- Email: [email protected]
About 4π-kink in a Josephson junction
27 December 1999
Physics Letters A 264 Ž1999. 324–327 www.elsevier.nlrlocaterphysleta
About 4p-kink in a Josephson junction V.P. Silin a
a,)
, A.V. Studenov
a
P.N. LebedeÕ Physical Institute, Russian Academy of Sciences, Moscow 117924, Russia Received 3 November 1999; accepted 18 November 1999 Communicated by V.M. Agranovich
Abstract The first analytical description of the 4p-kink Josephson vortex is presented for the model of the long Josephson junction with the sine-nonlinearity. The weakly nonlocal approach is used. This approach is efficient for the description of the vortices moving with the velocities which are close to the Swihart velocity. This new analytical result is obtained in the limit when the Josephson critical current density is not high enough. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 74.50.q r; 74.60.Ge Keywords: Low density of Josephson critical current; 4p-kink
Researchers are interested in 4p-kink vortex structures in a long Josephson junction ŽLJJ. because some experiments, for example w1x, point to the existence of multikinks carrying more then one magnetic flux quantum. The first theoretical result on this subject was published in Refs. w2,3x, where the analytical description of 4p-kink was obtained in terms of nonlocal electrodynamics of the LJJ with the sine nonlinearity. This vortex structure moves with definite velocity of propagation and carries two magnetic flux quanta. Shortly after Refs. w2,3x results of numerical investigations were published w4x. In Ref. w4x, the vortex moving with definite velocity was also obtained, as it was predicted in Refs. w2,3x. It should
) Corresponding author. Tel.: q7-095-1357808; fax: q7-0951357880; e-mail: [email protected]
be mentioned that the authors of Ref. w4x suggested using the model of weak nonlocality, which takes into account fourth spatial derivative in addition to the conventional sine-Gordon operator w5x. The analytical investigation of the 4p-kink structure was continued in terms of nonlocal generalization of Josephson electrodynamics with the sine nonlinearity in Ref. w6x. The results of this paper are important for our communication. Thus, by this year Ž1999. there have been two analytical results concerned with 4p-kink of Abrikosov–Josephson vortices ŽAJV’s. of nonlocal electrodynamics. The first result mentioned above describes the AJV in the LJJ between two bulk superconductors w2,3x. The second one representing the moving AJV in the LJJ between two superconducting electrodes of finite thickness Žsandwich geometry. was obtained in Ref. w6x. We use the latter
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 8 1 5 - 4
V.P. Silin, A.V. StudenoÕr Physics Letters A 264 (1999) 324–327
result in our communication. Before our 4p-kink will be presented, we ought to mention the paper w7x published in this year Ž1999., which has demonstrated a number of analytical solutions obtained in terms of the Aubry–Volkov ŽAV. model for the LJJ w8–11x. This model uses a triangular function instead of the conventional sine nonlinearity of the sineGordon equation and permits to obtain a simple analytical description of multikinks both in weak and strong nonlocal limits. Thus the 4p-kink vortex structures known today are an infinite set of Josephson vortices ŽJV. of the AV model w7x, and only two analytical solutions obtained in the model of strong nonlocality with the sine nonlinearity w2,3,6x. It should be emphasized that the latter results were obtained for the LJJ under the extraordinary condition when Josephson critical current density is high enough. In the present paper we obtain the first analytical solution of Josephson electrodynamics with the sine nonlinearity, which describes a travelling 4p-kink in the LJJ with small enough Josephson critical current density: "c 2 jc < j0 s
16p < e < l3
2 f 10 4lmy3 , m Arcm
Ž 1.
where l denotes the London depth. Our new description of the moving JV is derived by means of the nonlocal result of Ref. w6x, which is based on the following equation w12x for the phase difference w : sin w q
1 E 2w
v j2 E t 2
l y 2 ds
Hy`
where the value D and velocity Õ are defined from the equations:
cot
p ds
ž /
dzX
Ew
sinh p Ž zX y z . r2 d s
E zX
s 0, Ž 2 .
½
sinh "p Ž z y Õt . rD cos w p d srD x
s
q 2D
y
5
,
Ž 3.
(ž
(ž pl 2D
pl 2D
/
1 q 27 1r3
2
/
1r3
2
pl
1 q
y 27
ž / 2D
,
Ž 4. Õ 2 s v j2
2
D
cot 2
p ds
ž / ž / p
D
.
Ž 5.
