One-dimensional bright discrete solitons in media with saturable nonlinearity

Abstract

A problem of pulse propagation in a homogeneous nonlinear waveguide array with saturable nonlinearity is studied. The corresponding model equation is the discretized Vinetskii-Kukhtarev equation with neglected influence of diffusion of charge carriers. For periodic boundary conditions, exact homogeneous and oscillating stationary solutions are found. A wide instability region of the homogeneous, array-independent solution is identified. An approximate analytical solution for the bright one-dimensional discrete soliton where the energy is concentrated mainly in a few waveguides is obtained. The soliton stability is investigated both analytically and numerically and a cascade nature of the saturation mechanism is revealed.

Full text

One-dimensional bright discrete solitons in media with saturable nonlinearity
Milutin Stepić*and Detlef Kip
Institute of Physics and Physical Technologies, Technical University of Clausthal, D-38678 Clausthal-Zellerfeld, Germany
Ljupčo Hadžievski
Vinča Institute of Nuclear Sciences, P. O. Box 522, 11001 Belgrade, Serbia and Montenegro
Aleksandra Maluckov
Faculty of Sciences and Mathematics, University of Niš, 18000 Niš, Serbia and Montenegro
(Received 27 October 2003; published 23 June 2004)
A problem of pulse propagation in a homogeneous nonlinear waveguide array with saturable nonlinearity is
studied. The corresponding model equation is the discretized Vinetskii-Kukhtarev equation with neglected
influence of diffusion of charge carriers. For periodic boundary conditions, exact homogeneous and oscillating
stationary solutions are found. A wide instability region of the homogeneous, array-independent solution is
identified. An approximate analytical solution for the bright one-dimensional discrete soliton where the energy
is concentrated mainly in a few waveguides is obtained. The soliton stability is investigated both analytically
and numerically and a cascade nature of the saturation mechanism is revealed.
DOI: 10.1103/PhysRevE.69.066618 PACS number(s): 42.65.Tg, 05.45.Yv, 42.81.Dp, 42.82.Et
I. INTRODUCTION
A soliton is a type of normal mode of a special class of
nonlinear, usually infinite-dimensional mechanical systems
often described by integrable partial differential equations
such as Korteweg–de Vries, Klein-Gordon, or the nonlinear
Schrödinger equation. From a mathematical aspect, solitons
are a special nonlinear paradigm since they are associated
with Hamiltonians, i.e., conservative systems of a kind
which have action-angle variables, so-called integrable (or
completely integrable)nonlinear systems. Solitons are stable
with respect to collisions with both linear waves and other
solitons. However, there are a lot of dynamical systems in
nature which could be described by virtue of some noninte-
grable partial differential equation. Such systems also have
localized, so-called solitary wave solutions that can travel
without a change in shape, and a few integrals of motion
such as energy, momentum, and number of quanta (power).
These solitary wave solutions can exist in bounded media,
but often suffer from various kinds of instabilities. In the
optical community, it is very common to neglect these dif-
ferences and just to call these solitonlike pulses simply soli-
tons. Hereafter, we are going to follow this optical notation.
Many different sorts of solitons, which differ in their non-
linear physical mechanism, dimensionality, color, or coher-
ence, have been both theoretically predicted and experimen-
tally observed [1]. But, definitely, photorefractive solitons
attract the biggest attention both in the soliton and the optical
communication community. Due to the small optical power
required for their generation (microwatt level), it is very easy
to obtain them experimentally even with continuous-wave
lasers and standard optical equipment, and an almost full
control of the relevant parameters can be obtained in the
experiment. The formation time of these solitons can be as
short as milliseconds and below. Moreover, the magnitude of
the saturable nonlinearity of photorefractive crystals can be
easily driven by adjusting the applied external electrical
field.
There are few different classes of photorefractive solitons,
but of special interest are steady-state screening solitons that
were predicted and observed a few years ago [2,3]. The
physical mechanism which lies beyond the generation of
screening solitons is rather complicated and therefore we di-
rect the interested reader to some of the articles in which this
is explained in detail [4–6]. We shall just mention that this
mechanism includes several processes with a retarded tem-
poral response. Both charge separation and the consequent
generation of a space-charge electric field under the influence
of an external beam require a finite time which is propor-
tional to the dielectric relaxation time. Due to the charge
transportation over macroscopic distances via diffusion and
drift, this mechanism is also anisotropic and nonlocal.
