Embedded soliton dynamics in the asymmetric array of Josephson junctions
The dc-biased annular array of three-junction asymmetric superconducting quantum interference devices (SQUIDs) is investigated. The existence of embedded solitons (solitons that exist despite the resonance with the linear waves) is demonstrated both in the unbiased Hamiltonian limit and in the dc-bi...
Збережено в:
| Опубліковано в: : | Физика низких температур |
|---|---|
| Дата: | 2017 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2017
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/129514 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Embedded soliton dynamics in the asymmetric array of Josephson junctions / Ivan О. Starodub Yaroslav Zolotaryuk // Физика низких температур. — 2017. — Т. 43, № 6. — С. 828-834. — Бібліогр.: 40 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-129514 |
|---|---|
| record_format |
dspace |
| spelling |
Starodub, Ivan О. Zolotaryuk, Yaroslav 2018-01-19T20:34:45Z 2018-01-19T20:34:45Z 2017 Embedded soliton dynamics in the asymmetric array of Josephson junctions / Ivan О. Starodub Yaroslav Zolotaryuk // Физика низких температур. — 2017. — Т. 43, № 6. — С. 828-834. — Бібліогр.: 40 назв. — англ. 0132-6414 PACS: 05.45.Yv, 63.20.Ry, 05.45.–a, 03.75.Lm https://nasplib.isofts.kiev.ua/handle/123456789/129514 The dc-biased annular array of three-junction asymmetric superconducting quantum interference devices (SQUIDs) is investigated. The existence of embedded solitons (solitons that exist despite the resonance with the linear waves) is demonstrated both in the unbiased Hamiltonian limit and in the dc-biased and damped case on the current-voltage characteristics (CVCs) of the array. The existence diagram on the parameter plane is constructed. The signatures of the embedded solitons manifest themselves as inaccessible voltage intervals on the CVCs. The upper boundary of these intervals is proportional to the embedded soliton velocity. The authors acknowledge the financial support from the National Academy of Sciences of Ukraine (NAS of Ukraine) under the Project No. 0112U000053 and from the Project 1/30-2015 “Dynamics and topological structures in Bose–Einstein condensates of ultracold gases” of the Kyiv National University Branch Target Training Program at the NAS of Ukraine. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Свеpхпpоводимость, в том числе высокотемпеpатуpная Embedded soliton dynamics in the asymmetric array of Josephson junctions Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Embedded soliton dynamics in the asymmetric array of Josephson junctions |
| spellingShingle |
Embedded soliton dynamics in the asymmetric array of Josephson junctions Starodub, Ivan О. Zolotaryuk, Yaroslav Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| title_short |
Embedded soliton dynamics in the asymmetric array of Josephson junctions |
| title_full |
Embedded soliton dynamics in the asymmetric array of Josephson junctions |
| title_fullStr |
Embedded soliton dynamics in the asymmetric array of Josephson junctions |
| title_full_unstemmed |
Embedded soliton dynamics in the asymmetric array of Josephson junctions |
| title_sort |
embedded soliton dynamics in the asymmetric array of josephson junctions |
| author |
Starodub, Ivan О. Zolotaryuk, Yaroslav |
| author_facet |
Starodub, Ivan О. Zolotaryuk, Yaroslav |
| topic |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| topic_facet |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| publishDate |
2017 |
| language |
English |
| container_title |
Физика низких температур |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
The dc-biased annular array of three-junction asymmetric superconducting quantum interference devices (SQUIDs) is investigated. The existence of embedded solitons (solitons that exist despite the resonance with the linear waves) is demonstrated both in the unbiased Hamiltonian limit and in the dc-biased and damped case on the current-voltage characteristics (CVCs) of the array. The existence diagram on the parameter plane is constructed. The signatures of the embedded solitons manifest themselves as inaccessible voltage intervals on the CVCs. The upper boundary of these intervals is proportional to the embedded soliton velocity.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/129514 |
| citation_txt |
Embedded soliton dynamics in the asymmetric array of Josephson junctions / Ivan О. Starodub Yaroslav Zolotaryuk // Физика низких температур. — 2017. — Т. 43, № 6. — С. 828-834. — Бібліогр.: 40 назв. — англ. |
| work_keys_str_mv |
AT starodubivano embeddedsolitondynamicsintheasymmetricarrayofjosephsonjunctions AT zolotaryukyaroslav embeddedsolitondynamicsintheasymmetricarrayofjosephsonjunctions |
| first_indexed |
2025-11-25T20:35:32Z |
| last_indexed |
2025-11-25T20:35:32Z |
| _version_ |
1850526411412996096 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6, pp. 828–834
Embedded soliton dynamics in the asymmetric array
of Josephson junctions
Ivan О. Starodub and Yaroslav Zolotaryuk
Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine
14b vul. Metrologichna, Kyiv 03680, Ukraine
E-mail: starodub@bitp.kiev.ua; yzolo@bitp.kiev.ua
Received July 29, 2016 published online April 25, 2017
The dc-biased annular array of three-junction asymmetric superconducting quantum interference devices
(SQUIDs) is investigated. The existence of embedded solitons (solitons that exist despite the resonance with the
linear waves) is demonstrated both in the unbiased Hamiltonian limit and in the dc-biased and damped case on
the current-voltage characteristics (CVCs) of the array. The existence diagram on the parameter plane is con-
structed. The signatures of the embedded solitons manifest themselves as inaccessible voltage intervals on the
CVCs. The upper boundary of these intervals is proportional to the embedded soliton velocity.