The strongly nonlocal limit of the solution Ž3. l < d s has been discussed in Ref. w6x. In this communication the opposite limit is used: l 4 ds ,
Ž 6.
when, as it follows from the formulae Ž3. – Ž5., we have: 4
½
w s 4 arctan sinh " Ž z y Õt . 2
ds
3
l
ž ( /
Õ 2 s v j2 ld s 1 y
(ld
3 s
.
5
,
Ž 7. Ž 8.
On the other hand it is easy to understand that the limit Ž6. is that of weak nonlocality of Eq. Ž2., when it takes the form:
v j2 E t 2
E 2w q sin w s
Ez 2
´ 2 E 4w q
3 Ez 4
,
Ž 9.
where z s zr ld s , and ´ 2 s Ž d srl . is the small parameter at the highest derivative. Eq. Ž9. is some generalization of the conventional sine-Gordon equation which is the basic equation of the local theory of the LJJ Žcf. w4x.. It is important to underline that the formulae Ž7. and Ž8. represent the exact solution of the 4p-kink type of Eq. Ž9. determining weak nonlocal limit. To demonstrate further generalization of the results presented by the expressions Ž7.,Ž8. we use the general equation of nonlocal Josephson electrody-
(
where v j and l j denote the Josephson frequency and length, respectively, l s l2j rl, d s is the thickness of superconducting electrodes separated by a tunnel junction. The solution of Eq. Ž2. of the 4p-kink type has the following form: w6x:
w Ž z ,t . s 4 arctan
pl
D
1 E 2w
q`
325
V.P. Silin, A.V. StudenoÕr Physics Letters A 264 (1999) 324–327
326
namics of the LJJ between two bulk superconductors w13x: sin w q
s
1 E 2w
Using the general nonlocal theory of the LJJ presented in Refs. w2,3x, one can obtain the magnetic field H and total energy E corresponding to the 4p-kink Ž13.:
v j2 E t 2
l2j E 2 pl E z 2
q`
X
Hy` dz K
0
ž
< z y zX <
l
F0
H Ž x , z ,t . s
/
X
w Ž z ,t . ,
pll j
Ž 10 .
This limit usually gives us the sine-Gordon equation. However, when we take into account the small but finite ratio lrl j , we obtain Eq. Ž9. in the first approximation Žcf. w4,5x., where
z s zrl j ,
´ s
Žl
rl2j
..
"jc
Es
2 < e< q 2
Ž 12 .
And consequently the travelling 4p-kink has the form:
( ½ 5 ' 4 arctan ½ sinh " Ž z y Õt . l ( Ž 1 y Õ rc . 5 ,
,
ll j'3r2
j
3 2
2
2 s
q`
1
s
w s 4 arctan sinh " Ž z y Õt .
2 s
exp w y< x
( Ž1 y Õ rc . 3 2
2
2 s
,
Ž 15 . Ž 11 .
2
2
cosh Ž z y Õt . l j
lj 4 l.
3 2
3 2
=
where K 0 Ž x . is the McDonald function. The condition Ž1. corresponds to
2
( Ž1 y Õ rc .
½
Hy`
2 v j2 2
Ew l2j
ž / Ez
2
Ew
1
ž /
q Ž 1 y cos w .
Et
1 y 4
l2l2j
2
E 2w
ž /5 E z2
F 02 7 y Õ 2rcs2 6p 3ll j
( Ž1 y Õ rc . 3 2
2
2 s
F 02 p 3ll j
( Ž1 y Õ rc . 3 2
2
2 s
.
Ž 16 .
The conventional 2p-kink of local Josephson electrodynamics w5x has the well-known form:
Ž 13 .
(
w J s 4 arctan exp " Ž z y Õt . l j 1 y Õ 2rcs2 2 l
ž ( /
Õ 2 s cs2 Ž 1 y 23 ´ . s cs2 1 y
3 lj
½
- cs2 ,
Ž 14 .
where cs s v j l j denotes the Swihart velocity. Eqs. Ž13. and Ž14. give us the analytical description of the 4p-kink JV of the LJJ between the bulk superconductors in the traditional limit of the small enough Josephson critical current density Ž1.. Firstly, it should be stressed that the size of the vortex Ž13. f ll j is much larger than the London depth l, what allows us to consider Eq. Ž9. and to neglect higher derivatives in representing Eq. Ž10. in the differential approximation. On the other hand the spatial scale of the 4p-kink Ž13. is smaller than the Josephson length in accordance with the condition Ž11.. Secondly, we point out that the velocity Ž14. of the 4p-kink is smaller than the Swihart velocity Žcf. w4,5x., although the former is close to the latter one.
