~°)2. As for the Jost functions, under the conditions (31)-(32) they directly split into the "one-particle" expressions [1]

( (1)( t)

(iX.x)(exp(-z.) /

exp = 2cosh(z.)

exp(-i

.)J'

(35)

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133

where

z. = zn(x,t ) = 2,/.(x + 4 ~ . / - ~°)),.% -

(36)

~. = ~.(x, t) = 2~.x + 4 ( ~ - */2.)t + ~.,

(37)

in contrast with the case of SGE where only the combinations (9) were splitted. Substituting (35)-(37) into (33), (34) yields the evolution equations for the parameters of the slow solitons (n = 1, 2), cf. (17),

dxlu(x,t),4u(x,t)exp [i~.(/, t)]cosh2[z.(x,t)] sinh [ z . ( / , t ) ] } ,

d,.dt ='*/" Im{ dt = - ~ % R e dx{.°)

{/:

dxlu(x, t)14u(x, t) exp [iff.(x, t)] cosh [ 2. t)]

_,Re{f _~~ dxlu(x,t)14u(x,t)

dt

=

dt

eIm

/

(39)

'

xexp[i~"(x't)] } cosh[z.(x,t)] '

dxlu(x,t)14u(x,t) 1-2n"xtanh[z"(x't)l cosh [ 2. (x, t)]

(38)

(40) exp [i~'n(x, t)]},

(41)

and the same equations for the fast soliton's parameters (n = 3) where

n~=n, z3(x,t)=2n(x+4~t-x(°~), ~'3(x, t) = 2~x + 4(42 - "O=)t + q~. The expressions for the full collision-induced changes of the parameters A~1,2 ~---f~oo dt (d~l,2/dt) etc. (cf. (18)) imply, as in the preceding section, the two integrations. Following [9], it is convenient to perform the time integration first. To this end we insert (14) into the perturbation P[u] = luf4u (or P*[u] = lul4u *) inside (38)-(41) to obtain some different terms (cf. (19)). Selecting these terms as in [9], it is easy to realize that the equations (38)-(39) for the fast soliton are dominated by the term Pf[u] = 31usll4uf,

(42)

while those for the slow solitons are dominated by the other term P s i [ U ] ~-

61uelZlusllZusa.

(43)

In the first approximation the time integration concerns only the fast dependence on t related to uf (see (30)). If, e.g., we calculate changes of the slow solitons' parameters, this integration reduces to (cf. (20), (21))

f~_ lull 2 dt = 71/~, oo

and one should set t = - x / 4 ~ in all remaining functions of time.

(44)

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Yu.S. Kivshar and B.A. Malomed / Many-particle effects in nearly integrable systems

To proceed to the spatial integration we insert (14') into (42) and (43). Then the "slow" part of the, e.g., perturbation (43), lUsd2Us], is decomposed into a sum of some terms. As it can be seen from (38), (39), for the n th slow soliton (n = 1, 2) dominating is the term (lUsll2Us~), = 2(IG]12)3_n(Us~),,.

(43')

Inserting (42)-(44) and (43') into (38), (39) one can cast the final expressions into a simple form in the two particular cases: if n2 >> 7t

(45)

and if 72 = ~h =- 0.

(46)

In the case (45) the results are Af 1 =~

967~727 f g1(8), 967]7

zo~

=

A~=-,

1927372x 3 ~ gx(),

,471 = anz = 47 = 0,

(47) (48)

(49) (50)

where the odd function

g1(3) =

tanh3- sech23

(51)

(cf. (23)). As well as in the preceding section (see (23')), the parameter 3 characterizes the overlapping between the two slow solitons with centers located at the collision moment (t = 0) at the points x[ °) and X (20);

3 = 271( x(2°)

-

x~°)).

(51')

The results (47)-(50) can be readily verified to satisfy all the elementary conservation laws, the energy AE1, 2 and momentum AP1, 2 acquired by each of the slow solitons being AE x = 3271x Af I = 3 X 21%/]a?27x,(-1gl(3),

(52)

A E 2 = -32~12× A f 2 = AE1,

(53)

ae,

(54)

=

- 87,

=

- 3 x

AP2= --8Tl2af2= - a P 1.

(55)

According to (50), the third elementary integral of motion of each soliton, i.e. the charge N, = 47,

Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

135

("number of particles" [1]) does not change. Besides, we see from (55) that the fast soliton does not exchange the momentum with the slow ones. Comparing (47) and (48) we see the change of velocity to be much larger for the first soliton, i.e. that with the small amplitude (see (45)), than for the second one (cf. (25)). This is quite natural because the soliton's width is - ~-1, i.e. the first soliton is much wider, and the time of its interaction with the fast soliton is much larger too. Nonetheless, the changes of the energies of the both solitons are equal, as is seen from (53). Another important note is that, pursuant to (49), the change of the fast soliton's velocity is of the second order in ~-1 (cf. (47), (48) and (1)). Since this quantity is small it seems natural to calculate the perturbation-induced phase shifts of the fast soliton, using (40) and (41), x
0;

Aq~ = 32,~-1(~/2~/2 + ~/3/2 + 3~/2~2 sech28).

(56)

The results (47)-(55), as well as those (22)-(28) from the preceding section, are obviously three-particle. In the same time, the phase shift (56) contains, on a level with the three-partMe contribution A~123 vanishing at 3 --+ ~ , the perturbation-induced two-particle contribution from interaction of the fast soliton with the second slow soliton (it dominates over the contribution from interaction with the first slow sofiton due to (45)). We have also calculated the quantity a~x23 for the perturbation with arbitrary N > 2 in (29), N-2 N-k-1

At/~123 22N-lC~ -1 E k=0

E

d(N)'n2k'n2j-l~2(N-k-J)

~kj

'l

'12

ffl

"

(sech~) 2(N-k-j),

j=l

where N

N

(2k)!!(2j ( 2 k + 1 ) ! ! ( 2 j - 1)!!"

N o w we proceed to the case (46). As above, in this case the amplitudes do not change (see (50)), while the velocities change as follows: =

19204x

• .

9603~/ ~,,, A~I = --A~2 = C - - - - ~ g 2 [ o ) ,

(57) (58)

where the parameter 3 (characterizing the slow solitons' overlapping at the collision moment) this time is -- 20(x~20) - x~°)) (cf. (51')), and the odd function g2 (3) = [3(3 - tanh 3) coth 23 - 3 ] cosech 23

(59)

(cf. (23) and (51)). The energy exchanged between the fast soliton and the pair of slow ones can be easily calculated with the help of (57)-(59) similarly to (52), (53), while the momentum exchange is absent as above (see (55)). At last, the change of the charges is absent too.

