PhysicsLetters A 173 (1993) 172-178
North-Holland
PHYSICS LETTERS A
Localized modes in a chain with nonlinear on-site potential
Yuri S. K i v s h a r 1
Institut fiir Theoretische Physik I, Heinrich-Heine- UniversitiitDiisseldorf, W-4000 Dfisseldorf 1, Germany
Received 7 October 1992;revised manuscript received 30 November 1992;acceptedfor publication 30 November 1992
Communicatedby L.J. Sham
Localizedmodesare analysedfor a chain with nonlinear on-site potential usingan approximationbased on the discrete nonlinear Schr0dingerequation. It is pointed out that such localized modes exist in the parameter domains where the systemdisplays
modulational instability. Otherwise, nonlinear modes may appear in the form of dark-profile localized structures, and two types
of such structures are found. The decaytime of stronglylocalizednonlinear modesis also estimated.
1. Introduction
As is well known, spatially localized modes can exist in a linear lattice if they are supported by impurities [ 1 ]. The localized mode has a maximum at
the impurity site and it decreases exponentially as a
function of the distance from the impurity. Similar
localized modes may be also found for strongly anharmonic (but perfect) lattices in the form of the socalled intrinsic localized modes [ 2 ], in the latter case
the energy localization, which involves only a few
particles of the chain, is possible due to nonlinearity
itself.
Different properties of the intrinsic localized
modes have been discussed in a number of papers
(see, e.g. refs. [3-13] ) for one- and three-dimensional cases. The original model for the intrinsic localized mode [ 2 ] is a chain with anharmonic interatomic interaction, the so-called Fermi-Pasta-Ulam
( F P U ) model. It describes a one-dimensional lattice
in which each atom interacts only with its nearest
neighbors via a symmetric nonlinear potential. The
localized pattern is
u , ( t ) =A(..., 0, -- ½, 1, - ½, 0 .... ) cos tot,
where k2 and k4 are the nearest-neighbor harmonic
and quartic anharmonic force constants, and A is the
On leave from: Institute for Low Temperature Physics and
Engineering, 310164Kharkov, Ukraine.
] 72
mode amplitude. The approximation is better for
large values of (k4/k2)A 2. Some authors [ 4,7 ] have
proposed another variant of a stationary self-localized mode, the so-called even-parity localized mode,
which has been found to be extremely stable.
Another type of intrinsic localized modes may be
analysed for a model with nonlinearity introduced
through an on-site potential. The case of the quartic
on-site potential has been discussed in ref. [ 3 ], where
highly localized modes with the frequencies below
the frequency gap of the linear spectrum have been
briefly discussed. However, the linear spectrum of a
discrete chain always has an upper (cut-off) frequency, so that one may naturally expect to find localized modes with the frequencies above the cut-off
frequency similar to the odd-parity modes in the FPU
chain. The physically important problem related to
these localized nonlinear modes is to prove that they
are long-lived excitations which may contribute to
many properties of nonlinear discrete systems.
Another important problem is created by a comparison between nonlinear modes in discrete and
continuous models. As has been recently mentioned,
the intrinsic localized modes described by Sievers and
Takeno may be easily understood as a highly localized (discrete) version of the well-known envelope
solitons of the continuous nonlinear Schr6dinger
(NLS) equation [8,9]. Such kind of localized modes
appear for the case of the self-focusing nonlinearity
in the NLS equation. Nevertheless, soliton solutions
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PHYSICSLETTERSA
are also possible for the other case, i.e. for defocusing
nonlinearity, and they appear as dips in the stable
constant-wave (cw) background [ 14] (that is why
they are called "dark" solitons). It is clear that the
corresponding localized structures may also be possible in discrete chains when spatially localized modes
do not exist.
It is the purpose of this paper to answer the questions above taking a simple (but rather general for
various applications) model where nonlinearity is
introduced through a cubic and quartic on-site
potential.
The paper is organized as follows. In section 2 we
present our model which describes a chain of particles subjected to a cubic and quartic nonlinear onsite potential. As is shown, this model may be reduced to the discrete NLS equation in the case when
coupling between the particles in the chain is small.