PACS: 05.45.Yv Solitons;
63.20.Ry Anharmonic lattice modes;
05.45.–a Nonlinear dynamics and chaos;
03.75.Lm Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vor-
tices, and topological excitations.
Keywords: superconducting quantum interference devices, current-voltage characteristics (CVCs), Josephson
junctions.
1. Introduction
Nonlinear wave phenomena are ubiquitous in the Jo-
sephson junctions (JJs) [1,2]. It is a well established fact
that such excitations as breathers [3] and topological
solitons [4] can occur in systems that are based on the Jo-
sephson effect. This article will be focused on topological
solitons or, as they are known in physics of JJs, fluxons.
The origin of the term comes from the fact that a topologi-
cal soliton in the spatially extended Josephson system (ei-
ther a long Josephson junction or an array of small Joseph-
son junctions) carries the magnetic flux quantum,
0 = ( / ).e±Φ ± π Due to their remarkable structural and
dynamical stability, fluxons are widely discussed as qubits
[5] or as elements in the read-out process from qubits [6].
Relativistic time dilation in the array of three-junction
SQUIDs has been studied in Ref. 7.
In JJ arrays the soliton (fluxon) mobility is generally
obstructed due to the discreteness of the media [8,9]. As a
result, a fluxon couples with the small-amplitude wave. If
the inductive coupling between the junctions is small, the
free fluxon motion becomes impossible. However, the pos-
sibility of radiationless propagation of solitary waves in
discrete media for selected values of velocity has been re-
ported in the number of papers [10–20]. Also, free propa-
gation of the bound states of several discrete topological
solitons has been reported by Peyrard and Kruskal more
than 30 years ago [8]. This unexpected soliton mobility is a
part of the wider nonlinear phenomenon, known as embed-
ded solitons. Embedded solitons are spatially localized
solitary waves that exist for isolated parameter values (this
parameter can be the soliton velocity) despite the apparent
resonance with the linear waves of the system [21]. Apart
from the discrete media, they also exist in various continu-
ous systems like the sine-Gordon [22] and the double sine-
Gordon equations [23].
The array of three-junction SQUIDs, introduced in [7]
is governed by the discrete double sine-Gordon equation.
In this article we demonstrate the existence of embedded
solitons in this equation, study their properties in detail and
show how they manifest themselves in the current-voltage
characteristics of the array.
2. Main results
In this Section we present the model and report the
main results.
2.1. Model and equations of motion
We investigate the array of JJs, arranged in such a way
that the arms with one and two junctions alternate. The part
of this array is shown schematically in Fig. 1.
© Ivan О. Starodub and Yaroslav Zolotaryuk, 2017
mailto:starodub@bitp.kiev.ua
Embedded soliton dynamics in the asymmetric array of Josephson junctions
Thus, the elementary cell of such an array is an asymmetric
SQUID with two junctions in its left arm and one junction in
its right arm. The elementary cell of the array is obviously
asymmetric, but the whole array is not. The dynamics of
each junction is described by the Josephson phases, ( , ) ( )l r
n tϕ .