136

Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

Finally, the phase shifts for the fast soliton can be calculated in the present case by analogy with (56), A x (°) = 0,

Adp = 6 4 , 0 ~ - 1 ( ~ 2 + 0 2 / 2 + 30293(~)),

(60)

where g3 (~) = cosh 8. cosech38 (8 - tanh 8). It seems natural to confront the results (57)-(60) obtained upon the condition (46) with those of [9]. Indeed, in terms of the present paper, [9] treated of TPE in collision of the fast soliton (30) with the NSE breather that may be regarded as the system of two quiescent solitons (i.e. K = 0 in the present terms) with the close amplitudes: (7"/1 -- ~2) 2 << (TJ1 + T/2) 2"

(61)

The condition (61) evidently simulates (46), while the condition x = 0, opposite to (32), stipulates the difference between the two problems (in particular, the breather configuration essentially differs from the superposition of the two solitons, and the two-particle Jost functions do not split into the "one-particle" ones (35)). One of the results of [9] is that TPE, except for the phase shift (60), are absent in the first approximation (analogous to that dealt with here) for a perturbation from (29) with any N. Such a qualitative difference between the two problems may be interpreted as the manifestation of the "coherent" nature of the NSE breather. In this connection it should be emphasized that for the trivial "perturbations" with N = 0 and N = 1, which do not break the exact integrability of NSE (29), the consideration analogous to that developed above yields zero instead of (47)-(49) and (57)-(58). From the formal viewpoint, the reason is that TPE may be generated only by those terms from the full perturbation which involve both u r and Usl. However it can be readily verified that in the case of the conservative perturbation, i.e. real e in (29), the contribution from the terms of the type (42), (43) exists only for N > 2. The oddness of the functions (23), (51) and (59) also is a consequence of the conservative character of the perturbations treated in this paper. Nontrivial results were obtained in [9] for the dissipative perturbation corresponding to imaginary e in (29). Confronting these results with (57)-(59), we notice the essential difference between TPE caused by dissipative and conservative perturbations: the latter perturbations result in change of velocities while the former ones affect amplitudes. To conclude this section we note the strong difference between TPE for "relativistic" SGE kinks, treated in the preceding section, and TPE for "nonrelativistic" small-amplitude breathers equivalent to NSE solitons.

4. Perturbation-induced interaction between SGE kink and breather

In the two preceding sections we have separately considered TPE for kinks and small-amplitude breathers (or, equivalently, NSE solitons). Now it is natural to consider the effects involving both types of the "particles". However, the investigation of the interaction between a kink and two breathers or a breather and two kinks requires too lengthy calculations. In the same time, the more simple interaction between a kink and a breather may also be regarded as three-particle since a breather is a kink-antikink bound state.

Y~S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

137

Within the framework of the same perturbed equation (3) we shall consider the interaction of the fast kink (4)-(5) with a quiescent breather described by the unperturbed solution [1] U br ( X,

t)

= 4

(62)

arctan [ tan #. sin (cos/x. t - q~) sech ( x sin/, )],

/~ being the breather's amplitude (0
(63)

4.1. The small-amplitude breather First we shall dwell on the case when the breather's amplitude is small: /, << 1, i.e. (62) simplifies to (64)

Ubr(X, t) = 4/* sin (t -- ~) sech (~tx), and (63) takes the form ~kl,2 ~-

+~'+ih"=

(65)

+_½ +i/*/2.

The evolution of the breather's parameters is determined by the above equation (8) where i~, should be substituted by )~1,2 from (65), and (cf. (11)) Cl(t ) -- 2Xl/xex p [i(q~- t)],

C2(/) = C~'(t).

(66)

The standard condition 1 - V 2 << 1 for the fast kink's velocity provides for the splitting of the field potential (cf. (14)) [9]

U=Uf+Ubr+O(V-1).

(67)

Under the same condition, the quadratic combination (9) of the Jost functions is demonstrated in [9] to split into that made up of (15) and corresponding to the fast kink, and that corresponding to the breather and made up of the functions g'~l)(x, t) = a -z exp ( - # x / 2 ) [(1 + exp ( - 2/xx)) - i/,~/*(~/- 7*) exp ( - 2/~x)],

(68)

a//(2)(X, t) = A -lexp ( -- 3/~x/2)[b/* (1 + exp ( -- 2/~x)) --/~('/-- "t*)],

(69)

where A = 4cosh2/xx,

7 = exp [ i ( t - ~)].

The breather Jost functions (68), (69) are taken at the point h to X = X 2 are[l] ~/.t2(1) = [X/tl(1)]* '

~/t2(2) = _ [~/t(2)]*

= h 1

(see (65)); the functions corresponding

(70)

Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

138

So the evolution equation for the fast kink's velocity retains the form (16) where u(x, t) should be substituted by (67), and after inserting (66) and (68)-(70) into (8) the evolution equations for the breather's parameters' and "defined by (65) take the form dX'

N = -(c~/8)

£o dxsin[2u(x,t)][~sech(~x)cos(t-¢)-tanh(~x)sech(~x)sin(t-¢)], (71)

d~~" -(c~/8)

f o~

dxsin[2u(x,t)][sech(~x)cos(t-¢)+tanh(.x)sech(~x)sin(t-,)].

(72)

After inserting (67) into (71), (72) the time integration in the expressions for AX' and AX" is dominated by the same term that was principal in (19). The final expressions are A ~, = -- (64/3) ~rcu-1 sin (2¢) cosech (~r//~),

(73)

nX' = - (nv)/(4t~),

(74)

nX" = -n~/2.

The velocity W acquired by the breather due to the collision, and the change A# of the breather's amplitude are readily expressed through (74) with regard to (65), W = 2A~',

A ~ = 2A~".

