Dynamics of bound soliton states in regularized dispersive equations
The nonstationary dynamics of topological solitons (dislocations, domain walls, fluxons) and their bound states in one-dimensional systems with high dispersion are investigated. Dynamical features of a moving kink emitting radiation and breathers are studied analytically. Conditions of the breathe...
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irk-123456789-1173442017-05-23T03:02:47Z Dynamics of bound soliton states in regularized dispersive equations Bogdan, M.M. Charkina, O.V. Низкоразмерные и неупорядоченные системы The nonstationary dynamics of topological solitons (dislocations, domain walls, fluxons) and their bound states in one-dimensional systems with high dispersion are investigated. Dynamical features of a moving kink emitting radiation and breathers are studied analytically. Conditions of the breather excitation and its dynamical properties are specified. Processes of soliton complex formation are studied analytically and numerically in relation to the strength of the dispersion, soliton velocity, and distance between solitons. It is shown that moving bound soliton complexes with internal structure can be stabilized by an external force in a dissipative medium then their velocities depend in a step-like manner on a driving strength. 2008 Article Dynamics of bound soliton states in regularized dispersive equations / M.M. Bogdan, O.V. Charkina // Физика низких температур. — 2008. — Т. 34, № 7. — С. 713–720. — Бібліогр.: 40 назв. — англ. 0132-6414 PACS: 05.45.–a;05.45.Yv;75.40.Gb http://dspace.nbuv.gov.ua/handle/123456789/117344 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы Bogdan, M.M. Charkina, O.V. Dynamics of bound soliton states in regularized dispersive equations Физика низких температур |
| description |
The nonstationary dynamics of topological solitons (dislocations, domain walls, fluxons) and their
bound states in one-dimensional systems with high dispersion are investigated. Dynamical features of a
moving kink emitting radiation and breathers are studied analytically. Conditions of the breather excitation
and its dynamical properties are specified. Processes of soliton complex formation are studied analytically
and numerically in relation to the strength of the dispersion, soliton velocity, and distance between solitons.
It is shown that moving bound soliton complexes with internal structure can be stabilized by an external
force in a dissipative medium then their velocities depend in a step-like manner on a driving strength. |
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Article |
| author |
Bogdan, M.M. Charkina, O.V. |
| author_facet |
Bogdan, M.M. Charkina, O.V. |
| author_sort |
Bogdan, M.M. |
| title |
Dynamics of bound soliton states in regularized dispersive equations |
| title_short |
Dynamics of bound soliton states in regularized dispersive equations |
| title_full |
Dynamics of bound soliton states in regularized dispersive equations |
| title_fullStr |
Dynamics of bound soliton states in regularized dispersive equations |
| title_full_unstemmed |
Dynamics of bound soliton states in regularized dispersive equations |
| title_sort |
dynamics of bound soliton states in regularized dispersive equations |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2008 |
| topic_facet |
Низкоразмерные и неупорядоченные системы |
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http://dspace.nbuv.gov.ua/handle/123456789/117344 |
| citation_txt |
Dynamics of bound soliton states in regularized
dispersive equations
/ M.M. Bogdan, O.V. Charkina // Физика низких температур. — 2008. — Т. 34, № 7. — С. 713–720. — Бібліогр.: 40 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT bogdanmm dynamicsofboundsolitonstatesinregularizeddispersiveequations AT charkinaov dynamicsofboundsolitonstatesinregularizeddispersiveequations |
| first_indexed |
2025-07-08T12:04:06Z |
| last_indexed |
2025-07-08T12:04:06Z |
| _version_ |
1837080246283665408 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 7, p. 713–720
Dynamics of bound soliton states in regularized
dispersive equations
M.M. Bogdan and O.V. Charkina
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: bogdan@ilt.kharkov.ua
Received March 11, 2008
The nonstationary dynamics of topological solitons (dislocations, domain walls, fluxons) and their
bound states in one-dimensional systems with high dispersion are investigated. Dynamical features of a
moving kink emitting radiation and breathers are studied analytically. Conditions of the breather excitation
and its dynamical properties are specified. Processes of soliton complex formation are studied analytically
and numerically in relation to the strength of the dispersion, soliton velocity, and distance between solitons.