They are the phase difference between the phases of the
wave-functions of the superconductors on the different sides
of the junction. The superscripts l and r stand for the left
and right arms of the SQUID, respectively. The array is
studied under the resistively and capacitively shunted junc-
tion-model (RCSJ-model) [1]. Based on this model and with
the help of the Kirchhoff laws and the Josephson equations
the equations of motion of such an array have been derived
in Ref. 7. Under the assumption of the small loop size and
the single phase difference ( ) ( )= =l r
n n nϕ ϕ ϕ the final dimen-
sionless equations of motion read
1 1( 2 )n n n n+ −ϕ − κ ϕ − ϕ + ϕ +
2 sin sin = ,
1 2 2
n
n n
ϕ + η ϕ + +αϕ γ + η
= 1,2, , .n N (1)
This is the discrete double sine-Gordon equation (DDbSG).
The current model, according to [24], takes into account
only the self-inductance, while the mutual inductances of
the SQUIDs are neglected. The dimensionless parameters
are defined as follows
( )
0
( )
1= , = , = ,
2
r
c
l
J J c c
I
RC L I I
Φ
α κ η
ω π
(2)
( )
( )1 1 1= , = , = .
2 2 2
l
r cr
l c c
r l
ICC C I I
R R R
+ + +
Here = 2 /( )J ceI Cω is the Josephson plasma frequency
and the dimensionless time in Eq. (1) is normalized in the
units of 1
J
−ω , α is the dissipation parameter, 0Φ is the
magnetic flux quantum, JL is the elementary cell induct-
ance and γ is the dimensionless external bias current, nor-
malized to cI . Next, ,r lR , ,r lC and ( , )r l
cI are, respectively,
the resistance, capacitance and critical current of the right
or the left junction (marked by the sub(super)script «r» or
«l»). The parameter η measures the asymmetry of the
SQUID and is the ratio of the critical currents of the right
and left junctions of the SQUID. The discreteness param-
eter κ is responsible for the inductive coupling between
the cells.
The circular array is to be considered, thus, the boundary
conditions read = 4n n N Q+ϕ ϕ + π , where Q is the total topo-
logical charge, i.e., the total number of fluxons and anti-
fluxons trapped in the ring. In this article we restrict ourselves
only to the case of one fluxon in the array, hence = 1Q .
2.2. Soliton mobility in the hamiltonian limit
It is instructive to consider the DDbSG equation in the
Hamiltonian (dissipationless) limit = 0α and in the ab-
sence of the dc bias = 0γ . In this case Eq. (1) takes the
following form:
1 1( 2 ) = ( ),n n n n nV+ − ′ϕ − κ ϕ − ϕ + ϕ − ϕ (3)
2( ) = (1 cos ) 2 1 cos .
1 2 2
V ϕ ϕ η − ϕ + − + η
(4)
The periodic on-site potential ( ) = ( 4 )V Vϕ ϕ+ π has one
minimum within the interval [0, 4 ]ϕ∈ π if 0 < 1/2≤ η . The
height of the potential barrier that separates two minima at
= 0ϕ and = 4ϕ π is maximal if = 0η and decreases if η is
increased. An additional local minimum appears at = 2ϕ π
if > 1/2η . It remains local for any 1/2 <≤ η ∞, i.e.,
(2 ) > (0) = 0V Vπ .
The continuum version of Eq. (3) can be achieved when
κ →∞. The continuous double sine-Gordon equation has a
large number of applications [25], including the long Jo-
sephson junctions with the second harmonic in the current-
phase relation [26].
In this subsection we focus on the existence of the trav-
elling-wave topological solitons that propagate with the
constant shape and velocity. They must satisfy the follow-
ing condition: ( ) = ( ) ( )n t n t zϕ ϕ − ≡ ϕv . Thus, for these
solutions Eq. (3) can be rewritten as
[ ]2 ( ) ( 1) ( 1) 2 ( ) [ ( )] = 0 .z z z z V z′′ ′ϕ − κ ϕ + + ϕ − − ϕ + ϕv (5)
This is a differential-difference ODE with the delay and
advance terms. Analytical solution of such equations is not
possible. In some special cases it is possible to find its iso-
lated travelling-wave solutions [10,11]. Therefore we em-
ploy the numerical technique, developed in Refs. 27–29.
