- Email: [email protected]
Modulational instability of gap solitons in single-walled carbon nanotube lattices
Journal Pre-proof Modulational instability of gap solitons in single-walled carbon nanotube lattices Brantony Mozola, Conrad Bertrand Tabi, Timoléon Crépin Kofané
PII: DOI: Reference:
S0165-2125(19)30139-8 https://doi.org/10.1016/j.wavemoti.2020.102511 WAMOT 102511
To appear in:
Wave Motion
Received date : 22 April 2019 Revised date : 21 November 2019 Accepted date : 10 January 2020 Please cite this article as: B. Mozola, C.B. Tabi and T.C. Kofané, Modulational instability of gap solitons in single-walled carbon nanotube lattices, Wave Motion (2020), doi: https://doi.org/10.1016/j.wavemoti.2020.102511. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
Journal Pre-proof
of
Modulational instability of gap solitons in single-walled carbon nanotube lattices
1
p ro
Brantony Mozola1∗, Conrad Bertrand Tabi1†, and Timol´eon Cr´epin Kofan´e1,3‡ Department of Physics and Astronomy, Botswana International University of Science and Technology, Private Bag 16 Palapye, Botswana 3
Laboratoire de M´ecanique, D´epartement de Physique, Facult´e des Sciences, Universit´e de Yaound´e I,
Pr e-
B.P. 812 Yaound´e, Cameroun
al
January 13, 2020
Highlights
urn
• Nonlinear localized excitations are addressed in the framework of a one-dimensional diatomic carbon nanotube lattice.
• Two frequency modes are studied through the quasi-discrete approximation.
Jo
• The modulational instability analysis and soliton solutions are analyzed in the upper and lower optical frequency cutoff modes, and in the upper acoustic frequency cutoff regime. ∗
[email protected] (B. Mozola) Corresponding author: [email protected] (C. B. Tabi) ‡ [email protected] (T. C. Kofan´e) †
1
of
Journal Pre-proof
Abstract
p ro
Modulational instability and nonlinear localized excitations are addressed, in the framework of a one-dimensional diatomic carbon nanotube model, using the quasi-discrete approximation. Gap soliton solutions, based on the modulational instability criterion, are studied, where one considers
Pr e-
the solutions arising in the upper and lower optical frequency cutoff regimes, and in the upper acoustic frequency cutoff mode. Solutions are found as breathers and double breathers, and their response to interatomic interaction parameters is discussed. Vibrations of the CNTs from the two modes are compared based on their capability of carrying the amount of energy required for specific purposes, either in Microelectronics or in Nano-devices.
1
urn
al
Keywords: Carbon nanotubes; Modulational instability; Breathers; Gap solitons.
Introduction
Over the last two decades, carbon nanotubes (CNTs) have drawn considerable attention, and
Jo
significant amount of investigations has revealed their extraordinary, and sometimes unexpected mechanical, optical, chemical and structural properties [1, 2, 3]. One of the main research directions devoted to mechanical behaviours has also revealed that CNT properties fundamentally depend on structure, so that any uniform [4, 5] or nonuniform [6, 7] deformation of a CNT can considerably
2
Journal Pre-proof
affect its properties. Attention has been paid extensively to nonlinear excitations in graphene nanoribbons and their role in transporting and localizing energy has been reported [8]. In that
of
respect, CNTs, that are made by rolling up graphene sheet, may support nonlinear excitations and solitons. The existence of spatially localized nonlinear modes in the form of discrete breathers was
p ro
predicted in CNTs with chirality (m, 0) and (m, m) [9, 10]. Principally, three types of discrete breathers (DBs) were discussed, namely the longitudinal, radial, and twisting DBs. The structure and stability of stationary DBs in zigzag and armchair CNTs were studied by Doi and Nakatani [11], along with the involved structural physical processes and dynamical implications. By considering
Pr e-
a large radius of the CNT, a one-dimensional diatomic chain model was proposed by Savin and Kivshar [9, 10], and numerical studies were performed, using molecular dynamics simulations with a realistic interatomic potential, to confirm the intrinsic character of DBs in CNTs. The present work focuses on such a model.
Diatomic lattices are generalized versions of monoatomic chains, and their dispersion curve contains two frequency branches known as the acoustic branch and the optical branch. Between the
al
optical and the acoustic frequencies, there exist a gap where wave propagation of certain wavelengths is forbidden. More clearly, linear approximation is valid for small amplitude vibrations described
urn
as combinations of the linear modes within the phonon band. However, when vibration with large displacement takes place, nonlinearity of the interaction between atoms becomes dominant, leading to the so-called self-induced nonlinearity, and there appears some new effects in the system, mainly
Jo
the formation of gap solitons in the forbidden zone of the linear wave spectrum. Introduced by Chen and Mills and applied to nonlinear optical response of superlattices [12], the concept of gap solitons has gained interest in other physical contexts such as solid-state lattice vibrations [13, 14, 15], nonlinear optics [16, 17], biophysics [18, 19], electrical transmission lines [20, 21, 22], and so on.
3
Journal Pre-proof
The effect of cubic and quartic nonlinearities on gap solitons, in one-dimensional diatomic lattices, was addressed by Huang and Hu [13, 15] who further confirmed that such excitations can be
of
described by two coupled nonlinear Schr¨odinger (NLS) equations, each corresponding to a specific frequency mode. The nonlinear localized excitations corresponding to each of the modes have
p ro
been discussed in different physical situations [15, 18, 20], with instance on the different dynamical features of gap solitons. The model we consider in the present study, to the best of our knowledge, has been exploited only for numerical calculations until now, probably due to the complexity of the interatomic potentials.
Pr e-
One of the natural mechanisms leading to the formation of solitonic structures is modulational instability (MI). MI, which can be defined as an exponential growth of some modulation sidebands of a plane wave due to the balance between nonlinear and dispersive effects, appears in different physical settings, including nonlinear optics [23, 24, 25], plasma physics [26, 27, 28, 29], biological physics [30, 31, 32, 33, 34, 35, 37, 38], Bose-Einstein condensation [39, 40, 41], just to name a few. NLS breather solutions appear in general in regions of MI, i.e., where the product between
al
the dispersion and nonlinearity coefficients is positive, which mainly depends on the choice of right model parameters that constitute the nonlinear interatomic potentials. In that direction, an
urn
attempt was recently made to analyze the stability/instability of nonlinear waves in single walled CNTs under the effect of higher order interaction terms from the Brenner’s potential [42], which was found to strongly influence energy localization in the discrete context, through a discrete NLS
Jo
equation. Moreover, intense energy exchange between different parts of CNT and weak energy localization in the excited part of CNT were reported in the framework of the continuum shell theory, based on the NLS equations resulting from the multi-scale expansion procedure [43]. All the above clearly supports the idea that in one-dimensional lattices, weekly nonlinear excitations
4
Journal Pre-proof
with a large spatial extension, or envelope solitons, are governed by the NLS equation which is straightforwardly related to short wave wavepacket excitations [44].
of
In the rest of this paper, we first present the CNT model, via its Hamiltonian, in Section 2. In Section 3, the quasi-discrete approximation (QDA) is utilized to show that each of the modes,
p ro
optical and acoustic, is described by a NLS equation. Gap solitons are discussed in Section 4, and attention is given to breather solitons that are related to MI. The paper ends with some concluding remarks in Section 5.