It is shown that moving bound soliton complexes with internal structure can be stabilized by an external
force in a dissipative medium then their velocities depend in a step-like manner on a driving strength.
PACS: 05.45.–a Nonlinear dynamics and chaos;
05.45.Yv Solitons;
75.40.Gb Dynamic properties.
Keywords: nonlinear dynamics, soliton complex, kink, breather, strong dispersion.
1. Introduction
The soliton concept in applied science was formulated
35 years ago in the prominent review by Scott, Chu, and
McLaughlin [1]. Since the soliton search developed into
both well-established mathematical and physical theo-
ries. They cover a wide range of problems beginning from
complete integrability of nonlinear equations [2,3] up to
applications of the soliton concept for explanation of non-
linear phenomena in various fields of condensed matter
physics [4–7]. Topological defects and inhomogeneties
such as dislocations in crystals, domain walls and vortices
in magnets, quanta of magnetic flux (fluxons) in long
Josephson junctions, are a few examples of traditional
physical objects which are described in terms of solitons
in solid state physics.
Two pioneer works by Kosevich and Kovalev [8,9],
devoted to nonlinear dynamics of one-dimensional crys-
tals, initiated two novel directions in soliton investiga-
tions. In Ref. 8 the physical meaning of a self-localized
excitation was introduced for the first time. In the long
wave limit such an oscillating solitary wave corresponds
to the breather which is interpreted as the soliton–anti-
soliton bound state. The authors proposed a regular
asymptotic procedure to construct the self-localized
oscillation [8,10]. In the short-wave limit Kosevich and
Kovalev predicted the existence of a self-localized oscilla-
tion with a frequency above the upper edge of a linear exci-
tation spectrum. Later a high localization limit of these
high-frequency soliton states were studied and they called
the intrinsic localized modes or discrete breathers which
became a new concept in nonlinear lattice theory [11,12].
In the work [9], which concerned crowdion dynamics
in an one-dimensional anharmonic crystal, Kosevich and
Kovalev established, for the first time to our knowledge,
existence of supersonic and radiationless motion of topo-
logical solitons in a highly dispersive nonlinear medium.
The equations deduced in the work [9] generalize the
Boussinesq equation for the case of the sine–Gordon (SG)
and � 4-models:
u u u u u u F utt xx x x xx xxxx� � � � � �( ) ( )� � � 0, (1)
where the external force equals to either F u u( ) sin� , or
F u u u( ) � �3 , respectively. The equation (1) with the
sine force and � � 0
u u u u u utt xx x xx xxxx� � � � �� �2 0sin (2)
is known nowadays as the Kosevich–Kovalev equation
[13,14]. For the special choice of parameter � �� 3 2/ ,
© M.M. Bogdan and O.V. Charkina, 2008
Kosevich and Kovalev found an exact solution describing
the 2�-kink moving with an arbitrary velocity [9]. One
year later an integrable version of the equation was pro-
posed by Konno, Kameyama, and Sanuki [15]:
u u u u u uxt xx x xx xxxx� � � � �
3
2
02� � sin , (3)
and this fact could explain formally the existence of the
exact kink solution in Eq.(2). However significance of the
fact of radiationless motion of topological solitons in
highly dispersive media was realized after many years. In
1984 Peyrard and Kruskal showed numerically the exis-
tence of a stable moving 4�-soliton in the highly discrete
SG model [16]:
� � � � �� �
2
2 1 1 2
2
1
0
u
u u u
d
un
n n n n
sin , (4)
where d is discreteness parameter. They tried to explain
the formation of the bound state of two identical kinks by
exploiting the fact of the presence of the Peierls potential
in the lattice model. However, in work [17] it has been
found that the radiationless motion of such a soliton com-
plex can be described explicitly by the exact 4�-soliton
solution in the framework of the dispersive SG equation
with a fourth spatial derivative, i.e. Eq. (2) with � � 0:
u u u utt xx xxxx� � � �� sin 0 (5)
which is obtained as the long-wave limit of the Eq. (4).
Almost simultaneously the topological bound soliton
states were found numerically in the continuous nonlocal
SG model describing long Josephson junctions [18].