The solution is assumed to be periodic with rather large pe-
riod L that exceeds significantly all characteristic sizes of
the system and is sought in the form of the Fourier series
Fig. 1. (Color online) Schematic view of the SQUID array. Gray
colour stands for the superconducting electrodes and the insulat-
ing barriers are marked by red. Details of the full equivalent
scheme can be found in Ref. 7.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6 829
Ivan О. Starodub and Yaroslav Zolotaryuk
0
=0
2( ) ( ) sin ,
k
n
n
nzu z u z c
L
π +
∑ (6)
where the initial approximation 0 ( )u z is used in order to
enhance the initial guess and to enforce the appropriate
boundary conditions. It should be noted that the elementary
topological soliton in the DDbSG equation connects the
ground states = 0ϕ and = 4ϕ π. Coefficients nc are to be
determined. The integer part of L will effectively play the
role of the chain length in the original system (1). After
substituting the ansatz (6) into Eq. (5) and breaking the
interval [0, /2]z L∈ into k collocation points we obtain k
nonlinear algebraic equations for the coefficients =1{ }k
n nc .
These equations can be solved with the Newton–Raphson
method or any other appropriate scheme.
Currently it is a well-established fact [10–14,18–20] that
for the discrete Klein–Gordon equations of the type (3), the
continuous family of moving solitons turns into the discrete
finite set of monotonic | | const)(lim nn→∞ϕ → travelling kink
solutions with the velocities 0 1 2= { 0, , , }k≡ v v v v v .
Further on, all velocities that satisfy 0n ≠v will be called
sliding velocities since the kink slides along the lattice with
these velocities without any radiation. In the DSG equation
there are only non-mobile ( 0 = 0v ) monotonic solitons and
there is no sliding velocities for solitons with topological
charges = 1Q ± . There exist embedded solitons for higher
topological charges, = 2, 3,Q ± ± , which are, in fact,
bound states of several solitons or «soliton complexes»
[8,23] that appear due to destructive interference between
the elementary solitons. In general, everywhere away from
the sliding velocities, i.e., if n≠v v , the moving kinks are
non-monotonic, have oscillating asymptotic tails and are
often referred to as nanopterons.
Appearance of nanopterons is natural and its origin is
the resonance that happens due to the fact [8,30] that any
soliton which moves in the lattice governed by Eq. (3) with
velocity | | < 1v will always excite a small-amplitude wave
that moves with the same phase velocity. In other words,
the equation
2= ( ) = 1 4 sin ,
2L
qq qω + κv (7)
where ( )L qω is the spectrum of the small-amplitude waves
of Eq. (3), always has at least one real root if | |< 1v .
These solutions can be called discrete embedded
solitons, similarly to their continuous counterparts [21,22]
because they exist despite the resonance with the small-
amplitude waves and are «embedded» in the continuous
family of nanopterons. The spatial monotonicity of these
solutions, their localized nature, and, as a result their finite
energy also contribute to their importance.
Our numerical investigation of the DDbSG equation has
shown that embedded solitons can exist there as well. In
Fig. 2 we plot the dependence of the nanopteron amplitude
tail as a function of the velocity. We have fixed the cou-
pling constant to = 0.4κ and varied the asymmetry param-
eter η. One can observe that a slight change of η from
= 0.26η (red curve) to = 0.27η (blue curve) can bring a
significant change, namely a selected value of velocity for
which 0A → in an extremely sharp fashion (note the loga-
rithmic scale).
This is the sliding velocity mentioned above. The solu-
tion that corresponds to this velocity is spatially localized
embedded soliton with the monotonic asymptotical behavior
(see inset of Fig. 2). Away from this velocity we find
nanopterons or bound states of solitons and plane waves.
An example of a nanopteron is also plotted in the inset of
Fig. 2. For the sake of clarity only a small segment of ve-
locities was considered on this figure. No other sliding
velocities have been found outside of this segment.
From Fig. 2 one can notice that the dependences ( )A v
have sharp maxima. In fact, they are not maxima but sin-
gularities that appear because the numerical scheme is not
able to find a solution with the appropriate wavelength.
