Physica B 296 (2001) 251}258
Discrete gap breathers in chains with strong hydrogen bonding
A.V. Zolotaryuk *, P. Maniadis, G.P. Tsironis
Bogolyubov Institute for Theoretical Physics, 03143 Kyiv, Ukraine
Department of Physics, University of Crete and Foundation for Research and Technology } Hellas (Forth), P.O. Box 2208, 71003 Heraklion,
Crete, Greece
Abstract
We consider a diatomic chain of heavy ions coupled by hydrogen bonds which are su$ciently strong compared with
other interactions in the system. In this case, each proton in the hydrogen bond is subject to a single-minimum potential
resulting from its interaction with nearest-neighbor heavy ions through the Morse potential that contains soft anharmonicity. This diatomic chain of nonlinearly coupled masses admits discrete breather solutions in the gap of the phonon
spectrum. Simple analytical arguments accompanying explicit solutions that demonstrate the existence of the gap
breather with only one type of symmetry, namely the odd-parity pattern centered at a hydrogen-bonded proton, are
present. These arguments are supported by the numerically exact procedure using the anticontinuous limit. Some other
multi-breather solutions in the gap are also obtained exactly from the anticontinuous limit. 2001 Elsevier Science
B.V. All rights reserved.
PACS: 63.20.Ry; 63.20.Pw; 63.70.#h; 03.20.#i
Keywords: Hydrogen-bonded chains; Diatomic lattices; Discrete breathers; Anticontinuous limit
1. Introduction
Intrinsic localized modes or discrete breathers
(for a review see, e.g., [1]) are nonlinear collective
excitations that seem to play a very important role
in condensed matter physics. Interest in these
modes has been intensi"ed recently due to experimental generation and observation in some chemical compounds [2] and antiferromagnets [3],
coupled arrays of Josephson junctions [4,5], and
* Correspondence address: Department of Physics, University
of Crete, P.O. Box 2208, 71003, Heraklion, Crete, Greece. Fax:
#30-81-394201.
E-mail address: [email protected], [email protected]
(A.V. Zolotaryuk).
even possibly in myoglobin [6]. Another type of
condensed matter systems in which discrete
breathers could possibly exist and could be detected experimentally are hydrogen-bonded (HB)
systems such as ice, quasi-one-dimensional HB
crystals, and one-dimensional (1D) HB chains (for
a review see, e.g., [7]). The basic idea in the nonlinear model for proton dynamics in a HB chain stems
from the fact that the proton in each H-bond of the
chain can be constructed as the sum of two-body
ion}proton potentials, e.g., the Morse potentials
[8]. If this (asymmetric) two-body potential is
su$ciently strong compared to other interactions
(e.g., between heavy ions or with an external on-site
coupling), the resulting potential for the HB proton
has only one minimum and the H-bond in this case
is referred to as the strong H-bond [9]. Otherwise,
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 8 0 6 - 1
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A.V. Zolotaryuk et al. / Physica B 296 (2001) 251}258
this potential has two degenerate minima separated
by a barrier [8,10,11] and the breather solutions
have been studied before in the long-wavelength
approximation [12]. Schematically, a chain with
strong hydrogen bonding can be represented by the
diatomic sequence 2 }X}H}X}H}X}H}X} 2,
where X denotes a heavy ion and H a proton.
In general, a 1D anharmonic diatomic lattice
that has a gap in its phonon band, admits standing
(anharmonic gap modes) and moving (gap solitons)
solutions which were studied using di!erent
approximate techniques [13}20]. In particular, the
diatomic chain with realistic (asymmetric soft)
two-body nearest-neighbor potentials was studied
recently both analytically and numerically
[18,21,22]. However, on one side, the gap breather
solutions obtained in these studies have not been
treated in a rigorous way and therefore they cannot
be referred to as really exact and stable solutions,
and on the other hand, di!erent analytical procedures (like asymptotic expansion and others) are too
sophisticated to demonstrate which symmetry of
the intrinsic gap modes (gap breathers) does exist
being stable. Therefore some, possibly oversimpli"ed arguments or procedures should be developed
to gain a better understanding of the origin of the
existence and stability of the gap breathers (say, on
the level of linear and/or quadratic algebraic equations, like the equation explaining the structure of
the phonon band gap), before going to the numerically exact procedure of "nding the gap breathers
from the anticontinuous limit, discovered recently
in a series of papers [23}27]. Therefore the present
paper aims to "nd in chains with strong hydrogen
bonding the exact solutions for discrete gap
breathers and to prove their stability, using the
anticontinuous approach [26]. This numerical
work is accompanied by analytical arguments
based on the similarity [28] between the massimpurity mode [29,30] and intrinsic localized
modes.