These facts of existence of the multikink bound states in
discrete and continuous systems were generalized as a
universal phenomenon and led to the concept of the
soliton complexes formed by strongly interacting kinks in
highly dispersive media [19–21]. Physically such
two-kinks states correspond, e.g., to a moving defect con-
sisting of two neighboring dislocation half-planes, or to a
narrow 360� magnetic domain wall, which arises even in
the absence of magnetic field, or to a bound pair of
fluxons in a long Joshephson junction.
There are some approaches to explain mechanisms of
formation of the bound soliton states. The internal struc-
ture of the soliton complexes can be studied in detail in
models that lead to piecewise linear equations with strong
dispersion [20–22]. In this case the stationary states can
be constructed as a superposition of two quasi-solitons
possessing spatial periodic tails as asymptotics which
cancel each other exactly for the composed complex by
imposing some interference condition. Noting that these
bound solitons occur in resonance with the linear spec-
trum waves they called embedded solitons [23,24]. The
effect of the dispersion can be extracted already from the
dispersion relations of corresponding linearized equa-
tions [25,26]. However a principal circumstance for a
complex arising appears to be the influence of the strong
dispersion as a factor leading to complication of internal
structure of solitons beginning from a kink level [27]. A
taking into account of the interaction of such flexible
kinks allows to describe quantitatively conditions of a
soliton complex formation [19,21,28].
A picture of the soliton complex formation becomes
much more diverse when one considers internal dynamics
of kinks, nonstationary motion of complexes, the condi-
tions of their formation and stability depending on differ-
ent physical factors including the influence of dissipative
and external forces [29–31]. The present paper is devoted
to investigation of this circle of tasks concentrating on a
single kink propagation and especially on the bound
soliton states of both types, soliton complexes and breath-
ers, covering essentially nonlinear dynamics of the
strongly dispersive SG model.
The paper is organized as follows. Section 2 intro-
duces regularized dispersive equations and some their dy-
namical properties. Section 3 addresses the nonstationary
dynamics of a single 2�-kink in all the range of the disper-
sive parameter. Section 4 devoted to analysis of the com-
plex formation and its stability conditions. Section 5
deals with the breather dynamics. Section 6 addresses the
influence of dissipation and external forces on stabiliza-
tion of the soliton complexes with an internal structures.
Last section summarized obtained results.
2. The regularized dispersive SG equations
To investigate analytically and numerically the
nonstationary dynamics of kinks and their bound states
we use the regularized dispersive SG equation with a
fourth-order spatio-temporal derivative [19–21]:
u u u utt xx xxtt� � � �� sin 0 (6)
where � is a dispersive parameter. This equation has ad-
vantage in comparison with Eq. (5) because it does not
contain an artificial instability of states u � 0 2 4, ,� ��
with respect to a short-wave excitation. An idea of the
regularization of dispersive equations belongs to Bous-
sinesq who first proposed to use a mixed spatio-temporal
derivative instead of the fourth spatial derivative for the
shallow-water waves equations [4]. Such a replacement
was justified in the lattice theory by Rosenau [32] for
models with nonlinear interactions between atoms. The
Boussinesq’s idea was applied to the SG and double SG
equations with higher dispersion in Refs. [19–21]. At
present this approach is actively being used for analytical
description of discreteness effects [33–35]. With respect
to original discrete models an accuracy of this replace-
ment can be easy to estimate. In the long-wave limit
( )d �� 1 after introducing a coordinate x n/d� the second
714 Fizika Nizkikh Temperatur, 2008, v. 34, No. 7
M.M. Bogdan and O.V. Charkina
difference is replaced as u u u u un n n xx xxxx� �� �
�1 1 2 � ,
where � �1 12 2/ d . If one expresses a second derivative
from Eq. (5) as u u u uxx tt xxxx� � �� sin and inserts it in
the fourth derivative and keeps terms which are linear
with respect to�, one obtains the equation:
u u u u utt xx xxtt xx� � � � �sin (sin )� � 0. (7)
Hence to approximate Eq. (5) by Eq. (6) one needs to take
into account the term with ��(sin )u xx . It would be ex-
pected that a form of static kinks would be different for all
the Eqs. (4)–(7), depending on a value of �. Curiously,
it appears that Eq. (5) does not possess a static 2�-kink
solution satisfying boundary conditions u( )�� � 0 and
u( )� � 2� at all [21]. At the same time exact static kink
and moving complex solutions exist simultaneously in
Eq. (6). Therefore a lot of problems of kink and complex
dynamics can be solved analytically in the framework of this
equation. In particular the spectral problem for linear excita-
tions of the static kink has been solved completely [14,29].
