On a mechanism of soliton bunching in Josephson junctions

Volume 105A, number 1,2 PHYSICS LETTERS 1 October 1984

ON A MECHANISM OF SOLITON BUNCHING IN JOSEPHSON JUNCTIONS

V.I. KARPMAN and N.A. RYABOVA I Z M I R A N , Academic City, Moscow Region, 142092, USSR

Received 25 July 1984

The interaction of solitons in long Josephson junctions governed by the perturbed sine-Gordon equation is investigated. The theory of milton bunching presented by Karpman et al. is refined, generalized, and supported by new numerical mate- rial. It is demonstrated that soliton reflections from the junction ends may play a significant role in stabilization of the dis- Lance between the solitons in a bunched state.

The soliton (or fluxon) bunching in Josephson junctions, investigated in a number of analog and numerical studies (e.g. refs. [1-9] ) and recently ob- served experimentally [10], is of great interest both for theory and applications. The mentioned papers show that the bunching is described by the perturbed sine-Gordon equation

~Ott -- ~Oxx + sin ~ = - - f - - a~pt +/3~Oxx t . (1)

Here all quantities are dimensionless, being normal- ized as in refs. [5,6]. The first term in the rhs of eq. (1) is the bias current, and the second and third terms are losses due to the shunting and the surface resistiv- ity, respectively.

The bunching takes place for fluxons with the same topological charge (polarity). The unperturbed sine-Gordon solitons with equal topological charges repel each other. Thus, the bunching appears to be due to the terms in the rhs of eq. (1). Two theories of bunching have been proposed. One of them, dis- cussed in refs. [11,12] with reference to ref. [ 13] *x, is connected with the third derivative term in eq. (1). It does not work at/3 = 0, whereas the bunching is ob- served in analog and numerical studies also in this case (e.g., refs. [1,2,5,6,9] ). The other mechanism proposed in refs. [5,6] is based on the fact that the bias current and dissipation effects may significantly change the nature of soliton interaction. According

*1 Ref. [13] has not been available to u~

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to this theory, the bunching may exist if/3 = 0 and it is that case for which the theory has been developed in the mentioned papers.

In the present paper we give a more general and refined treatment of the theory described in refs. [5,6]. In particular, we extend it to the case/3 :/: 0. This gives the possibility to investigate the role of ei- ther of the two dissipative terms, with a and/3. In particular, we show that if a = 0 and/3 4: 0, the bunching may not exist. (Generally, in real systems, both terms may be important.) The role of so]iron reflections from the junction ends is also investigated and shown to be important for the bunching.

Our basic approach is similar to that of refs. [5,6]. The approximate two-soliton solution of eq. (1) may be written as

tp(x, t) = ~(Zl) + ~o(z2) + ~Ol2(X , t ) , (2)

where ~ ( Z n ) is the one-soliton solution of eq. (1) investigated in refs. [12,14,15] and

Zn = [ x - X n ( O ] (1 - -1/2 (3)

(n = 1,2). If the distance between the solitons is sufficiently large, the variables x n and v n satisfy the equations [5,6]

d u n / d t = - - 4 ( - 1 ) n a n ( 1 - 02)3/2

X exp[-r(1 - 02) -1/21 - OtOn(1 - 02)

- - ½/30 n + ~ nOnf(1 - v2) 3/2 , (4)

0.3750601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


d x n / d t = o n + 2(1 - o 2) e x p [ - r ( 1 - o2) -1/2] , (5)

where o n is the topological charge (o n = -+ 1 for kink and antikink), and

o = ( v 1 +u2)/2 , r = x l ( t ) - x 2 ( t ) > O , (6)

i.e. o(t) is the mean velocity and r(t) the distance between the solltons. The applicability conditions of eqs. ( 2 ) - ( 5 ) are analysed in ref. [6]. Here we give only two main conditions,

r(t) >> (1 - 02) 1/2 , I(o + p/4)p l ¢ 1 - o 2 , (7)

where p = v 1 - o 2 is the relative velocity. The coeffi- cients ct,/3 and f are assumed to be small. From eq. (4) it follows that the mean velocity o(t) at o 1 = o 2 = o satisfies the approximate equation

dv /d t = F(o ) + O(tx 2 , /32 , f2 , a p 2 , f p 2 ) , (8)