Model
Pr e-
2
The model adopted in the present paper was introduced in Refs. [9, 10], where the CNT is considered to have a radius R. Considering a chirality (m, 0), the Hamiltonian for the vibration of the lattice of carbon atoms is written as H=
X X 1 n
l
2
M (u˙ 2n,l,0
+
u˙ 2n,l,1 )
+ Pn,l ,
(1)
al
where the three indices (n, l, k) define the structure of the CNT, with (n, l) defining an elementary cell (n = 0, ±1, ..., l = 1, 2, ..., m), and k = 0, 1 being the atom index in the cell. l is unlimited
urn
for a planar sheet of carbon atoms. M = 12 × 1.6603 · 10−27 kg ≈ 12 amu (atomic mass units) is the mass of a carbon atom. un,l,k = (xn,l,k (t), yn,l,k (t), zn,l,k (t)) is the radius-vector that gives the spatial coordinates of the carbon atom with index (n, l, k) at time t. Carbon atoms interact in the
Jo
lattice according to the potential energy Pn,l , given by Pn,l = P (u−1,l,1 , un,l−1,1 , un,l+1,0 ) = V(u1 , u2 ) + U(u1 , u2 , u3 ) + W(u1 , u2 , u3 , u4 ),
(2)
where V = V (un,l,0 , un,l,1 ) + V (un,l,1 , un+1,l,0 ) + V (un,l,1 , un,l+1,0 ) is the term describing interactions between pairs of atoms with the coordinates u1 and u2 . U = U (un−1,l,1 , un,l,0 , un,l−1,1 ) + 5
Journal Pre-proof
U (un−1,l,1 , un,l,0 , un,l,1 ) + U (un,l−1,1 , un,l,0 , un,l,1 ) + U (un,l,0 , un,l,1 , un,l+1,0 ) + U (un,l,0 , un,l,1 , un+1,l,1 ) + U (un,l+1,0 , un,l,1 , un+1,l,0 ) is the potential due to the deformation of the angles between the links u1 u2
of
and u2 u3 , and, finally, the term W = +W (un,l,1 , un,l,0 , un−1,l,1 , un,l−1,1 )+W (un,l,1 , un,l,0 , un,l−1,1 , un−1,l,1 )+ W (un−1,l,1 , un,l,0 , un,l,1 , un−1,l,1 ) + W (un,l,0 , un,l,1 , un,l+1,0 , un,l,0 ) + W (un,l,0 , un,l,1 , un+1,l,0 , un,l+1,0 ) +
p ro
W (un+1,l,0 , un,l,1 , un,l,0 , un,l+1,0 ) is the energy related to the variation of the effective angles between the planes u1 u2 u3 and u2 u3 u4 . More explicitly, the potentials U, V and W can be written as n
o2 V(u1 , u2 ) = D exp[−α(ρ − ρ0 )] − 1 ,
(3)
Pr e-
where ρ = |u2 − u1 |. D = 4.9632eV is the energy of the valent coupling, and ρ0 = 1.418˚ A is the static length;
U(u1 , u2 , u3 ) = v (cos ϕ − cos ϕ0 )2 ,
(4)
where cos ϕ = (u3 − u2 , u1 − u2 )(|u3 − u2 | · |u2 − u1 )−1 , and cos ϕ0 = cos(2π/3) = −1/2; W(u1 , u2 , u3 , u4 ) = t [1 − (v1 , v2 )(|v1 | · |v2 |)−1 ],
(5)
al
where v1 = (v2 − v1 ) × (v3 − v2 ) and v2 = (v3 − v2 ) × (v4 − v3 ). The rest of the model parameters α, v ,and t were determined from the phonon frequency spectrum of a plane of carbon atoms [45, 46]
urn
and were adopted in Refs. [9, 10]. A one-dimensional diatomic lattice of CNT can be obtained from the above-described generalized model if the radius is considered as R → ∞. In such a special case,
Jo
the Hamiltonian (1) reduces to its simplified form written as X 1 2 2 H= M (u˙ 2n−1 + u˙ 2n ) + V1 (ρ2n ) + V2 (ρ2n−1 ) , 2 n
(6)
where ρn = un+1 − un . The anharmonic interatomic potentials are given by V1 (w) =D(e−αw − 1)2 ,
" #2 2 n o2 a (w) 1 (w + ρ /2) 1 1/2 0 − V2 (w) =2D e−α[a+ (w) −ρ0 ] − 1 + 2v + + 4v p − , a+ (w) 2 2 a+ (w) 6
(7)
Journal Pre-proof
8
30 = 1.7889 Å-1
7
= 1.7889 Å-1
= 2.0 Å-1 = 2.5 Å
6
4 3
15 10
p ro
2
of
V2(w)
V1(w)
= 2.5 Å-1
20
5
5
1 0 -1
= 2.0 Å-1
25
-1
(a) 0
1
2
3
0 -5
4
w
(b) 0
5
w
Pr e-
Figure 1: The graphs above shows the potentials, V1 and V2 given by the expressions on Eq. (7) with D = 4.9632 eV and ρ0 = 1.418˚ A. (a) The well becomes narrower as α increases. (b) An asymmetric double well is obtained for α > 2. with a± (w) = (w + ρ0 /2)2 ± 3ρ20 /4. The above potentials are represented in Fig. 1, where panel (a) shows V1 (w) which is a Morse potential, with D being the energy of the valent coupling and α, the width of the the potential. Both D and α are constants determining the strength and the curvature
al
of the potential, respectively. V2 (w), represented in Fig. 1(b), is an asymmetric potential with a Morse plateau for w → +∞. When α changes, the width of the Morse potential gets reduced.
urn
However, the modified potential of Fig. 1(b) splits into an asymmetric double-well potential which is obviously related to the anharmonic interactions of the network and may bring about important nonlinear effects in the dynamics of the CNT model. We should however stress that some recent works have been done, where higher-order nonlinear effective potential for single walled carbon
Jo
nanotubes were obtained by using the Brenner’s potential [42]. In order to study discrete effects on longitudinal wave formation, the author used the rotating wave approximation and obtained a discrete NLS equation. However, this differs from our context because we consider both transverse and longitudinal dynamics, which, not only complicates the study of solitons in CNTs, but also
7
Journal Pre-proof
gives the room for new phenomena to be observed. This further motivates the use of the QDA, which is appropriate to qualitatively and quantitatively study the input of both nonlinear and
Amplitude equation
p ro
3
of
discrete effects in the model under consideration.
The Hamiltonian (6) contains complicated anharmonic interatomic potentials which may prevent direct exact solutions to be found. This further explains why most of the works devoted to the studied model have been essentially numerical. In order to proceed with the QDA, we expand the
Pr e-
potentials V1 and V2 of Eq.(7) in Taylor’s series up to the fourth order, i.e., 1 1 1 1 1 1 V1 (w) ' I2 w2 + I3 w3 + I4 w4 and V2 (w) ' J2 w2 + J3 w3 + J4 w4 , 2 3 4 2 3 4 2 (3−αρ ) 2 27v 6Dα3 14Dα4 1 0 2 , J3 = 3Dα2M + Dα where I2 = 2Dα , I = − , I = , J = − 3 4 2 M M M M ρ0 2ρ2 0
J4 =
Dα3 (7α2 ρ20 −54αρ0 −9) 4M ρ0
+
486v M ρ60
(8) 81v M ρ60
and
depend stronly on the model parameters, especially α, D and ρ0 .
The coefficients with indexes 2, 3 and 4 are respectively the harmonic, the cubic and quartic
al
interatomic constants. Moreover, together with the values of M , D and ρ0 , values for the rest of parameters were found from the phonon spectrum frequency of a planar lattice of carbon atoms as
urn
α = 1.7889˚ A−1 , v = 1.3143eV and t = 0.2 eV. The unit system that is used here, i.e., amu, ˚ A and eV, brings about a time unit (t.u.), with 1t.u.≈ 1.021 × 10−14 s. Thanks to Hamilton’s equations, the following equations for the longitudinal (u2n−1 → vn ) and transverse (u2n → wn ) motions are M v¨n = I2 (wn+1 − vn ) − J2 (vn − wn ) + I3 (wn+1 − vn )2 − J3 (vn − wn )2
Jo
obtained:
(9a)
− I4 (wn+1 − vn )3 − J4 (vn − wn )3 ,
M w¨n = −I2 (wn − vn−1 ) + J2 (vn − wn ) − I3 (wn − vn−1 )2 + J3 (vn − wn )2 + I4 (wn − vn−1 )3 + J4 (vn − wn )3 , 8
(9b)
Journal Pre-proof
12
14
16
(b)
(a)
(c) 14
12
10
12
10 8
10 8 8 6
of
6
6
4 4 2
4
2
+ -
-0.5
0
0.5
+
-
0 -1
1
-0.5
qa/
0
0.5
1
p ro
0 -1
2
+
0 -1
qa/
-0.5
-
0
0.5
1
qa/
Pr e-
Figure 2: The dispersion curve(ω vs q) for a diatomic lattice. Blue represents the optical modes and red for the acoustical modes. The parameters are M = 12 amu, D = 4.9632 eV,v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV. To each of the panels, corresponds a value of α so that (a) is plotted for α = 1.7889˚ A−1 , (b) for α = 2.2˚ A−1 and (c) for α = 2.5˚ A−1 . The linear dispersion relation can be obtained by considering linear terms in Eqs.(9a) and (9b). Using the plane wave solution ansatz for vn and wn we find s r 2 qa 2 , ω± (q) = (I2 + J2 ) ± (I2 + J2 ) − 4I2 J2 sin 2
(10)
where ω is the frequency, q is the wave number and a is the lattice spacing. The (−) sign corresponds
al
to the acoustic frequency and the (+) sign to the optical frequency as shown in Fig. 2. At wave
urn
number q = 0, the eigenfrequency spectrum has a lower cutoff ω− for acoustic branch and an p upper cutoff ω+ for the optical branch is w+ (q = 0) = 2(I2 + J2 ). When q = π/a, there exists √ a frequency gap between the upper cutoff of the acoustic branch, ω− (q = π/a) = 2J2 and the √ lower cutoff of the optical branch, ω+ (q = π/a) = 2I2 so that the width of the frequency gap √ √ is ∆ω = 2I2 − J2 . With increasing α, one notices that the gap between the two frequencies
Jo
also increases. In Fig. 2(a), where α = 1.7889˚ A−1 , we have a gap ∆ω = 0.452t.u.−1 , while for ˚−1 (see Fig. 2(b)), the gap is ∆ω = 0.878t.u.−1 . Additionally, for α = 2.5˚ α = 2.2A A−1 , the frequency gap in Fig. 2(c) is ∆ω = 1.91t.u.−1 . In the frequency gap, no linear wave can propagate, but in the nonlinear context, some waves may propagate and their oscillatory frequency may lie in 9
Journal Pre-proof
the forbidden band of the continuum wave spectrum. The effects of nonlinearity and dispersion can be investigated by assuming trial solutions to
2 (2) 3 (3) un (t) = u(1) n,n + un,n + un,n =
∞ X j=1
of
Eqs. (9) in the form (j)
j un,n±1 ,
(11)
p ro
where is a small but finite parameter denoting the relative amplitude of the excitations and (j)
un,n = u(j) (ξn , τ ; φn ), ξn = (na − λt) and τ = 2 t are two multiple-scale variables which are known as slow variables. λ is a parameter to be determined by a solvability condition. The phase of the carrier wave φn = qnl − ωt is the fast variable and is considered fully discrete. (j)
Pr e-
Introducing solutions (11) into Eqs. (9) and comparing expressions in similar powers in lead (j)
to a hierarchy of equations about vn,n and wn,n (j = 1, 2, 3, ...) as follows:
∂2 (j) (j) (j) (j) + 2I2 vn,n − I2 wn,n + I2 wn,n−1 = Mn,n , ∂t2
with
(1) Mn,n = 0,
al
h 2 i ∂ 2 (1) ∂ (1) (1) (1) (1) 2 (1) vn,n − I2 a wn,n−1 + I3 wn,n − vn,n − wn,n−1 − vn,n , ∂t∂ξn ∂ξn 2 ∂2 a2 ∂ 2 (1) ∂ 2 (2) ∂ (2) 2 ∂ (3) (1) v − 2 +λ w + w vn,n + I2 −a Mn,n = 2λ ∂t∂ξn n,n ∂t∂τ ∂ξn2 ∂ξn n,n−1 2 ∂ξn2 n,n−1 ∂ (1) (1) (2) (1) (2) − 2I3 wn,n−1 − vn,n wn,n−1 − vn,n − a w ∂ξn n,n−1 # " 3 3 (1) (1) (2) (1) (1) (1) (1) (2) + I4 wn,n − vn,n + wn,n−1 − vn,n , − vn,n wn,n − vn,n + 2I3 wn,n and
Jo
.... ..