5
' p q 2 arctan
(
= sinh " Ž z y Õt . l j 1 y Õ 2rcs2
½
5,
Ž 17 .
which gives us the following formulae for the magnetic field and energy w5x: HJ s
F0
(
2pll j 1 y Õ 2rcs2
(
exp w y< x
=
(
cosh Ž z y Õt . l j 1 y Õ 2rcs2 EJ s
F 02
(
4p 3ll j 1 y Õ 2rcs2
.
,
Ž 18 .
Ž 19 .
We see from Eq. Ž15. that the vortex Ž13. bears two magnetic flux quanta, what differs it from the
V.P. Silin, A.V. StudenoÕr Physics Letters A 264 (1999) 324–327
conventional Josephson vortex Ž17. carrying one magnetic flux quantum. The spatial size along the z-direction of our new 4p-kink Ž13. differs from that of the traditional 2p-kink Ž17. by the multiplier 3r2 . This leads to the quantitative distinction in the formulae for magnetic fields H and HJ as well as in the expressions for energies E and EJ . However, the most important difference between the 4p-kink Ž13. and the conventional Josephson vortex Ž17. is in the velocities of their motion. The velocity of the 2p- kink can vary over a wide range Ž0 - Õ - cs ., whereas the velocity of the 4p-kink is the constant defined by Eq. Ž14.. Eqs. Ž13., Ž15. and Ž16. for the 4p-kink and Eqs. Ž17. – Ž19. for the 2p-kink which are written in terms of the velocity Õ, are alike, but this analogy is perfunctory. Moreover, Eq. Ž9. with the small parameter at the highest derivative must be seriously examined to find more exact description for the Josephson vortex of the 2p-kink type. It is necessary to underline once more that the expression Ž13. represents the exact solution of Eq. Ž9.. Conversely, the formula Ž17. may be used to describe some approximate solutions of Eq. Ž9., which are valid when Ž1 y Õ 2rcs2 . is not small enough. Our solution Ž13.,Ž14. of Eq. Ž9. is the first analytical solution describing a moving 4p-kink JV in the case of the LJJ with the low enough density of the Josephson critical current jc Ž1.. Naturally this solution is to be used only for the vortices moving with the velocity value which is close to the Swihart one. It should be emphasized that the solution Ž13. describes a spatially monotonous JV as well as the vortices of w2,3,6x do. One might expect that this
327
solution can be used as a base to obtain nonmonotonous 4p-kink vortices connected with the Cherenkov trapping of Swihart waves into the moving Josephson vortices w7,14x.
'
Acknowledgements This work was partially supported by the Scientific Council on High-Temperature Superconductors ŽProject No. 99002. and the Government Program in Support of the Leading Scientific Schools ŽProject No. 96-15-96750..
References w1x B. Dueholm, O.A. Levring, J. Mygind, Phys. Rev. Lett. 46 Ž1981. 1299. w2x Yu.M. Aliev, V.P. Silin, Phys. Lett. A 117 Ž1993. 295. w3x Yu.M. Aliev, V.P. Silin, JETP 77 Ž1993. 142. w4x G.L. Alfimov, V.M. Eleonsky, N.E. Kulagin, N.V. Mitzkevich, Chaos 3 Ž1993. 405. w5x A. Barone, G. Paterno, Physics and Applications of the Josephson Effect, Wiley, New York, 1982. w6x V.P. Silin, S.A. Uryupin, JETP 81 Ž1995. 1179. w7x A.S. Malishevskii, V.P. Silin, S.A. Uryupin, Phys. Lett. A 253 Ž1999. 333. w8x S. Aubry, P.J. Le Daeron, Physica D 7 Ž1983. 240. w9x S. Aubry, J. Phys. C 16 Ž1983. 2497. w10x A.F. Volkov, Physica C 183 Ž1991. 177. w11x A.F. Volkov, Physica C 192 Ž1992. 306. w12x G.L. Alfimov, A.F. Popkov, Phys. Rev. B 52 Ž1995. 4503. w13x Yu.M. Aliev, V.P. Silin, S.A. Uryupin, Supercond.: Phys. Chem. Technol. 5 Ž1992. 230. w14x A.S. Malishevskii, V.P. Silin, S.A. Uryupin, Pis’ma Zh. Eksp. Teor. Fiz. 69 Ž1999. 318.
