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arXiv:0907.5564v1 [cond-mat.mes-hall] 31 Jul 2009
Discrete solitons in electromechanical resonators
M. Syafwan,
∗
H. Susanto, and S. M. Cox
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
We consider a parametrically driven Klein–Gordon system describing micro- and nano-devices,
with integrated electrical and mechanical functionality. Using a multiscale expansion method we
reduce the system to a discrete nonlinear Schr¨odinger equation. Analytical and numerical calcula-
tions are performed to determine the existence and stability of fundamental bright and dark discrete
solitons admitted by the Klein–Gordon system through th e discrete Schr¨odinger equation. We show
that a parametric driving can not only destabilize onsite bright solitons, but also stabilize intersite
bright discrete solitons and onsite and intersite dark solitons. Most importantly, we show that t here
is a range of values of the d riving coefficient for which dark solitons are stable, for any value of
the coupling constant, i.e. oscillatory instabilities are totally suppressed. Stability windows of all
the fundamental solitons are presented and approximations to the onset of instability are derived
using perturbation theory, with accompanying numerical results. Numerical integrations of the
Klein–Gordon equation are performed, confirming the relevance of our analysis.
PACS numbers: 05.45.-a, 63.20.Pw, 85.85.+j
I. INTRODUCTION
Current advances in the fabrication and control of elec-
tromechanical systems on a micro and nanoscale bring
many technological promises [1]. These include efficient
and highly sensitive sensors to detect stresses, vibrations
and forces at the atomic level, to detect chemical signals,
and to perform signal proces sing [2]. As a particular
example, a nanoelectromechanical system (NEMS) can
detect the mass of a single atom, due to its own very
small mass [3, 4].
On a fundamenta l level, NEMS with high frequency
will allow research on quantum mechanical effects. This
is because NEMS, as a miniaturization of microelec-
tromechanical systems (MEMS), can contain a macro-
scopic number of atoms, yet still require quantum me-
chanics for their proper description. Thus, NEMS can
be considered as a natural playground for a study of
mechanical systems at the quantum limit and quantum-
to-classica l transitions (see, e.g., Ref. 5 and references
therein).
Typically, nanoelectr omechanical devices comprise an
electronic device coupled to an extremely high frequency
nanoresonator. A large number of arrays of MEMS and
NEMS resonators have recently been fabricated experi-
mentally (see, e .g., Ref. [6 ]). One direction of resea rch
in the study of such arrays has focuse d on intrinsic local-
ized modes (ILMs) o r dis c rete breathers. ILMs can be
present due to parametric instabilities in an array of os-
cillators [7]. ILMs in dr iven arrays of MEMS have been
observed experimentally [8, 9, 10].
Motivated by a recent experiment of Buks and
Roukes [6] that succeeded in fabricating and e xciting an
∗
On leave from Department of Mathematics, Faculty of Mathe-
matics and Natural Sciences, Andalas University, Limau Manis,
Padang, Indonesia 25163
array of MEMS and measuring oscillatio ns of the res-
onators, here we consider the equation [11]
¨ϕ
n
= D∆
2
ϕ
n
− [1 − H cos(2ω
p
t)]ϕ
n
± ϕ
3
n
, (1)
which governs the oscillation amplitude of such an array.
Equation (1) is a simplified model of that discussed in
Ref. 12, subject to an assumption that the piezoelectric
parametric drive is applied directly to each oscillator [13].
The variable ϕ
n
represents the oscillation amplitude of
the nth oscillator from its e quilibrium position, D is a
dc electrostatic near e st-neighbor coupling term, H is a
small ac component with frequency 2ω
p
responsible fo r
the parametric driving, ∆
2
ϕ
n
= ϕ
n+1
−2 ϕ
n
+ϕ
n−1
is the
discrete Lapla c ian, the dot denotes the der ivative with re-
sp e c t to t, and the ‘plus’ a nd ‘minus’ signs of the c ubic
term corres po nd to a ‘softening’ and ‘stiffening’ nonlin-
earity, respectively. Here, we assume ideal oscillators, so
there is no damping present. The creation, stability, a nd
interactions of ILMs in (1) with low damping and in the
strong-coupling limit, have been investigated in Ref. 11.
Here, we extend that study to the case of small coupling
parameter D.
In performing our analysis of the governing equation
(1), we introduce a small parameter ǫ ≪ 1, and assume
that the following scalings hold:
D = ǫ
2
3C, H = ∓ǫ
2
3γ, ω
p
= 1 ∓ ǫ
2
3Λ/2.
We then expand each ϕ
n
in powers o f ǫ, with the leading-
order term being of the form
ϕ
n
∼ ǫ
ψ
n
(T
2
, T
3
, . . . )e
−iT
0
+
ψ
n
(T
2
, T
3
, . . . )e
iT
0
, (2)
where T
n
= ǫ
n
T . Then the terms at O(ǫ
3
e
−iT
0
) in (1)
yield the following equation for ψ
n
(see Refs. 14, 15 for
a related reduction method):
− 2i
˙
ψ
n
= 3C∆
2
ψ
n
∓ 3γ
ψ
n
e
±i3ΛT
2
± 3|ψ
n
|
2
ψ
n
, (3)
where the dot now denotes the derivative with re-
sp e c t to T
2
. Correction terms in Eq. (2) are of order
2
O(ǫe
±i(k+1)T
0
, ǫ
3
e
±i(k−1)T
0
), k ∈ Z
+
. A justification of
this rotating wave type approximation can be obtained
in, e.g ., Ref. 16.