Bright solitons in saturable bulk media are already well
described [7–11]. The biggest differences from solitons in
Kerr media are that photorefractive solitons exhibit a stable
self-trapping behavior in both transverse dimensions [8]and
an inelastic character of collisions between them leading to
the fission and fusion of solitons [7,9].
Although many problems in the nonlinear dynamics of
spatially extended systems involve continuous media, there
are many inherent discrete systems such as Davydov’s model
for transport of energy in biophysical systems [12], models
describing the optical pulse propagation in arrays of coupled
optical waveguides [13–16], the model of Scheibe aggre-
gates [17], and others. The major physical parameter in these
systems is the interelement distance. The localized states in
discrete systems arise through the balance between nonlin-
earity and linear coupling effects among the adjacent poten-
tial wells. The effect of discreteness may significantly change
the stability properties of localized states [13,18], collapse
*Email address: milutin.stepic@tu-clausthal.de
PHYSICAL REVIEW E 69, 066618 (2004)
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dynamics [19–21], and other features as compared to con-
tinuous systems. Among the nonlinear discrete systems,
probably the most interesting (from a practical point of view)
are those in nonlinear optics [15,22]. Such discrete optical
solitons are found both in arrays of coupled waveguides
[13–16]and fibers [22].
The nonlinear waveguide array was introduced in soliton
theory 15 years ago [13]. It is suggested that these arrays
possess a great potential for various applications such as op-
tical interconnects, beam deflectors, and modulators as well
as nonlinear all-optical switches and amplifiers. The first ex-
perimental observation of discrete spatial solitons in nonlin-
ear waveguide arrays with Kerr nonlinearity was reported
only five years ago [23]. Soon thereafter, waveguides with a
negative diffraction were obtained which enable defocusing
of light and paved the way to the discovery of the discrete
diffraction managed spatial solitons [24]. Discrete gap soli-
tons in modulated (binary)waveguide arrays were predicted
[25]and discrete solitons in two-dimensional (2D)networks
of nonlinear waveguide arrays have been proposed [26].
Also, the influence of long-range interactions on the nonlin-
ear localized modes in 2D photonic crystal waveguides has
been investigated [27]. Optically induced 1D and 2D photo-
nic lattices have been created by virtue of plane-wave inter-
ference, and discrete photorefractive solitons in such systems
were numerically and experimentally obtained very recently
[28].
However, as of yet, there are neither explicit results for
the dependence of these localized discrete structures in a
saturable media on the system parameters nor information on
their linear stability. The objective of our paper is to inves-
tigate the existence and stability of 1D bright discrete screen-
ing solitons. Our findings could be interesting not only for a
particular application in nonlinear optics but also for differ-
ent discrete biophysics and solid-state physics systems with
the same type of nonlinearity. This paper is organized in the
following manner: the basic evolution equation is defined in
Sec. II, analytical and numerical results concerning homoge-
neous solution are collected in Sec. III, corresponding results
related to the soliton solution are placed in Sec. IV, while the
conclusions are given in Sec. V.
II. MODEL EQUATION
The evolution equation of bright 1D optical spatial soli-
tons in bulk photorefractive media, based on the Vinetskii-
Kukhtarev model [4](with the neglected diffusion term), can
be written as [6]
i
⳵
U
⳵␰
+1
2
⳵
2U
⳵
s2−
␤
U
1+兩U兩2=0, 共1兲
where U=I/Idis a normalized slowly varying envelope of
the electric field of the light wave, Iis the intensity of the
beam while Idis the so-called dark irradiance, and
␰
=z/kx0
2
is a dimensionless coordinate along which the beam propa-
gates. Here x0is an arbitrary spatial width and k=2
␲
neo/␭0
is the wave number with the unperturbed extraordinary re-
fractive index neo and light wavelength ␭0. We use s=x/x0as
a normalized transverse coordinate and
␤
=共kx0neo兲2r33E0/2
is a positive parameter (r33 is the electro-optic coefficient,
E0⬇V/L, where Vis a constant bias voltage and Lis the
width of the photorefractive crystal along the xdirection).
The crystal is so oriented that its ferroelectric caxis coin-
cides with the xdirection. It is also assumed that the incident
laser beam is polarized along the caxis (extraordinary refrac-
tive index)and that the applied electric field E0has a com-
ponent only in the same direction.