(75)

It is easy to verify that the results (73)-(75) meet the energy and momentum conservation. The energy A E and momentum Ap transferred by the fast kink to the breather are AE = 32(A•" --/~AX') - 32/~AX',

(76)

Ap = 32/~ (AX' +/xA~") -- 32/~A~' = AE

(77)

(A E and A p ought to approximately coincide because the fast kink is relativistic). As we see, the above results (73)-(77) essentially depend on the value of the breather's phase , at the collision moment (the analogous dependence occurs in [9]). The simple analysis of (64) and (73) reveals that the fast soliton loses energy provided sin(2¢) > 0, i.e. if the kink and antikink inside the breather are moving to meet each other, and it gains energy in the opposite case. It is interesting to note that, as is seen from (28), the same is true for the collision of the fast kink with the pair of more slow kinks if the relative distance between them 181 < In(1 + ~ - ) (for the small-amplitude breather (64) the analogous quantity always is _< 1). Another noteworthy fact is the exponential smallness of the results (73)-(77) in the parameter/~-1 (the same was revealed in [14] when calculating the energy radiated from the breather (64) under the action of the same perturbation as in (3)). As it is seen from (71), (72), the reason for this smallness is that the breather's size 1 is large: 1 - / ~ - 1 (see (64)). If we consider the interaction of the fast kink with a nonsoliton (dispersive) wide wavetrain, which is locally similar to the small-amplitude breather [9], we shall analogously obtain the results exponentially small in the wavetrain's width. 4.2. The weakly bound breather Now we proceed to the case opposite to that treated in the preceding section: - ~r/2 - / ~ << 1.

(78)

Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

139

The condition (78) implies the breather's binding energy [1] E b = --8~ 2

(79)

to be small. As was shown in [9], the collision of the weakly bound breather with a fast kink in the presence of the dissipative perturbation may result (in the first order of PT) in the break-up of the breather into a kink-antikink pair. It was also remarked in [9] that, with the same accuracy, this inelastic process cannot be generated by a conservative perturbation. In the present subsection we shall consider the same process for the perturbed SGE (3). As in [9], we shall confine ourselves by the first PT order, but we shall take into account the first order with respect to I,-1 while the assertion of [9] implies the zeroth order in this small parameter. The breather solution (62) with regard to (78) takes the form Ubr ( X,

t) =

4 sgn (sin ( ~ - ~'t )) arctan [ T(t ) sech x ],

(8o)

where we have introduced the quantity [9, 17] T( t ) = ~'-x sgn (sin ( • - ~t )) cos ( ko - ~t ),

(8a)

which evolves according to the equation [9, 17] dT = V/1 _ ~.2T 2

(82)

a-7

In (80), (81) we have redefined the phase constant: '/" --- q~+ ~r/2, in order to accord the notation with [9]. The eigenvalues (63) corresponding to (78) are ~kl.2 =

"q-X' "{- iX"=

+ f / 2 + i / 2 + d)(~z).

(83)

The evolution equations for the parameters X' and X" ensue from (8) where one should substitute [9]

C l ( t ) = - 2 2 , a i f - 1 e x p [ i ( ~ - ft)],

C2(t ) = C{'(t)

(84)

(cf. (66)). Under the usual condition 1 - V z << 1 the Jost function combination (9) splits, as above, into those corresponding to the fast kink and to the breather, the latter being calculated in [9]:

Q l ( x , t) = Q2(x, t) = (cosh x ) { 2 [ T 2 ( t ) + cosh 2 x ] ) -1

(85)

Inserting (67), (80) and (84), (85) into (8) we use (19)-(21) to cast the evolution equations into the form d(X') 2 dt = (16'/(3p))Tc°sh2 t [c°sh2 t - T2(t)]

× [cosh4 / + T4(t) - 6T2(/) cosh 2 t]. [cosh2 / + T2(t)] -5, d~" d----7-= - 2 X ' [ 1 + r ( / ) ] dh' dt (cf. the analogous equation (I1.12) from [9]).

(86) (87)

Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

140

In the present problem the most interesting TPE is not the energy and momentum exchange as in the preceding ones, but the possibility of breaking the breather (80)-(81) into a kink-antikink pair [9]. However, one should keep in mind that the escaping kink and antikink emit radiation due to their overlapping [9]. According to [9], the total emitted energy Eem - c 2, and if it is larger than the total kinetic energy Ek~n of the escaping kink and antikink, these two "particles" will again merge into a breather, i.e. breaking will be unobservable. If the kink and antikink inside the breather overlap weakly at the collision moment t = 0, i.e. if Icos'/'l - 1, or, in terms of T(t), To2 = T2(0) -- ~ - 2 (see (80) and (81)), the estimate for Ekin, following from (86), is

e~.-

C - 1 - -7 C~ ~1Tol

(88)

In the same time it is clear that, for a "typical" initial condition which admits breaking, Ekan -- ~2 [9], so that we obtain from (88) ~ ~ c/v, i.e. Ekin -- c2/v 2 << gem - c2" So we infer that in the case IT01 - ~-1 breaking is unobservable. In the same time, if the overlapping inside the breather is essential at the collision moment, i.e.

1 < T2 << ~-2

(89)

(in other words, ~ 2 ~ COS2 ~ << 1, see (81)), the analogous arguments enable us to bring the condition Eem < Ekin to the form

vlT01 z c -1.

(90)

At last, if the overlapping at the collision moment is strong, i.e. (91)

IT01 << 1 (or cos 2 ~/" << ~2 in terms of ~), the condition analogous to (90) is

(92)

p ~ IZolc -x.

So breaking may take place upon the conditions (89) or (91) supplemented by, respectively, (90) or (92). It is interesting to note that the same conditions (89) or (91) provide for the observability of breaking in the case of the dissipative perturbation [9]. As well as in [9], these conditions mean that, during the collision, (82) reduces to T ( t ) = TO+ t.

(93)

Substituting (93) into (86) yields the final result A(~ '2) = ( 1 6 , / 3 v ) T o f _ °° dtcosh2t[cosh2t + ( T O+ t) 2] oo

× [cosh4 t + (TO+ t ) 4 _ 6(To + t)2cosh2 t] [cosh2 t + (TO+ t)2]-5

(94)

(cf. the analogous result (11.17) from [9]). Note that, according to (94), A(~kt2) is the odd function of T0.

YttS. Kioshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

141

Breaking takes place if A(?~'2) < - ~ 2 / 4 (see (83)). The threshold value of the binding energy, i.e. the maximum value which makes breaking possible [9], can be found from (79), (83) and (94) [gtm. I =

32[A(X'2) I.

(95)

If IEbl < [Ethrl the collision breaks the breather into the kink-antikink pair described by the unperturbed solution [1] (cf. (6)) u k~ ( X, t ) -----4 arctan ( W - ' [ fl exp ( - W t / 2 ) - f1-1 exp ( W t / 2 )]/cosh x },

(96)

W being the small relative velocity of the kink and antikink. This velocity together with the constant fl are determined by the formulae of [9]

w2 = 4141za(h,2)[- ~.2], =

cos:

(97) = w

rg.