This is a consequence of the boundary conditions and these
singularities depend on the length L . For the velocity that
lies between the two adjacent singularities the nanopteron
solution has a certain fixed number of small-amplitude
wavelength that fit into the interval [ /2, /2]L L− . The
wavelength equals 2 /qπ where the wavenumber q is the
root of Eq. (7). In the case of several roots there exist sev-
eral nanopteron solutions with different wavelengths. As
one changes v , passes the singularity and enters the next
interval the number of fitted wavelengths changes.
If one of the pair of parameters, κ or η is fixed and an-
other one is varied, the critical parameter value of the vari-
able parameter can be found, below which there is no slid-
ing velocities. This is demonstrated in Fig. 3(a), where the
Fig. 2. (Color online) Dependence of the tail amplitude A on the
moving soliton velocity for = 30L , = 0.4κ , = 0.26η (curve 1,
red), = 0.27η (curve 2, blue) and = 0.28η (curve 3, black). The
inset shows the soliton profile for = 0.192302v (red) and
= 0.1992v (black). Respective position of these solutions on the
( )A s curve are marked by ().
830 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6
Embedded soliton dynamics in the asymmetric array of Josephson junctions
dependence of the sliding velocity 1v is plotted as a function
of the asymmetry parameter η for the different values of κ .
Interestingly, the embedded solitons can exist for the strong-
ly discrete arrays. The existence diagram on the plane ( , )κ η ,
presented in Fig. 3(b), generalizes the above result.
It is possible to draw a line on this plane that separates
the area without sliding velocities (and, consequently, with
no embedded solitons) from the area where one sliding
velocity exists. For the DDbSG equation we were able to
find only one sliding velocity. In some models there can be
more of them [12,13].
2.3. Current-voltage characteristics
Now we focus on the more realistic situation when the
array is dissipative ( > 0α ) and dc-biased ( 0γ ≠ ). The cur-
rent-voltage characteristics (CVCs) provide the necessary
information about the JJ array dynamics and can be meas-
ured experimentally. We have varied the bias and comput-
ed the average voltage drop
=1 0
1 1= ( ) .lim
tN
n
tn
V t dt
N t→∞
′ ′ϕ∑ ∫ (8)
If there is a fluxon in the array that moves with the constant
velocity v, the average voltage drop will be = 4 /V Nπv .
The numerically obtained CVCs are given in Fig. 4.
The coupling constant and the array size were fixed to
= 0.5κ and = 30N , respectively, while we have changed
the asymmetry parameter η and the damping constant α. It
appears that there exists a significant difference between
the CVCs if the system supports embedded soliton and
when it does not. The numerically computed CVCs are
shown by the different markers while the solid line corre-
sponds to the CVC that was obtained from the continuum
approximation [36]:
1/22
24 4 4= = 1 ( ) ,cV
N N
−
∞
π π α κ κ +Φ η πγ
v (9)
1 2( ) = 1 arctanh .
1 22 (2 1)
η
Φ η +
+ ηη η+
This formula is very similar to the well-known McLaugh-
lin–Scott result for the fluxon motion in the ordinary long
JJ [4]. It is reduced to the McLaughlin–Scott formula in the
both limits: 0η→ and η→∞. The continuum version is
plotted here mainly as a reference point, but appears to
work surprisingly well even in this strongly discrete case.
First of all we focus on the principal differences between
fluxon mobility in discrete and continuous media. The
fluxon velocity in the continuous JJ is defined uniquely by
the ratio /γ α and, as one can see in the CVCs in Fig. 4, the
respective characteristics are continuous functions that pass
through the origin. There are two main differences in the dis-
crete case: (i) the CVC does not pass through the origin;
(ii) one continuous branch is replaced by the set of roughly
parallel branches. The first difference is the natural conse-
quence of the discreteness, because some finite bias is always
needed to overcome the lattice pinning. The second difference
is caused by the boundary conditions and by the fluxon cou-
pling with the linear modes [30–34]. While moving along the
array, the fluxon excites the linear modes and forms a bound
state that propagates with the same velocity. The wavelength
of the linear mode is given by Eq. (7). Because of the periodic
boundary conditions the phase locking in the array would
occur if the finite number of the Josephson phase oscillations
fits into one cycle of the fluxon journey along the array. The
phase locking means that the system settles on the periodic
attractor that corresponds to the fluxon restoring its shape
and position completely after circumventing the array. In
other words, a certain number of the linear modes wave-
length should be fitted in the array and the different branches
of the CVC in Fig. 4 correspond to the different number of
these wavelengths.