Thus Eq. (6) has an exact static kink solution for arbitrary�,
which coincides with a kink of the usual SG equation:
u x x2 4�( ) exp( )� arctan . (8)
The kink solution of Eq. (7) can be find in an implicit
form using the first integral:
du
dx
u/
u
u/�
�
�
2 2
1
1 22sin( )
cos
cos ( )
�
� . (9)
It appears that even when the discreteness parameter d �1
and hence � �1 12/ , the static kink solutions for the dis-
crete equation (4) and continuous Eqs. (6) and (7) differ
very slightly (see Fig.1). This justifies the use Eq.(6) in-
stead Eqs. (5) and (7) to explain qualitatively a majority
of effects which are inherent in the discrete model (4) but
in reality arise due to the higher-order dispersion.
The equation (6) can be derived from the Lagrangian:
L u u u u dxt x xt� � � � ��
1
2
2 12 2 2[ ( cos )]� . (10)
Using the expression (10) it is easy to find the first
integrals, total energy and momentum:
E u u u u dxt xt x� � � � ��
1
2
2 12 2 2[ ( cos )]� , (11)
P u u u dxx t xxt� �
��
�
� ( )� . (12)
Note that first two terms in the Eq. (11) give the kinetic
energy therefore the higher-order dispersion in the regu-
larized equations contributes to the kinetic energy
whereas in the case of Eq. (5) it produces an additional
contribution to the potential energy [21].
Spectrum of linear excitations for Eq. (6) can be found
exactly for both cases of a homogenous ground state and
in the presence of the static kink (8) [14,29]. The disper-
sion relation for continuous waves takes the form:
� �( ) ( ) ( )k k / k� � �1 12 2 . (13)
This spectrum has the peculiarity of being bounded in fre-
quency not only from below but also from above. This
property makes it similar to the spectrum of the initial dis-
crete model (4). Moreover, it simply coincides with the
spectrum of the SG model with a nonlocal interaction
[18]. In the case of a kink there exists a discrete spectrum
of internal modes of oscillations [29], the number of
which becomes infinite when � � 1 while the continuous
spectrum degenerates to one frequency �0 1� .
At last it is remarkable that Eqs. (5) and (6) have exact
solutions describing a moving 4�-soliton complexes. For
Eq. (6) the moving bound state of strongly coupled kinks
has a form:
u x t
x V t
l
4
0
0
8�( , ) exp�
��
�
��
�
�
��
�
�
�
�
�
�arctan . (14)
The velocityV0 of such a complex, its effective width l0 and
its energy E0 are specified functions of the parameter�:
V0 1
3 3
( )�
� �
� � � , l V0 0
2 1 43� ( ) /� ,
E l
l
0 0
1 032
9
� �
�
�
�
�
�
�� . (15)
In next two sections we discuss dynamical properties
of a single kink and specify conditions of the soliton com-
plex formation.
Dynamics of bound soliton states in regularized dispersive equations
Fizika Nizkikh Temperatur, 2008, v. 34, No. 7 715
–20 –10 0 10 20
–1
0
1
2
3
4
5
6
7
n
u
(n
)
Fig. 1. Comparing static kink profiles for a discrete equation
(4) and Eqs. (6) and (7) for d �1 (� =1/12). Continuous solu-
tions are undistiguishable.