F ( v ) = ¼ zrof(1 02) 3/2 av(1 - 02 1 - - - ~ / 3 o . ( 9 )

Eq. (8) coincides with the first.order equation for the velocity o f one sollton governed by eq. (1), when the second soliton is absent. The plots o f F ( v ) for different values o f f are shown in fig. 1. From it one may see that eq. (8) describes a relaxation of the mean velocity v(t) to the constant limit value v, which is a root of the function F(o) , 0 < Iv, [ < 1. It is also seen that there is only one real root and that the signs of

to ~ !

o.s

o,e

o,4

o.2

l l

Fig. 2. Plots o f o . versus f at different values o f ~ and 3. 1: a = 0.08, 3 = O; 2: ,', = O, 3 = 0.02; 3: o~ = 0.08, 3 = 0.02.

o, and o f are the same. At/3 = 0, the quanti ty v, has the simple analytical expression

o, = (~rof/4a)[1 + (uf/4o0 2] - 1 / 2 , (10)

which, naturally, coincides with the limit velocity of a single soliton governed by eq. (1) at/3 = 0 (cf. ref. [12]). The plots showing the dependence of v, o n f at fixed a and/3 are given in fig. 2.

From eq. (4) it also follows that

d p / d t = 8(1 - o2) 3/2 e x p [ - r ( 1 - 02) -1/2]

-- a p -- ~/3[1 + 3v2/(1 -- 02)] p

- [3pu/(1 -- 02)] do/dr

+ O(ot 2, 3 2 , f 2, o t p 2 , f p 2 ) . (11)

T

0,2-

i , i , i , i , J ,

-t.o -o.s -0.6 -~ -02 , r

Fig. 1. The f u n e t i o n F ( u ) at a = 0.08, 3 = 0.02 and different values o f f . 1: f = 0.5; 2: f = 0.4; 3: f = 0.3; 4: f = 0.2; 5: f = 0 . 1 .

At sufficiently large t, one can put in (11) v = v, = const, and, correspondingly, do~dr = 0. Besides that, one can show that at large t the term dp/dt in eq. (11) may be neglected. Then, expressing p through r f rom the obtained equation and substituting this into the equation dr /d t = p, which follows from (5), one has

r(t) = (1 - 02) 1/2 ln[87-1(1 - v2)(t + t l ) ] , (12)

where 7 is the effective dissipative coefficient at large t ,

7 = a +1]/311 + 3o2/(1 - v2)1 , (13)

and t 1 is an integration constant (which may be both positive and negative). Substituting (12) in to (11) one can see that the neglect o f d p / d t is justified if (t + t l ) 7 >> 1. At/3 = 0, one obtains f rom (12) and (13) the corresponding result o f refs. [5,6].

Now we assume t 1 > 0 and write expression (12) in the form

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r( t ) = r , + (1 - 02) 1/2 ln(1 + t / t l ) , (14)

where

r , = (1 -- o2) 1/2

X {ln[87-2(1 2 - o,)] + l n (T t l ) ) . (15)

On the other hand, t 1 is expressed through r . as follows:

t 1 = (7/8)(1 - v.2) -1 exp[(1 - v 2 . ) - l / Z r . l . (15a)

From (14) it follows that r ~ r . if

7 t l ~ l , l / 7 ~ t ' ~ t 1 , (16)

i.e. if conditions (16) are satisfied the distance between the solitons is approximately constant, which may be considered as "almost bunching". (The condition 7t >> 1 is necessary for the assumption v -~ v. . ) From (15) it follows that

r . > ( r , )mi n = (1 - 02) 1/2 ln[87-2(1 - o2)1 . (17)

The rhs of relation (17) gives a good estimation for r . if ln(Ttl) is not too large.

The logarithmic law (14) is to be compared with the linear one (r ~ t) for the unperturbed solitons at large t. We see that the effects described by the rhs of eq. (1) significantly reduce the soliton divergence. The larger the parameter 7 t l , which depends on initial conditions, the smaller the growth of the distance r(t). An explicit estimation of 7t 1 can be made in the following example.