Fig. 3. (Color online) (a) The value of the sliding velocity 1v as a
function of the asymmetry parameter η for different values of
the coupling constant: = 1κ (curve 1), = 0.5κ (curve 2) and
= 0.25κ (curve 3). (b) The existence diagram on the parameter
plane ( , )κ η . One sliding velocity exists in the green area while
there is no sliding velocities in the white area. All results were
obtained for = 30L .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6 831
Ivan О. Starodub and Yaroslav Zolotaryuk
Now we discuss the signatures of the embedded
solitons on the CVCs. In Fig. 4(a) the case of = 0.1η is
considered, and, according to the previous subsection,
there is no embedded solitons in the hamiltonian limit for
this value of the asymmetry parameter. On contrary, for
= 0.6η (Fig. 4(b)) and for = 1.5η (Fig. 4(c)) there exist one
embedded soliton with the sliding velocity 1 = 0.399493v
and 1 = 0.469944v , respectively. For details see Fig. 3. In
the first case the CVC branches are distributed almost uni-
formly. In the second and third cases one can easily spot
the significant gap IVI IVIV V V− +≤ ≤ which will be called
inaccessible voltage interval (IVI). This IVI appears more
pronounced as η increases or when α is decreased. More-
over, if α is decreased significantly, its lower boundary,
equals zero. The upper boundary, IVIV + tends to the value
14 /Nπv , which is the voltage drop produced by the em-
bedded soliton (marked by the red vertical line). We have
reduced α and computed the detuning 1= | 4 / | .IVIv N V +ν π −
For the lowest value of dissipation which we have used in
our numerical analysis, = 0.005α , the detuning value
reached 310 .−ν
From these data one can make the following conclu-
sion. If in the hamiltonian limit the JJ array supports the
embedded soliton that moves with velocity v, in the weak-
ly driven and weakly dissipative array this will be the dom-
inant attractor of the system. Thus, the average voltage drop
in this limit is defined by the sliding velocity and is far from
zero. If the system does not support embedded solitons, its
behavior in the limit 0α → , 0γ → is different: the CVC
ends up close to the origin, as shown in Fig. 4(a). However,
it does not pass the origin due to discreteness.
The small-wave radiation intensity
2
/2( ) = ( )e ,i t
NI t dt
+∞
− Ω
−∞
Ω ϕ∫ (10)
that appears during the fluxon motion around the array is
shown in Fig. 5. Here the fast Fourier transform (FFT) is
taken for the /2N th junction of the array. The time interval
0 fin[ , ]t t for FFT was chosen in such a way that the fluxon
motion did not contribute to the power spectrum. We have
focused on the array parameters that correspond to Fig. 4(c)
and for the dissipation value = 0.02α . Three points on the
CVC branch that ends at 0.2IVIV +
were chosen for the
power spectrum analysis. Since the fluxon velocity is close
to the sliding velocity 1 = 0.469944v , the time for the
fluxon to travel around the array is 1/ 60N v . Therefore,
we have taken fin 0 = 50t t− .
We take the CVC branch that is adjacent to the upper
bond of the IVI. If the bias is decreased we move down the
branch, and the amount of the radiated energy decreases
significantly (note the log scale). The peak of the emitted
waves lies within the linear band (7).
Fig. 4. (Colour online) Current-voltage curves for = 0.5κ , = 30N ,
= 0.05α (black ), = 0.02α (red ), = 0.01α (blue ) and
= 0.1η (a), = 0.6η (b) and = 1.5η (c). The blue solid lines corre-
spond to the respective CVC in the continuum limit. The red verti-
cal lines in (b), (c) are given by 14 /Nπv , where 1v is the sliding
velocity for the respective value of η. The inset in the panel (c)
shows the details of CVCs in the neighbourhood of the sliding
velocity for = 0.02α (), = 0.01α () and = 0.005α ().
Fig. 5. (Сolor online) Radiation intensity of the plane waves in the
JJ array with the parameters that correspond to Fig. 4(c) and with
= 0.02α . The respective values of the dc bias and voltage are:
= 0.05γ , 0.205V (curve 1, black), = 0.044γ , 0.201V
(curve 2, blue) and = 0.0423γ , 0.199V (curve 3, red).