3. Dynamics of a kink in the dispersive SG model
Internal oscillations of the static kink of the regular-
ized Eq. (6) have been studied theoretically and main fea-
tures of its nonstationary motion have been revealed
numerically [14,29–31]. Here we present analytical ap-
proaches to the kink dynamics. One of them consists in
application of a perturbation theory for the case of a weak
dispersion (small�). In this limit dynamical properties of
a kink would be expected to be similar to those in the
usual SG equation. The latter has a moving kink u z2�( )
obtained from the expression (8) by the Lorentz transfor-
mation of coordinates: z x Vt / V� � �( ) 1 2 . Therefore
one can seek a solution of Eq. (6) in the form:
u x t u z u z( , ) ( ) ( , )� �2 1�
(16)
where
� � �( )t Vx / V1 2 and a small addition function
u1 to the kink form obeys the linearized equation:
�
�
�
�
�
�
�
�
�
� � � �
�
�
�
�
�
� �
2
2 1 1 1 2 11
2
L u u u
z
uzz
cosh
� �� �( ( ) )u z u xxtt2 1 . (17)
In the first approximation one has to neglect the term
u1 in the right-hand side of Eq. (17). Then it is easy to find
a partial solution of the equation:
�u z
z
z
z
z
( )
sinh
cosh cosh
� �
�
�
�
�
�
�� 3
2
, �
�
�
�
V
V
2
2 21( )
. (18)
A g e n e r a l s o l u t i o n c a n b e w r i t t e n a s u z1( , )
�
� ��u z v z( ) ( , )
where v z t( , ) is a solution of a homoge-
neous part of Eq. (17) (without the right-hand side). This
solve a evolution problem of the SG kink in the dispersive
system for the case of small �. Really suppose that at the
initial moment u z u z( , ) ( )0 2� � and u z1 0 0
( , ) � . It means
that v z u z( , ) ( )0 � �� and v z
( , )0 0� . Owing to the knowl-
edge of eigenfunctions of operator L, one can solve com-
pletely the initial problem for the function v z t( , ):
v z t
k
k kz z k kz( , ) cos( )(cos tanh sin )�
�
� �
��
�
��
�
8
1
1
1
2
2
� ��
�
�
�
�
�
1
2
2 2
1 9 2
cosh
tanh
�
� �
k
k k k dk. (19)
This addition to the kink form describes decaying oscilla-
tions of the effective kink width which correspond to the
SG quasimode. The addition to the stationary reverse
kink width !
( ) is easily found from Eq. (19):
"!
�
�
( ) ( , )� �
�
�
�
��
�
�u z
k
k
z z1 0
2
28
1
1
�
� ��
�
�
�
�
�
cos( )
cosh
tanh
1
2
2 2
1 9
2
2k
k
k k k dk
�
� � (20)
and its temporal behavior (Fig. 2) repeats entirely the
kink velocity modulation found numerically [29]. The
power spectrum of the oscillation reveals Rice’s fre-
quency value � �R /� 2 3 [36].
Thus the perturbation theory predicts that the initial
SG kink has to evolve into a steady moving profile
u z u z u zK ( ) ( ) ( )� �2� � . However it is known [21] that
the equation for stationary waves
u u uzz zzzz� � �� sin 0, (21)
does not possess an exact solution for a moving 2�-kink al-
though one can find formally first terms in asymptotic se-
ries for such a solution, which coincide with Eq. (18). The
paradox is solved by noting that the solution u zK ( ) can be
expressed as superposition of two �-kinks in a form
u z z iK ( ) exp� ��
�
�
�
�
� �
�
�
�
�
�
� �2 1
2
3arctan
�
�
� ��
�
�
�
�
� �
�
�
�
�
�
�2 1
2
3arctan exp
�
�z i (22)
which prompts the ansatz for an adiabatic approach to the
2�-kink dynamics. The nonstationary evolution of the
kink at small� reduces to decaying collective oscillations
of the effective kink width and velocity with a consequent
growth of a kink steepness and a slow energy loss due to
the radiation emission. With increasing the dispersive
parameter the notable oscillating kink tail appears and this
phenomenon can be described by the following ansatz:
u z t u z a z k z vtKr K( , ) ( ) [ tanh( )]sin( ( ))� � � �1 0 , (23)
716 Fizika Nizkikh Temperatur, 2008, v. 34, No. 7
M.M. Bogdan and O.V. Charkina
0 5 10 15 20 25 30
–0,75
–0,50
–0,25
0
0,25
0,50
0,75
1,00
!
!
(
)/
(0
)
Fig. 2. Decaying oscillations of the reverse effective kink
width during motion in the case of small �.
where the second term corresponds to radiation on the
wake of the kink. We have carried out a numerical model-
ing of the dynamics of kinks and soliton complexes (de-
tails of the numerical scheme can be found in Ref. 31).
Results of the simulations for small � are in a good rela-
tion with expression (22) and (23) and confirm entirely
theoretical predictions. For large enough parameter� and
the initial velocity V in a moving kink emits the breather as
shown in Fig. 3.