The optical pulse propagation in 1D equidistant nonlinear
waveguide arrays with saturable nonlinearity can be mod-
eled, within the nearest-neighbor approximation and with ne-
glected influence of diffusion of charge carriers, by virtue of
the following discrete version of the Vinetskii-Kukhtarev
equation:
i
⳵
Un
⳵␰
+1
2h2共Un+1 +Un−1 −2Un兲−
␤
Un
1+兩Un兩2=0, 共2兲
where Unis the wave function in the nth nonlinear element
共n=1,...,N兲with 共UN+1 =U1兲for the case of periodic
boundary conditions, h=共L−Nd兲/Nx0is the normalized dis-
tance between two elements, and drepresents the width of a
single waveguide (usually a few microns). This equation in
fact represents a system of linearly coupled nonlinear ordi-
nary differential equations which are not integrable in the
general case. It possesses a hidden Hamiltonian structure
i
⳵
Un/
⳵␰
=
␦
H/
␦
Un
*, where His the Hamiltonian of the sys-
tem and the asterisk denotes a complex conjugation. Here
H=兺n关
␤
ln共1+兩Un兩2兲+兩Un−1−Un兩2/2h2兴is the first con-
served quantity of the system, while the second one is a
number of quanta (power)P=兺n兩Un兩2. In the small-
amplitude limit 兩Un兩2Ⰶ1, this equation passes into the well-
known 共1+1兲discrete nonlinear Schrödinger equation with
Kerr nonlinearity. It means that with Eq. (2), under proper
conditions, one could also describe various real discrete
structures such as a chain of tightened atoms [20,29], the
model of dynamics in globular proteins and some molecular
crystals [12,30], arrays of Josephson junctions [31], polarons
in condensed-matter physics [32], and pulse propagation in
short fiber arrays (where dispersion can be neglected)[22].
Hereafter, we have restricted our study to a planar homo-
geneous array of waveguides in a photorefractive SBN61
共Sr0.61Ba0.39Nb2O6兲crystal, which possesses excellent opti-
cal properties and large nonlinear electro-optic coefficients
[33]. Permanent waveguides in SBN61 can be fabricated by
ion implantation [33]. Usually SBN61 crystals are both a few
mm long (zdirection)and wide (xdirection). The unper-
turbed extraordinary refractive index for SBN61 is neo
=2.35, while the relevant electro-optic coefficient is r33
=280 pm/V. This crystal is sensitive to both blue and
green light. The arbitrary scaling length x0is set to 8
␮
m.
III. HOMOGENEOUS SOLUTION
The stationary solutions are of significant importance be-
cause they represent some of the available attractors of the
system. Within a linear theory, it shows up that both the
exponential perturbations’ growth rate and the threshold
amplitude for the onset of instability depend on the wave
STEPIĆet al. PHYSICAL REVIEW E 69, 066618 (2004)
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number (and/or on the nonlinear frequency shift)of the
stationary solution. There are a lot of reports about the linear
stability of these stationary solutions in discrete
[13,15,18,21,27,29,34–37], continuous [19,38], as well as in
continuous-discrete systems [39,40]. For any set of the sys-
tem’s parameters
␤
and hthere exist 3Npossible stationary
solutions such as homogeneous, oscillatory, soliton, and mul-
tisoliton solutions. Here we investigate the existence and sta-
bility of the simplest homogeneous and soliton solution.
We have found the following exact oscillatory wave solu-
tion of Eq. (2):
Un
osc共n,
␰
兲=±
冑
␤
+2h−2 sin2共Kh/2兲−
␯
␯
−2h−2 sin2共Kh/2兲e−i
␯␰
+iKnh,共3兲
where Kis a discrete wave number and
␯
is the nonlinear
frequency shift. The amplitude of this solution is real if
2h−2 sin2共Kh/2兲⬍
␯
⬍
␤
+2h−2 sin2共Kh/2兲and the corre-
sponding existence region is of width
␤
. It is also possible to
find an exact, homogeneous solution
Uhom =A0e−i
␯␰
=±
冉
␤
−
␯
␯
冊
1/2e−i
␯␰
.共4兲
This simple array-independent solution is a special case of
the oscillatory one, where K=0 is taken. If 0⬍
␯
艋
␤
, its
constant amplitude is real and, opposite to the corresponding
solution in Kerr-like media, this one has a limited existence
range due to the saturable nature of the nonlinearity.