urn
(2) Mn,n = 2λ
∂2 (j) (j) (j) (j) , + 2J2 wn,n − J2 vn,n + J2 vn,n+1 = Nn,n 2 ∂t 10
(12)
(13a) (13b)
(13c)
(14)
Journal Pre-proof
with
p ro
of
(1) Nn,n = 0, (15a) h 2 i ∂2 ∂ (1) (1) (2) (1) (1) (1) 2 (1) Nn,n = 2λ wn,n + J2 a vn,n+1 + J3 vn,n − wn,n − vn,n+1 − wn,n , (15b) ∂t∂ξn ∂ξn 2 ∂2 ∂ (2) a2 ∂ 2 (1) ∂2 (2) 2 ∂ (1) (3) w − 2 +λ wn,n + J2 a v + v Nn,n = 2λ ∂t∂ξn n,n ∂t∂τ ∂ξn2 ∂ξn n,n+1 2 ∂ξn2 n,n+1 (2) ∂ (1) (1) (2) (1) (1) (2) (1) (2) + 2J3 vn,n − wn,n vn,n − wn,n − 2J3 vn,n+1 − wn,n vn,n+1 − wn,n − a v ∂ξn n,n+1 (15c) 3 (1) (1) (1) 3 (1) + J4 vn,n+1 − wn,n − wn,n + J4 vn,n
.... ..
Pr e-
The rest of the calculation procedure amounts in collecting terms in different powers of the small parameter . We should however keep in mind that we have obtained, in the linear approximation, the optical and acoustical frequencies, which implies that both of the modes should be considered in the treatment of the problem related to Eqs.(12)-(15).
3.1
Acoustic Mode Excitations
al
For the acoustic mode, Eqs. (12) and (14)are rewritten as
urn
∂2 (j) (j) (j) (j) ˆ n,m Lw = J2 Mn,n + Mn,n+1 + + 2J Nn,n , 2 ∂t2 ∂2 (j) (j) (j) (j) + 2I v = I w + w 2 2 n,n−1 + Mn,n , n,n n,n ∂t2
(16a) (16b)
ˆ is defined by where the operator L ∂2
∂t2
+ 2I2
Jo
(j) ˆ n,m Lw =
∂ 2
∂t2
(j) (j) (j) (j) . + 2J2 wn,n − I2 J2 wn,n+1 + wn,n−1 + 2wn,n
(17)
Eqs. (16a) and (16b) correspond to their acoustic frequency expressions. The order j = 1 leads to linear equations, and solutions are found as −
−
(1) (1) wn,n = A− (ξn , τ )eiφn + c.c. and vn,n = BA− (ξn , τ )eiφn + c.c.,
11
(18)
Journal Pre-proof
where A− (ξn , τ ) is an envelope function to be determined later, φ− n = qna − ω− (q)t, and B = −iqa
) − I2 ω(1+e . The frequency ω− (q) given by (10) is related to the acoustic dynamical regime. 2 −2I 2 −
I2 J2 a sin(qa) 2 ), ω− (2I2 +2J2 −2ω−
of
− For the case j = 2, a solvability condition is obtained as λ = vg− = − ∂ω = ∂q
which is the group velocity, upon which solutions are obtained, after making the change of variable
p ro
ξn = ξn− = (na − vg− t), as −
(2) (2) = B0 wn,n = C1 A2− (ξn , τ )e2iφn + c.c. and vn,n
− − ∂ A− eiφn + B1 A2− e2iφn + c.c., − ∂ξn
(19)
J2 )2 − 4I2 J2 sin2 (qa)}1/2 .
Pr e-
2 −2I ) 2 −2I ) −(2iλω− I2 (1+e−iqa ))+I2 ae−iqa (ω− F1 J2 (1+e2iqa )−G1 (4ω− 2 2 C1 I2 (1+e−iqa )2 +F1 where B0 = , B = , with C = 2 2 2 2 2 2 [2q]) , 1 1 (ω− −2I2 )2 4ω− −2I2 (4ω− −ωq− [2q])(4ω− −ωq+ h i h i 2 F1 = I3 (1 − B)2 − (e−iqa − B)2 , G1 = J3 (B − 1)2 − (B eiqa − 1)2 , and 2ωq± = (I2 + J2 ) ± {(I2 +
At the third order j = 3, the solvability condition leads to the evolution equation for A− (ξn , τ ) in the form i
∂A− 1 ∂ ∂ + P− − − A− + Q− |A− |2 A− = 0, ∂τ 2 ∂ξn ∂ξn
al
with the coefficients P− and Q− being given by
(20)
Jo
urn
2 2 2 − 2J2 ) − 2I2 J2 − 2I2 )(ω− 2(2I2 + 2J2 − 4ω− )(Vg− )2 − a2 (ω− P− = , 2 2ω− (2I2 + 2J2 − 2ω− ) h i (21) 2 I3 (σ1 − σ2 ) + J3 (α1 − α2 ) − 3β1 + β2 Q− = , 2 2ω− (2I2 + 2J2 − 2ω− ) h i h i n h iqa 2 ∗ ∗ iqa −iqa iqa and α1 = B1 J2 (1 + e ) (1 − B ) , α2 = J2 B (1 + e ) + (1 + e ) , β1 = I4 J2 (1 + e ) 1 − 2 2 2 2 io nh i o 2 B (1−B)+ eiqa −B (e−iqa −B) , β2 = J4 B −1 (B −1)+ eiqa B −1 (eiqa B −1) (ω− −2I2 ) , h i h i σ1 = J2 B1 (1 + 2B ∗ )(1 − e2iqa ) , and σ2 = J2 C1 2 sin(qa) + B ∗ (e−2iqa − 1) .