Writing ψ
n
(T
2
) = φ
n
(T
2
)e
±i3Λ/2T
2
, we find that
Eq. (3) becomes
−
2
3
i
˙
φ
n
= C∆
2
φ
n
∓ Λφ
n
∓ γ
φ
n
± |φ
n
|
2
φ
n
. (4)
Then, taking T
2
=
2
3
ˆ
T , we find that the equation
above becomes the parametric driven discrete nonlinear
Schr¨odinger (DNLS) equation
i
˙
φ
n
= −C∆
2
φ
n
± Λφ
n
± γ
φ
n
∓ |φ
n
|
2
φ
n
; (5)
here the dot denotes the derivative with respect to
ˆ
T .
The softening and stiffening nonlinearity of (1) corre-
sp ond, respectively, to the so-called focusing and defo-
cusing nonlinearity in the DNLS (5).
In the absence of parametric driving, i.e., for γ = 0,
Eq. (5) is known to admit bright and dark solitons in the
system with fo c using and defocusing no nlinearity, respec-
tively. Discrete bright solitons in such a system have been
discussed before, e.g. in Refs. 17, 18, 19, where it was
shown tha t one-excited-site (onsite) so lito ns are stable
and two-excited-site (intersite) solitons are unstable, for
any coupling constant C. Undriven discrete dark solitons
have also been examined [20, 21, 22, 23, 24]; it is known
that intersite dark solitons are always unstable, for any
C, and onsite solitons are stable only in a small window
in C. Furthermore, an onsite dark solito n is unsta ble due
to the presence of a quartet of complex eigenvalues , i.e.,
it suffers oscillatory instability.
The parametrically driven DNLS (5) with a focusing
nonlinearity and finite C has been considered briefly in
Ref. 25, where it was shown that an onsite bright dis-
crete soliton can be destabilized by parametric driving.
Localized excitations of the continuous limit of the para-
metrically driven DNLS, i.e. (5) with C → ∞, have been
considered by Barashenkov et al. in a differe nt context
of applications [26, 27, 28, 29, 3 0, 31, 32]. The same
equation als o applies to the study of Bose–Einstein con-
densates, describing the so-called long bosonic Jos e phson
junctions [33, 34].
In this paper, we consider (1) with either softening
or stiffening nonlinearities, which admit bright or dark
discrete solitons, respectively. The existence a nd stabil-
ity of the fundamental onsite and intersite excitations
are discussed through the reduced equation (5). Eq. (5)
and a corresponding eigenvalue problem are solved nu-
merically for a range of values of the coupling and driv-
ing constants, C and γ, giving stability windows in the
(C, γ) pla ne. Analytical approximations to the bound-
aries of the numerically obtained stability windows are
determined through a perturbation analysis for small C.
Fro m this a nalysis, we show, complementing the r esult
of Ref. 25, that parametric driving can stabilize intersite
discrete bright solitons. We also show that parametric
driving can even stabilize dark solitons, for any c oupling
constant C. These findings, which are obtained from the
reduced equation (5), are then confirmed by direct nu-
merical integrations of the or iginal governing equation
(1).
The present paper is organized as follows. In Sec. II we
present the existence and stability analysis of onsite and
intersite bright solitons. Analysis of dark solitons is pre-
sented in Sec. III. Confirmatio n of this analysis, through
numerical simulations of the Klein–Gordon system (1), is
given in Sec. IV. Finally, we give conclusions in Sec. V.
II. BRIGHT SOLITONS IN THE FOCUSING
DNLS
In this section we first consider the existence and sta-
bility of bright solitons in the fo c us ing DNLS equation.
For a static solution of (5) of the form φ
n
= u
n
, where
u
n
is real-valued and time-independent, it follows that
− C∆
2
u
n
− u
3
n
+ Λu
n
+ γu
n
= 0. (6)
Once such discrete so lita ry-wave solutions of (5) have
been fo und, their linear stability is determined by solv-
ing a corresponding eigenvalue problem. To do so, we
introduce the linearization ansatz
φ
n
= u
n
+ δǫ
n
,
where δ ≪ 1, and substitute this into (5), to yield the
following linearized equa tion at O(δ):
i˙ǫ
n
= −C∆
2
ǫ
n
− 2|u
n
|
2
ǫ
n
− u
2
n
ǫ
n
+ Λǫ
n
+ γǫ
n
. (7)
Writing ǫ
n
(t) = η
n
+ iξ
n
, we then obtain from Eq. (7)
the eige nvalue problem
˙η
n
˙
ξ
n
= H
η
n
ξ
n
, (8)
where
H =
0 L
+
(C)
−L
−
(C) 0
and the operators L
−
(C) and L
+
(C) are defined by
L
−
(C) ≡ − C∆
2
− (3u
2
n
− Λ − γ),
L
+
(C) ≡ − C∆
2
− (u
2
n
− Λ + γ).
The stability of the solution u
n
is then determined by the
eigenvalues of (8). If we denote these eigenvalues by iω,
then the solution u
n
is stable only when Im(ω) = 0 for
all eigenvalues ω.
We note that, because (8) is linear, we may eliminate
one of the eigenvectors, for instance ξ
n
, to obtain an alter-
native expression of the eigenvalue problem in the form
L
+
(C)L
−
(C)η
n
= ω
2
η
n
≡ Ωη
n
. (9)
In view of the relation Ω = ω
2
, it follows that a soliton
is unstable if it has an eigenvalue with either Ω < 0 or
Im(Ω) 6= 0.