4. Kinks interaction and formation of soliton
complexes
An analytical approach to the description of the
soliton-complex formation in dispersive equations was
proposed in [21,29]. It is based on the use of the collective
variable ansatz which is constructed by taking into ac-
count the translational and internal degrees of freedom of
a soliton as well as interactions between solitons and
solitons with radiation. Now using results of previous
section we can specify the form of ansatz:
u x t u R u R f twb K K b( , ) ( ) ( ) ( , )( tanh( ))� � � � � �# # # #1 .
(24)
Here first two terms are kinks superposition and the last
term describes a small-amplitude breather f tb ( , )# �
� � � �a t k /sin( ( )) cosh( ( ))$ # # % # #0 0 or radiation emit-
ting. It turns out that the condition of the complex forma-
tion of the closely sited solitons can be found from the en-
ergy expression of the pair of strongly interacting solitons
without taking into account the breather or radiation
[19,21,28]. Now we use this approximation for the de-
scription of the regularized SG system. So we suppose
that the complex dynamics can be considered in the
framework of the soliton ansatz
u x t R Rkk ( , ) (exp( )) (exp( )))� � � �4 4arctan arctan# # (25)
which is prompted by the form of a generalization of the
exact solution in Eq. (14). Here # !� �( ( ))x X t and X t( ),
!( )t , and R t( ) are functions of time. Functions !( )t and
X t( ) describe the changing of the effective width of
solitons and their translational motion, respectively. The
function corresponds to the changing separation between
solitons, which is defined obviously as L R/� 2 !. Let the
distance between solitons be small. Inserting the ansatz
into Eqs. (11) and (12), we find the effective Lagrangian for
two interacting solitons in the strongly dispersive medium:
L R RRt t
t� � �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� �16
12 36
2
3
2
3
2
2
2
2
!
!
� � !
!
!( X t
2 1�
&
'
(
)(
)
�
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
1
3
1
3
1
5 2 18
2
3
2 2 2 2R R t
!
� !
!
�
�
�
�
�
�
�
�
�
&
'
(
)(
� �
�
�
�
�
�
�
�
�
�
�
�
�
� � �
�
�
�
�
�
�
R RR X
R
t t t
2
2
3 2
27
90
2
3
2
2
3
1
7
5
�
! ! �
�
*
+
(
,(
*
+
(
,(
.
(26)
Analysis of Eq. (26) shows that the soliton complex is sta-
ble for high values of its velocity and a small distance be-
tween composite kinks due to the effective kinks attrac-
tion. For value of velocity much larger than the velocity
of stationary motion the complex also dissociates in the
manner shown in Fig. 4.
Dynamics of bound soliton states in regularized dispersive equations
Fizika Nizkikh Temperatur, 2008, v. 34, No. 7 717
0 50 100 150 200
–8
–6
–4
–2
0
2
4
6
8
110 120 130 140
–0,3
0,0
0,3
x
u
(x
,t
)
1
Fig. 4. Decay of a soliton complex for � = 1, Vin = 0.9 and
t1 = 500. The first kink moves with a constant velocity V1= 0.152.
Behind the second kink are breather modes. The inset shows
the spatial modulation of the field between kinks on an ex-
panded scale.
0 10 20 30 40 50 60
0
2
4
6
x
u
(x
,t
)
1
Fig. 3. A fast kink evolution with generating a breather on its
wake for � �1 6/ and Vin � 0.86.
5. Breather properties in the regularized dispersive
SG equation
The form of static breather can be found analytically as
an asymptotic series using the Kosevich–Kovalev scheme
of construction of the self-oscillation solution [8]:
u x t A x t B x t C x t( , ) ( ) sin ( ) sin ( ) sin .� � � �� � �3 5 �
(27)
For a main harmonics with a frequency�which is close to
the linear spectrum lower edge, i.e. for % �� � --1 12 ,
one obtains the following effective equation:
" ". . . �. . .tt xx xxtt� � � � �
1
8
0
2
. (28)
Here a complex function .( , )x t determines the solution
u x t x t( , ) Re( ( , ))� . in the first approximation with re-
spect to the small parameter %. Seeking the solution of
Eq. (28) in the form . �� f x i t( ) exp( ) we derive the non-
linear ordinary equation:
( ) ( )1 1
1
8
02 2 3� � � � ��� �f f fxx , (29)
which gives a coordinate dependence of the harmonic am-
plitude as a usual soliton profile:
f
x
�
4%
!cosh
, !