12
Journal Pre-proof
3.2
Optical mode
Here, we consider the high-frequency optical mode excitations and Eqs. (16a) and (16b) can be rewritten as
(j) Nn,n+1
(22a) (22b)
p ro
(j) Nn,n
of
∂2 (j) = I2 + + + 2J2 Mn,n , ∂t2 2 ∂ (j) (j) (j) (j) + 2J2 wn,n = J2 vn,n + vn,n+1 + Nn,n , 2 ∂t
ˆ (j) Lv n,n
with j = 1, 2, 3.... Through a procedure similar to the one used for the acoustic mode, we solve the equations for the optical mode in a sequence. The following substitution can then be made: (j)
(j)
(j)
(j)
Pr e-
a → −a, I2 → J2 , I3 → J3 , I4 → J4 , wn,n → vn,n , wn,n±1 → vn,n±1 . Additionally, Eqs. (12)-(13) (j)
(j)
(j)
(j)
can be transformed into Eqs. (14)-(15) through the permutations Mn,n → Nn,n , Nn,n → Mn,n , and (j)
(j)
Mn,n+1 → Nn,n−1 . The same procedure is followed and we get solutions for different orders. Linear (1)
(1)
wave equations are also obtained from the first order j = 1, where Mn,n = Nn,n = 0. This leads to the solutions
+ (1) (1) e + (ξn , τ ) eiφ+n + c.c, vn,n = A+ (ξn , τ )eiφn + c.c. and wn,n = BA
al
(23)
J2 (1+eiqa ) e where φ+ n = qna − ω+ t, B = − ω 2 −2J2 and A+ (ξ, τ ) an envelope function to be determined. ω+ (q) +
urn
is the linear optical frequency of the dispersion relation (10). For j = 2, the solvability condition gives λ = vg+ =
∂ω+ ∂q
=
I2 J2 a sin(qa) 2) ω+ (2I2 +2J2 −2ω+
and ξn = ξn+ =
(na − vg+ t), leading to the solutions
e1 = where B
Jo
(2) (2) e1 A2+ (ξn , τ )e2iφ+n + c.c. and wn,n e0 ∂ A+ eiφ+n + C e1 A2− e2iφ+n + c.c., vn,n =B =C ∂ξn e 1 (4ω 2 −2J2 ) Fe1 I2 (1+e2iqa )−G + 2 −ω 2 [2q])(4ω 2 −ω 2 [2q]) (4ω+ q− + q+
e0 = ,C
Fe0 2 −2J ω+ 2
e1 = and C
e1 2J2 (1+eiqa )2 +Fe1 B . 2 −2J 4ω+ 2
(24)
(3)
For j = 3, we get the third-order approximation equations. The solvability condition for vn,n
13
Journal Pre-proof
(3)
and wn,n yields the NLS equation for A+ (ξn+ , τ ) i
∂A+ 1 ∂ ∂ + P+ + + A+ + Q+ |A+ |2 A+ = 0, ∂τ 2 ∂ξn ∂ξn
of
where
(25)
Pr e-
p ro
2 2 2 − 2J2 ) − 2I2 J2 − 2I2 )(ω+ )(Vg− )2 − a2 (ω+ 2(2I2 + 2J2 − 4ω+ , P+ = 2 ) 2ω+ (2I2 + 2J2 − 2ω+ h i (26) 2 J3 (e σ1 − σ e2 ) + I3 (e α1 − α e2 ) − 3βe1 + βe2 Q+ = , 2 ) 2ω+ (2I2 + 2J2 − 2ω+ h i h i n h iqa 2 ∗ ∗ iqa −iqa iqa e e e and α e1 = B1 I2 (1 + e ) (1 − B ) , α e2 = I2 B (1 + e ) + (1 + e ) , β1 = J4 I2 (1 + e ) 1 − 2 2 2 2 o nh i io e (1− B)+ e eiqa − B e (e−iqa −B) , βe2 = I4 B e −1 (B −1)+ eiqa B e −1 (eiqa B e −1) (ω 2 −2J2 ) , B − h i h i e ∗ (e−2iqa − 1) . From these equations, e ∗ )(1 − e2iqa ) , and σ e1 (1 + 2B e2 = I2 C1 2 sin[qa] + B σ e1 = I2 B the envelope function A− (ξn− , τ ) and A+ (ξn+ , τ ) evolve according to the NLS equation in a unified form i
∂A± 1 ∂ ∂ + P± ± ± A± + Q± |A± |2 A± = 0 ∂τ 2 ∂ξn ∂ξn
(27)
al
whose coefficients P± and Q± depend on the CNT model parameters. In order to return to the
have
urn
± ± 2 original variables we let A± (ξn± , τ ) = (1/)F± (x± n , t) and since (na − Vg t) = xn and τ = t we
∂F± ∂ 2 F± i + P± ±2 + Q± |F± |2 F± = 0. ∂t ∂xn
(28)
4
Jo
Solutions for Eqs.(28) will be discussed in the next sections.
Nonlinear localized modes and gap solitons in CNTs
In general, solutions whose amplitudes are governed by Eq.(28) depends on the sign of the product P± × Q± . To this effect, let us assume plane solutions for Eq.(28) to be of the form F± (t, x± n) = 14
Journal Pre-proof
±
2
iQ± F0 t i(Kxn −Ω± t) , with K F0 (x± , and subjected to a small perturbation δF± (t, x± n )e n ) = (U0 + iV0 )e
and Ω± being, respectively, the wavenumber and frequency of the perturbation. The use of the 2 Kcr = (K P± ) 1 − 2 , K 2
2
q
2Q± . P±
p ro
Ω2±
of
standard calculations of MI leads to the nonlinear dispersion relation
with the critical wavenumber of the perturbation being Kcr = F0
(29)
For the plane wave to be
Pr e-
unstable under modulation Eq.(29) must satisfy the condition Ω2± < 0, which leads to the condition s 2Q± K < Kcr = F0 . (30) P± This clearly shows that for P± Q± > 0, the amplitude-modulated envelope is unstable. Therefore, depending on the sign of the product P± × Q± , there are two kinds of solutions which are the pulse soliton solution and the kink soliton solution. The pulse soliton solution is generally obtained for P± Q± > 0. The corresponding generalized expression for such a soliton is given by =
P± Q±
1/2
n o 1 ±2 ± ± ± 0 ± |P± |η0 t − φ0 η0 xn − xn − vg t exp i , 2
η0± sech
(31)
al
F± (x± n , t)
the general form F± (x± n , t)
=
urn
where η0± is a free constant. On the other hand, when P± Q± < 0, the dark soliton is obtained in
P± − Q±
1/2
n o 1 ±2 ± ± ± 0 ± |P± |η0 t − φ0 η0 xn − xn − vg t exp i . 2
η0± tanh
(32)
Jo
In what follows, regions of instability are discussed along with the subsequent analytical solutions for specific values of the wavenumber q falling into the forbidden/gap zone of the frequency spectrum. √ Usually, the instability is characterized by the MI growth rate Γ = −Ω2 , which is plotted for each of the studied cases.
15
p ro
of
Journal Pre-proof
0.5
Pr e-
Figure 3: The panels show the growth rate of MI versus the perturbation wavenumber K and the parameter α in the case of the upper cutoff mode, i.e., q = 0. Different values of D are considered such that panel (a) is plotted for D = 2.5 eV, panel (b) for D = 3.6 eV and panel (c) for D = 4.9 eV, with the rest of parameters being fixed as M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV. 500
0
1
D = 4 eV D = 4.932 eV D = 5.4 eV
400
0.5
300 200
-0.5
D = 2.5 eV D = 3.6 eV D = 4.9632 eV
0
Q-
0
-0.5
P-
Q+ P+
-1
P+
Q+
100
-100 -1.5
-300 D = 2.5 eV D = 3.6 eV D = 4.9632 eV
0
0.5
(a) 1
1.5
2
-500
(b)
1
1.5
urn
(Å-1 )
-1.5
-400
al
-2
-2.5
-1
-200
(Å-1 )
2
2.5
-2 0.6
(c) 0.8
1
1.2
1.4
1.6
1.8
(Å-1 )
Jo
Figure 4: Panel (a) shows the product P+ Q+ for the upper cutoff frequency, i.e., q = 0, versus the Morse parameter α, for different values of the parameter D. Panel (b) also represents P+ Q+ , but for the lower cutoff frequency, i.e., q = π/a. Panel (b) depicts the behaviors of the product P− Q− for the upper acoustical frequency, i.e., ω− (q = π/a), versus the parameter α and different values of D. The rest of the parameters are such that M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV.
16
2
Journal Pre-proof
4.1
The optical upper cutoff mode
This mode corresponds to the wavenumber value q = 0 (see Fig. 2), which leads to vg+ = 0, ω+ (q = J √ I3 (I2 −J2 )+I2 (3J4 +I4 ) 1+ I 2 a2√ I2 J2 + 2 √ 0) = 2I2 + 2J2 , xn = na = xn , P+ = and Q+ = . The 2(I2 +J2 )
(I2 +J2 )
of
4(I2 +J2 )
2(I2 +J2 )
instability growth rate related to this case is shown in Fig. 3 versus the perturbation wavenumber
p ro
K and the Morse parameter α. For small values of the valent coupling energy D, the instability region is large as shown in Fig. 3(a). For some high values of K, there are regions of α where the plane wave solution is stable. For high values of α, i.e., α = 1.5˚ A−1 , restricted values of K are expected to give rise to instability. When D increases to 3.6 eV (see Fig. 3(b)), the stability
Jo
urn
al
Pr e-
region broadens and this persists when D = 4.9632 eV. If the parameters fall into the detected
Figure 5: Space-time evolution of the upper optical cutoff breather (33a) and (33b). Panels (a1) and (b1) have been recorded for α = 1.25˚ A. Panels (a2)-(b2) have been obtained for α = 1.3˚ A, and ˚ panels (a3)-(b3) have been obtained for α = 1.35A, with D = 2.5eV, M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ A, and t = 0.2 eV. the presented solutions correspond to positive values of product P+ Q+ picked from Fig. 4. 17
Journal Pre-proof
4
8
2
v n (t)
2
v n (t)
w n (t)
0
v n (t)
w n (t)
6
w n (t)
1
4
0
2
-1
0
-2
-2
-6
-2
-8
-3
(a)
-10 -5
0
of
-4
(b)
-4 -5
5
0
n
5
(c) 0
5
n
p ro
n
-4 -5
Figure 6: Solutions of Fig. 5 at different instant: (a) t = 0, (b) t = 0.5 and (c) t = 0.7 with D = 2.5 eV, α = 1.25˚ A, M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ A, and t = 0.2 eV. zones of instability, the plane wave will be said to be unstable under modulation. Otherwise, the
Pr e-
plane wave solution will remain stable under modulation. In other words, the product P+ Q+ plays a fundamental role in finding solutions for the model under our study. Therefore, for this specific case, P+ Q+ is plotted in Fig. 4(a) and shows positive regions whose intervals depend on α. As in Fig. 4, increasing the value of D reduces regions of α where bright solitons are expected and the generalized solutions
urn
al
1/2 n o P+ vn (t) = sech η0+ (n − n0 ) a cos Ω1 t − φ+ 0 , Q+ 1/2 n o n o P+ + + + J2 1 − η0 a sech η0 (n − n0 ) a tanh η0 (n − n0 ) a wn (t) = − 2η0 I2 Q+ n o + × sech η0 (n − n0 ) a cos Ω1 t − φ+ 0 2η0+
(33a)
(33b)
exist, where Ω1 = 12 P+ (η0+ )2 − ω+ (q = 0), with η0+ being any real constant, and 0 ≤ φ+ 0 < 2π. Otherwise, when the condition for MI is not satisfied, i.e., P+ Q+ < 0, solutions vn (t) and wn (t) are 1/2 n o P+ tanh η0+ (n − n0 ) a cos Ω2 t − φ+ vn (t) = − 0 , Q+ 1/2 n o P+ + J2 wn (t) = − 2η0 − 1 + aη0+ sech2 η0+ (n − n0 ) a I2 Q+ n o + × tanh η0 (n − n0 ) a cos Ω1 t − φ+ 0 .