3
A. Analytical calculations
Analytical calculations of the existence and stability
of disc rete solitons can be carried out for small co upling
constant C, using a perturbation analysis. This analysis
exploits the exact solutions of (6) in the uncoupled limit
C = 0, which we denote by u
n
= u
(0)
n
, in which each u
(0)
n
must take one of the three va lues given by
0, ±
p
Λ + γ. (10)
Solutions of (6) for small C can then be calculated ana-
lytically by writing
u
n
= u
(0)
n
+ Cu
(1)
n
+ C
2
u
(2)
n
+ ··· .
In studying the stability problem, it is natural to also
expand the eigenvector η
n
and the eigenvalue Ω in powers
of C, as
η
n
= η
(0)
n
+ Cη
(1)
n
+ O(C
2
), Ω = Ω
(0)
+ CΩ
(1)
+ O(C
2
).
Upo n substituting this expansio n into Eq. (9) and iden-
tifying c oefficients of successive powers of the small pa-
rameter C, we obtain from the equations at O(1) and
O(C) the results
h
L
+
(0)L
−
(0) − Ω
(0)
i
η
(0)
n
= 0, (11)
h
L
+
(0)L
−
(0) − Ω
(0)
i
η
(1)
n
= f
n
η
(0)
n
, (12)
where
f
n
= −(∆
2
+2u
(0)
n
u
(1)
n
)L
−
(0)−L
+
(0)(∆
2
+6u
(0)
n
u
(1)
n
)+Ω
(1)
.
(13)
In the uncoupled limit, C = 0, the eigenvalue problem
is thus simplified to
Ω
(0)
= L
+
(0)L
−
(0), (14)
from which we conclude that there are two possible eigen-
values, given by
Ω
(0)
C
= Λ
2
− γ
2
, Ω
(0)
E
= 4(Λ + γ)γ,
which correspond, respectively, to the solutions u
(0)
n
= 0
(for all n) and u
(0)
n
= ±
√
Λ + γ (for all n).
We begin by considering bright soliton solutions, for
which u
n
→ 0 as n → ±∞. This then implies that (for
C = 0) the eigenvalue Ω
(0)
C
has infinite multiplicity; it
generates a corresponding continuous spectrum (phonon
band) for finite positive C. To investigate the significance
of this continuous spectrum, we intro duce a pla ne wave
expansion
η
n
= ae
iκn
+ be
−iκn
,
from which one obtains the dispers ion relation
Ω = (2C(cos κ − 1) −Λ)
2
− γ
2
. (15)
FIG. 1: A sketch of the dynamics of th e eigenvalues and the
continuous spectrum of a stable onsite bright soliton in the
(Re(Ω),Im(Ω)) plane. The arrows indicate the direction of
movement as the coupling constant C increases. Note that
a soliton is unstable if there is some Ω with either Ω < 0 or
Im(Ω) 6= 0.
This in turn shows that the continuous band lies between
Ω
L
= Λ
2
− γ
2
, when κ = 0, (16)
and
Ω
U
= Λ
2
− γ
2
+ 8C(Λ + 2C), when κ = π. (17)
Fro m the continuous spectrum analysis above, it can
be concluded that an instability can only be caused by
the dynamics of discrete spectrum.
1. Onsite bright solitons
The existence and stability of a single excited state, i.e.
an onsite bright soliton, in the presence of a parametric
driving has been considered in Ref. 25. For small C, the
soliton is given by [25]
u
n
=
√
Λ + γ + C/
√
Λ + γ + O(C
2
), n = 0,
C/
√
Λ + γ + O(C
2
), n = −1, 1,
O(C
2
), otherwise,
(18)
and its eigenvalue by
Ω
E
= 4(Λ + γ)γ + 8γC + O(C
2
). (19)
It was shown in Ref. [25] that the configuration (18),
which is known to be stable for any value of C when
γ = 0, can be destabilized by parametric driving. Fur-
thermore, it was shown that there are two mechanisms of
destabilization, as sketched in Fig. 1. The two instabil-
ity scenarios are determined by the relative positions of
Ω
(0)
E
and Ω
(0)
C
, as we now summarize. First, we note that
there is a threshold value, γ
th
= Λ/5, at which the two
leading-order eigenvalues coincide, so that Ω
(0)
E
= Ω
(0)
C
.
4
For γ > γ
th
, upon increasing C from C = 0, the in-
stability is caused by the co llision of Ω
E
with Ω
U
; tak-
ing Ω
E
= Ω
U
then yields the corresponding approximate
critical value
γ
1
cr
= −
2
5
Λ −
4
5
C +
1
5
p
9Λ
2
+ 56CΛ + 96C
2
. (20)
For γ < γ
th
, by contrast, the instability is caused by the
collision of Ω
E
with an eigenvalue bifurcating from Ω
L
.
In this case, the critical value of γ can be approximated
by taking Ω
E
= Ω
L
, giving
γ
2
cr
= −
2
5
Λ −
4
5
C +
1
5
p
9Λ
2
+ 16C(Λ + C). (21)
Together, γ
1
cr
and γ
2
cr
give approximate boundaries of the
instability region in the (C, γ)-plane.
FIG. 2: A s Fig. 1, but for a stable intersite bright soliton.