In order to investigate the linear stability of the homoge-
neous solution, we follow a standard procedure and intro-
duce a small, complex, array-dependent, in-phase perturba-
tion around this solution Un=共A0+
␦
n兲exp共−i
␯␰
兲, where
␦
n共
␰
兲=关a共
␰
兲+ib共
␰
兲兴exp共2i
␲
n/N兲and 共兩
␦
n兩ⰆUhom=A0兲. Sub-
stitution into Eq. (2), after linearization with respect to the
small perturbations and use of Fourier’s transform a,b
⬀e−i
␻␰
, leads to the following dispersion relation:
␻
2=4
h2sin2
冉
␲
N
冊
冋
1
h2sin2
冉
␲
N
冊
−
␯
冉
1−
␯
␤
冊
册
.共5兲
In the small-amplitude regime, this result coincides (after a
simple adjustment of notations and for wave number q=0)
with the dispersion relation [Eq. (10)] in Ref. [35], where a
general approach to modulational instability of discrete non-
linear systems with cubic nonlinearity is described. A suffi-
cient condition for the homogeneous solution to become
modulationally unstable is
␻
2⬍0 which, together with the
fact that N⬇100 and h⬇1, results in the next instability
frequency band
sin2共
␲
/N兲
h2⬍
␯
⬍
␤
−sin2共
␲
/N兲
h2.共6兲
In the very narrow frequency bands ⌬
␯
=sin2共
␲
/N兲/h2at the
ends of the existence interval, this solution is stable with
respect to the given form of the perturbation. Note that in
these regions, instabilities might occur under some other
kind of perturbations. This result is valid in the limit
4 sin2共
␲
/N兲/
␤
h2→0.
The dispersion relation in Eq. (5)defines the instability
growth rate spectra for the frequency band in Eq. (6).In
order to confirm our analytical results, we have numerically
solved Eq. (2)by a sixth-order Runge-Kutta procedure with
regular checking of the conserved quantities Pand H. For
the initial conditions 共
␰
=0兲in the numerical calculations, we
have used the perturbed stationary homogeneous solution
[Eq. (4)] in the form
Un=Uhom
冋
1+
⑀
cos
冉
2
␲
n−N/2−1
N
冊
册
,
where
⑀
is a small parameter (here we choose ␧=0.001). The
growth rates ⌫共
␻
=⍀+i⌫兲are numerically estimated from
the early stage of the time evolution of the electric field
across the waveguide array.
An example of the time evolution of the electric field
across a waveguide array with 41 elements and the nonlinear
frequency shift
␯
=2 is shown in Fig. 1, illustrating the de-
velopment of the modulation instability. The values of the
system parameters in this work (except in Fig. 6)are
␤
=18.2 (we choose the blue light from an argon-ion laser with
a wavelength of 488 nm)and h=0.5, which correspond to
an external applied electrical field E0=4.5 kV/cm and an
interelement distance of 4
␮
m. The numerically estimated
(circles)and analytically calculated values (dashed lines)of
the growth rates ⌫in the central waveguide are plotted over
the instability region, which is given by Eq. (6)for ⍀=0 and
for arrays with N=5, 15, and 41 waveguides in Fig. 2. The
agreement between the numerical and analytical results is
fairly good with only small discrepancies for the array with
N=5 waveguides in the region of medium values of
␯
.
This result indicates the presence of exponentially grow-
ing modes in the system, giving no predictions about the
subsequent nonlinear evolution stage. It is shown that dis-
crete systems with Kerr nonlinearity, instead of a collapse
behavior that is observed in the multidimensional continuous
case [19], exhibit a quasicollapse behavior leading to the
formation of localized structures in the form of discrete soli-
tons [34]. The Kerr nonlinearity is just the small-amplitude
limit of the saturable nonlinearity, therefore a similar quasi-
collapse process and existence of bright 1D discrete screen-
FIG. 1. Example of the time evolution of the electric field across
a waveguide array with N=41. System’s parameters are
␤
=18.2,
h=0.5, while
␶
=
␰
/2h2.
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ing solitons, where the energy is localized only in between a
few central waveguides of the nonlinear array, are fully ex-
pected.