Jo
given by
2η0+
18
(34a)
(34b)
Journal Pre-proof
In Fig. 5, we have plotted solutions (33). The component vn is described by a single breather soliton, while wn displays features of double alternating breathers. Moreover, with increasing the
of
value of α, the temporal frequency of both solutions increases. We should also stress, from Fig. 6,
Pr e-
p ro
that the amplitude of vn is lower than that of wn .
Figure 7: The panels show the growth rate of MI versus the perturbation wavenumber K and the parameter α in the case of the lower cutoff mode, i.e., q = π/a. Different values of D are considered such that panel (a) is plotted for D = 3.2 eV, panel (b) for D = 4.932 eV and panel (c) for D = 5.4 eV, with the rest of parameters being fixed as M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV.
The optical lower cutoff mode
al
4.2
This case corresponds to q = π/a and we get ω+ (q = π/a) = √I4 . 2I2
2I2 , vg+ = 0, P+ =
2 √ a I2 J2 2 2I2 (J2 −I2 )
and
The MI features for the lower optical cutoff regime are summarized in Fig. 7, where the
urn
Q+ =
√
growth rate of MI is also plotted versus α and the perturbation wavenumber K. Obviously, MI is expected in this case only for small values of K, while the whole instability domain is very sensitive to the change in D. For D = 3.2 eV, the features of Fig. 7(a) show that very small values of α
Jo
do not support instability compared to its high values. Unstable intervals of α get reduced when D increases, as depicted in Figs. 7(b) and (c), with values 0.4932 eV and 5.4 eV. Consequently, the region of modulational stability gets expanded, giving more chance to the plane wave to keep its initial characteristics. However, the regions where Γ is positive strongly depend on the positive 19
Journal Pre-proof
sign of the product P+ Q+ which supports the general breather solutions
n
2
wn (t) = 2(−1) a
(η0+ )2
J2 2(I2 − J2 )
P+ Q+
P+ Q+
1/2
1/2
n o sech η0 (n − n0 ) a cos Ω2 t − φ+ 0 ,
(35a)
n o n o sech η0+ (n − n0 ) a tanh η0 (n − n0 ) a cos Ω1 t − φ+ 0 ,
of
vn (t) =
2(−1)n η0+
(35b)
p ro
where Ω2 = 21 P+ (η0+ )2 − ω+ (q = π/a). When the product P+ Q+ is negative, general solutions are given by
1/2 n o P+ vn (t) = − tanh η0 (n − n0 ) a cos Ω2 t − φ+ 0 , Q+ 1/2 n o J2 P+ n 2 + 2 wn (t) = 2(−1) a (η0 ) − sech2 η0+ (n − n0 ) a cos Ω1 t − φ+ 0 . 2(I2 − J2 ) Q+
Pr e-
2(−1)n η0+
(36a) (36b)
It is important to notice that the behavior of the product P+ Q+ depicted in Fig. 4(b) confirms the comportment of the growth rate of instability, and breather solitons are possible only for some values of α ≤ αcr . Solutions (35) expected from such regions of instability are depicted in Fig. 8, where vn shows single breathers and wn displays symmetrical double breathers. The amplitude of
al
vn is higher than that of wn and the temporal frequency of the obtained waves also increases with
4.3
urn
D.
The acoustic upper cutoff mode
In this case, as previously, q = π/a and vg− = 0, ω− =
√ 2J2 , P− =
a2 J2 √ 2 2J−2
and Q− =
J4 (I2 −J2 ) √ . 2I2 2J−2
In general, the growth rate of MI, that is plotted in Fig. 9, shows only one breast of instability.
Jo
Very small values of the perturbation wavenumber are at the onset of unstable waves and this gets more pronounced with increasing D. The later also causes the region of instability to delocalize from high values of α to its small values. This corresponds of course to the behavior displayed by Fig. 4(c), where the product P− Q− is plotted versus the Morse potential parameter α. When its 20
Pr e-
p ro
of
Journal Pre-proof
Figure 8: Solutions vn (t) and wn (t) for the lower optical cutoff frequency, ω+ (q = π/a), given by Eqs.(35). Panels (aj)j=1,2,3 correspond to vn and panels (bj)j=1,2,3 picture wn . The first column (a1)-(b1) is plotted for α = 1.2˚ A−1 , (a2)-(b2) for α = 1.25˚ A−1 and (a3)-(b3) for α = 1.4˚ A−1 . These solutions are related to the case P+ Q+ > 0 of Fig. 4(b), with the remaining parameters being M = 12 amu, D = 4 eV,v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV.
al
values are positive, the interplay between nonlinear and dispersive effects will give rise to solitonic
by n
vn (t) =2(−1)
urn
structures, or breather, whose some exact expressions related to the model studied here are given
J2 a(η0− )2 I2
P− Q−
1/2
2(−1)n η0−
Jo
wn (t) =
n o n o sech η0− (n − n0 ) a tanh η0 (n − n0 ) a cos Ω3 t − φ− 0 ,
P− Q−
(37a)
1/2
n o sech η0− (n − n0 ) a cos Ω3 t − φ− 0 ,
(37b)
where Ω3 = 21 P− (η0− )2 − ω− (q = π/a). On the other hand, for negative values of the product P− Q− , the above solutions become n
vn (t) =2(−1)
J2 a(η0− )2 I2
1/2 n o P− 2 − − sech η0 (n − n0 ) a cos Ω1 t − φ− 0 , Q− 21
(38a)
p ro
of
Journal Pre-proof
Figure 9: The panels show the growth rate of MI versus the perturbation wavenumber K and the parameter α in the case of the acoustical upper cutoff mode, i.e., q = π/a. Different values of D are considered such that panel (a) is plotted for D = 2.5 eV, panel (b) for D = 3.6 eV and panel (c) for D = 4.932 eV, with the rest of parameters being fixed as M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV.
1/2
Pr e-
wn (t) =
2(−1)n η0−
P− − Q−
n o tanh η0− (n − n0 ) a cos Ω3 t − φ− 0 .
(38b)
When the MI is satisfied, values of parameters collected from Fig. 4 lead to the solutions (37), depicted in Fig. 10. Here, the component vn is described by symmetrical double breather, while wn oscillates in the form of single breathers. However, one notices in this particular case that the
al
amplitude of solitonic structures in lower panels are higher than those of the double breather. In Fig. 10, columns from left to right correspond to different values of the Morse parameter α. Like
urn
in all the previous cases, the number of solitons, with respect to time, increases with α. The study of breathers, and other types of localized excitations, has been done extensively in CNTs. Kinoshita et al. [47] studied the formation of DBs in zigzag and armchair CNTs, and concluded that due to their structure, breathers could be excited in zigzag CNTs and not in armchair
Jo
ones. Moreover, if existent, breathers and DBs were found to be unstable, and it was suggested that an intensive strain could suppress any wave instability [11]. Most of the factors that may affect DBs formation in systems like CNTs may arise from the highly nonlinear character of the interatomic potentials which, combined to dispersive effects, may bring about the formation of 22
Pr e-
p ro
of
Journal Pre-proof
Figure 10: Solutions (37) corresponding to the upper acoustical cutoff frequency ωq=π/a in the case P− Q− > 0 shown in Fig. 4(c). Panels (aj)j=1,2,3 represent vn (t), while (bj)j=1,2,3 give wn (t), where (a1)-(b1) are plotted for α = 1.4˚ A−1 , (a2)-(b2) for α = 1.7889˚ A−1 and (a3)-(b3) for α = 1.9˚ A−1 , the rest of parameters being M = 12 amu, D = 2.5 eV, v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV. exotic wave patterns. Together with such effects, energy localization may take place, which is
al
the main essence and advantage of studying the formation of solitonic structures in CNTs. This
urn
was reported by Doi and Nakatani [48] who obtained long-alive vibrational modes via molecular dynamics simulations and showed that energy localization in CNTs is due to nonlinearity of atomic interactions. Besides, all the interaction considered here arise from the nonlinearity originating from the interaction between excited atoms and the breathers obtained from the various cutoff
Jo
modes seem to be inherent to the studied system while, as already observed by Shimada et al. [49], indubitably, stronger nonlinearity excites ILMs with both higher frequency and longer lifetime. As witnessed from Figs. 5, 8 and 10, the wave solutions obtained are very sensitive to the interatomic potential parameters and depending on their values, the energy they transport may be qualitatively
23
Journal Pre-proof
and quantitatively modified for specific nanotechnological purposes.