Section 3 is devoted to the spatially localized nonlinear modes of the chain. Small-amplitude modulations of a nonlinear wave in the discrete chain are
investigated to show that spatially localized modes
exist in the parameter domains where the system displays modulational instability. The lifetime of
strongly localized nonlinear modes is also estimated
analytically. In section 4 we obtain dark-profile localized structures which may be considered as a
highly discrete version of the well-known dark solitons. We present two types of such nonlinear structures, the so-called "cut-off" and "noncut-off" kinks.
Section 5 concludes the paper.
of the potential. Linear oscillations of the chain of
frequency co and wavenumber q are described by the
dispersion relation
co2=coo2+ 4 D sin2(½qa) ,
(2)
a being the lattice spacing. As shown by eq. (2), the
linear spectrum has a gap coo and it is limited by the
cut-off frequency comax=(co2+4D) 1/2 due to
discreteness.
Analysing slow temporal variations of the wave
envelope, we try to keep the discreteness of the primary model completely. In fact, this is possible only
under the condition o) 2 >> D, i.e. when the coupling
force between the particles is weak. Looking for a solution in the form
un =0n + ~n exp(-icoot) + ¥~*exp (ico0t)
+ ~n exp( - 2icoot) + ~*~exp(2icoot) +...,
(3)
we will assume the following relations (similar to the
continuum case, see, e.g. ref. [15 ] ), #n ~ ¢2, ~ ~ ¢2,
~ ~ ¢, and also the following relations between the
model parameters, D ~ 2, 0)2 ,., 1, ~, fl,-, 1, d/dt~¢ 2.
It is clear that this choice of parameters corresponds
to large values of to 2 (we may simply deride all the
terms by the frequency gap value).
Substituting eq. (3) into eq. ( 1 ) and keeping only
the lowest order terms in ¢, we obtain the equation
for ¥.,
2icoo~/~ + D ( ~ + 1 +~n-1 - 2~/n)
- 2a(~n ~ + ~*~n) - 3ill ~n 12~n = 0 ,
(4)
and two algebraic relations for 0. and ~n,
2. Model and the discrete NLS equation
2ct
~ . ~ - co-~ i~/~
We consider the dynamics of a one-dimensional
chain made of atoms with unit mass, harmonically
coupled with their neighbors, and subjected to a
nonlinear cubic and quartic on-site potential. Denoting by u,(t) the displacement of atom n, its equation of motion is
12
where D is the coupling constant, coo the frequency
of small amplitude on-site vibrations in the substrate
potential, a and p are the anharmonicity parameters
2
(5)
The results (4) and (5) are generalizations of the
well-known results for the continuum case [ 15].
Thus, the final discrete NLS equation is
igtn +K(gtn+~ + ¥ n _ 1 - 2 ~ n ) +AI ~. 12~. = 0 ,
(6)
where
fi-D(un+~ +un_~ - 2 u . ) +co~u~+au~ + flu~ =0,
(1)
a
, ~.~ 3co-~oo~,..
K=D/2~,
2= ~
l
(10a2/3COo2-3/~).
(7)
Equation (6) is used below to analyse different types
of localized modes in the chain. We would like to
point out again that the assumption of slow variation
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PHYSICS LETTERS A
of the envelopes in time, as well as the neglecting of
higher-order harmonics to derive (6), assume that
the gap frequency 090 is large with respect to the other
frequencies in the system, i.e. o902>> 4D, and to 2 >>
OtUo, flu02, Uo being the wave amplitude. The first
condition is valid in a weakly dispersive system where
mo is close to ogma~,while the second one means that
the nonlinearity is not large. These are the usual conditions to get the NLS equation, but in the lattice,
the condition o92 >> 4D means also that discreteness
effects are considered strong pointing out the interest in the discrete modes localized on a few particles.
3. Spatially localized nonlinear modes
[ t 2 - 2 K s i n ( Q a ) sin(qa) ]2
= 4 K sin2(½Qa) cos (qa)
× [4Ksin2(½Qa) cos (qa)-22q/2 ]
for the wavenumber Q and frequency 12 of the linear
modulation waves. In the long wavelength limit,
when Qa << 1 and qa << 1, eq. (10) reduces to the
usual expression obtained for the continuous NLS
equation [ 16 ].
Equation (10) determines the condition for the
stability of a plane wave with wavenumber q in the
lattice. Contrary to what would be found in the continuum limit, the stability depends on q. An instability region appears only if
2 cos(qa) > 0 .