Physics Letters A 264 Ž1999. 324–327 www.elsevier.nlrlocaterphysleta
About 4p-kink in a Josephson junction V.P. Silin a
a,)
, A.V. Studenov
a
P.N. LebedeÕ Physical Institute, Russian Academy of Sciences, Moscow 117924, Russia Received 3 November 1999; accepted 18 November 1999 Communicated by V.M. Agranovich
Abstract The first analytical description of the 4p-kink Josephson vortex is presented for the model of the long Josephson junction with the sine-nonlinearity. The weakly nonlocal approach is used. This approach is efficient for the description of the vortices moving with the velocities which are close to the Swihart velocity. This new analytical result is obtained in the limit when the Josephson critical current density is not high enough. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 74.50.q r; 74.60.Ge Keywords: Low density of Josephson critical current; 4p-kink
Researchers are interested in 4p-kink vortex structures in a long Josephson junction ŽLJJ. because some experiments, for example w1x, point to the existence of multikinks carrying more then one magnetic flux quantum. The first theoretical result on this subject was published in Refs. w2,3x, where the analytical description of 4p-kink was obtained in terms of nonlocal electrodynamics of the LJJ with the sine nonlinearity. This vortex structure moves with definite velocity of propagation and carries two magnetic flux quanta. Shortly after Refs. w2,3x results of numerical investigations were published w4x. In Ref. w4x, the vortex moving with definite velocity was also obtained, as it was predicted in Refs. w2,3x. It should
) Corresponding author. Tel.: q7-095-1357808; fax: q7-0951357880; e-mail: [email protected]
be mentioned that the authors of Ref. w4x suggested using the model of weak nonlocality, which takes into account fourth spatial derivative in addition to the conventional sine-Gordon operator w5x. The analytical investigation of the 4p-kink structure was continued in terms of nonlocal generalization of Josephson electrodynamics with the sine nonlinearity in Ref. w6x. The results of this paper are important for our communication. Thus, by this year Ž1999. there have been two analytical results concerned with 4p-kink of Abrikosov–Josephson vortices ŽAJV’s. of nonlocal electrodynamics. The first result mentioned above describes the AJV in the LJJ between two bulk superconductors w2,3x. The second one representing the moving AJV in the LJJ between two superconducting electrodes of finite thickness Žsandwich geometry. was obtained in Ref. w6x. We use the latter
0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 8 1 5 - 4
V.P. Silin, A.V. StudenoÕr Physics Letters A 264 (1999) 324–327
result in our communication. Before our 4p-kink will be presented, we ought to mention the paper w7x published in this year Ž1999., which has demonstrated a number of analytical solutions obtained in terms of the Aubry–Volkov ŽAV. model for the LJJ w8–11x. This model uses a triangular function instead of the conventional sine nonlinearity of the sineGordon equation and permits to obtain a simple analytical description of multikinks both in weak and strong nonlocal limits. Thus the 4p-kink vortex structures known today are an infinite set of Josephson vortices ŽJV. of the AV model w7x, and only two analytical solutions obtained in the model of strong nonlocality with the sine nonlinearity w2,3,6x. It should be emphasized that the latter results were obtained for the LJJ under the extraordinary condition when Josephson critical current density is high enough. In the present paper we obtain the first analytical solution of Josephson electrodynamics with the sine nonlinearity, which describes a travelling 4p-kink in the LJJ with small enough Josephson critical current density: "c 2 jc < j0 s
16p < e < l3
2 f 10 4lmy3 , m Arcm
Ž 1.
where l denotes the London depth. Our new description of the moving JV is derived by means of the nonlocal result of Ref. w6x, which is based on the following equation w12x for the phase difference w : sin w q
1 E 2w
v j2 E t 2
l y 2 ds
Hy`
where the value D and velocity Õ are defined from the equations:
cot
p ds
ž /
dzX
Ew
sinh p Ž zX y z . r2 d s
E zX
s 0, Ž 2 .
½
sinh "p Ž z y Õt . rD cos w p d srD x
s
q 2D
y
5
,
Ž 3.
(ž
(ž pl 2D
pl 2D
/
1 q 27 1r3
2
/
1r3
2
pl
1 q
y 27
ž / 2D
,
Ž 4. Õ 2 s v j2
2
D
cot 2
p ds
ž / ž / p
D
.
Ž 5.