Conclusion
of
5
The present paper was devoted to the investigation of gap envelope solitons, as consequence of MI,
p ro
in single-walled CNTs. As proposed by Savin and Kivshar [9, 10], when the radius of the nanotube is large, the dynamics of the system becomes similar to that of a one-dimensional diatomic lattice. This case was considered here and, under the linear approximation, two frequency modes have been detected: the acoustical and the optical frequency modes. In order to specifically study each of the
Pr e-
modes, the QDA has been used, and amplitude equations have been found to be NLS equations, whose the generalized criterion for MI has been derived. Values for the wavenumber related to the upper and lower cutoff frequencies have been used, and the MI gain has been plotted for each of the cases. To start, the optical upper cutoff regime has been considered, and solutions for the corresponding NLS equation have been discussed relatively to the MI gain. In general, solutions have been found as single and double asymmetric breathers, which have shown to be very sensitive
al
to the change in the Morse potential width α. Calculations have been extended to the optical lower cutoff mode and the acoustical upper cutoff regime. Most of the results have converged to
urn
the fact that the interaction potential parameters highly affect the parametric expansion of the growth rate of instability which reduces with increasing the value of the valent coupling energy D. We should stress that the valent coupling energy and the Morse potential width α appear
Jo
in nonlinear and linear terms. Their values then contribute to balance nonlinear and dispersive effects, the straightforward consequence being the emergence of modulated waves. In the studied model, due to such effects, the transverse and longitudinal degrees of freedom have been found to support single breathers and symmetric double breathers in the case of the optical lower cutoff
24
Journal Pre-proof
mode, and the inverse in the case of the upper acoustical cutoff regime. Important dynamical consequences of discrete breather have been addressed in other configurations of CNTs such as
of
zigzag CNTs, where the nonlinearity has been found to be strong in the optical mode because of the in-plane vibrations. The discrete breathers specific to zigzag CNTs are strongly dependent on
p ro
their structure and need to be characterized through MI. This is to be addressed in another work. Moreover, as said so far, when the longitudinal vibrations are considered, the Brenner’s potential can be simplified for the motion to be studied in a one-dimensional context, ie., after neglecting the radial degree of freedom [42]. It would be interesting to explore the dynamics of such a model when
nonlinear and dispersive effects.
Acknowledgements
Pr e-
both longitudinal and radial degrees of freedom are involved, which could bring about interesting
The work by CBT is supported by the Botswana International University of Science and Technology under the grant DVC/RDI/2/1/16I (25). CBT thanks the Kavli Institute for Theoretical
al
Physics (KITP), University of California Santa Barbara (USA), for invitation.
urn
References
[1] R. H. Baughman, A. A. Zakhidov and W. A. de Heer WA., Science 297 (2002) 787. [2] P. Avouris, Z. Chen and V. Perebeinos, Nat. Nanotechnol. 2 (2007) 605.
Jo
[3] M. S. Dresselhaus, G. Dresselhaus and P. Avouris, Carbon nanotubes: synthesis, structure, properties and applications. Berlin: Springer-Verlag (2001). [4] R. Narula, N. Bonini, N. Marzari and S. Reich, Phys. Rev. B 85 (2012) 115451.
25
Journal Pre-proof
[5] A. L. Kitt, V. M. Pereira, A. K. Swan and B. B. Goldberg, Phys. Rev. B 85 (2012) 115432. [6] P. P. Azar, N. Nafari and M. R. R. Tabar, Phys. Rev. B 83 (2011) 165434.
p ro
[8] A. V. Savin and Y. S. Kivshar, EPL 89 (2010) 46001.
of
[7] J. A. Baimova, S. V. Dmitriev, K. Zhou and A. V. Savin, Phys. Rev. B 86 (2012) 035427.
[9] A. V. Savin and Y. S. Kivshar, EPL 82 (2008) 66002.
[10] A. V. Savin and Y. S. Kivshar, Fiz. Nizk. Temp. 34 (2008) 695.
Pr e-
[11] Y. Doi and A. Nakatani, Lett. Mater. 6 (2016) 49
[12] W. Chen and D. L. Mills, Phys. Rev. Lett. 58 (1987) 160. [13] G. Huang, S.-Y. Lou and M. G. Velarde, Int. J. Bifurc. Chaos 6 (1996) 1775. [14] S. A. Kiselev and A. J. Sievers, Phys. Rev. B 55 (1997) 5755.
al
[15] G. Huang and B. Hu, Phys. Rev. B 57 (1998) 5746.
urn
[16] A. De Rossi, C. Conti and S. Trillo, Phys. Rev. Lett. 81 (1998) 85. [17] T. Iizuka and Y. S. Kivshar, Phys. Rev. E 59 (1999) 7148. [18] C. B. Tabi, H. P. F. Ekobena, A Mohamadou and T. C. Kofan´e, Phys. Scr. 83 (2011) 035802.
Jo
[19] C. B. Tabi, E. Tankou and A. Mohamadou, Chaos Solit. Fract. 95 (2017) 187. [20] B. Z. Essimbi and I.V. Barashenkov, J. Phys. Soc. Japan 71 (2002) 448. [21] B. Z. Essimbi and I.V. Barashenkov, J. Phys. D: Appl. Phys. 35 (2002) 1438.
26
Journal Pre-proof
[22] B. Z. Essimbi and D. J¨ager, J. Phys. D: Appl. Phys. 39 (2006) 390. [23] A. Mohamadou, B. E. Ayissi, and T. C. Kofan´e, Phys. Rev. E 74 (2006) 046604.
of
[24] C. G. L. Tiofack, F. II Ndzana, A. Mohamadou, and T. C. Kofan´e, Phys. Rev. E 97 (2018)
p ro
032204.
[25] M.-R. Soumo Tchio, S. Abdoulkary and A. Mohamadou, Phys. Scr. 94 (2019) 035207 [26] S. A. El-Tantawy, N. A. El-Bedwehy and W. M. Moslem, J. Plasma Phys. 79 (2013) 1049.
Pr e-
[27] C. S. Panguetna, C. B. Tabi and T. C. Kofan´e, Commun. Nonlinear Sci. Numer. Simul. 55 (2018) 326.
[28] C. S. Panguetna, C. B. Tabi and T. C. Kofan´e, Phys Plasmas 24 (2017) 092114. [29] C. B. Tabi, C. S. Panguetna and T. C. Kofan´e, Physica B 545 (2018) 70.
al
[30] R. Y. Ondoua, C. B. Tabi, H. P. Ekobena, A. Mohamadou and T. C. Kofan´e, Eur. Phys. J. B 86 (2012) 374.
urn
[31] H. P. F. Ekobena, C. B. Tabi, A. Mohamadou and T. C. Kofan´e, J. Phys.: Condens. Matter 23 (2011) 375104.
[32] C. B. Tabi, J. C. F. Mimshe, H. P. F. Ekobena, A. Mohamadou and T. C. Kofan´e, Eur. Phys.
Jo
J. B 86 (2013) 374.
[33] J. C. F. Mimshe, C. B. Tabi, H. Edongue, H. P. F. Ekobena, A. Mohamadou and T. C. Kofan´e, Phys. Scr. 87 (2013) 025801. [34] S. E. Madiba, C. B. Tabi, H. P. F. Ekobena and T. C. Kofan´e, Physica A 514 (2019) 298. 27
Journal Pre-proof
[35] I. Ma¨ına, C. B. Tabi, A. Mohamadou, H. P. F. Ekobena and T. C. Kofan´e, Chaos 25 (2015) 043118.
of
[36] C. B. Tabi, I. Ma¨ına, A. Mohamadou, H. P. F. Ekobena and T. C. Kofan´e, Physica A 435 (2015) 1.
p ro
[37] A. S. Et´em´e, C. B. Tabi and A. Mohamadou, Commun. Nonl. Sci. Num. Simul. 43 (2017) 211. [38] C. B. Tabi, A. S. Et´em´e, A. Mohamadou and T. C. Kofan´e, Chaos Solitons and Fractals 123 (2019) 116.
Pr e-
[39] E. Wamba, A. Mohamadou, and T. C. Kofan´e, Phys. Rev. E 77 (2008) 046216. [40] R. Tamilthiruvalluvar, E. Wamba, S. Subramaniyan and K. Porsezian, Phys. Rev. E 99 (2019) 032202.
[41] I. A. Bhat, T. Mithun, B. A. Malomed, and K. Porsezian, Phys. Rev. A 92 (2015) 063606.
al
[42] E. Parasuraman, Wave Motion 83 (2018) 188.
urn
[43] V.V. Smirnov, L.I. Manevitch, M. Strozzi and F. Pellicano, Physica D 325 (2016) 113. [44] A.Tsurui, Prog. Theor. Phys. 48, 1190(1972) [45] R. Al-Jishi and G. Dresselhaus, Phys. Rev. B 26 (1982) 4514.