IV. SOLITON SOLUTION
As for Kerr-like media [34], it is possible to find an ap-
proximate expression for such a narrow discrete photorefrac-
tive soliton. Namely, from Eq. (2), in the symmetric case
Un−m=Un+m,共m=1,2,...兲with 兩Un兩Ⰷ兩Un±1兩Ⰷ兩Un±2兩one
can get
Un=±
冑
␤
+1
h2−
␯
␯
−1
h2
e−i
␯␰
,
Un±m=Un
冋
2h2
冉
␤
+1
h2−
␯
冊
册
m共7兲
for the wave functions in a dominant element and its neigh-
bors. The result for the neighbors is valid only under the
restriction that 兩Um兩2→0. Such localized states are possible
neither in the small-amplitude regime nor in the oversatura-
tion regime. Figure 3 depicts the intensity distribution along
the central part of the nonlinear waveguide array for
␯
=5,
␤
=18.2, and for three different values of the waveguide dis-
tance h. The symbols denote the intensity in each waveguide,
while lines represent just a guide to the eye. In the case h
=1.5, the waveguides are well separated and there is almost
no coupling between them resulting in a discrete soliton with
the energy almost completely concentrated in the central
waveguide. Decreasing the distance will increase the cou-
pling, thus small satellites in the first neighbors form and the
discrete photorefractive soliton starts to spread. By further
reducing the distance, more and more elements of the array
are excited, thus the localized structure becomes wider. Ob-
viously, in the saturation region, where these solitons become
wider, it is not justified to neglect the amplitude of the first
neighbors in comparison to the central element’s amplitude.
By solving Eq. (2)for the case
␯
=h−2, with the following
corrected amplitude ratios A0⬎A1ⰇA2→0 one can get the
next approximate expressions for the pulse amplitudes in the
three central elements of the nonlinear array:
A0=±
冑
␤
2h4−2±
␤
h2冑4+
␤
2h4
2,
A1=
␤
h2A0
1+A0
2.共8兲
We applied a Vakhitov-Kolokolov [38]criterion which
gives the answer about the system’s stability with respect to
small longitudinal perturbations. With an assumption that the
system’s total power Pis shared only between three
waveguides, one may obtain
P=1+2h4共
␤
+h−2 −
␯
兲2
2h4共
␯
−h−2兲共
␤
+h−2 −
␯
兲.共9兲
According to [38,27], this localized structure is stable with
respect to the small longitudinal perturbations if the power P
is a monotonically decreasing function for any value of the
frequency
␯
. This is both a necessary and sufficient condition
for the stability in the discrete system [21]. The analytical
form for the power Pof discrete screening solitons placed on
the center of the lattice as a function of the nonlinear fre-
quency shift
␯
is presented in Fig. 4(a)(solid line). As the
power is a monotonically decreasing function of
␯
, one
might conclude that bright 1D discrete screening solitons are
stable with respect to small perturbations. In the small-
amplitude regime, this result confirms a conclusion about the
stability of the corresponding nonlinear mode (odd, unstag-
gered, and symmetric)from Ref. [37], where the stability of
strongly localized modes was investigated by virtue of a di-
rect linear analysis. In order to study dynamics of the dis-
crete systems in media with saturable nonlinearity, it is nec-
essary to use numerical simulation because Eq. (2)is not
integrable in a general case. These numerical results are es-
pecially valuable in the deep saturation regime where our
approximate theoretical solution fails. The numerically cal-
culated power is given in Fig. 4(a)(dotted line). The agree-
ment with our analytical results is fairly good, except in the
big amplitude regime. However, despite a small bend near
FIG. 2. Comparison of the numerical (circles)and analytical
(dashed line)results for the growth rate spectra ⌫=Im共
␻
兲for the
nonlinear waveguide arrays with N=5, 15, and 41 elements.
FIG. 3. Intensity patterns of discrete solitons for three different
values of the waveguide distance h. The symbols denote the inten-
sity in each waveguide while lines represent a guide to the eye.
STEPIĆet al. PHYSICAL REVIEW E 69, 066618 (2004)
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the left asymptote 共
␯
=h2兲, the curve is still monotonically
decreasing, thus confirming the soliton stability.
We also explore the behavior of discrete solitons in the
regime of saturation in detail. In Fig. 4(b), where the depen-
dence of the soliton amplitude on the frequency for the cen-
tral element and its first and second neighbors is presented,
one can notice a gradual transition to the saturation regime.
The central element goes first into saturation (for
␯
⬇4),
while the amplitude in its neighbors keeps rising monotoni-
cally until
␯
⬇1.5, when the first neighbors go in the satura-
tion too. This cascade saturation mechanism can explain the
observed bend of the numerical curve for
␯
⬇4 in Fig. 4(a).
