Bifurcation picture and stability of the gap and out-gap discrete solitons
The dynamics of a quaternary fragment of a discrete system of coupled nonlinear oscillators with modulated frequency parameters is investigated and the stability of its gap and out-gap soliton-like excitations is studied.
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2007
|
| Назва видання: | Физика низких температур |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/127822 |
| 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: | Bifurcation picture and stability of the gap and out-gap discrete solitons / L. Kroon, M.M. Bogdan, A.S. Kovalev, and E.Yu. Malyuta // Физика низких температур. — 2007. — Т. 33, № 5. — С. 639-642. — Бібліогр.: 9 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-127822 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1278222017-12-29T03:03:21Z Bifurcation picture and stability of the gap and out-gap discrete solitons Kroon, L. Bogdan, M.M. Kovalev, A.S. Malyuta, E.Yu. Письма pедактоpу The dynamics of a quaternary fragment of a discrete system of coupled nonlinear oscillators with modulated frequency parameters is investigated and the stability of its gap and out-gap soliton-like excitations is studied. 2007 Article Bifurcation picture and stability of the gap and out-gap discrete solitons / L. Kroon, M.M. Bogdan, A.S. Kovalev, and E.Yu. Malyuta // Физика низких температур. — 2007. — Т. 33, № 5. — С. 639-642. — Бібліогр.: 9 назв. — англ. PACS: 05.45.–a, 63.10.+a, 42.79.Gn http://dspace.nbuv.gov.ua/handle/123456789/127822 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Письма pедактоpу Письма pедактоpу |
| spellingShingle |
Письма pедактоpу Письма pедактоpу Kroon, L. Bogdan, M.M. Kovalev, A.S. Malyuta, E.Yu. Bifurcation picture and stability of the gap and out-gap discrete solitons Физика низких температур |
| description |
The dynamics of a quaternary fragment of a discrete system of coupled nonlinear oscillators with modulated
frequency parameters is investigated and the stability of its gap and out-gap soliton-like excitations is
studied. |
| format |
Article |
| author |
Kroon, L. Bogdan, M.M. Kovalev, A.S. Malyuta, E.Yu. |
| author_facet |
Kroon, L. Bogdan, M.M. Kovalev, A.S. Malyuta, E.Yu. |
| author_sort |
Kroon, L. |
| title |
Bifurcation picture and stability of the gap and out-gap discrete solitons |
| title_short |
Bifurcation picture and stability of the gap and out-gap discrete solitons |
| title_full |
Bifurcation picture and stability of the gap and out-gap discrete solitons |
| title_fullStr |
Bifurcation picture and stability of the gap and out-gap discrete solitons |
| title_full_unstemmed |
Bifurcation picture and stability of the gap and out-gap discrete solitons |
| title_sort |
bifurcation picture and stability of the gap and out-gap discrete solitons |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2007 |
| topic_facet |
Письма pедактоpу |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/127822 |
| citation_txt |
Bifurcation picture and stability of the gap and out-gap discrete solitons / L. Kroon, M.M. Bogdan, A.S. Kovalev, and E.Yu. Malyuta // Физика низких температур. — 2007. — Т. 33, № 5. — С. 639-642. — Бібліогр.: 9 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT kroonl bifurcationpictureandstabilityofthegapandoutgapdiscretesolitons AT bogdanmm bifurcationpictureandstabilityofthegapandoutgapdiscretesolitons AT kovalevas bifurcationpictureandstabilityofthegapandoutgapdiscretesolitons AT malyutaeyu bifurcationpictureandstabilityofthegapandoutgapdiscretesolitons |
| first_indexed |
2025-07-09T07:48:21Z |
| last_indexed |
2025-07-09T07:48:21Z |
| _version_ |
1837154754837348352 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5, p. 639–642
Letters to the Editor
Bifurcation picture and stability of the gap and out-gap
discrete solitons
L. Kroon
Department of Physics, Chemistry and Biology, Linkoping University, Linkoping SE-581 83, Sweden
M.M. Bogdan and A.S. Kovalev
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: kovalev@ilt.kharkov.ua
E.Yu. Malyuta
Electrophysical Scientific and Technical Center, 28 Chaikovskogo Str., Kharkov 61002, Ukraine
Received March 12, 2007
The dynamics of a quaternary fragment of a discrete system of coupled nonlinear oscillators with modu-
lated frequency parameters is investigated and the stability of its gap and out-gap soliton-like excitations is
studied.