3.1. Modulational instability
As is well known, nonlinear physical systems may
exhibit an instability that leads to a self-induced
modulation of the steady state as a result of an interplay between nonlinear and dispersive effects. This
phenomenon, referred to as modulational instability, has been studied in continuum models (see, e.g.,
refs. [ 16-19] ) and, only recently, in a discrete model
[ 20 ]. As has been pointed out, modulational instability is responsible for energy localization and formation of localized pulses.
For the NLS equation (6), derived in the single
frequency approximation, modulational instability
in the lattice can be easily analyzed. Equation (6)
has an exact cw solution
~u.( t ) = ~¢oexp (iO.),
O.=qan-ogt,
(8)
where the frequency to obeys the nonlinear dispersion relation
(11)
For positive 2 and a given q, e.g. q< n/2a, a plane
wave will be unstable to modulations in all this region provided ~/~ > 2K/2.
3.2. Structure of localized modes
One of the main effects of modulational instability
is the creation of localized pulses (see, e.g., ref. [21 ] ).
For the model under consideration it means that for
2> 0 the small q region is unstable, and, therefore,
nonlinearity can induce the formation of localized
modes in the gap of the linear spectrum (o92< o92).
Such a localized mode can be obtained directly from
the discrete NLS equation (6) following the method
of ref. [4]. We seek highly localized odd-parity solutions to the NLS equation (6) in the form
~/n(t)=Aexp(iOt)fn, where we take f o = l , f_n=f~
assuming [f~ I <<f~ for [n I > 1. Two equations at n = 0
and n = 1 then yield
~/n(t)=Aexp(it2t)(...,O,~l, 1, r/, 0, ...),
og= 4K sin 2( ½qa ) - 2gt2 .
(9)
The linear stability of the wave (8), (9) can be investigated by looking for a solution of the form
~/n(t) = (~Uo+bn) exp(i0~ + i z ~ ) ,
where b , = b , ( t ) and X,=X,(t) are assumed to be
small in comparison with the parameters of the cartier wave. In the linear approximation two coupled
equations for these functions yield the dispersion
relation
174
(10)
(12)
where the values inside parentheses are the amplitudes at successive sites, and
~r~= -- ~ 4 2,
r/=g/~a2<<
1 .
(13)
The mode (12), (13) is indeed highly localized provided K < < , ~ 4 2 (see fig. la). For 2>0, such nonlinear modes cannot exist for O92>o92m~,, because, according to eq. (10), a cw solution at these frequencies
is stable.
In the case when 2 = - 1 2 l < 0, the result of the
Volume 173, number 2
(a)
,!,
(b)
iT
3.3. Lifetime of localized modes
i
Fig. 1. Diagrammatic representation of the spatially localized
nonlinear modes in the chain with nonlinear on-site potential:
(a) low-frequency mode, (b) high-frequency mode.
modulational instability analysis is just reverse: The
small q region is stable whereas modulational instability occurs for q> n/2a (see eq. ( 11 ) ). As a result,
the localized modes (12) do not exist, but instead,
the possible spatially localized structure is (see fig.
lb)
~n(t)=Aexp(ig2t)( .... 0,
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PHYSICS LE'VrERS A
-r/, 1, -r/, 0,...),
(14)
where ~I=K/I21A 2 is assumed to be small, and
t 2 = 4 K + I21A 2 lies above the cut-off frequency.
It is interesting to compare the results obtained for
model (6) with those of the exactly integrable discrete Ablowitz-Ladik equation [22] (cf. eq. ( 6 ) ) ,
i~,, + K(¥~+ ~+ ~,,_ ~- 2~,)
+21 ~', 12(q/,+~ + ~ _ ~ ) = 0 .
(15)
Models (6) and (15) have the same linear properties and lead to the same NLS equation in the continuum limit provided 2 ' = ½2, however their nonlinear properties are very different. For model (15),
the dispersion relation for modulations is
[ t 2 - 2 (K+R~u2) sin(Qa) sin(qa)]2
= 16 (K+2~/2) sin2(½Qa) cos2(qa)
x [ (K+2g/2) sin2 (½Qa) - 2 ~ 2 ]
instead of eq. (10). Therefore, for this model, the
modulational instability does not depend on q, as for
the continuum NLS equation. For Q<Q* determined by sin2(½Q*a)=2¥2/(K+2~/2), all the carrier waves are unstable, while for Q > Q* they are stable. As a consequence, for a fixed positive value of
2, the Ablowitz-Ladik model (15) can have simultaneously two types of nonlinear modes localized
either above or below the linear spectrum band [ 20 ].