2. Intersite bright solitons
The next natural fundamental so lution to be cons id-
ered is an intersite bright soliton, i.e., a two-excited-
site disc rete mode. In the uncoupled limit, the mode
structure is of the for m u
(0)
n
= 0 for n 6= 0, 1 and
u
(0)
0
= u
(0)
1
=
√
Λ + γ. Using a perturbative expansion,
one can show further that the s oliton is given by
u
n
=
√
Λ + γ +
1
2
C/
√
Λ + γ + O(C
2
), n = 0, 1,
C/
√
Λ + γ + O(C
2
), n = −1, 2,
O(C
2
), otherwise.
(22)
To study the stability of the intersite bright soliton
above, let us co ns ider the O(1) equation (11). Due to
the presence of two non-zero excited sites at C = 0, the
soliton (22) has at leading or der the double eigenvalue
Ω
(0)
E
= 4(Λ+γ)γ, with corresponding eig e nvectors η
(0)
n
=
0 for n 6= 0, 1, η
(0)
0
= 1, and η
(0)
1
= ±1.
The co ntinuation of the eigenvalue Ω
(0)
E
above for
nonzero coupling C can be obtained from Eq. (12) by ap-
plying a solvability condition. The Fredholm alternative
requires that f
n
= 0 for all n, from which we immediately
deduce that the double eigenvalue splits into two distinct
eigenvalues, which are given as functions of C by
Ω
1
= 4(Λ + γ)γ + 4γC + O(C
2
), (23)
and
Ω
2
= 4(Λ + γ)γ − 4(Λ + γ)C + O(C
2
). (24)
As is the case for onsite discrete solitons, intersite
bright solitons can also become unstable. The mecha-
nism of the ins tability is again determined by the relative
positions of Ω
(0)
E
and Ω
(0)
C
, as sketched in Fig. 2. Per-
forming an analysis co rresponding to that in Ref. [25],
we find that the two mechanisms of destabilization for
an onsite discrete soliton also occur here. The two sce-
narios have corresponding critical values of γ, which are
given as functions of C by
γ
1
cr
= −
2
5
Λ +
2
5
C +
1
5
p
9Λ
2
+ 52ΛC + 84C
2
, (25)
γ
2
cr
= −
2
5
Λ −
2
5
C +
1
5
p
9Λ
2
+ 8ΛC + 4C
2
. (26)
We emphasize, as is apparent from the sketch shown
in Fig. 2, that there is another possible mechanism of
destabilization for γ < γ
th
, namely when Ω
2
becomes
negative. The third critical choice of parameter values is
then obtained by setting Ω
2
= 0, i.e.
γ
3
cr
= C. (27)
B. Comparisons with numerical calculations
We have solved the steady-state equation (6) numeri-
cally using a Newton–Raphson method, and analyzed the
stability of the numerical solution by solving the eigen-
value problem (8). In this section, we compare these
numerical results with the analytica l calculations of the
previous section. For the sake of simplicity, we set Λ = 1
in a ll the illustrative examples.
1. Onsite bright solitons
Comparisons between numerical calculations and ana-
lytical approximations for the case o f onsite bright soli-
tons have been fully prese nted and discussed in Ref. [25].
For the sake of completeness, we reproduce the results of
Ref. [25] for the (in)stability domain of onsite solitons in
the (C, γ) plane in Fig. 3. Approximations (20) and (21)
are also shown there.
2. Intersite bright solitons
For the stability of intersite bright solitons, we start
by examining the validity of our analytical prediction for
5
C
γ
0 1 2 3 4
−0.2
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1
1.2
Λ=1
FIG. 3: (Colour online) The (in)stability region of onsite
bright solitons in (C, γ) space. For each value of C and γ, the
corresponding colour indicates the maximum value of |Im(ω)|
(over all eigenvalues ω) for the steady-state solution at that
point. Stability is therefore indicated by the region in which
Im(ω) = 0, namely the dark region. White dashed and dash-
dotted lines give the analytical approximations (20) and (21),
respectively.
0 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
C
|ω|
Λ=1
FIG. 4: Comparison between the critical eigenvalue of inter-
site bright solitons obtained numerically (solid lines) and its
analytical approximation (dashed lines). The upper and lower
curves correspond, respectively, to γ = 0.5 and γ = 0.1, ap-
proximated by Eq. (24), whereas the middle one corresponds
to γ = 0.18, approximated by Eq. (23).
the eigenvalue associated with the phase mode as given
by Eqs. (23) and (24). In Fig. 4, we present a compar-
ison between the analytical approximation and the nu-
merics for some representative values of γ (specifically
γ = 0.1, 0 .18, 0.5). This figure reveals the relative ac-
curacy of the small-C approximations, and we conclude
that their range of validity is wider for smaller values
of γ.
Next we turn to a description of the eigenvalue struc-
ture of this intersite configuration for the three values of
γ given above; this is shown in Fig. 5, where the left and
right panels respectively present the structur e just befo re
and just after the first collision that results in the mode
instability. We now describe results in more deta il for
the three values of γ in turn.
For γ = 0.1, when C = 0 the eigenvalues ω lie in the
gap betwe e n the two parts of the continuous spectrum,
and the instability is caused by a collision between the
critical eigenvalue and its twin at the origin (see the top
panels o f Fig. 5). For γ = 0.18 , the eigenvalues ω also
lie in the gap between the two parts of the c ontinuous
sp e c trum, but the instability in this case is due to a col-
lision between one of the eigenvalues and the inner edge
of the continuous spectrum at ω = ±
√
Ω
L
(see the mid-
dle panels of Fig. 5). In contrast to the two cases a bove,
for γ = 0.5 the eigenvalues lie beyond the continuous
sp e c trum, and the instability is caused by a collision be -
tween the critical eigenvalue and the outer boundary at
ω = ±
√
Ω
U
(see the bottom panels of Fig . 5). All the
numerical results presented here are in accordance with
the sketch shown in Fig. 2.