Let initial conditions be

o1(13 ) = 0 2 0 ) = o , . (18)

Then the system of equations (11) and dr[d t = p for- mally describe the motion of a particle with unit mass under the action of a repulsive force [the first term in the rhs of eq. (11)] and friction 7P. The time scale t o of the action of the repulsive force is

t o = ½ expt½r0(1 - o2)-1/21(1 - 02) -112 , (19)

where r 0 = r(0). Assume that 7t o ~ 1. Then the fric- t ion may be neglected at t ~ to, and integration o f the equations of motion gives [6]

r(t) = r 0 + 2(1 - 02) 1/2 l n [ch ( t / t o ) ] . (20)

The interval 0 < t < t o ~< 7 -1 may be considered as the first stage of motion. In the second stage,

,),-1 ~ t ~ t 1 , (21)

the main role is played by the friction term and integration of the equations of motion gives

r = r . + C e -Tt , (22)

where r . and C are constants of integration. In the time interval t o ,~ t '~ 3'- 1, the solutions (20) and (22) can be approximately sewed together. Therefore one may substitute (20) into the equation d r / d t

= 7(% - r) which is equivalent to (22). As a result we have

r . ~ 2(7t0)-1(1 - 02) 1/2 + r 0 . (23)

Interval (21) in which, according to (22), r -+ r . , may be considered as the stage of bunching. Estimation (23) is valid under the conditions 7t 0 ,~ 1 and (18). Substituting (23) into (15a), one obtains for this case

t 1 ~ (T /g ) (1 -- 02) - 1

X exp[2(Tt0) -1 + r0(1 _ %]" 2-,-1/211 • (24)

We see that if 7t 0 < 1, the value of 7t 1 is very large. Eqs. (22) - (24) are the extensions of results of refs. [5,6] to # 4: 0.

If 7t 0 >~ 1 and (or) condition (18) does not hold, estimations (23) and (24), generally speaking, are not valid. However, one can still use expressions ( 1 4 ) - (17) at 7(t + t I >> 1.

The above-stated is illustrated in fig. 3. The full

" t

6

5

0

- . : . . . . : . . . . . i . : : . . . . . . . . . . . . . ::::

. . . . . . . . . . - e

, , , . . . . . . , , ,

100 200 ,lO0 ~O0 50O ~00 700 800 9OO 1ooo t

Fig. 3. Plots of r versus t at vl(0) = 0.77, v2(O) = 0.93,r(0) = 1.75 and f = 0.1644. a: numerical solution of eq. (1) at a = 0.08 and 0 = 0.02 with be (25); b: numerical solution of eq~ (4) and (5) at a = 0.08 and 0 = 0; c: numerical solution of eq. (1) at a = 0.08 and 13 = 0 with be (25); d: numerical solution of eq. (1) with periodic be; e: numerical solution of cq~ (4) and (5) at a = 0, ~ = 0.02; f: numerical solution of eq. (1) at a = 0 and # = 0.02 with bc (25); g: plot of expression (12) at ~ = 0, B = 0.02 and tz = - 1 1 0 , v . = 0.93.

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line b represents the numerical solution of the system (4), (5) at parameters indicated in the caption. Though here 7t 0 is not too small (Tt 0 ~ 0.5), the stage of bunching, where r ~- r . ~ 5.75, is reached rather soon and takes place with a good accuracy during a large time interval. The dots constituting the line c are obtained from the numerical solution of eq. (1) at the same parameters as for eqs. (4), (5) and the boundary

nditions (bc) corresponding to the full reflection cq fr-om the ends a and b of the junction

¢x(a, t) = O, ~ox(b, 0 = 0 . (25)