(98)

The sign of sin if' played an important role in the above consideration (see, e.g., (81)). The qualitative picture of the inelastic kink-breather collision for the two opposite signs is illustrated by figs. la and lb (where it is implied that the break-up of the breather takes place, i.e. A(?~'2) < --~2/4). If A(~,n) > 0, the above consideration may describe the converse process of fusion of the kink-antikink pair (96) into the breather (80), (81) due to the collision with the fast kink. In this situation one should find TO with the aid of (98) and then find A(X'2) from (94). The parameter ~ of the arising breather can be found from (97). In particular, the threshold velocity for the fusion according to (97) is: W ~ = 161A (~,'2)1. It should be noted here that, in absence of the third kink, the conservative perturbation in (3) may cause the fusion of the pair into the breather on account of energy emission into the continuous spectrum [9]. On contrary to that radiative fusion, the mechanism considered in the present paper is nonradiative: the excessive energy (to be shed that the pair may merge into the breather) is accepted by the fast kink. The change AX,, of the parameter ~" can be found from equation (87) where one should substitute ?~'(t) as the solution of the preceding equation (86). The quantity A ~,,, determines the velocity of the breather (or of the kink-antikink pair inertia center): Vbr ----2 A?~". The energy and momentum exchanged between the fast kink and breather are also expressed in terms of AX,,. However all these effects are of less interest because, as one sees comparing (87) and (86) with regard to (89) or (91), IA?~"I << I,a~,'l. To conclude this section we would like to note that the results (94)-(98) essentially depend on TO, i.e. on the value of q' (see (81)). The crucial dependence on xo is as well inherent in the results obtained for the dissipatively perturbed SGE [9]. However the difference is that the present results are not sensitive to the fast kink's topological charge while in [9] the charge played a significant role.

5. Inelastic interaction of two SGE breathers

In this section we shall consider the collision between two fast SGE breathers. Since each breather is a "two-particle" bound state, this collision may be interpreted as a four-particle interaction. First of all, the collision of two small-amplitude breathers is, in the first approximation, equivalent to the collision of two NSE solitons [4, 15], and the latter collision is known to be purely elastic in the first order of PT [9]. Of course, in reality the collision of small-amplitude SGE breathers results in some inelastic effects, but, as

142

Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems a

/v.

t
1£

kink

breather

-W/2 ~

WI2

kink-antikink pair

/

t~-O

kink

t
kink

breather

kink-antikink pair

kink

Fig. 1. The inelastic kink-breather collision: a) sgn (sinxo) = 1; b) sgn (sin'/') = - 1.

well as those treated in subsection 4.1, they are exponentially small in the breathers' amplitudes. The collision between a small-amplitude breather and a weakly bound one may result in the more interesting inelastic interaction, namely, the break-up of the latter breather into a kink-antikink pair. This interaction, induced by the same perturbation as in (3), was investigated in [9]. Here we shall concentrate on the collision of two weakly bound breathers (80)-(81). As we shall demonstrate below, different nontrivial modes of inelastic interaction are possible in this situation. In the rest reference frame of one breather the second breather has the form [1] u ~b,) ( z ) = 4 arctan [tan/*2 sin (cos 1'2 z - ¢2 )sech (sin 1'2 z )],

(99)

where z = ( x - t ) / ~ - V 2 , and V is the relative velocity of the breathers (we set V = 1 in all the expressions except for 1 - V2). The parameters/~2 and ~2 in (99) are the amplitude and initial phase of the second breather in its rest reference frame (see (62)). One can derive the evolution equations for the parameters of the first (quiescent) breather, following the lines of the preceding section with the difference

Yu.S. Kivsharand B.A. Malomed/ Many-particleeffects in nearly integrablesystems

143

that the kink potential (4) in the integrals (20)-(21) should be replaced by (99) to yield

dxsin[2u~br)(x,t)] =0,

(lOO)

_ ~ d x sin: [u~br)(x, t)] = 16/3~,

(lOl)

f?

o~

if cos 2 g'2 - 1, i.e. unless the second (fast) breather satisfies the conditions (89) or (91) in its rest reference frame at the collision moment. So, comparing (101) and (21), we see that in this case the evolution equations for the quiescent breather differ from those (86), (87) only in the multiplier 2 before rhs of (86). This result is quite obvious: it means that, from the quiescent breather's viewpoint, the fast breather with small overlapping between internal kinks at the collision moment is, in the first approximation, equivalent to two independent fast kinks. The most interesting case is that when the internal kinks are essentially overlapping inside the both breathers at the collision moment; then the interaction is four-particle indeed. In this case calculating the integrals in lhs of (100), (101) and substituting them together with (84), (85) into (8) results in evolution equations of a rather lengthy explicit form. Therefore we shall write only the eventual expression for the change of the parameter ~,'1 (see (63), (65)) of the first (quiescent) breather (cf. (94)),

A( ~ti2) = ( {/g )[ F1 ( T02) G1 (TOl) -

oF2(T02)

(102)

G2 ( TOl)],

where we have introduced the functions

=

r2(ro) =

+ z):' cos , z

+ z)"]

tof_2 dz cosh z [cosh 4 z + ( T O+ z ) ' -

z+

+ z)"]-",

(lO3)

6(to + z)2 cosh2 z]

×[cosh2z--(rO+z)~lIcosh~z+(ro+Z)~]-4,

(104)

Cl( o)= Tof_2dzcosh2zIcosh2z-( o + Z) ×[cosh'lz -t-(TO+ z ) 4 - 6 ( r o G2(;

o) =

o~

+z)2cosh2z][cosh2z -t-(r 0 +z)2] -5,

(105)

dzcosh z [cosh 8 z + (T o + z)8 + 70(T o + z)4 cosh 4 z

- 28(TO+ z)2 cosh6 z _ 28(TO+ z)6 cosh 2 z ][cosh2 z + (T O+z )2] - 5,

(106)

and the sign function o --- sgn (sin'/' D sgn (sin'/'2)

(107)

(the quantities F1/2 and 8F2 are equal to the integrals in lhs of (101) and (100)). The parameters T0x and To2 in (102) and '/'1 and '/'2 in (107) are the standard parameters (see (80), (81)) defined for each breather in its rest reference frame. Note that, according to (103) and (104), F2/F 1 ~ 0 when I T021 ~ o¢, i.e. only the first term from rhs of (102) survives in this limit to recover the formula (94) with doubled rhs, as it is

144

Y~S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

4<0

k/ 2 nd

breather

Jet

breather

4st

breather

2 nd

breather

Fig. 2. The elastic breather-breather collision.

seen from (103) and (105). As was explained above, this circumstance corresponds to the evident fact that the fast breather with small overlapping of the internal kinks is equivalent to two fast independent kinks. If TOE is not large the marked difference of (102) from (94) is that F 2 is the even function of To1, on contrary to the odd function F r The quantity A(A'2) for the second breather can be obtained from (102) by means of the obvious transposition To1 ~ T02. If

4A (X'~) < --~',,z (n = 1,2)

~

V

t~o

L/ 2 nd breather

~ s t breather

t>o kink-antikink pair

A 2 nd

2 nd b~-eather

V

breather

t
k/

{ st breather V ~-~ t~O

kink-antikink pair

2 nd

breather

Fig. 3. The breather-breather collision resulting in the break-up of one breather into a kink-antikink pair: a) sgn(sin'/q)~ 1; b) sgn(sinxOl) = - 1.