2. The diatomic model and localization length of
gap breathers
Let the heavy ions and the protons (light masses)
in a HB chain, label according to the sequence
2, Q
,q
,Q , q , Q
,q
, where
L\ L\ L L L> L> 2
Q is the displacement of the nth ion and q , the
L
L
displacement of the nth proton from their
equilibria. We denote by and , the phonon
frequencies of the heavy and the light masses at
these equilibria, respectively. These two lattice
displacement "elds can be replaced by one "eld
2, u , u , u ,2 with local characteristic
H\ H H>
frequency according to the relations u "Q and
H
H
L
" if j"2n, and u "q and " if
H
H
L
H
j"2n#1. Then the equations of motion of this
system can be written in the form of one discrete
equation as
uK "[=(u !u )!=(u !u )],
H
H
H>
H
H
H\
j"0,$1,2,
(1)
where =(r) is an asymmetric two-body potential of
the standard type and normalized by =(0)"1;
the overdot and the prime denote the di!erentiation with respect to time t and the relative distance
r, respectively. In the case of the Morse potential we
have
=(r)"(\/2)[1!exp(!r)], '0.
(2)
The phonon band of the system is obtained directly from the linearized equation of motion (1): it
has the gap (2 ))(2 splitting the band
into the lower (acoustic) and the upper (optical)
branches [see below Eq. (17) at "1]. Here we will
deal with strongly localized vibrational modes the
amplitude of which has an exponential asymptotic
behavior. Imposing for the amplitudes of oscillations to have at nP$R the exponential factor
exp(!n/
),pL, 0(p(1, with being the
localization length of the mode and substituting
this ansatz into the linearized equation (1), one
obtains the relation
/ "1#$((1!p)(1!/p),
"m/M" / ,
(3)
that couples the breather frequency with the
localization parameter . This relation has two
(lower and upper) branches that merge at the critical point "!1/ln as shown in Fig. 1. The
upper (lower) branch bifurcates from the out-ofphase oscillations of light (heavy) masses and
A.V. Zolotaryuk et al. / Physica B 296 (2001) 251}258
Fig. 1. Two (upper and lower) branches of frequency as
functions of localization length within phonon gap that merge
at critical point . Region 1 corresponds to acoustic band,
region 2 to phonon gap and region 3 to optical band.
standing heavy (light) masses. Note that for stronger localization than the exponential one, like the
breathers with compact support in the continuum
limit [31], the region *
(see Fig. 1), in which
the gap breather solutions are possible, would be
extended to the left. However, for our qualitative
arguments given below, Eq. (3) can be adopted as
a `working curvea with the inequality (
e\H"p to be valid.
(heavy or light) particles oscillate symmetrically
and out-of-phase with large amplitudes. The rest
particles of the chain are assumed to perform
small-amplitude oscillations. We denote the
breather mode with a "xed ion or proton by light
even-parity (LE) or heavy even-parity (HE), respectively. Using the rotating wave approximation in
this way, we get three equations with respect to the
three amplitudes: one for the central particles and
the other two for the decaying light and heavy tails
of the chain. The solution to these equations,
including Eq. (3), appears to be very simple, so that
its behavior with varying the breather frequency
within the phonon band gap can be treated in
very simple terms.