Numerical calculations of the stability of intersite
bright solitons, for a re latively large range of C and γ,
give us the stability domain of the bright solitons in the
two-parameter (C, γ) plane, which is presented in Fig. 6.
We use colours to represent the maximum of |Im(ω)| as a
function of C and γ; thus solitons are stable in the black
region. Our analytical predictions for the occurre nce of
instability, given by Eqs. (25)–(27), are also shown, re-
sp e c tively, by dashed, dotted, and dash-dotted lines.
III. DARK SOLITONS IN THE DEFOCUSING
DNLS
In this section we consider the existence and stability of
onsite and intersite dark solitons for the defocusing DNLS
equation. Then a static (real-valued, time-independent)
solution u
n
of (5) sa tisfies
− C∆
2
u
n
+ u
3
n
− Λu
n
− γu
n
= 0. (28)
In contrast to bright solitons, where u
n
→ 0 as n → ±∞,
dark solitons have u
n
→ ±
√
Λ + γ as n → ±∞.
To examine the s tability of u
n
, we again introduce the
linearization ansatz φ
n
= u
n
+ δǫ
n
, where again δ ≪ 1.
Substituting this ansatz into the defocusing equation (5 ),
writing ǫ
n
(t) = η
n
+ iξ
n
, and linear iz ing in δ, we again
find
˙η
n
˙
ξ
n
=
0 L
+
−L
−
0
η
n
ξ
n
= H
η
n
ξ
n
, (29)
but where the operators L
±
(C) are now defined as
L
−
(C) ≡ − C∆
2
+ (3u
2
n
− Λ − γ),
L
+
(C) ≡ − C∆
2
+ (u
2
n
− Λ + γ).
The eigenvalue problem above can be simplified further
as for the focusing case, to the alternative form
L
+
(C)L
−
(C)η
n
= ω
2
η
n
= Ωη
n
. (30)
6
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−1
−0.5
0
0.5
1
Re(ω)
Im(ω)
(a)γ = 0.1, C = 0.05
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−1
−0.5
0
0.5
1
Re(ω)
Im(ω)
(b)γ = 0.1, C = 0.3
−1.5 −1 −0.5 0 0.5 1 1.5
−0.2
−0.1
0
0.1
0.2
0.3
Re(ω)
Im(ω)
(c)γ = 0.18, C = 0.05
−1.5 −1 −0.5 0 0.5 1 1.5
−0.2
−0.1
0
0.1
0.2
0.3
Re(ω)
Im(ω)
(d)γ = 0.18, C = 0.18
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−0.2
−0.1
0
0.1
0.2
0.3
Re(ω)
Im(ω)
(e)γ = 0.5, C = 0.05
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
−0.2
−0.1
0
0.1
0.2
0.3
Re(ω)
Im(ω)
(f)γ = 0.5, C = 0.2
FIG. 5: The structure of the eigenvalues of intersite bright solitons in the complex plane for three values of γ, as indicated in
the caption of each plot. Left and right panels depict the eigenvalues of stable and unstable solitons, respectively.
Performing a stability analysis as before, we find the dis-
persion re lation for a dark soliton to be
Ω = (2C(cos κ − 1) − (Λ + 2γ))
2
− Λ
2
, (31)
and so the continuous band lies be tween
Ω
L
= 4(Λ + γ)γ, when κ = 0, (32)
and
Ω
U
= 4(Λ + γ)γ + 8C(Λ + 2γ + 2C), when κ = π. (33)
7
C
γ
0 0.1 0.2 0.3 0.4 0.5 0.6
−0.1
0
0.1
0.2
0.3
0.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Λ=1
FIG. 6: (Colour online) As Fig. 3, but for intersite b right
solitons. Our analytical approximations, given by Eqs. (25),
(26) and (27), are shown as white dash-dotted, dotted, and
dashed lines, respectively.
A. Analytical calculations
To study the eigenvalue(s) of the dark soliton analyt-
ically, we again expand η
n
and Ω in powers of C, and
hence obtain from (30), at O(C
0
) and O(C
1
), respec-
tively, the equations
h
L
+
(0)L
−
(0) − Ω
(0)
i
η
(0)
n
= 0, (34)
and
h
L
+
(0)L
−
(0) − Ω
(0)
i
η
(1)
n
= f
n
, (35)
with
f
n
= (Q
n
+ Ω
(1)
)η
(0)
n
, (36)
where
Q
n
= (∆
2
− 2u
(0)
n
u
(1)
n
)L
−
(0) + L
+
(0)(∆
2
− 6u
(0)
n
u
(1)
n
).
(37)
We next investigate the eigenvalues of both intersite and
onsite modes.
1. Onsite dark solitons
With errors of order C
2
, an onsite dark soliton is given
by
u
n
=
−
√
Λ + γ, n = −2, −3, . . .,
−
√
Λ + γ +
1
2
C/
√
Λ + γ, n = −1,
0, n = 0,
√
Λ + γ −
1
2
C/
√
Λ + γ, n = 1 ,
√
Λ + γ, n = 2, 3, . . ..
(38)
For this configur ation,
L
+
(0)L
−
(0) =
Λ
2
− γ
2
, n = 0,
4(Λ + γ)γ, n 6= 0 .