YmS. Kioshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

145

the nth breather decays into a kink-antikink pair (96), the pair's parameters being determined, as above, by (97)-(98). Thus we see that, depending on the values of the breathers' parameters T01, T02, ~'1, ~'2, o, three different modes (channels) of the interaction are possible: the both breathers survive (fig. 2); one survives and one decays (fig 3; cf. fig. 1); the both decay (fig. 4). As it is illustrated by these drawings, the result of the inelastic collision essentially depends on sgn (sin ~/',). At last, the same formulae (102)-(107) together with (97), (98) enable use to describe the inelastic process in which any of the breathers is replaced by a kink-antikink pair (96).

6. The energy emission in the kink-kink collision

In the above sections we employed the so-called adiabatic approximation neglecting the energy emission into the continuous spectrum (CS). In this section we shall consider the perturbation-induced emission that accompanies kink-kink or kink-antikink collision. Though this effect is not three-particle, it is of obvious interest from a general viewpoint [9, 10]. In the "nonrelativistic" case when the relative velocity 2W of the colliding kinks is small, W<< 1, the emission was calculated in [9]. Here we shall dwell on the opposite case v = (1 - W2) -1/2 >> 1. The CS energy density is [1] p(X) = (rrX2)-l(1 + 4X2)ln(1 - Ib(X)12) -1 = (~rX2) -a(1 + 4~,2)lb(X)l 2,

(108)

where b(~) is the complex reflection coefficient that determines the CS scattering data [1], and it is implied Ib(~)l 2 << 1. In the case of the equation (3) the evolution equation for b(~) is (see (I1.25) of [9]) dfl-Tb(h) = - i ( X + 1/4~,)b(X) + i ( , / 4 ) f =_ × ([ ~(2)*(x, t; X ) ] 2

dxsin[2u(x,t)]

[ ~(1)*(x, t; X)]2},

(109)

'I'a'2)(x, t; h) being the CS Jost functions [1]. In the case v >> 1 these functions are

-

t; X) )

• a)'(x, t; X)

exp [ i ( X / 2 - 1/8X)x]

= ( 0 ) + 2vol(~,- i v ) - l e x p ( _ z l / 2 +

vt/2)

t)

+ ( 02/2 v) (X - i / 4 v) -1 exp ( - z2/2 - vt/2) (

\

x, t

)

(110)

where ~1(112)(x, t) are the one-particle Jost functions (15), 01 and o 2 being the kinks' topological charges, zx, 2 ~. v(x T-t). We assume the emission to be absent at the moment t = - o o . Then the total emitted energy density is determined by (108) where one should substitute b(~,) = b(h, t = + o0). So, integrating

146

Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

_ _ _ L _ _ v

t
2 nd

V

4st

breather

breather

- ','~/2 ~

V

l

WJ2 t>O

4st kink-antikink pair

2 nd kink-antikink

pair

t
V

2 nd breather

~st breather

t~O

4st !:ink-antikink pair

2 nd kink-antikink pair

A V_ ~nd

breather

t
l/

~st breather

-W2/2~

"-Y

~st kink-antikink pair

W2/2

----~

k~_

t~O

2 nd kink-a~tikink pair

V _

-l/ 2 nd breather

t
~st breo_ther t>O 27,

~st kink-antikink pair

2 nd

'.-:ink-~_nti'4ink pair

Fig. 4. The breather-breather collision resulting in the break-up of the both breathers: a) s g n ( s i n ~ l ) = s g n ( s i n ~ 2 ) = 1: b) s g n ( s i n ~ l ) = - s g n ( s i n ~ 2 ) = 1; c) s g n ( s i n #1 ) = - s g n ( s i n x~2 ) = - 1; d) s g n (sin xv1 ) = sgn ( s i n xP2) = - 1.

Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

147

(109) yields the expression (cf. (II.27) of [9])

p()k) = (~'h2)-l(1 + 4)k2)(,2/16)f~oodtf;oodxsin[2u(x,t)] X exp [i()~ + 1 / 4 ) ~ ) t ] ( [ ~2)*(x, t; )~)]2- [ ,/,~l)*(x, t; )~)]2} 2.

(111)

The further calculations are facilitated by the fact that, due to 1, >> 1, we may, as above, take the field potential in the form (14), i.e. as the sum of the potentials of separate kinks. Inserting this "splitted" potential and (110) into (111), we obtain after rather tedious calculations

~2 p()k) = ~-~)k-2(1 + 4)k2) ~i~J()k),

(112)

where J(•) =

IQ(~')I2" sinh2 [(~r/2p)(X - ol°2/4X)] X6(X2 + ~,2)2(~2 + 1/16p2)2sinh2 (~rX/~,) sinh 2 (~r/aX~,) '

(113)

Q(~,) being the complex polynom Q ( ~ ) = ~x0/4 - llih9v/12 - ~8v2 _ i~7v3/3 - 5~k61,4/4 + 7i~,5p5/12 + 5~4v4/16 - iX3~,3/48 + X2~,2/64 - lliX~,/(3 × 21°) - 2-12.

(114)

Note that, according to (113) and (114), J(~) possesses the symmetry property J(h) = J(1/4X).

(115)

The energy spectral density (108) is defined as density in the spectral parameter scale. The physical density 8 ( k ) should be determined as the density in the scale of the wavenumber k = X - l/4X,

(116)

i.e. e ( k ) = p(X) dX

-~ =

16~r3,2

91tl 4 J ( h ) .