�
��
2
2
2
1
1
�
�
�
. (30)
However one can see a new feature of the breather,
which consists in vanishing the effective width depend-
ence on the amplitude% in the limit � � 1. In fact it ap-
pears that in this case the amplitude of breather is not al-
ready a constant but a slowly time-oscillating function.
This results in the main frequency splitting and a complex
breather behavior showing in Fig. 5. Such a behavior is
similar to dynamical properties of breathers in discrete ant
nonlocal SG models [37,38]. At last have we found that a
single breather motion is accompanied by a small breather
bursting process and emitting radiation as shown in Fig. 6.
As one has seen in previous sections the excitation of
breather modes plays a crucial role in the kink and soliton
complex dynamics in the case of a strong dispersion.
6. Stabilization of soliton complexes by driving forces
in dissipative media
Finally, we have investigated the influence of external
forces and dissipation on the dynamics of soliton com-
plexes. For this purpose we add the dissipation term /u t
and a driving force f 0 to the right-hand side of Eq. (6)
u u u u u ftt t xx xxtt� � � � �/ � sin 0. (31)
The term f 0 in the right-hand side corresponds, for exam-
ple, to the bias current in a long Josephson junction. The
result of a numerical modeling are presented in Fig. 7 for
4�-complex profiles and in Fig. 8 for their step-like ve-
locity dependences on the driving force strength (one can
compare this result with the velocity-force dependence
for a single 2�-kink in the discrete SG model [39]). Pa-
rameters are chosen as follows: / = 0.1 and six sequential
values of f 0 from –0.1 to –0.35. It turns out that the driv-
ing force under conditions of dissipation permits stabili-
zation not only of the soliton complex but also of its «ex-
cited» states with internal structures. For waves of
stationary profile, the derivatives u t and u xare propor-
tional to each other, and both have the form of closely
spaced double peaks. These derivatives are directly re-
lated to experimentally measurable quantities, in particu-
lar, the voltage U u t0 and magnetic field H u x0 in the
case of a long Josephson junction, and in a crystal with
dislocations the derivative u xdetermines the elastic de-
formation of the medium. In conclusion we note that the
possibility of observing multisoliton excitations in long
718 Fizika Nizkikh Temperatur, 2008, v. 34, No. 7
M.M. Bogdan and O.V. Charkina
100 110 120 130 140
–2
–1
0
1
2
x
u
(x
,t
)
Fig. 5. A half-period evolution of a static breather at � � 0.9.
20 40 60 80 100
–4
–3
–2
–1
0
1
2
3
4
x
u
(x
,t
)
Fig. 6. Two moving breather profiles divided of a half-period
in time at � � 0.9.
Josephson junctions was demonstrated quite some time
ago [40].
7. Summary
Thus we have studied the nonstationary dynamics and
interactions of topological solitons (kinks) in one-dimen-
sional systems with a strong dispersion. Analytical ap-
proach has been proposed for investigation of dynamical
features of a single kink motion which accompanied by
emitting radiation and small-amplitude breathers. Collec-
tive coordinate ansatz has been also proposed to study
processes of soliton complex formation in relation to the
strength of the dispersion, soliton velocity, and distance
between solitons. The breather solution has been con-
structed in a small amplitude-limit and its internal oscilla-
tion and propagation in the dispersive medium have been
investigated in detail. It has been shown that theoretical
results are in good relation with numerical simulations
and quantitatively explain them. It is demonstrated that
stable bound soliton states with complex internal
structure can propagate in a dissipative medium owing to
their stabilization by external forces.
The results obtained can be used for explanation and de-
scription of new effects in the dynamics of topological
solitons in highly dispersive media — in particular, disloca-
tions in nonideal lattices, fluxons in Josephson junction sys-
tems, and magnetic domain walls in anisotropic magnets.
Acknowledgement
This work is dedicated to the memory of Arnold
Markovich Kosevich who was the teacher for us in sci-
ence and life.
We are grateful for a support to a joint scientific project
No.24-02-a of NAS of Ukraine and RFBR, and to joint
French-Ukrainian project in the framework of scientific co-
operation between NAS of Ukraine and CNRS of France.
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