(39)
Fro m Eq. (34), we then deduce that at C = 0 the
eigenvalues of onsite discrete dark solitons are given by
Ω
(0)
C
= 4(Λ + γ)γ, which becomes the continuous band
for nonzero C, and Ω
(0)
E
= Λ
2
− γ
2
.
The continuation of the eigenvalue Ω
(0)
E
for nonzero C
can be ca lc ulated from Eq. (35). The coefficient of η
(1)
n
in this case is given by
L
+
(0)L
−
(0) − Ω
(0)
=
0, n = 0,
4Λγ − Λ
2
+ 5γ
2
, n 6= 0.
(40)
The solvability condition for (35) then requires that f
0
=
(4Λ − Ω
(1)
)η
(0)
0
= 0. Setting η
(0)
0
6= 0, we deduce that
Ω
(1)
= −4Λ. Hence the eigenvalue of an onsite dark
soliton for small C is given by
Ω = Λ
2
− γ
2
− 4ΛC + O(C
2
). (41 )
Initially, i.e. for C = 0, the relative positions of the
eigenvalue and the continuous spectrum can be divided
into two cases, according to whether γ ≷ γ
th
= Λ/5.
When C = 0 and γ < γ
th
(γ > γ
th
) the eigenva lue
(41) will be above (below) the c ontinuous spectrum, as
sketched in Fig. 7. These relative positions determine
the instability mechanism for an onsite dark soliton, as
we now describe.
FIG. 7: A s Fig. 1, but for a stable onsite dark soliton.
For γ < Λ/5, the instability is due to a collision be-
tween the eigenvalue (41) and Ω
U
, which approximately
occurs when γ = γ
1
cr
, where
γ
1
cr
= −
2
5
Λ −
8
5
C +
1
5
p
9Λ
2
− 28ΛC − 16C
2
; (42)
note that this critical value is meaningful only when C ≤
9Λ/(14 + 2
√
85). For γ > Λ/5, the instability is caused
by the eigenvalue (41) becoming negative, which occurs
when γ = γ
2
cr
, where
γ
2
cr
=
p
Λ
2
− 4ΛC; (43)
8
this value is meaningful only when C ≤ Λ/4.
Furthermore, if we include terms up to O(C
2
), we ob-
tain
Ω = Λ
2
− γ
2
− 4ΛC + 4C
2
+ O(C
3
) (44)
as the eige nvalue of an onsite disc rete dark solito n. Using
this expression, we find the c ritical value of γ indicating
the ons e t of instability to be
γ
1
cr
= −
2
5
Λ −
8
5
C +
1
5
p
9Λ
2
− 28ΛC + 4C
2
, (4 5)
for γ < 0.2Λ and
γ
2
cr
=
p
Λ
2
− 4ΛC + 4C
2
, (46)
for γ ≥ 0.2Λ.
2. Intersite modes
Intersite discrete dark solitons are given, with errors of
O(C
2
), by
u
n
=
−
√
Λ + γ, n = −2, −3, . . .,
−
√
Λ + γ + C/
√
Λ + γ, n = −1,
√
Λ + γ − C/
√
Λ + γ, n = 0,
√
Λ + γ, n = 1, 2, . . ..
(47)
Starting from Eq. (34), we then find
L
+
(0)L
−
(0) = 4(Λ + γ)γ (48)
for all n, from which we deduce that there is a single
leading-order eigenvalue, given by Ω
(0)
= 4(Λ + γ)γ,
with infinite multiplicity. This eigenvalue then ex pands
to form the continuous spectrum for nonzero C.
Because a localized structure must have an eigenvalue,
we infer that an eigenvalue will bifurcate from the lower
edge of the continuous spectrum. This bifurcating eigen-
value may be calculated from Eq. (35). Because
L
(0)
+
(0)L
(0)
−
(0) − Ω
(0)
= 0 (49)
for a ll n, the solvability condition for Eq. (35) requires
f
n
= 0 for all n. A simple calculation then yie lds
f
n
=
(
4Λ + 16γ + (2Λ + 4γ)∆
2
+ Ω
(1)
η
(0)
n
, n = −1, 0,
(2Λ + 4γ)∆
2
+ Ω
(1)
η
(0)
n
, n 6= −1, 0.
(50)
Taking η
(0)
n
= 0 for n 6= −1, 0 leaves the two nontrivial
equations
(8γ + Ω
(1)
)η
(0)
−1
+ (2Λ + 4γ)η
(0)
0
= 0,
(8γ + Ω
(1)
)η
(0)
0
+ (2Λ + 4γ)η
(0)
−1
= 0,
from which we see that η
(0)
−1
= ±η
(0)
0
. Thus we obtain two
possibilities for the O(C) contribution to the eigenvalue,
given by
Ω
(1)
1
= −(12γ + 2Λ), Ω
(1)
2
= 2Λ − 4γ.
Hence the eigenvalues bifurcating from the lower edge of
the co ntinuous spectrum are given by
Ω
1
= 4(Λ + γ)γ − (12γ + 2Λ)C + O (C
2
), (51)
and
Ω
2
= 4(Λ + γ)γ + (2Λ −4γ)C + O(C
2
). (52)
FIG. 8: A s Fig. 1, but for a stable intersite dark soliton.
A simple analysis shows that Ω
2
< Ω
L
only when γ >
Λ/2. The sketch in Fig. 8 then illustrates that instability
is cause d by Ω
1
becoming negative. This consider ation
gives the critical γ as a function of the coupling constant
C to be
γ
cr
= −
1
2
Λ +
3
2
C +
1
2
p
Λ
2
− 4ΛC + 9C
2
. (53)
When there are two eige nvalues (Ω
1
and Ω
2
), Ω
2
de-
creases more slowly than Ω
1
, in s uch a way that for
γ > Λ/2 the instability is still caused by Ω
1
becoming
negative.