PACS: 05.45.–a Nonlinear dynamics and chaos;
63.10.+a General theory;
42.79.Gn Optical waveguides and couplers.
Keywords: linear gap, nonlinear gap, stability, modulated systems.
1. Introduction
A conception of the gap (Bragg) soliton has appeared
for the first time in nonlinear optics [1,2]. These solitons
can exist in nonlinear systems with spatial periodicity of
some material parameters and possess the frequencies in
the gap of the spectrum of linear excitations. The interest
to the problem is caused by the fact that the group velocity
of linear waves, linear pulses and solitons tends to zero at
the boundary of the gap. The velocity of optical pulse is
reduced significantly in material domains with modulated
parameters which are inserted into the optical fiber, and
this effect could be used in nonlinear optical devices.
Solitons with the frequencies outside of the gap, the
out-gap solitons, were first studied in [3]. This type of em-
bedded solitons has frequencies inside the band of linear
excitations and nonzero asymptotics at infinity. Therefore
in contrast to the gap soliton the out-gap one has infinite
norm. The gap- and out-gap solitons in discrete systems
with alternating atom or spin characteristics, e.g. in di-
atomic lattices and two-sublattice magnets, were investi-
gated by many authors. A lot of obtained results were sim-
ilar to those for extended systems, while some finite-size
models were solved exactly in special cases. However two
important questions remain still open: (i) how the gap
soliton transforms into the out-gap one at the boundary of
the frequency gap of linear waves («linear gap»), and (ii)
what are stability properties of the gap and out-gap so-
litons.
It is well-known [4] that many aspects of soliton dy-
namics of nonlinear systems can be elucidated in the
framework of models with a finite number of degrees of
freedom. The simplest discrete modulated system which
permits of an existence of analogues of the gap and
out-gap excitations represents the ring of four coupled
nonlinear oscillators with alternating frequency parame-
ters [5]. To date this model is of a great interest for
low-temperature physics due to a topical problem of mag-
netic molecular nanoclusters [6]. It is known that in sys-
tems of finite size or with a finite number of degrees of
freedom quasi-solitons appear in the bifurcation manner
beginning from the moment when the energy or system
© L. Kroon, M.M. Bogdan, A.S. Kovalev, and E.Yu. Malyuta, 2007
parameters, in particular a depth of the frequency modula-
tion � [5], exceed the threshold values. The scenario of the
birth of the analogue of the gap soliton in the quaternary
model contains two bifurcations: at � c �1707. , where
some new excitations appear in «nonlinear gap» (see be-
low) and at �* .�1750, where analogues of the gap and
out-gap modes appear. This bifurcation pattern was quali-
tatively depicted in the inset of Fig. 3 in [5]. For the first
time these two bifurcations were discovered numerically
by L. Kroon [7] who informed the authors of [5] about his
results before the publication [5]. The exact bifurcation
picture represented numerical results of [7] is shown in
Fig. 1. The main goal of this paper is to reveal details of
the scenario of transformation of the gap soliton analogue
into out-gap soliton one and to analyse a stability of these
nonlinear excitations.
2. The model
Nonlinear dynamics of a ring consisting of four cou-
pled anharmonic oscillators (or classical spins with num-
bers n = 1,2,3,4) with a periodic modulation of the fre-
quency parameter is considered in the framework of
discrete nonlinear Schr�dinger equation (DNLSE) [5]:
i n
n
n n n n n� ( ) | |
( )� � � � � � � �� � � � �� �0 1 1
2 0. (1)
The corresponding Hamiltonian of the model has the
form
� � � �H
n
n n n n n n
n
� � � �� � [ ( ) / ]
( * *� � � � � � � �
0
2
1 1
4
2 ,
(2)
with canonical conjugated variables { }i n� , { }*� n and
� �
0
( )n
a
( )
( )� �
0
n
b
when index n is an odd (even)
number. In addition to the Hamiltonian (2) also the norm
(excitations number), defined by � �N n
n
� � 2
, is a con-
served quantity for (1). The ratio � � �
�b a (we suppose
for definiteness � � 1) reflects the depth of modulation.