For the LO mode centered at the site n"0, the
central proton is assumed to oscillate with large
amplitudes: q "a cos( t), where the amplitude
a is to be determined. The rest of the heavy and
light particles are assumed to oscillate in a symmetric way with an exponential decay:
Q "(!1)L>ApL\ cos( t), n"1,2,2,
L
Q "(!1)LAp\L cos( t), n"0,!1,!2,2,
L
q "(!1)LapL cos( t), n"$1,$2,2,
L
3. Analytical arguments obtained from similarity
between impurity and intrinsic gap modes
To simplify analytical calculations as much as
possible, we restrict ourselves, in Eq. (1), to the
symmetric quartic potential =(r)"r/2#r/4
with the anharmonicity parameter being either
positive (hard anharmonicity) or negative (soft
anharmonicity). Similarly to the mass-impurity
mode, we assume that only one mass (a proton or
a heavy ion), supposed to be a center of the breather
with odd-parity symmetry, performs large-amplitude oscillations, whereas the rest masses oscillate
with small amplitudes, so that they approximately
obey the linearized equations of motion for which
Eq. (3) is valid. We denote each of these localized
modes by light odd-parity (LO) and heavy oddparity (HO), respectively, for a proton-centered and
an ion-centered breather. For the breathers with
even-parity symmetry, we suppose the central (light
or heavy) particle to be "xed, whereas the lateral
253
(4)
so that the LO pattern schematically can be represented as
2; pa,!pA;!pa, A; a ; A,!pa;!pA, pa;2.
(5)
The two equations of motion for the central and
one of the lateral (e.g., the right one) particles with
n"0 can be written approximately in the form
qK K2 [Q !q #q (3Q !q )],
Q$ K [q !2Q #q #q (q !3Q )],
(6)
where only the coordinate q is assumed to have
large-amplitude values. The rest of the linearized
equations of motion (1) yields relation (3) and the
amplitude ratio
a
/ !2
p\!1
"! "
.
A
1!p
/ !2
(7)
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A.V. Zolotaryuk et al. / Physica B 296 (2001) 251}258
Using Eqs. (3) and (7), from the equations of motion
(6) we get the solution for the central proton
2(p\!p)
a "
3[1#p! / !3(1!p) /2 ]
(8)
as a function of the breather frequency and the
amplitude ratio
A
/ !2!3a /2
.
"! a
2#9a /2
(9)
Similarly, for the HO pattern we assume the
ansatz which schematically can be represented as
2; pa, pA;!pa,!pA; a; A , a;
!pA,!pa; pA, pa;2.
(10)
Therefore the solution for the HO pattern can be
obtained from Eqs. (7) to (9) by the substitution
a PA , aA, and .
Consider now the LE mode centered at a "xed
heavy ion and assume that the lateral protons perform large-amplitude out-of-phase oscillations, i.e.,
Q ,0 and q "!q "a cos( t), where the
\
amplitude a is to be determined. The LE ansatz
can be represented schematically as
2; pa,!pA;!pa, pA;!a , 0, a ;
!pA, pa; pA,!pa;2.
(11)
In the way similar to the LO pattern, one obtains
the solution for this case. It is given by the quadratic equation
3
1
/ !2
8
#
# a # a 3!
9 2 1!p\
16
! p\"0
27
(12)
and the relations
A
/ !2!3a /2
,
" a
p(1#9a /4)
(13)
a
/ !2
1!p
" "
.
A
1!p\
/ !2
(14)
Finally, the ansatz for the HE mode is given by
q ,0, Q "!Q "A cos( t) with the ampli
tude A to be determined, and the sequence
2; pA,!pa;!pA, pa;!A , 0, A ;
!pa, pA; pa,!pA;2.
(15)
The solution for this mode is obtained from
Eqs. (12) and (13) by the same substitution as in the
case of the odd-parity modes.
In order to "nd which of the four solutions
obtained above is right one in the case of soft
anharmonicity, we accept the rule that such a solution bifurcating from the upper or lower gap edge,
must continuously pertain the signs of all its amplitudes A, a, and A or a along the curve (3) and
pass the region of strong localization, i.e., the critical point . Note that Eqs. (7) and (14) keep the
sign of A/a within the whole curve, but this is not
a case for the other equations that determine the
breather solutions.
First consider the solution for the LO mode.