(117)

As one sees from (117) with regard to (115) and (116), the physical density satisfies the evident condition which follows from the symmetry arguments: d~(k) = 8 ( - k), i.e. equal portions of energy are radiated to the left and to the right. The dependence ~ ( k ) is plotted in figs. 5a and 5b for the cases ola 2 = - 1 and o l a 2 = + 1 corresponding to the, respectively, opposite and equal polarities of the two kinks. The maxima of the spectral density lie at the two symmetric points (see fig. 5) k-- ±k~-

+i,

(118)

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Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

I~(k) ~ rrl~X

-"max

0

"]~Max

¢:max

k

+kma x

k

Fig. 5. The emitted energy spectral density vs. the wavenumber k: a) olo 2 = - 1 ; b) olo 2 = 1.

corresponding to X ~ 1, and X - (4~) -t. The maximum value of do(k) is the same for the both cases " a " and " b " " domax = e ( -4- k m a x ) - ,21p-4.

(119)

In the case " a " the value do0= ~ ( k = 0) is, according to (113)-(117), (120)

OX~O~ ~2/.,-6,

while in the case " b " do0= 0 (fig. 5b). The function N(k) expontially falls at k 2 >> ~2 (i.e. X2 >> 1,2 or

X2 << ~-2): (121)

8 - exp ( - ~ r l k [ / ~ , ) ,

while in the range 1 << k 2 << p2 (corresponding to 1 << X2 << u2 o r ~,-2 << X2 << 1) it has the power asymptotic form d ° - C 2 k l 0 / p 14"

(122)

The total emitted energy Eem can be found from (112)-(117), gem =

f-2

do(k)dk=

_ O ( ; k ) d X = E l - a x o 2 E 2,

(123)

YmS. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

149

where

E1 = c2v-3A + O(¢zv-5),

(124)

E2 = £ 2 p - 5 n 4. ~ ( ~ 2 p - 7 )

(125)

(computing the integral (123) is facilitated by (115)). The exact expressions for the numbers A and B are adduced in appendix A. Their approximate values are A = 8.78, B = 1.10. The interesting property of (123)-(125) is that, in the first nonvanishing approximation with respect to g-1, the total emitted energy does not depend on the relative polarity OlO2 of the two kinks. This is something similar to the famous Pomeranchuk theorem about the asymptotic equality of the total sections for scattering of particles on particles and particles on antiparticles [18]. The accurate results (123)-(125) are in accordance with the simple estimates following from (118)-(122). Indeed, according to (121) and (122), Ee m _ ¢2fo-VklOv-14dk _ £ 2 p - 3 ,

which confirms (124). As in (125), it can be estimated as 8 o • kmax - c2v -5 (see (118) and (120)). The difference between the cases of the opposite and equal polarities is essential in the long-wave range k 2 .<<1, i.e. h2 _ 1 (compare figs. 5a and 5b). This accords with the known fact that in the case of small relative velocity W << 1, when the emitted energy is concentrated in the long-wave range, the quantity Eem crucially depends on OlO2 [9]. At last, the momentum density q(~) of the emitted field can be found to [1] q(X) = (~'Jk2)-1(1 - 4X2)lb(X)l 2.

(126)

As is seen from comparing (126) and (108), the explicit expression for q(~) is (cf. (112))

q ( ~ ) = - ~ ~- 2(1 - 4~2) v~g4J ( ~ ) ,

(127)

J(?Q being defined in (113). The physical density ~ ( k ) is related to q(X) as d~(k) to p(~) (see (116) and (117)) ~ ( k ) = (1 + 1/4X2)-lq(X) =

167r3c2 (1 - 4~ 2) J(X), 9v14 (1 + 4 ~ 2)

(128)

where we have used the expression (127) for q(A). According to (116) and (115), the momentum density (128) possesses the obvious symmetry property =

which provides the total emitted momentum to be zero.

Acknowledgements The authors thank S.V. Manakov for useful discussions and A.M. Kosevich for attention to the work.

Yu.S. Kivsharand B,A. Malomed/ Many-particleeffects in nearly integrablesystems

150

Appendix A

The numbers A and B in (124) and (125) are 27~

6

8~ 2

A = --if- Y'. Dk/Ek(Tr),

6

B = ---if- E DklEk-l(~r),

k=l

k~l

where

In(P)=

fo

xn e-PX ~ d x (x2 + 1)2

dp" (sinp[psip-cip]+cosp[pcip+sip]},

_ ( - 1 ) "+x d"

2

ci

x

=

-

f ° ~ .at--T-, cost

sint six = Si x - ~~r = - fxO~dt----T-,

and D 1 = ( 7 / 1 2 ) 2, D4 = 71/72,

D E = 1 6 9 / 2 4 X 32,

D 5 = D1 = (7/12) 2,

D 3 ---.37/24, D 6 = 1/16.

N o t e added in proof

After this paper has been completed, the authors have revealed the earlier paper by M.J. Ablowitz, M.D. Kruskal and J.F. Ladik, SIAM J. Appl. Math. 36 (1979) 428 (to be referred to as AKL), that is devoted to the numerical investigation of the kink-anti-kink collision within the framework of eq. (3). It seems natural to compare our analytical results (123)-(125) with the numerical ones of AKL. Those results actually represent the total emitted energy Eem as a function of the initial velocity Vi of the colliding kinks. Evidently, the relative velocity W we deal with is W = 2Vi/(1 + Vi2). Unfortunately, in the range 1 - W 2 << 1, where our results are literally applicable, the numerical error in the AKL's results seems to be larger than genuine Eem. The first reliable data pertain to Vi = 0.4, c being 0.1. The corresponding value of 1 - W 2 is 0.52 which does not seem to be sufficiently small. Nevertheless, the accordance between the analytical and numerical results proves to be good even at this value of Vi. Indeed, according to AKL, the final velocity of the colliding kinks is V~= 0.396 which yields the numerically computed emitted energy (Eem)num= 0.0330; in the same time (123)-(125) yield E 1 = 0.0333; E 2 = 0.0021, i.e. the analytically calculated emitted energy is (E~m)~ ~ = 0.0354, so the relative error is only 0.068. Even for Vi = 0.2, when the kinks are slow rather than fast, the accordance is not very bad: (Eem)num = 0.055; (Eem)ana 1 = 0.076. It seems quite natural that the accordance deteriorates with the dimension of W. Nevertheless, for very small W the accordance between (123)-(125) and numerical results suddenly becomes very good again. The value of Eem corresponding to W = 0 determines, according to [9], the threshold value of W for the inelastic process of fusion of the kink-antikink pair into a breather, Wt~ = ½Eem. This quantity has been calculated in [9]: Wt~ = 4.95c 2, the numerical constant having been elicited from [10]. In the same time

Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

151

(123)-(125) yield E~r~= 9.88¢ 2 at W = 0, i.e. W2 = 4.94c 2, which is in striking accordance with the above relation. So we infer that the formulae (123)-(125) with oxo2 = - 1 , though analytically derived under the condition 1 - W 2 << 1, furnish a good fit for the total emitted energy in all the range 0 < W 2 < 1.