B. Comparison with numerical computations
1. Onsite dark solitons
We now compare our analytical results with co rre-
sp onding numerical ca lculations. As for bright solitons,
for illustrative purposes we set Λ = 1.
We s tart by checking the validity of our analytical ap-
proximation for the critical eigenvalue associated with
the pha se mode. As explained above, the change in
the position of the eigenvalues relative to the continu-
ous spectrum at C = 0 occurs at γ = 1/5. Therefore we
consider the two values γ = 0.1 and γ = 0.6, represent-
ing both cases. Figure 9 depicts a co mparison between
our a nalytical res ult Eq. (41) and the numerical compu-
tations, from which we conclude that the prediction is
9
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
C
|ω|
Λ=1
FIG. 9: Comparisons between the critical eigenvalue for on-
site dark solitons obtained numerically (solid lines) and an-
alytically using Eq. (41) (dashed lines) for γ = 0.1 (upper
curves) and γ = 0.6 (lower curves). An approximation that
explicitly includes the next term in expansion Eq. (44) is also
plotted (dotted lines).
quite accurate for small C. The accuracy can be im-
proved if one includes further orders in the perturbative
expansion Eq. (44), and this improvement is shown in
the same figure by the dotted line.
The eigenvalue structure of onsite dark solitons is de-
picted in Fig. 10; left and right panels refer respectively
to conditions just before and just after a collision result-
ing in an instability.
As sketched in Fig. 7, for γ < 1/5 the instability is
caused by a collision between the eigenva lue and one edge
of the continuous spectrum. On the other hand, when
γ ≥ 1/5 the instability is caused by a collision between
the eigenvalue and its twin at the origin (see the bottom
panels of Fig. 10).
We now proceed to evaluate the (in)stability region of
this solution in (C, γ) space. Shown in Fig. 11 is again
the maximum of the imaginary part of the eigenvalue, to-
gether with our approximation to the (in)stability bound-
ary. The white dashed line represents Eq. (43), corre-
sp onding to the instability caused by the collision with
the continuous spectr um. Equa tio n (42) is represented
by the white dash-dotted line, which corresponds to the
other instability mechanism. In addition, pink dashed
and dash-dotted lines show, resp e c tively, Eq. (46) and
Eq. (45), where a better analytical approximation is ob-
tained.
An important observation from the figure is that there
is an interval of values of γ in which the ons ite dark
soliton is always stable, fo r any value of the coupling
constant C. This indicates that a parametric driving can
fully suppress the oscillatory instability reported for the
first time in [21].
2. Intersite dark solitons
Now we examine intersite dark solitons.
Firstly, Fig. 12 shows the analytical prediction for the
critical eigenvalue, given by Eq. (51), compared to nu-
merical results. We see that the approximation is excel-
lent for small C and its range of validity is wider for larger
values of γ. The eigenvalue structure of this configuration
is shown in Fig. 13 for the two values γ = 0.1, 0.8. The
mechanism of instability explained in the section above
can be seen clearly in the top panels of Fig. 13.
It is interesting to note that a pa rametric driving
can also fully suppress the oscillatory instability of an
intersite dark soliton. As shown in the bottom pan-
els of Fig. 13, there are values of the parameter γ fo r
which no instability-inducing collision ever occurs. The
(in)stability region of this configuration is s ummarized
in Fig. 14, where we see that for any C and γ > 0.3 an
intersite dark soliton is always stable. Our analytical pre-
diction for the onset of instability is given by the dashed
line in tha t figure. We observe that for relatively small
C, the prediction of Eq. (51) is re asonably close to the
numerical results.
IV. DISCUSSION
In the sections above we discus sed the existence and
the stability of localized modes through our reduced
DNLS equation (5). In this section, we confirm the
relevance of our findings through solving numerically
the original time-dependent equation (1). We use a
Runge–Kutta integration method, with the initial con-
dition ϕ
n
= 2ǫu
n
and ˙ϕ
n
= 0, where u
n
is the static
solution of the DNLS (5) and ǫ is the small parameter
of Sec. I. Throughout this section, we use the illustrative
value ǫ = 0.2.
Shown in the left and right panels of Fig. 15 are the
numerical evolution of a s table and unstable onsite bright
soliton, respectively. From the right panel of the figure,
we note that a parametric driving seems to destroy an
unstable so lito n. This observation is similar to the cor-
responding observation for the dynamics of an unstable
soliton in the DNLS equation (5) rep orted in Ref. 25.
In Fig. 16 we present the numerical evolution of in-
tersite brig ht solitons for the same parameter values as
those in Fig. 5, corresponding to each of the instability
scenarios. From the panels in this figure, we see that
the typical dynamics of the ins tability is in the form of
soliton destruction or discharge of a traveling breather.