The strongly nonlocal limit of the solution Ž3. l < d s has been discussed in Ref. w6x. In this communication the opposite limit is used: l 4 ds ,
Ž 6.
when, as it follows from the formulae Ž3. – Ž5., we have: 4
½
w s 4 arctan sinh " Ž z y Õt . 2
ds
3
l
ž ( /
Õ 2 s v j2 ld s 1 y
(ld
3 s
.
5
,
Ž 7. Ž 8.
On the other hand it is easy to understand that the limit Ž6. is that of weak nonlocality of Eq. Ž2., when it takes the form:
v j2 E t 2
E 2w q sin w s
Ez 2
´ 2 E 4w q
3 Ez 4
,
Ž 9.
where z s zr ld s , and ´ 2 s Ž d srl . is the small parameter at the highest derivative. Eq. Ž9. is some generalization of the conventional sine-Gordon equation which is the basic equation of the local theory of the LJJ Žcf. w4x.. It is important to underline that the formulae Ž7. and Ž8. represent the exact solution of the 4p-kink type of Eq. Ž9. determining weak nonlocal limit. To demonstrate further generalization of the results presented by the expressions Ž7.,Ž8. we use the general equation of nonlocal Josephson electrody-
(
where v j and l j denote the Josephson frequency and length, respectively, l s l2j rl, d s is the thickness of superconducting electrodes separated by a tunnel junction. The solution of Eq. Ž2. of the 4p-kink type has the following form: w6x:
w Ž z ,t . s 4 arctan
pl
D
1 E 2w
q`
325
V.P. Silin, A.V. StudenoÕr Physics Letters A 264 (1999) 324–327
326
namics of the LJJ between two bulk superconductors w13x: sin w q
s
1 E 2w
Using the general nonlocal theory of the LJJ presented in Refs. w2,3x, one can obtain the magnetic field H and total energy E corresponding to the 4p-kink Ž13.:
v j2 E t 2
l2j E 2 pl E z 2
q`
X
Hy` dz K
0
ž
< z y zX <
l
F0
H Ž x , z ,t . s
/
X
w Ž z ,t . ,
pll j
Ž 10 .
This limit usually gives us the sine-Gordon equation. However, when we take into account the small but finite ratio lrl j , we obtain Eq. Ž9. in the first approximation Žcf. w4,5x., where
z s zrl j ,
´ s
Žl
rl2j
..
"jc
Es
2 < e< q 2
Ž 12 .
And consequently the travelling 4p-kink has the form:
( ½ 5 ' 4 arctan ½ sinh " Ž z y Õt . l ( Ž 1 y Õ rc . 5 ,
,
ll j'3r2
j
3 2
2
2 s
q`
1
s
w s 4 arctan sinh " Ž z y Õt .
2 s
exp w y< x
( Ž1 y Õ rc . 3 2
2
2 s
,
Ž 15 . Ž 11 .
2
2
cosh Ž z y Õt . l j
lj 4 l.
3 2
3 2
=
where K 0 Ž x . is the McDonald function. The condition Ž1. corresponds to
2
( Ž1 y Õ rc .
½
Hy`
2 v j2 2
Ew l2j
ž / Ez
2
Ew
1
ž /
q Ž 1 y cos w .
Et
1 y 4
l2l2j
2
E 2w
ž /5 E z2
F 02 7 y Õ 2rcs2 6p 3ll j
( Ž1 y Õ rc . 3 2
2
2 s
F 02 p 3ll j
( Ž1 y Õ rc . 3 2
2
2 s
.
Ž 16 .
The conventional 2p-kink of local Josephson electrodynamics w5x has the well-known form:
Ž 13 .
(
w J s 4 arctan exp " Ž z y Õt . l j 1 y Õ 2rcs2 2 l
ž ( /
Õ 2 s cs2 Ž 1 y 23 ´ . s cs2 1 y
3 lj
½
- cs2 ,
Ž 14 .
where cs s v j l j denotes the Swihart velocity. Eqs. Ž13. and Ž14. give us the analytical description of the 4p-kink JV of the LJJ between the bulk superconductors in the traditional limit of the small enough Josephson critical current density Ž1.. Firstly, it should be stressed that the size of the vortex Ž13. f ll j is much larger than the London depth l, what allows us to consider Eq. Ž9. and to neglect higher derivatives in representing Eq. Ž10. in the differential approximation. On the other hand the spatial scale of the 4p-kink Ž13. is smaller than the Josephson length in accordance with the condition Ž11.. Secondly, we point out that the velocity Ž14. of the 4p-kink is smaller than the Swihart velocity Žcf. w4,5x., although the former is close to the latter one.