The transformation � ��n n
, � � �
0 0
( ) ( )n n
is invok-
ed in normalizing the coupling � to unity. We will discuss
the stationary states of the form � ��
n nt( )( ) � �
� �exp ( )i t� with real amplitudes � n . In the linear limit
(� n
0) the spectrum of normal modes contains only 4
frequencies for inphase and antiphase oscillations with
�min,max �� � � �[( ) ( ) ] /� � � �a b b a�
2 16 2, and for
the gap boundaries solutions with � �� a and with
� �� b . In the limit of a long chain these frequencies do
not change but the domains ( , )min� �a and ( , )max� �b
transform into two bands of the spectrum with the gap
( , )� �a b . That is why we will call in our simple model
this domain as a linear gap. In nonlinear case the frequen-
cies of aforesaid four «main nonlinear modes» decrease
and «linear gap» transform into «nonlinear gap» with
� �a b a b a bN N, , ,( ) � � � 2 (see the lines (a) and (b) in Fig.
1 and Fig. 2). The frequencies �a and �b at the bound-
aries of the gap correspond to the antiphase oscillations
( )� �0 0 and ( )0 0� � , respectively, where the zeros indi-
cate immovable particles and the thickness of the arrows
characterizes the relative amplitude of the oscillations.
The frequencies for nonlinear oscillations depend not
only on the parameter �, but also on the amplitude and,
hence, implicitly on the norm N and the energy E defined
from the Hamiltonian (2). The «spectral» dependence
E E N� ( ) of the system is uniquely determined by the
characteristic � �� ( )N due to the fulfillment of the re-
lation � � dE dN/ for monochromatic oscillations. The
most interesting for us are the upper boundary of the non-
linear gap (line b in Fig. 1) and the analogue of the gap
and out-gap modes (line e in Fig. 2), bifurcating from the
boundary.
640 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5
L. Kroon, M.M. Bogdan, A.S. Kovalev, and E.Yu. Malyuta
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
0 1 2 3 4 5 6 7 8 9
N
�
b
a c
d
1
2
4
3
Unstable
Stable
Fig. 1. Stationary solutions in the (�, N) plane for � �175. .
Solid (dotted) lines represent stable (unstable) regions of the
solutions.
–4
–2
0
2
4
6
8
10
0 5 10 15 20 25
a
b
d
c
e
N
�
Unstable
Stable
1
2 3
4
5
Fig. 2. Stationary solutions in the (�, N) plane for � � 25. ,
which is the threshold for the linear stability of the gap and
out-gap solitons.
3. The stability problem
The stability of the solution � �
n t( )( ) is analyzed
by adding the perturbation � �n nt i t( ) exp ( )� � ��
� � n i t* *exp ( )� to its time-independent amplitude � n .
Linearization of Eq. (1) around the stationary solution
yields the eigenvalue problem
( )
( )� � � � � � � � �� � � � � � �� �0
2
1 1
22
n
n n n n n n n� ,
( )
( )� � � � � � � � �� � � � � �� �0
2
1 1
22
n
n n n n n n n� .
(3)
The linear stability of the stationary solution is equivalent
to all eigenfrequencies � being real.
(i) The stability of b-mode. This problem can be solved
analytically. Introducing the notations A a� �� �, B �
� �� �b , � � �0 2� � �( )a b and the width �
� �B A
� �( )� �1 0 of the linear gap, the solution for the upper
boundary of the «nonlinear» gap (b-mode) can be written
as { } { , , , }( )� n
b B B� �0 0 [5] and the eigenvalues � are
found to satisfy the equation
� � � �2 2 2 4 2 28 8 16 0( )( ( ) )� � � � � �A A AB . (4)
The root � � 0 corresponds to «phase mode» and the
roots � � � A are real, whereas the remaining eigenvalues
� � � � � � � � � �( ) ( ) ( )A A AB2 2 28 2 8 4 8 2 (5)
are complex whenever A A A( )3 16 32 0� � �� . Denoting
�� � � � �16 16 16 32 33 � �( ) ( ) the criterion leads to the
oscillatory instability [2] in the region 0 � � �� �A � � ,
which appears through Krein collisions [8] and manifests
itself through resonances of the internal modes. One may
note from Fig. 1 that there exists two windows of instabil-
ity for b-mode. The first one is bounded by the bifurca-
tion point 1 of gap e-mode and the bifurcation point 2 in
Fig. 1. For large � this is the bifurcation point for unstable
d-mode (Fig. 2). A second interval developing for
� � �4 3 3( ), is ruled out by the constraint B A� � �� 0
and is bounded by the point 3 for the bifurcation of
c-mode { }( )� n
c � � � �{ , , , }A B A B [5] at the fre-
quency � �� a (the low boundary of the linear gap). If
AB � �2 0 is satisfied two of the eigenvalues (5) are not
real. Applying the dependence � ��b aN N( ) � � � 2 of
frequency on norm N , inequality gives an interval
� �� �b N( ) ( )� � � �0
22 2� , for which the solutions
{ }( )� n
b is unstable, if and only if � ��
2 2 c . For large �
(see Fig. 2) c-mode is stable only in the vicinity of
point 3. The lower boundary of nonlinear gap corre-
sponds to a-mode which is linearly stable for all values of
parameter �. (ii) The stability of the gap and out-gap-
modes. First of all we notice that our investigations of the
gap solitons in the systems with 4, 6, 8, 10, 12, 16 partic-
les have shown that the dependence of the gap and
out-gap solution frequency on the norm changes qualita-
tively in the same manner when the number of particles
grows or the parameter � grows. The following result is
the most important: the dependence � �� ( )N does not
embed into the lower zone of linear waves spectrum and
remains inside the nonlinear gap of the spectrum, but it
transforms essentially at the frequency of the lower
boundary of the linear gap (� � 4 in Fig. 2). At this fre-
quency value the transformation of the gap soliton into
the out-gap one takes place. There are not analytical ex-
pressions for the gap and out-gap solitons, except the
case � � 2 5. [9]. We studied the stability of these excita-
tions numerically in the framework of Eqs. (3) with the
use of numerical solutions for � n . The results are shown
in Fig. 3. There exists the window of the Krein
(oscillatory) instability of the gap and out-gap solitons
analogues with nonzero imaginary part of parameter �.