Addendum Since submitting the present paper for publication, the authors have had sufficient time to obtain new results on three-particle and radiative effects in perturbed sine-Gordon and nonlinear SchrSdinger equations. Here we give a brief compendium of these results which constitutes a natural supplement to the basic part of the paper.

1. E n e r g y emission in collision o f two N S E sofitons

In section 6 we have calculated the total energy Eem emitted during the collision of two SGE kinks in presence of the perturbation (3). Here we give an expression for the energy emitted by two NSE solitons (30) colliding in presence of the perturbation (29) with N = 2. We consider the case when the solitons' amplitudes ,/ are equal, and their velocities +4~ are large: ~ >> 7/. The final result is ECrn = A¢ 2,17 + ¢2~5,12exp ( - ¢r~/~i ) ( n I COS A~b + B 2 sin A~ ),

(Ad.1)

where A~ is the phase difference between the two solitons at the collision moment and the numerical constants are: A ~ 690.5; B 1 = 1200.5; B2 = 173.3 (cf. (123)-(125)). It is noteworthy that, with the exponential accuracy, in the considered case ~ >> , / t h e expression (Ad.1) does not depend on A~. This is analogous to the "Pomeranchuk theorem" for SGE kinks. Besides energy, the emitted waves also carry the charge ( " n u m b e r of particles"). The total emitted charge is Nero -- Eem/4~ 2. A full account of this problem, including explicit expressions for the radiation spectral density analogous to (117) will be published shortly in the Russian journal Plasma Physics. . A coupled system o f two S G E According to the paper by M.B. Mineev et al. (J. Low Temp. Phys. 45 (1981) 497), a pair of long weakly coupled Josephson junctions is described by a system of equations, Utt - - U x x "~ sin u = eG~, vt, - vxx + sinv = CUxx,

(Ad.2)

¢ << 1 being the coupling constant. In the first approximation the "u-kink" solution to (Ad.2) has the form u = 4o arctan (e-Z), z = (x - Vt)/Crl-

v -- 2¢o (1 - V 2) -1 sgn z [ z cosh z - sinh z . log (2 cosh z )], V2.

The system (Ad.2) is a convenient object for studying many-particle and radiative effects. First let us consider the collision between a u-kink with a velocity V > 0 and polarity a and a pair of v-kinks with

152

Yu.S. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

velocities V1 = - V2, V1 > 0, and polarities 01, 2. Assuming v = (1 - V2) -1/2 >> v1 = (1 - 1/2) -1/2 >> 1, we obtain the following expressions for the "three-particle" charges A v and A vI (cf. (22), (24) and (28)): Av = --~oWrrv 3 sinh8 sech2&

Av 1 ~ --Av >> Av2,

(Ad.3)

where ~ = vlx0, and x o is the distance between the centers of the v-kinks at the moment of the collision with the u-kink (cf. (23')). Consideration of other three-particle effects developed in the basic part of the paper can be readily reproduced for the system (Ad.2). Collision between the u-kink and the small-amplitude o-breather (64) results in the change of the kink's velocity: AV ~-~ E OqT 2 V - 3 (1

-

V 2)3/2

sin q~sech (rr/21,tV),

(Ad.4)

o being its polarity, and in the change of the breather's parameters )¢, )¢' (see (75)): A~: = - ( , o r r 2 / 4 t t V 3) sinq~ seth (~r/2/W),

A)¢' =/~(1 + V) A)~'.

(Ad.5)

Note that, contrarily to (73), (74), the expressions (Ad.4), (Ad.5) are valid for any value of V (not only for 1 - V 2 << 1). Collision between the small-amplitude u- and v-breathers (64) with the amplitudes /~1,2, moving with velocities V1.2 = + V, 1 - V 2 << 1, results in A~k'1 = -,rr2V/1 - V2/~ -1 sin~b2 cos #~1 seth (~r/2/.tl) sech (~r/2/.t2) ,

A}K' 1' = 2/, 1 AX'~,

(Ad.6)

where ~1,2 are internal phases of the breathers at the collision moment, and expressions for A~,'2, A2¢~can be obtained from (Ad.6) by the change of indices 1 ~ 2. The results (Ad.3)-(Ad.6) can be readily verified to satisfy energy and momentum conservation. At last, we can calculate radiative energy loss accompanying collision between u- and v-kinks moving with the velocities +_ V. It is important that, contrarily to the problem considered in section 6, here we do not require the kinks be fast, i.e. the expressions below are valid for any value of V. The spectral density of the emitted radiation is (cf. (117)) e , ( ~. ) = (, 2~r3/16 V 4 ) ( 1 --

V 2 )2

cosech2 ( ~rg 1/2) sech2 ( 7r× 2/2),

e2 (~., V) = ~f~(1/ax, V) = d°,(X, - V).

(Ad.7) (Ad.8)

Here xl, 2 ( ~ / 1 - - V 2/2V)[~(1 _+ V) + (1 -Y- V)/4h], and the indices "1" and "2" in (Ad.7), (Ad.8) and below refer to the radiation emitted in u- and c-subsystems. The total emitted energy can be calculated for any V; the corresponding expression takes a simple form in the cases 1 - V 2 << 1: =

(Eem) 1 = (Eem) 2 = (c2/~r3/2)(1 - V2) -1/2,

(ad.9)

(cf. (123)-(125)) and V 2 << 1: (Eem)l = (Eem)2 = ~ 3 V ~ 2 V - 7 / 2 e x p ( - ' z ' / V ) .