Since the expression in the square brackets of
Eq. (8) is negative for all the gap frequencies , the
LO pattern can exist only if (0 (soft anharmonicity). This mode bifurcates from the upper gap
edge, where it has the asymptotics given by
a P2(1!p)/3(1!) and A/a P!(1!p)/
2(1!)(0. Next, it follows from Eq. (7) that
A/a(0 and therefore A(0 and a'0 if a '0.
These signs are kept up to some point on the lower
branch, overcoming the central region of strong
localization. In particular, at the gap middle where
" # , the exact solution a "4/
3(2#3), A"!a /2, and a"a /2, demon
strates strong localization of the LO mode.
The similar analysis of the equations which are
obtained from Eqs. (7) to (9) as described above
shows that the HO solution bifurcating from the
lower edge fails before reaching the middle point
along curve (3). Moreover, it exists only for hard
anharmonicity and its pattern coincides with that
found by Chubykalo and Kivshar [16,17], but it
seems to be unstable because it cannot reach the
middle point.
In the case of the LE solution that bifurcates
from the upper edge, we "nd from the quadratic
equation (12) the asymptotics a P!(1!p)/
A.V. Zolotaryuk et al. / Physica B 296 (2001) 251}258
3(1!)(0 as pP1 (therefore (0) and from
Eq. (13) A/a P(1!p)/2(1!)'0. Since
a/A(0 [see Eq. (14)], a implies the inequalities
A'0 and a(0. However, these signs change on
the way along the upper branch before reaching the
critical point . Therefore, the LE mode is not
acceptable.
The similar analysis of the HE solution bifurcating from the lower edge and obtained from Eqs. (12)
to (14) by the substitution described above, yields
the asymptotics A P(1!p)/3(1!)'0 and
a/A P!(1!p)/2(1!)(0 as pP1, and the
inequality a/A'0. Consequently, this pattern
exists if the anharmonicity is hard and A (0
implies A'0 and a'0; this also agrees with the
pattern found by Chubykalo and Kivshar [16,17].
In the gap middle, the solution is A "4/3'0
and a/A "!(1#)/(3#)(0. This solution
is well de"ned and its signs are persistent along the
upper branch; therefore it is expected to be stable if
the anharmonicity is hard.
255
the LO pattern survives. Here we support this
statement using the Newton method in the numerical calculation of the gap breathers which is based
on the concept of the anticontinuous (AC) limit
[23}27]. In order to introduce the AC in our system, we rewrite the equation of motion (1) in terms
of the relative displacement "eld r "u !u
H
H>
H
[26]:
rK "!(# )=(r )
H
H
H>
H
#[=(r )# =(r )]
(16)
H
H\
H>
H>
with the new parameter , 0))1. Then Eq. (16)
can be thought as a Klein}Gordon system with
nonlinear dispersion and the nearest-neighbor
coupling, . In the limit P0, we obtain the system
of uncoupled nonlinear oscillators and in the other
4. Exact breather solutions obtained from the
anticontinuous limit
Thus, from analysis given above, we expect that
in the case of the realistic Morse potential (2), only
Fig. 2. Phonon dispersion relation as a function of parameter .
Acoustic band (k; ) is bounded by solid lines and optical
\
band (k; ) by the dashed lines. At "0 both bands merge at
>
frequency "( # ( "0.1 and "0.7).
Fig. 3. Exact breather solution (a) and its Floquet analysis (b)
for the chain with "0.1 and "0.7. This solution corres
ponds to asymmetric even-parity breather centered at a proton
(LO mode). The Floquet analysis illustrates stability of this
breather solution.
256
A.V. Zolotaryuk et al. / Physica B 296 (2001) 251}258
limit P1, we have the original system we want to
study.
Using the standard Newton method, "rst we can
"nd the breather solution for small values of and
then continue it up to "1. Unfortunately, due to
resonances with the phonons, this procedure fails
and the reason of this can be illustrated on the
dispersion relation of the system (16) given as
a function of " # $( #2 cos(k) # ,
!