The nonlinear modes (12), (14) are strongly localized excitations, so that most of the energy is concentrated on a central particle. Let us consider lowfrequency modes (12) for definiteness. According to
the results presented above, for such a mode the amplitude of the particle at site n = 0 is large, but its frequency is small, o9~ o9o-AA 2 << 09o. Other particles,
i.e. those for n ~ 0, are at rest or they oscillate with
small amplitudes corresponding to the gap frequency
o90. It means that the excitation of the neighboring
particles by the particle at site n = 0 is nonresonant
and the energy transfer from it to other particles of
the chain is small. Nevertheless, for the full equation
( 1 ) the localized modes ( 12 ) or (14) are not exact
stationary solutions, and a part of the energy is emitted during each period of the mode oscillations. To
estimate the energy lost per one oscillation period,
we note that for the particles with the numbers
n = + 1 in the case of a strongly localized mode the
following equation is valid,
iil +o92ul ~DUO,
(16)
where the right-hand side plays the role of an external force, F ( t ) = Duo (t). The energy AE transmitted
by this force to an oscillator per period may be calculated as
T
2
AE= ½1!F(t) exp(iogot)[
that yields the estimate A E ~ (TDA)2~D2/2. Because the total energy of an anharmonic lowfrequency oscillation at site n = 0 may be estimated
as E~o9~/22, finally we arrive at the estimate
AE/E~
(D/o9o2)2<< 1 .
(17)
The result (17) means that the localized nonlinear
mode (12) excited in the chain (1) decays exponentially slowly into radiation, i.e. it is indeed a longlived excitation of the anharmonic lattice.
4. Dark-profile localized structures
As is well known, eq. (6) in the continuum limit
has two different kinds of soliton solutions, bright
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1 February 1993
and dark ones. The bright solitons are similar to the
spatially localized modes discussed above, whereas
up to now localized structures which are similar to
dark solitons have not been discussed yet. It is the
purpose of this section to present two types of these
structures in the lattice.
First, we note that for positive 2, the cw solution
is stable only for q> ½n, so that dark-profile structures are possible, for example, near the cut-off frequency tOm= 4K. Substituting
(a)
~/, = ( - 1 ) " ~ ( x , t) exp (itomt)
into eq. (6), where the slowly varying envelope ~ i s
found to be a solution of the continuous NLS equation, it is easy to obtain the dark soliton solution in
the continuum approximation,
~u, = ( - 1 )".4 tanh(Ax) exp( - i / 2 t ) ,
(18)
w h e r e / 2 = 4 K - 2 A 2, and x = nav/K is considered as
continuous variable.
Looking now for the similar structures in the discrete NLS equation (6), we find that they are possible, for example, in the form
~/, =A e x p ( - i / 2 t )
× ( .... 1, - 1, l, - ~ 1 , 0 , ~,, - 1, 1,...),
(19)
where/2 = 4 K - 2A 2, and
~1 = 1 - - A t ,
A 1=K/,;L42<< 1 .
(20)
The structure (19) is a solution of the NLS equation
with better accuracy for smaller A~, and it is a kinklike excitation with the width localized on a few particles in the lattice (see fig. 2a). Because the frequency/2 coincides with the cut-off frequency of the
nonlinear spectrum, we call these solutions "cut-off"
kinks.
Another type of dark-profile nonlinear localized
structures which may be also described analytically
in continuum as well as discrete models, is realized
in the case when the mode frequency is just at the
middle of the spectrum band, i.e. the wavenumber
is equal to n/2a. In this case we may separate the
particles in the chain into two groups, odd and even
ones, and describe their dynamics separately, introducing new variables, i.e. ¥ , = v . , for n=2k, and
~ . = w., for n = 2 k + 1. The main idea of such an approach is to use the continuum approximation for
176
Fig. 2. Dark-profde localized structures: "cut-off" (a) and "noncut-off'' (b) kinks.
two envelopes, v, and w, (see ref. [23] ).