(Ad.10)

Note that the expression (Ad.7)-(Ad.10) do not depend on the polarities of the kinks. The emitted energy (Ad.9), c ~ to (123)-(125), grows with the decrease of 1 - V 2. However, its ratio to the total energy 1 6 / ~ / 1 - V 2 of the two kinks remains small - e 2. The expression (Ad.10)

Yu.S. Kiosharand B.A. Malomed/ Many-particle effects in nearly integrablesystems

153

demonstrates that in the limit V 2 << 1 the emitted energy is much smaller than the kinks' kinetic energy 8V 2, i.e. the collision cannot result in binding the kinks into a bound state, while in the problem described by (3) such a process is possible provided olo2 = - 1 [9, 10]. 3. A coupled system of two N S E

For nonrelativistic small-amplitude breathers the system (Ad.2) turns into iu t+ Uxx + 2 l u l 2 u = c v ~ ,

(Ad.ll)

iv t + o~, + 2lvl2v = c u ~

(here u and v are not the same as in (Ad.2); the system (Ad.11) is conservative provided E be real). First of all, in the adiabatic approximation a collision between a u-soliton and a v-soliton with equal amplitudes 771 = 72 =77 and with velocities + 4~ (see (30)) results in the phase shifts (Ax0)I=

(Ax0)2

= 0,

A~I = A~b2= -- (~r 3c~2/872) cos Aq~tanh (~r~/27) sech2 (~r~/27),

(Ad.12)

Aq~ being the phase difference between the solitons at the collision moment. The validity of (Ad.2) is not restricted by any condition on ~/~/. The spectral density of the energy emitted during the collision of the two solitons can be found for any value of ~/7. However, the corresponding expressions are rather lengthy. Here we shall only display the expression for the total emitted energy in the case ~ >> 7 (cf. (Ad.9)), (Eem)l = (Eem)2-- 98.62,27~ 2,

(Ad.13)

which is independent of A~. Note that (Ad.13) grows with increase of ~, but its ratio to the total solitons' energy remains small - c 2. Contrarily to the analogous problem for the kink-kink collision described by (Ad.2), here radiative losses may result in binding two solitons into a bound state. An estimate for the threshold (maximum) velocity which admits binding is ~ c27. In other physical problems (e.g., in a problem describing propagation of Davydov solitons in two coupled molecular chains; see, e.g., A.S. Davydov, Sov. Phys. Usp. 25 (12) (1982) 808) there also occurs the system iu t+ uxx + 2 l u l 2 u = co,

(Ad.14)

iv, + vx~ + 2[vl2v = ~u,

where c should again be real. The essential difference between (Ad.ll) and (Ad.14) is that in the present case the interaction between u- and v-solitons is nontrivial even in the adiabatic approximation. Considering, as above, the collision between the solitons with amplitudes 71 = 72 ~-- 7 and velocities + 4~, we find the following results: a ~ = -z172 = (c~r2/16~) sin a,~. sech2 (~r~/2n),

AV~ = AV2 = - (~/7) An.

(Ad.15)

The system (Ad.14) possesses four integrals of motion: total energy, total charge, u-momentum and v-momentum. It is easy to verify that (Ad.15) conserves all these integrals.

Yu.s. Kivshar and B.A. Malomed/ Many-particle effects in nearly integrable systems

154

4. Edge effects in S G E A t last, it is noteworthy to mention that in various physical problems a S G E occurs (e.g., (3)) defined o n a half-axis x > 0 with the b o u n d a r y conditions: (a) u x l x = 0 = 0 or (b) u l x = 0 = 0 (see, e.g., R.M. D e Leonardis, S.E. Trullinger and R.F. Wallis, J. Appl. Phys. 53 (1982) 699). It is clear that the interaction of a soliton with the edge will seem as an interaction of two symmetric solitons in a complete S G E system since the edge can be obviously replaced by the continuation (a) u ( - x ) = u ( x ) or (b) u ( - x ) = -u(x). F o r instance, reflection of the small-amplitude breather (64), moving with velocity V (1 - V 2 << 1), from the edge results in the change of the breather's amplitude and velocity: A/~ = _+(c~rz/2tt)(1 - VZ)sin(2q~)sech z (~r/2/~), q~ being pertain, radiative in Phys.

A V = ½(1 - V2)a/ZA/~,

(Ad.16)

the value of the breather's internal phase at the reflection moment. The two signs in (Ad.16) respectively, to the b o u n d a r y conditions (a) and (b). Further results on many-particle and effects stipulated by the presence of the edge are set forth in the paper which will shortly appear Lett. A.

References [1] V.E. Zaldaarov, S.V. Manakov, S.P. Novikov and L.P. Pitaevsky, The Theory of Solitons (Nauka, Moscow, 1980). [2] D.J. Kaup, SIAM J. Appl. Math. 31 (1976) 121. [3] V.I. Karpman, Sov. Phys. JETP Letters 25 (1977) 271; Physica Scripta 20 (1979) 462. [4] D.J. Kaup and A.C. Newell, Proc. Roy. Soc. A 361 (1978) 413; Phys. Rev. B 18 (1978) 5162. [5] K.A. Gorshkov, L.A. Ostrovsky and E.N. Pelinovsky, Proc. IEEE 82 (1974) 1511. [6] K.A. Gorshkov and L.A. Ostrovsky, Physica 3D (1981) 428. [7] J.P. Keener and D.W. McLaughlin, J. Math. Phys. 18 (1977) 2008; Phys. Rev. A16 (1977) 777. [8] D.W. McLanghlin and A.C. Scott, Appl. Phys. Lett. 30 (1977) 545; Phys. Rev. A 18 (1978) 1652. [9] B.A. Malomed, Physica 15D (1985) 374, 385; also in: Proc. Second Int. Workshop "Nonfinear and Turbulent Processes" (Kiev, 1983), (Gordon and Breach, New York, 1984). [10] M. Peyraxd and D.K. Campbell, Physica 9D (1983) 32. [11] A.M. Kosevich and Yu.S. Kivshar, Sov. J. Low. Temp. Phys. 8 (1982) 644. [12] V.I. Karpman, E.M. Maslov and V.V. Solov'ev, Zhurn. Exp. Teor. Fiz. (Sov. Phys. JETP) 84 (1983) 289. [13] J.K. Perring and T.H.R. Skyrme, Nucl. Phys. 31 (1962) 550. R.F. Dashen, B. Hasslacher and A. Neveu, Phys. Rev. D 10 (1974) 4114; (1975) 3424. R. Rajaraman, Phys. Rep. C 21 (1975) 229. P.I. Candrey, J.C. Eilbeck and J.D. Gibbon, Nuovo Cimento B 25 (1975) 497. [14] B.A. Malomed, Phys. Lett. A 102 (1984) 83. [15] A.C. Newell, J. Math. Phys. 19 (1978) 1126. [16] B.A. Malomed, Teor. Mat. Fiz. (Russian journal Theoretical and Mathematical Physics) 51 (1982) 34. [17] K. Nozaki, Phys. Rev. Lett. 49 (1982) 1883. [18] S.S. Schweber, An Introduction to Relativistic Quantum Field Theory (Row, Peterson and Co., Evanston and Elmsford, 1961).