832 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6
Embedded soliton dynamics in the asymmetric array of Josephson junctions
3. Conclusions
In this paper we have demonstrated that the array of
asymmetric three-junction SQUIDs, which is described
by the discrete double sine-Gordon equation, supports
embedded solitons. This equation is also used to describe
the parallel array of the superconductor–ferromagnet–super-
conductor (SFS) or superconductor–ferromagnet–insulator–
superconductor (SFIS) junctions, that are governed by the
biharmonic current-phase relation
,1 ,2( ) = sin sin 2 .c c cI I Iϕ ϕ+ ϕ
Embedded solitons are spatially localized excitations
that exist despite the resonance with the small-amplitude
waves of the underlying system. They exist for the isolated
set of velocities. The important parameter of the model is
the ratio η of the critical currents of the left and right arms
of the SQUID, which we call the asymmetry parameter.
These embedded solitons manifest themselves on the cur-
rent-voltage characteristics of the JJ array as inaccessible
voltage intervals (IVI), i.e., the forbiden values of voltage.
The upper bound of such an IVI tends to 4 /Nπv as 0α → ,
where v is the embedded soliton velocity in the hamiltoni-
an limit. The lower bound of the IVI tends to zero if the
asymmetry parameter is large enough and dissipation is
small enough.
This phenomenon should not be confused with the series
of results on the models with the non-zero Peiers–Nabarro
barrier [16,39,40]. For the discrete embedded solitons the
Peierls–Nabarro barrier is non-zero [12,13], however, this
does not prevent the free soliton propagation.
The symmetric SQUID array or, equivalently, the array
of parallel shunted small Josephson junctions [35] is de-
scribed by the DSG equation. It has been demonstrated
both theoretically [8,12,14,35] and experimentally [37] that
the radiationless motion of the coupled state of several
solitons is possible for the selected set of their velocities.
This phenomenon has been treated analytically in the qua-
si-continuum approximation in Refs. 22, 23, but it takes
place even in the sufficiently discrete array ( < 1κ ) as well.
In the limit η→∞ the double SG equation becomes the
ordinary SG equation with the period 2π, thus, the above-
mentioned result of the bound state of two 2π solitons is
the special case of the kink mobility of the DDbSG equa-
tion in the limit η→∞.
Acknowledgements
The authors acknowledge the financial support from the
National Academy of Sciences of Ukraine (NAS of
Ukraine) under the Project No. 0112U000053 and from the
Project 1/30-2015 “Dynamics and topological structures in
Bose–Einstein condensates of ultracold gases” of the Kyiv
National University Branch Target Training Program at the
NAS of Ukraine.
1. A. Barone and G. Paterno, Physics and Applications of the
Josephson Effect, Wiley, New York (1982).
2. A.V. Ustinov, Physica D 123, 315 (1998).
3. E. Trías, J.J. Mazo, and T.P. Orlando, Phys. Rev. Lett. 84,
741 (2000); P. Binder, D. Abraimov, A. V. Ustinov, S. Flach
and Y. Zolotaryuk, Phys. Rev. Lett. 84, 745 (2000).
4. D.W. McLaughlin and A.C. Scott, Phys. Rev. A 18, 1652
(1978).
5. D.V. Averin, K. Rabenstein, and V.K. Semenov, Phys. Rev.
B 73, 094504 (2006).
6. A. Fedorov, A. Shnirman, G. Schon, and A. Kidiyarova-
Shevchenko, Phys. Rev. B 75, 224504 (2007).
7. M. Nishida, T. Kanayama, T. Nakajo, T. Fujii, and N.
Hatakenaka, Physica C 470, 832 (2010).
8. M. Peyrard and M.D. Kruskal, Physica D 14, 88 (1984).
9. O.M. Braun and Y.S. Kivshar, Phys. Rep. 306, 2 (1998).
10. S. Flach, Y. Zolotaryuk, and K. Kladko, Phys. Rev. E 59,
6105 (1999).
11. V.H. Schmidt, Phys. Rev. B 20, 4397 (1979).
12. A.V. Savin, Y. Zolotaryuk, and J.C. Eilbeck, Physica D 138,
265 (2000).