Jo
[46] T. Aizawa, R. Souda, S. Otani, Y. Ishizawa and C. Oshima, Phys. Rev. B 42 (1990) 11469. [47] Y. Kinoshita, Y. Yamayose, Y. Doi, A. Nakatani and T. Kitamura, Phys. Rev. E 77 (2008) 024307.
[48] Y. Doi and A. Nakatani, Procedia Eng. 10 (2011) 3393. 28
Journal Pre-proof
[49] T. Shimada, D Shirasaki, Y. Kinoshita, Y. Doi, A. Nakatani and T. Kitamura, Physica D 239
Jo
urn
al
Pr e-
p ro
of
(2010) 407.
29
Journal Pre-proof Highlights: • Nonlinear localized excitations are addressed in the framework of a one-dimensional diatomic carbon nanotube lattice. • Two frequency modes are studied through the multiple-scale approximation combined with the quasi-discrete approximation.
Jo
urn
al
Pr e-
p ro
of
• The modulational instability analysis is combined to the study of soliton solutions in the upper and lower optical cutoff modes, and in the upper acoustic cutoff regime.
Journal Pre-proof Conflicts of Interest Statement:
of
The authors of the manuscript “Modulational instability of gap solitons in single-walled carbon nanotube lattices”, submitted to Wave Motion, certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or nonfinancial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript. On behalf of all the co-authors:
Department of Physics & Astronomy
p ro
Conrad Bertrand TABI, PhD Botswana International University of Science and Technology Private Mail Bag 16 Palapye, Botswana
Jo
urn
al
Pr e-
Email: [email protected] or [email protected]
PII: DOI: Reference:
S0165-2125(19)30139-8 https://doi.org/10.1016/j.wavemoti.2020.102511 WAMOT 102511
To appear in:
Wave Motion
Received date : 22 April 2019 Revised date : 21 November 2019 Accepted date : 10 January 2020 Please cite this article as: B. Mozola, C.B. Tabi and T.C. Kofané, Modulational instability of gap solitons in single-walled carbon nanotube lattices, Wave Motion (2020), doi: https://doi.org/10.1016/j.wavemoti.2020.102511. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
Journal Pre-proof
of
Modulational instability of gap solitons in single-walled carbon nanotube lattices
1
p ro
Brantony Mozola1∗, Conrad Bertrand Tabi1†, and Timol´eon Cr´epin Kofan´e1,3‡ Department of Physics and Astronomy, Botswana International University of Science and Technology, Private Bag 16 Palapye, Botswana 3
Laboratoire de M´ecanique, D´epartement de Physique, Facult´e des Sciences, Universit´e de Yaound´e I,
Pr e-
B.P. 812 Yaound´e, Cameroun
al
January 13, 2020
Highlights
urn
• Nonlinear localized excitations are addressed in the framework of a one-dimensional diatomic carbon nanotube lattice.
• Two frequency modes are studied through the quasi-discrete approximation.
Jo
• The modulational instability analysis and soliton solutions are analyzed in the upper and lower optical frequency cutoff modes, and in the upper acoustic frequency cutoff regime. ∗
[email protected] (B. Mozola) Corresponding author: [email protected] (C. B. Tabi) ‡ [email protected] (T. C. Kofan´e) †
1
of
Journal Pre-proof
Abstract
p ro
Modulational instability and nonlinear localized excitations are addressed, in the framework of a one-dimensional diatomic carbon nanotube model, using the quasi-discrete approximation. Gap soliton solutions, based on the modulational instability criterion, are studied, where one considers
Pr e-
the solutions arising in the upper and lower optical frequency cutoff regimes, and in the upper acoustic frequency cutoff mode. Solutions are found as breathers and double breathers, and their response to interatomic interaction parameters is discussed. Vibrations of the CNTs from the two modes are compared based on their capability of carrying the amount of energy required for specific purposes, either in Microelectronics or in Nano-devices.
1
urn
al
Keywords: Carbon nanotubes; Modulational instability; Breathers; Gap solitons.
Introduction
Over the last two decades, carbon nanotubes (CNTs) have drawn considerable attention, and
Jo
significant amount of investigations has revealed their extraordinary, and sometimes unexpected mechanical, optical, chemical and structural properties [1, 2, 3]. One of the main research directions devoted to mechanical behaviours has also revealed that CNT properties fundamentally depend on structure, so that any uniform [4, 5] or nonuniform [6, 7] deformation of a CNT can considerably
2
Journal Pre-proof
affect its properties. Attention has been paid extensively to nonlinear excitations in graphene nanoribbons and their role in transporting and localizing energy has been reported [8]. In that
of
respect, CNTs, that are made by rolling up graphene sheet, may support nonlinear excitations and solitons. The existence of spatially localized nonlinear modes in the form of discrete breathers was
p ro
predicted in CNTs with chirality (m, 0) and (m, m) [9, 10]. Principally, three types of discrete breathers (DBs) were discussed, namely the longitudinal, radial, and twisting DBs. The structure and stability of stationary DBs in zigzag and armchair CNTs were studied by Doi and Nakatani [11], along with the involved structural physical processes and dynamical implications. By considering
Pr e-
a large radius of the CNT, a one-dimensional diatomic chain model was proposed by Savin and Kivshar [9, 10], and numerical studies were performed, using molecular dynamics simulations with a realistic interatomic potential, to confirm the intrinsic character of DBs in CNTs. The present work focuses on such a model.
Diatomic lattices are generalized versions of monoatomic chains, and their dispersion curve contains two frequency branches known as the acoustic branch and the optical branch. Between the
al
optical and the acoustic frequencies, there exist a gap where wave propagation of certain wavelengths is forbidden. More clearly, linear approximation is valid for small amplitude vibrations described
urn
as combinations of the linear modes within the phonon band. However, when vibration with large displacement takes place, nonlinearity of the interaction between atoms becomes dominant, leading to the so-called self-induced nonlinearity, and there appears some new effects in the system, mainly
Jo
the formation of gap solitons in the forbidden zone of the linear wave spectrum. Introduced by Chen and Mills and applied to nonlinear optical response of superlattices [12], the concept of gap solitons has gained interest in other physical contexts such as solid-state lattice vibrations [13, 14, 15], nonlinear optics [16, 17], biophysics [18, 19], electrical transmission lines [20, 21, 22], and so on.
3
Journal Pre-proof
The effect of cubic and quartic nonlinearities on gap solitons, in one-dimensional diatomic lattices, was addressed by Huang and Hu [13, 15] who further confirmed that such excitations can be
of
described by two coupled nonlinear Schr¨odinger (NLS) equations, each corresponding to a specific frequency mode. The nonlinear localized excitations corresponding to each of the modes have
p ro
been discussed in different physical situations [15, 18, 20], with instance on the different dynamical features of gap solitons. The model we consider in the present study, to the best of our knowledge, has been exploited only for numerical calculations until now, probably due to the complexity of the interatomic potentials.
Pr e-
One of the natural mechanisms leading to the formation of solitonic structures is modulational instability (MI). MI, which can be defined as an exponential growth of some modulation sidebands of a plane wave due to the balance between nonlinear and dispersive effects, appears in different physical settings, including nonlinear optics [23, 24, 25], plasma physics [26, 27, 28, 29], biological physics [30, 31, 32, 33, 34, 35, 37, 38], Bose-Einstein condensation [39, 40, 41], just to name a few. NLS breather solutions appear in general in regions of MI, i.e., where the product between
al
the dispersion and nonlinearity coefficients is positive, which mainly depends on the choice of right model parameters that constitute the nonlinear interatomic potentials. In that direction, an
urn
attempt was recently made to analyze the stability/instability of nonlinear waves in single walled CNTs under the effect of higher order interaction terms from the Brenner’s potential [42], which was found to strongly influence energy localization in the discrete context, through a discrete NLS
Jo
equation. Moreover, intense energy exchange between different parts of CNT and weak energy localization in the excited part of CNT were reported in the framework of the continuum shell theory, based on the NLS equations resulting from the multi-scale expansion procedure [43]. All the above clearly supports the idea that in one-dimensional lattices, weekly nonlinear excitations
4
Journal Pre-proof
with a large spatial extension, or envelope solitons, are governed by the NLS equation which is straightforwardly related to short wave wavepacket excitations [44].
of
In the rest of this paper, we first present the CNT model, via its Hamiltonian, in Section 2. In Section 3, the quasi-discrete approximation (QDA) is utilized to show that each of the modes,
p ro
optical and acoustic, is described by a NLS equation. Gap solitons are discussed in Section 4, and attention is given to breather solitons that are related to MI. The paper ends with some concluding remarks in Section 5.