Indeed, the amplitude of the central element of the discrete
soliton grows with the increase of the power Puntil it
reaches the saturation level. The further increase of Pis an
outcome of the growing amplitudes of the first neighbors. It
is plausible to expect that the amplitudes of the second
neighbors, etc., behave in a similar manner. Thus increasing
of Pdoes not lead to a continuous energy localization into a
single waveguide and its decoupling from the rest of the
array as in the case of the discrete media with cubic nonlin-
earity. Instead, it leads to a widening of the localized struc-
ture. Moreover, by replacing
␤
=18.2 and h=0.5 in Eq. (8)
we obtain A0=4.545, A1=0.955 while the corresponding nu-
merical values are A0=4.439, A1=0.976.
Figure 5 depicts a comparison between analytical and nu-
merical results for the possible stationary states of the dis-
crete screening soliton, which is placed at the center of the
array consisting of N=101 elements. For intermediate fre-
quencies [Fig. 5(a)], narrow discrete solitons are formed. By
lowering the soliton amplitude, our numerical simulation
suggests a widening of the localized structure [Fig. 5(b)].It
is interesting to mention that for the same initial conditions
in the small-amplitude Kerr regime, numerics revealed that
both localized and oscillatory solutions are possible. In this
region, the slopes of the curves are very small and practically
all corresponding soliton solutions are marginally stable.
In Fig. 6, a typical example of a discrete photorefractive
soliton propagation along an array with N=101 is presented.
Our focus is on the central part of the array. Similar patterns
are achieved in a wide interval of
␯
for discrete solitons
given by Eq. (7). Note that a qualitatively similar behavior,
where the input beam evolves into a stable discrete soliton, is
obtained with initially narrow Gaussian, sech, and nearly
rectangular pulses (which are more natural in the experi-
ment). We would like to underline that the parameters in Fig.
6(a waveguide distance of 3
␮
m and E0=9 kV/cm)are
very close to the practically achievable values in the crystal
SBN61. For the higher values of
␯
, the oscillatory solutions
are also observed.
FIG. 4. (a)Power of discrete screening soliton Pas a function of
nonlinear frequency shift
␯
共
␤
=18.2,h=0.5兲.(b)Amplitude in the
central waveguide and its first and second neighbors vs frequency
␯
for the same parameters as in (a).
FIG. 5. Amplitude of discrete screening soliton 共
␤
=18.2,h
=0.5兲vs waveguide number nfor (a)
␯
=10 and (b)
␯
=17. The
numerical results are given by circles while the analytical predic-
tions are given by squares.
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V. SUMMARY
In conclusion, the discrete nonlinear evolution equation
which describes pulse propagation in a planar homogeneous
array of waveguides in media with a saturable nonlinearity is
studied. In the case of periodic boundary conditions, two
exact (homogeneous and oscillatory)and one approximate
(solitonlike)stationary solution are found. A linear stability
analysis of the homogeneous solution is performed and ana-
lytical expressions for the corresponding instability thresh-
olds and the growth rate spectra are calculated. The growth
rate depends on the nonlinear frequency of the solution. Our
results show the existence of two cutoff frequencies and a
wide instability region in between, where the homogeneous
solution is modulationally unstable. In addition, the instabil-
ity thresholds and the growth rate spectra are numerically
calculated for discrete systems with a different number of
elements and compared with the corresponding analytical re-
sults. Here a very good agreement between numerical and
analytical results is found. It is also demonstrated, both ana-
lytically and numerically, that such nonlinear waveguide ar-
rays support stable bright one-dimensional discrete spatial
solitons. The high amplitude region is studied numerically
and a cascade mechanism of saturation in the nonlinear array
is found. Finally, the authors would like to emphasize that
these explicit analytical and numerical results can be inter-
esting not only for a particular application in nonlinear optics
but also for various discrete biophysics and solid-state phys-
ics systems with the same saturable nonlinearity.
ACKNOWLEDGMENTS
This work was funded by the German Federal Ministry of
Education and Research (BMBF, Grant No. DIP-E6.1)and
INTAS (Contract No. 01-0481). The work of Lj.H. and A.M.
(and partially M.S.)was supported by the Ministry of Sci-
ence, Development and Technologies of the Republic Serbia,
Project 1964. We are grateful to Dr. Wesner for a critical
reading of this manuscript.
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