But Im � tends to zero while �
2.5 (large depth of mod-
ulation). This justifies the expectation that the gap and
out-gap solitons are stable in the large modulated sys-
tems. In the inset of Fig. 3 the transformation of the win-
dow of instability is presented: it lies inside the linear
gap, the frequencies of the window and its width decrease
with the growth of parameter �.
4. Conclusion
The analogues of gap- and out-gap solitons have been
studied in the quaternary fragment of discrete modulated
nonlinear system of coupled oscillators. It has been de-
monstrated that transformation of such monochromatic
Bifurcation picture and stability of the gap and out-gap discrete solitons
Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 641
0
0.1
0.2
0.3
1.7 1.8 1.9 2,0 2.1 2.2 2.3 2.4 2.5
Maximum of imaginary part
�
�
�
4.7 4.8 4.9 5.0�
0.01
0
–0.01
4
1
2
3
Fig. 3. The development of the Krein instability for the gap
and out-gap breathers versus �: 2.47 (1), 2.48 (2), 2.49 (3),
�2.5 (4). For � � 235. the oscillatory instability enters the gap
regions ( )�� 4 and the resonance vanishes finally at � � 25.
( )� � 5 . Close to the threshold of the stability the maximum
value of the imaginary part of � shows a linear scaling in �
(and �).
soliton-like solutions and their stability depend essen-
tially on the value of the modulating parameter �. After
two bifurcations at � �1707. and � �1750. the unified de-
pendence of the soliton frequency �on the norm N for gap
and out-gap solitons is formed. The gap soliton trans-
forms into out-gap one at the lower boundary of the «li-
near gap» of the spectrum, while the dependence � �� ( )N
for these excitations is situated above the lower boundary
of «nonlinear gap». In the region 175 2 5. .� �� there exists
the window of the oscillatory instability of the soliton so-
lution, but for � � 2.5 the gap and out-gap solitons are
stable for all the frequencies.
We would like to thank Magnus Johansson for valu-
able discussions. Financial support from the Royal Swe-
dish Academy of Sciences is gratefully acknowledged.
1. D. Mills and J. Trullinger, Phys. Rev. B36, 947 (1987);
W. Chen and D.L. Mills, Phys. Rev. Lett. 58, 160 (1987).
2. Yu.S. Kivshar and G.P. Agrawal, Optical Solitons, Aca-
demic Press, Amsterdam (2003).
3. J. Coste and J. Peyraud, Phys. Rev. B39, 13096 (1989).
4. A.M. Kosevich and A.S. Kovalev, Introduction in Nonlinear
Physical Mechanics, Naukova dumka, Kiev (1989) (in
Russian).
5. M.M. Bogdan, A.S. Kovalev, and E. Malyuta, Fiz. Nizk.
Temp. 31, 807 (2005) [Low Temp. Phys. 31, 613 (2005)].
6. W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999).
7. L. Kroon, Private Communication (2003).
8. J.E. Howard and R.S. MacKay, J. Math. Phys. 28, 10036
(1987).
9. M. Johansson, Private Communication (2005).
642 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5
L. Kroon, M.M. Bogdan, A.S. Kovalev, and E.Yu. Malyuta
|