We have also examined the dynamics of onsite dark
solitons in the Klein–Gor don system (1). Shown in
Fig. 17 is the numerical evolution of a solution with the
eigenvalue structure illustrated in Fig. 10. The instabil-
ity of an unstable onsite dark soliton typically manifests
itself in the form of oscillations in the loc ation of the
soliton center about its initial position (top right panel)
10
−1.5 −1 −0.5 0 0.5 1 1.5
−1
−0.5
0
0.5
1
Re(ω)
Im(ω)
(a)γ = 0.1, C = 0.02
−1.5 −1 −0.5 0 0.5 1 1.5
−1
−0.5
0
0.5
1
Re(ω)
Im(ω)
(b)γ = 0.1, C = 0.2
−3 −2 −1 0 1 2 3
−0.5
0
0.5
Re(ω)
Im(ω)
(c)γ = 0.6, C = 0.01
−3 −2 −1 0 1 2 3
−0.5
0
0.5
Re(ω)
Im(ω)
(d)γ = 0.6, C = 1
FIG. 10: The eigenvalue structure of on-site dark solitons for several values of γ and C, as indicated in the caption of each
panel.
C
γ
0 0.5 1 1.5 2
−0.2
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Λ=1
FIG. 11: (Colour online) The (in)stability region of onsite
dark solitons in the two-parameter ( C, γ) space. The white
and pink dashed lines respectively give the analytical approx-
imations Eq. (43) and (46). White and pink dash-dotted lines
show Eqs. (42) and (45); note that these curves are indistin-
guishable in this p lot.
0 0.05 0.1 0.15 0.2
0
0.5
1
1.5
2
2.5
C
|ω|
Λ=1
FIG. 12: Comparisons between the critical eigenvalue for in-
tersite dark solitons obtained numerically (solid lines) and
analytically (dashed lines) using Eq. (51), for two values of γ.
The upper curves correspond to γ = 0.8, and the lower ones
to γ = 0.1.
11
−3 −2 −1 0 1 2 3
−0.5
0
0.5
Re(ω)
Im(ω)
(a)γ = 0.1, C = 0.05
−3 −2 −1 0 1 2 3
−0.5
0
0.5
Re(ω)
im(ω)
(b)γ = 0.1, C = 0.5
−8 −6 −4 −2 0 2 4 6 8
−1
−0.5
0
0.5
1
Re(ω)
Im(ω)
(c)γ = 0.8, C = 0.5
−8 −6 −4 −2 0 2 4 6 8
−1
−0.5
0
0.5
1
Re(ω)
Im(ω)
(d)γ = 0.8, C = 2
FIG. 13: The eigenvalue structure of intersite dark solitons with parameter values as indicated in the caption for each panel.
C
γ
0 0.5 1 1.5
−0.2
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1
Λ=1
FIG. 14: (Colour online) As Fig. 11, but for an intersite dark
soliton. The white dashed line is our analytical approximation
Eq. (53).
or oscillations in the width of the soliton (bottom right
panel).
Finally, we illustrate the dynamical behavior of an un-
stable intersite dark soliton in Fig. 18, fro m which we
see that the instability makes the soliton travel. This
dynamics is similar to that reported in Ref. 20.
V. CONCLUSION
In this paper, we have considered a parametrica lly
driven Klein–Gordon system describing nanoelectrome-
chanical systems. Using a multiscale expansion method
we have reduced the system to a parametrically driven
discrete nonlinear Schr¨odinger equation. Analytical and
numerical calc ulations have been performed to deter-
mine the existence and stability of fundamental bright
and da rk discrete solitons in the Klein–Go rdon system
through use of the Schr¨odinger equation. We have shown
that the presence of a parametric driving can destabilize
an onsite bright soliton. On the other ha nd, a parametr ic
driving has also been shown to stabilize intersite bright
and dark discrete solitons. We even found an interva l
in γ for which a discrete dark soliton is stable for any
value of the coupling constant, i.e. a parametric driving
can suppress oscillatory instabilities. Stability windows
12
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
FIG. 15: (Colour online) The spatio-temporal evolution of an onsite bright soliton governed by the original time-dependent
parametrically driven Klein–Gordon system (1), with ǫ = 0.2 and γ = 0.1. The left and right panels show a stable and unstable
soliton, at C = 0.1 and C = 1, respectively.
for all the fundamental solitons have been presented and
approximations using perturbation theory have been de-
rived to accompany the numerical r e sults. Numerical
integrations of the original Klein–Gordon system have
demonstrated that our analytical and numerical inves-
tigations of the discrete nonlinear Schr¨odinger equation
provide a useful guide to behavior in the original system.
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13
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
(a)γ = 0.1, C = 0.05
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
(b)γ = 0.1, C = 0.3
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
(c)γ = 0.18, C = 0.05
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
(d)γ = 0.18, C = 0.18
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.5
0
0.5
(e)γ = 0.5, C = 0.05
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
(f)γ = 0.5, C = 0.2
FIG. 16: (Colour online) As Fig. 15, but for an intersite bright soliton, with parameter values as indicated in the caption for
each panel. The initial profile in each panel corresponds to the same parameters as in Fig. 5.
14
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
(a)γ = 0.1, C = 0.02
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
(b)γ = 0.1, C = 0.2
t
n
0 500 1000 1500
10
20
30
40
50
60
70
80
90
100
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
(c)γ = 0.6, C = 0.01
t
n
0 500 1000 1500
10
20
30
40
50
60
70
80
90
100
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
(d)γ = 0.6, C = 1
FIG. 17: (Colour online) As Fig. 15, but for on-site dark solitons. The parameter values are as in Fig. 10.
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
t
n
500 1000 1500
−10
−8
−6
−4
−2
0
2
4
6
8
10
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
FIG. 18: (Colour online) As Fig. 15, but for an intersite dark soliton with γ = 0.1. The left panel shows the evolution of a
stable dark soliton with C = 0.05, while the right panel shows the evolution of an unstable dark soliton with C = 0.5.