Looking now for solutions in the vicinity of the
point q = n/2a, we may use the following ansatz,
V2k = ( -- 1 )kV( 2k, t) exp( - i t o l t ) ,
to2k+l=(--1)kw(2k+l,t)exp(--itolt),
(21)
where col = 2K is the frequency of the wavelength-four
linear mode, assuming that the functions V(2k, t)
and W ( 2 k + 1, t) are slowly varying in space. Substituting eqs. (21 ) into eq. (6), we finally get the
system of two coupled equations,
i O-z7
V, + 2aKO-z--W+2[ VI2V=0,
ot
(22 )
ox
OW
OV
i-~-{ - 2aK-~x +21W[2W=O
(23)
where the variable x is treated as continuous. Analysing localized structures, we look for stationary solutions of eqs. (22), (23) in the form
( V, W)oc (f~,f2) exp(i/2/),
(24)
assuming, for simplicity, the functionsf~ and f2 to he
real. Then, the stationary solutions of eqs. (22) and
(23) are described by the system of two ordinary
differential equations of the first order,
dfl = _/2f2 + 2 f 23,
dz
(25)
dA =/2f~ - 2 f 3 ,
(26)
dz
Volume 173, number 2
PHYSICS LETTERS A
where z = x / 2 a K . Equations (25), (26) represent the
dynamics of a Hamiltonian system with one degree
of freedom and the conserved energy,
E = - ½~tf~ + f D + ~ t f 4+ f D ,
and they may be easily integrated with the help of
the auxiliary function g=f~/f2, for which the following equation is valid
( d g / d z ) 2 = t o 2 f f 2 2 ( l + g 2 ) 2 + 4 , ~ F , ( l + g 4) .
(27)
Different kinds of solutions of eq. (27) may be characterized by different values of the energy E [23].
On the phase plane (f~, f2) soliton solutions correspond to the separatrix curves connecting a pair of
neighboring saddle points (0, fo), (0, -fo), (fo, 0),
or ( -fo, 0), where f 2 = ~2/2. Calculating the value
of E for these separatrix solutions, E = - ~ 2 / 4 ~ . , it is
possible to integrate eq. (27) in elementary functions and to find the soliton solutions,
g ( z ) = exp( _+x / ~ 2 z ) .
f22 =
(28)
exp( -T-x/-2g2z) [2 cosh (x/~g2z) + x/~]
2;t cosh (2x/~2z)
'
fl =gf2.
5. Conclusions
In conclusion, nonlinear localized modes have been
investigated analytically for the case of a chain where
nonlinearity is introduced through an on-site potential. Under special conditions, the model has been
shown to be described by a discrete nonlinear
SchrSdinger equation. The existence of spatially localized modes in the discrete model is consistent with
the results of the modulational instability analysis
for the discrete NLS equation. In the case when
modulational instability does not occur, the localized modes may appear in the form of dark-profile
nonlinear structures, a discrete analog of dark solitons. The decay time of spatially localized modes in
a chain with nonlinear on-site potential has been calculated to be exponentially small provided coupling
between the particles in the chain is described by a
small parameter. Therefore, in spite of the fact that
the primary model is not exactly integrable, such localized modes are long-lived objects which can make
important contribution to dynamical and statistical
properties of nonlinear discrete chains.
(29)
The solutions (28), (29), but for negative £2, exist
also for a defocusing nonlinearity when A< 0.
The results (28), (29) together with (24) and
(21 ) give the shapes of the localized structures in the
discrete nonlinear lattice. The whole localized structure represents two kinks in the odd and even oscillating modes which are composed to have opposite polarities (see the envelopes in fig. 2b).
Highly localized nonlinear structures in the lattice
corresponding to the solutions (28), (29) may be
also found and one of these structures has the following form
¥,, =A exp(-ig2t)
X (..., l, O, - 1 , O, ~2, ~2, 0, - 1 , O, 1.... ) ,
1 February 1993
(30)
where £2=2K-AA 2 is the frequency at the middle
of the nonlinear spectrum, and ~2=1-A2, A2=
K/22A 2 << 1 (see the whole structure in fig. 2b). The
approximation is better for small values of the parameter d2.
Acknowledgement
Discussions with M. Peyrard, K.H. Spatschek, and
S. Takeno are gratefully acknowledged. The author
thanks the Institut f'tir Theoretische Physik I for hospitality. This work has been supported by the Alexander yon Humboldt-Stiftung through a research
fellowship.
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