13. V.M. Karpan, Y. Zolotaryuk, P.L. Christiansen, and A.V.
Zolotaryuk, Phys. Rev. E 66, 066603 (2002).
14. A. Aigner, A. Champneys, and V. Rothos, Physica D 186,
148 (2003).
15. Y. Zolotaryuk, J.C. Eilbeck, and A.V. Savin, Physica D 108,
81 (1997).
16. I.V. Barashenkov, O.F. Oxtoby, and D.E. Pelinovsky, Phys.
Rev. E 72, 035602(R) (2005).
17. B.A. Malomed, J. Fujioka, A. Espinosa-Cerón, R.F.
Rodríguez, and S. González, Chaos 16, 013112 (2006).
18. O. Oxtoby, D.E. Pelinovsky, and I.V. Barashenkov,
Nonlinearity 19, 217 (2006).
19. S.V. Dmitriev, A. Khare, P.G. Kevrekidis, A. Saxena, and L.
Hadzievski, Phys. Rev. E 77, 056603 (2008).
20. G.L. Alfimov, E.V. Medvedeva, and D.E. Pelinovsky, Phys.
Rev. Lett. 112, 054103 (2014).
21. A. Champneys, B. Malomed, J. Yang, and D. Kaup, Physica
D 152–153, 340 (2001).
22. A. Champneys and Y.S. Kivshar, Phys. Rev. E 61, 2551
(2000).
23. M.M. Bogdan, A. Kosevich, and G.A. Maugin, Wave Motion
34, 1 (2001).
24. R.D. Bock, J.R. Phillips, H.S.J. van der Zant, and T.P.
Orlando, Phys. Rev. B 49, 10009 (1994).
25. C.A. Condat, R.A. Guyer, and M.D. Miller, Phys. Rev. B 27,
474 (1983); D.K. Campbell, M. Peyrard, and P. Sodano,
Physica D 19, 165 (1986).
26. E. Goldobin, D. Koelle, R. Kleiner, and A. Buzdin, Phys.
Rev. B 76, 224523 (2007).
27. D. Hochstrasser, F. Mertens, and H. Büttner, Physica D:
Nonlinear Phenomena 35, 259 (1989).
28. J.C. Eilbeck and R. Flesch, Phys. Lett. A 149, 200 (1990).
29. D. Duncan, J. Eilbeck, H. Feddersen, and J. Wattis, Physica
D 68, 1 (1993).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6 833
Ivan О. Starodub and Yaroslav Zolotaryuk
30. A.V. Ustinov, M. Cirillo, and B.A. Malomed, Phys. Rev. B
47, 8357 (1993).
31. S. Watanabe, H.S.J. van der Zant, S.H. Strogatz, and T.P.
Orlando, Physica D 97, 429 (1996).
32. O.M. Braun, B. Hu, and A. Zeltser, Phys. Rev. E 62, 4235
(2000).
33. A.V. Ustinov, M. Cirillo, and B.A. Malomed, Phys. Lett. A
183, 383 (1993).
34. A.V. Ustinov, M. Cirillo, Britt H. Larsen, V.A. Oboznov,
P. Carelli, and G. Rotoli Phys. Rev. B 51, 3081 (1995).
35. A.V. Ustinov, B.A. Malomed, and S. Sakai, Phys. Rev. B 57,
11 691 (1998).
36. Y. Zolotaryuk and I.O. Starodub, Phys. Rev. E 91, 013202
(2015).
37. J. Pfeiffer, M. Schuster, A.A. Abdumalikov, and A.V.
Ustinov, Phys. Rev. Lett. 96, 034103(4) (2006).
38. A.A. Golubov, M.Y. Kupriyanov, and E. Ilichev, Rev. Mod.
Phys. 76, 411 (2004).
39. M. Speight and Y. Zolotaryuk, Nonlinearity 19, 1365 (2006).
40. S.V. Dmitriev, P.G. Kevrekidis, N. Yoshikawa, and D.J.
Frantzeskakis, Phys. Rev. E 74, 056609 (2006).
834 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 6
1. Introduction
2. Main results
2.1. Model and equations of motion
2.2. Soliton mobility in the hamiltonian limit
2.3. Current-voltage characteristics
3. Conclusions
Acknowledgements
|