5
' p q 2 arctan
(
= sinh " Ž z y Õt . l j 1 y Õ 2rcs2
½
5,
Ž 17 .
which gives us the following formulae for the magnetic field and energy w5x: HJ s
F0
(
2pll j 1 y Õ 2rcs2
(
exp w y< x
=
(
cosh Ž z y Õt . l j 1 y Õ 2rcs2 EJ s
F 02
(
4p 3ll j 1 y Õ 2rcs2
.
,
Ž 18 .
Ž 19 .
We see from Eq. Ž15. that the vortex Ž13. bears two magnetic flux quanta, what differs it from the
V.P. Silin, A.V. StudenoÕr Physics Letters A 264 (1999) 324–327
conventional Josephson vortex Ž17. carrying one magnetic flux quantum. The spatial size along the z-direction of our new 4p-kink Ž13. differs from that of the traditional 2p-kink Ž17. by the multiplier 3r2 . This leads to the quantitative distinction in the formulae for magnetic fields H and HJ as well as in the expressions for energies E and EJ . However, the most important difference between the 4p-kink Ž13. and the conventional Josephson vortex Ž17. is in the velocities of their motion. The velocity of the 2p- kink can vary over a wide range Ž0 - Õ - cs ., whereas the velocity of the 4p-kink is the constant defined by Eq. Ž14.. Eqs. Ž13., Ž15. and Ž16. for the 4p-kink and Eqs. Ž17. – Ž19. for the 2p-kink which are written in terms of the velocity Õ, are alike, but this analogy is perfunctory. Moreover, Eq. Ž9. with the small parameter at the highest derivative must be seriously examined to find more exact description for the Josephson vortex of the 2p-kink type. It is necessary to underline once more that the expression Ž13. represents the exact solution of Eq. Ž9.. Conversely, the formula Ž17. may be used to describe some approximate solutions of Eq. Ž9., which are valid when Ž1 y Õ 2rcs2 . is not small enough. Our solution Ž13.,Ž14. of Eq. Ž9. is the first analytical solution describing a moving 4p-kink JV in the case of the LJJ with the low enough density of the Josephson critical current jc Ž1.. Naturally this solution is to be used only for the vortices moving with the velocity value which is close to the Swihart one. It should be emphasized that the solution Ž13. describes a spatially monotonous JV as well as the vortices of w2,3,6x do. One might expect that this
327
solution can be used as a base to obtain nonmonotonous 4p-kink vortices connected with the Cherenkov trapping of Swihart waves into the moving Josephson vortices w7,14x.
'
Acknowledgements This work was partially supported by the Scientific Council on High-Temperature Superconductors ŽProject No. 99002. and the Government Program in Support of the Leading Scientific Schools ŽProject No. 96-15-96750..
References w1x B. Dueholm, O.A. Levring, J. Mygind, Phys. Rev. Lett. 46 Ž1981. 1299. w2x Yu.M. Aliev, V.P. Silin, Phys. Lett. A 117 Ž1993. 295. w3x Yu.M. Aliev, V.P. Silin, JETP 77 Ž1993. 142. w4x G.L. Alfimov, V.M. Eleonsky, N.E. Kulagin, N.V. Mitzkevich, Chaos 3 Ž1993. 405. w5x A. Barone, G. Paterno, Physics and Applications of the Josephson Effect, Wiley, New York, 1982. w6x V.P. Silin, S.A. Uryupin, JETP 81 Ž1995. 1179. w7x A.S. Malishevskii, V.P. Silin, S.A. Uryupin, Phys. Lett. A 253 Ž1999. 333. w8x S. Aubry, P.J. Le Daeron, Physica D 7 Ž1983. 240. w9x S. Aubry, J. Phys. C 16 Ž1983. 2497. w10x A.F. Volkov, Physica C 183 Ž1991. 177. w11x A.F. Volkov, Physica C 192 Ž1992. 306. w12x G.L. Alfimov, A.F. Popkov, Phys. Rev. B 52 Ž1995. 4503. w13x Yu.M. Aliev, V.P. Silin, S.A. Uryupin, Supercond.: Phys. Chem. Technol. 5 Ž1992. 230. w14x A.S. Malishevskii, V.P. Silin, S.A. Uryupin, Pis’ma Zh. Eksp. Teor. Fiz. 69 Ž1999. 318.