0)k)
(17)
and depicted in Fig. 2 for "0.1 and "0.7 . As
can be seen from this "gure, in the AC limit (P0)
all the normal modes of the system have the same
frequency, namely "( # . Indeed, as the
coupling increases, the frequencies change continuously to form the two (acoustic and optical)
bands and when "1, we retrieve the well-known
dispersion law for the diatomic lattice. However, at
"0 there is no a gap in the phonon spectrum in
which we could initially "nd a breather solution.
Therefore, in order to avoid the resonance with the
phonons at "0, we consider at each lattice site
[in Eq. (16)] a local mass m "1#(!1)L, where
L
is some small quantity. This modi"cation of
Eq. (16) creates a gap in the phonon spectrum even
in the limit "0. Now it is easy to "nd the breather
solution in this gap and then to continue it for
non-zero values of . After some iterations, we can
slowly set "0 and continue the breather
solution up to "1, getting the breather solution
for the original system. After we have found the
breather solution for some frequency, we vary the
frequency with small steps and using the Newton
method, we "nd the breather solution for any
frequency within the phonon gap. The breather
solution and the Floquet stability analysis are presented in Fig. 3, and in Fig. 4, we can see the time
evolution of the breather solution. Using this
method, one can "nd not only single breathers, but
also multi-breather solutions of the system. Thus,
Fig. 5 depicts the `101a multi-breather (according
to the notation given by Aubry [24}26]), and
Fig. 6 illustrates its time evolution.
Fig. 4. Time evolution of the breather solution shown in Fig. 3.
A.V. Zolotaryuk et al. / Physica B 296 (2001) 251}258
257
5. Summary and outlook
Fig. 5. Exact `101a multi-breather solution (a) and its
Floquet analysis (a) for the with "0.1 and "0.7. As
illustrated by the Floquet analysis, two pairs of eigenvalues
collide and escape from the unit circle, so that the solution
becomes unstable.
The diatomic chain with an asymmetric interatomic coupling of the standard type (Morse,
Lennard}Jones, etc.) that contains soft anharmonicity seems to be an appropriate theoretical
model for one-dimensional hydrogen-bonded
systems in which the H-bond is su$ciently strong
compared to other interactions in the chain (like
the coupling between the nearest heavy ions or an
interaction with a substrate), so that the total
on-site potential for the HB proton appears to have
a single equilibrium position. This diatomic chain
of nonlinearly coupled masses is known to admit
discrete breather solutions in the gap of the linear
phonon band [21,22,32,33]. However, yet these
solutions have not obtained in the framework of
the AC approach [23}26], the procedure which
presently is considered as a rigorous method to "nd
discrete breathers and to investigate their stability
properties.
We have also developed the simplest analytical
procedure that allows us to "nd which symmetry of
possible gap breather ansatzen can be accepted.
Fig. 6. Time evolution of the multi-breather shown in Fig. 5.
258
A.V. Zolotaryuk et al. / Physica B 296 (2001) 251}258
This approach is based on the similarity between
the linear mass-impurity mode and the possible
intrinsic gap modes. We impose an exponential
decay of the gap modes, "nd the relation (3) between the gap frequency and the localization length
using it as a `working curvea, and next allow only
one or two particles at the breather center to perform large-amplitude oscillations. As a result, we
get the set of three simple algebraic equations
which can be solved explicitly. Finally, we accept
the rule the breather solution is appropriate if while
moving along curve (3) starting at the upper or
lower gap edge, it reaches and passes the region of
strong localization near the critical point at the
middle of the gap, not changing all the amplitude
signs. Thus, in our system, the odd-parity pattern
centered at a HB proton (we call it the LO mode) is
shown to be an appropriate solution using such an
approach. This result has been supported by the
numerically exact procedure using the AC limit.
Some other multi-breather solutions in the gap
have also been obtained exactly from the AC
limit.
Acknowledgements
The work was supported from INTAS grant No.
96-0156 of the European Community. The authors
are grateful to S. Aubry for helpful discussions and
valuable suggestions. We also thank Prof. E.N.
Economou for numerous discussions and fruitful
collaboration over the course of many years on
topics related to the subjects of nonlinearity and
localization and wish to dedicate this work to him
on the occasion of his 60th birthday.
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