It is seen that lines b and c are rather close to each other, which shows that eqs. (4) and (5) give, to some extent, a good description o f soliton interaction. One can see, however, a slight discrepancy between the curves b and c at large t (t > 800). This may be under- stood if one takes into account a change of the nature of the interaction between solitons in the course of the reflection process, which is not described by eqs. (4) and (5), valid for the soliton mot ion in the infinite interval. It is known that the reflection of a soliton from a junction terminal is accompanied by a change of its topological charge. When the leading soliton of the bunched pair is reflected while the second one is not yet, there is an attraction between them instead of repulsion. During this t ime interval the distance between the solitons reduces and that may compensate or even surpass a slow growth of r(t) described by (14). Therefore, a plateau or even a slight declination of r(t) instead of logarithmic growth may appear due to the reflections from the junction terminals. The height of the plateau, i.e. the distance between bunched solitons, is approximately equal to r , defined by eq. (15). This idea is supported by results o f numerical solution o feq . (1) with the periodic bc ~Ox(a , t) = ~x(b, t), which corresponds to soliton propagation in a ring and imitate to some extent an infinite line, if the size of the bunched pair is sufficiently smaller than the ring length. The curve d which corresponds to the periodic bc and the same parameters and initial conditions as for b and c, shows an increase of r(t) similar to that shown by curve b.

The dots forming the line a are obtained from the numerical solution of eq. (1) with the same initial conditions and be (25), but fl #: 0 (/~ < a). It is seen again that it does not show an increase ofr ( t ) . Therefore, in this case the reflections also compensate the logarithmic growth of r(t).

However, if 3,t 1 is insufficiently large, the compen- sation does not take place. As an example, consider the case a = 0,/3 = 0.02 a n d f = 0.1644 which gives o. = 0.93 and 3' = 0.14. The full line e represents the solution of eqs. (4), (5) and the dots constituting line f are obtained from the numerical solution o f eq. (1) with the same parameters and bc (25). The curves e and f are very close to each other showing the same growth of r(O up to the end of computat ion time t

103 . Evidently, there is no bunching here. It is seen that at t > 300, both curves, e and f, are very close to the dashed line g which is a plot o f eq . (12) with t 1

- 1 1 0 and o. = 0.93. This is connected with the fact that here the parameter r . , defined by eq. (15), loses its sense because 7 t 1 < 0. Accordingly, the logarithmic growth described by asymptotic expression (12) is too big to be compensated by the short attraction periods in the course of the reflections.

In addition, it is worth mentioning that as far as the topological charge o changes after a reflection, only those dots in fig. 3 are shown which correspond to the instants when bo th solitons are sufficiently far from the ends and, consequently, have equal topological charge. The picture showing the soliton pair in the course of a reflection process is presented in fig. 4. It may be seen how the soliton pair completely re-

t

X

Fig. 4. Plot of ~ox(t, x) obtained from the numerical solution of eq. (1) with bc (25) at ul(0) = 0.77, u2(0) = 0.93, r(O) = 1.75, a = 0,08, # = 0.02 and f = 0.1644 in space-time in- tervals -8 < x < 10, 1000.8 < t < 1020.8. An illustration of the reflection of a bunched soliton pair from a junction end and subsequent total recovery of the pair with the change of the sign of the topological charge.

75


stores itself after the reflection, changing only the sign of its topological charge. We are not discussing here other consequences of the reflections, in particular the transformation of a flttxon-antifluxon pair into the bunched system of fluxons with equal topological charges [16,9].

In conclusion, the theory of soliton buching in Josephson junctions [5,6] has been generalized, refined, and supported by new numerical experiments. In the infinite junction with bias current and dissipation, the distance between solitons with equal topological charge slowly increases according to eq. (12), if ~(f + t l ) >~ 1. This may be called "logarithmic bunching". In the junctions with terminals, the soliton reflections may reduce the logarithmic growth of the distance if ~/t 1 is sufficiently large. In this case the ex- istence of soliton bunches, stable in a long time interval, is possible and the distance between bunched sol- irons is approximately equal to r . [eq. (15)]. If, however, ~,t 1 is small enough, in particular, if t 1 < 0, the logarithmic growth of r(t) is too strong to be compensated by the reflections. Some estimations show that t 1 decreases with an increase of 3' and (or) the mean velocity o. of a soliton pair.

References

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On a mechanism of soliton bunching in Josephson junctions