Model
Pr e-
2
The model adopted in the present paper was introduced in Refs. [9, 10], where the CNT is considered to have a radius R. Considering a chirality (m, 0), the Hamiltonian for the vibration of the lattice of carbon atoms is written as H=
X X 1 n
l
2
M (u˙ 2n,l,0
+
u˙ 2n,l,1 )
+ Pn,l ,
(1)
al
where the three indices (n, l, k) define the structure of the CNT, with (n, l) defining an elementary cell (n = 0, ±1, ..., l = 1, 2, ..., m), and k = 0, 1 being the atom index in the cell. l is unlimited
urn
for a planar sheet of carbon atoms. M = 12 × 1.6603 · 10−27 kg ≈ 12 amu (atomic mass units) is the mass of a carbon atom. un,l,k = (xn,l,k (t), yn,l,k (t), zn,l,k (t)) is the radius-vector that gives the spatial coordinates of the carbon atom with index (n, l, k) at time t. Carbon atoms interact in the
Jo
lattice according to the potential energy Pn,l , given by Pn,l = P (u−1,l,1 , un,l−1,1 , un,l+1,0 ) = V(u1 , u2 ) + U(u1 , u2 , u3 ) + W(u1 , u2 , u3 , u4 ),
(2)
where V = V (un,l,0 , un,l,1 ) + V (un,l,1 , un+1,l,0 ) + V (un,l,1 , un,l+1,0 ) is the term describing interactions between pairs of atoms with the coordinates u1 and u2 . U = U (un−1,l,1 , un,l,0 , un,l−1,1 ) + 5
Journal Pre-proof
U (un−1,l,1 , un,l,0 , un,l,1 ) + U (un,l−1,1 , un,l,0 , un,l,1 ) + U (un,l,0 , un,l,1 , un,l+1,0 ) + U (un,l,0 , un,l,1 , un+1,l,1 ) + U (un,l+1,0 , un,l,1 , un+1,l,0 ) is the potential due to the deformation of the angles between the links u1 u2
of
and u2 u3 , and, finally, the term W = +W (un,l,1 , un,l,0 , un−1,l,1 , un,l−1,1 )+W (un,l,1 , un,l,0 , un,l−1,1 , un−1,l,1 )+ W (un−1,l,1 , un,l,0 , un,l,1 , un−1,l,1 ) + W (un,l,0 , un,l,1 , un,l+1,0 , un,l,0 ) + W (un,l,0 , un,l,1 , un+1,l,0 , un,l+1,0 ) +
p ro
W (un+1,l,0 , un,l,1 , un,l,0 , un,l+1,0 ) is the energy related to the variation of the effective angles between the planes u1 u2 u3 and u2 u3 u4 . More explicitly, the potentials U, V and W can be written as n
o2 V(u1 , u2 ) = D exp[−α(ρ − ρ0 )] − 1 ,
(3)
Pr e-
where ρ = |u2 − u1 |. D = 4.9632eV is the energy of the valent coupling, and ρ0 = 1.418˚ A is the static length;
U(u1 , u2 , u3 ) = v (cos ϕ − cos ϕ0 )2 ,
(4)
where cos ϕ = (u3 − u2 , u1 − u2 )(|u3 − u2 | · |u2 − u1 )−1 , and cos ϕ0 = cos(2π/3) = −1/2; W(u1 , u2 , u3 , u4 ) = t [1 − (v1 , v2 )(|v1 | · |v2 |)−1 ],
(5)
al
where v1 = (v2 − v1 ) × (v3 − v2 ) and v2 = (v3 − v2 ) × (v4 − v3 ). The rest of the model parameters α, v ,and t were determined from the phonon frequency spectrum of a plane of carbon atoms [45, 46]
urn
and were adopted in Refs. [9, 10]. A one-dimensional diatomic lattice of CNT can be obtained from the above-described generalized model if the radius is considered as R → ∞. In such a special case,
Jo
the Hamiltonian (1) reduces to its simplified form written as X 1 2 2 H= M (u˙ 2n−1 + u˙ 2n ) + V1 (ρ2n ) + V2 (ρ2n−1 ) , 2 n
(6)
where ρn = un+1 − un . The anharmonic interatomic potentials are given by V1 (w) =D(e−αw − 1)2 ,
" #2 2 n o2 a (w) 1 (w + ρ /2) 1 1/2 0 − V2 (w) =2D e−α[a+ (w) −ρ0 ] − 1 + 2v + + 4v p − , a+ (w) 2 2 a+ (w) 6
(7)
Journal Pre-proof
8
30 = 1.7889 Å-1
7
= 1.7889 Å-1
= 2.0 Å-1 = 2.5 Å
6
4 3
15 10
p ro
2
of
V2(w)
V1(w)
= 2.5 Å-1
20
5
5
1 0 -1
= 2.0 Å-1
25
-1
(a) 0
1
2
3
0 -5
4
w
(b) 0
5
w
Pr e-
Figure 1: The graphs above shows the potentials, V1 and V2 given by the expressions on Eq. (7) with D = 4.9632 eV and ρ0 = 1.418˚ A. (a) The well becomes narrower as α increases. (b) An asymmetric double well is obtained for α > 2. with a± (w) = (w + ρ0 /2)2 ± 3ρ20 /4. The above potentials are represented in Fig. 1, where panel (a) shows V1 (w) which is a Morse potential, with D being the energy of the valent coupling and α, the width of the the potential. Both D and α are constants determining the strength and the curvature
al
of the potential, respectively. V2 (w), represented in Fig. 1(b), is an asymmetric potential with a Morse plateau for w → +∞. When α changes, the width of the Morse potential gets reduced.
urn
However, the modified potential of Fig. 1(b) splits into an asymmetric double-well potential which is obviously related to the anharmonic interactions of the network and may bring about important nonlinear effects in the dynamics of the CNT model. We should however stress that some recent works have been done, where higher-order nonlinear effective potential for single walled carbon
Jo
nanotubes were obtained by using the Brenner’s potential [42]. In order to study discrete effects on longitudinal wave formation, the author used the rotating wave approximation and obtained a discrete NLS equation. However, this differs from our context because we consider both transverse and longitudinal dynamics, which, not only complicates the study of solitons in CNTs, but also
7
Journal Pre-proof
gives the room for new phenomena to be observed. This further motivates the use of the QDA, which is appropriate to qualitatively and quantitatively study the input of both nonlinear and
Amplitude equation
p ro
3
of
discrete effects in the model under consideration.
The Hamiltonian (6) contains complicated anharmonic interatomic potentials which may prevent direct exact solutions to be found. This further explains why most of the works devoted to the studied model have been essentially numerical. In order to proceed with the QDA, we expand the
Pr e-
potentials V1 and V2 of Eq.(7) in Taylor’s series up to the fourth order, i.e., 1 1 1 1 1 1 V1 (w) ' I2 w2 + I3 w3 + I4 w4 and V2 (w) ' J2 w2 + J3 w3 + J4 w4 , 2 3 4 2 3 4 2 (3−αρ ) 2 27v 6Dα3 14Dα4 1 0 2 , J3 = 3Dα2M + Dα where I2 = 2Dα , I = − , I = , J = − 3 4 2 M M M M ρ0 2ρ2 0
J4 =
Dα3 (7α2 ρ20 −54αρ0 −9) 4M ρ0
+
486v M ρ60
(8) 81v M ρ60
and
depend stronly on the model parameters, especially α, D and ρ0 .
The coefficients with indexes 2, 3 and 4 are respectively the harmonic, the cubic and quartic
al
interatomic constants. Moreover, together with the values of M , D and ρ0 , values for the rest of parameters were found from the phonon spectrum frequency of a planar lattice of carbon atoms as
urn
α = 1.7889˚ A−1 , v = 1.3143eV and t = 0.2 eV. The unit system that is used here, i.e., amu, ˚ A and eV, brings about a time unit (t.u.), with 1t.u.≈ 1.021 × 10−14 s. Thanks to Hamilton’s equations, the following equations for the longitudinal (u2n−1 → vn ) and transverse (u2n → wn ) motions are M v¨n = I2 (wn+1 − vn ) − J2 (vn − wn ) + I3 (wn+1 − vn )2 − J3 (vn − wn )2
Jo
obtained:
(9a)
− I4 (wn+1 − vn )3 − J4 (vn − wn )3 ,
M w¨n = −I2 (wn − vn−1 ) + J2 (vn − wn ) − I3 (wn − vn−1 )2 + J3 (vn − wn )2 + I4 (wn − vn−1 )3 + J4 (vn − wn )3 , 8
(9b)
Journal Pre-proof
12
14
16
(b)
(a)
(c) 14
12
10
12
10 8
10 8 8 6
of
6
6
4 4 2
4
2
+ -
-0.5
0
0.5
+
-
0 -1
1
-0.5
qa/
0
0.5
1
p ro
0 -1
2
+
0 -1
qa/
-0.5
-
0
0.5
1
qa/
Pr e-
Figure 2: The dispersion curve(ω vs q) for a diatomic lattice. Blue represents the optical modes and red for the acoustical modes. The parameters are M = 12 amu, D = 4.9632 eV,v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV. To each of the panels, corresponds a value of α so that (a) is plotted for α = 1.7889˚ A−1 , (b) for α = 2.2˚ A−1 and (c) for α = 2.5˚ A−1 . The linear dispersion relation can be obtained by considering linear terms in Eqs.(9a) and (9b). Using the plane wave solution ansatz for vn and wn we find s r 2 qa 2 , ω± (q) = (I2 + J2 ) ± (I2 + J2 ) − 4I2 J2 sin 2
(10)
where ω is the frequency, q is the wave number and a is the lattice spacing. The (−) sign corresponds
al
to the acoustic frequency and the (+) sign to the optical frequency as shown in Fig. 2. At wave
urn
number q = 0, the eigenfrequency spectrum has a lower cutoff ω− for acoustic branch and an p upper cutoff ω+ for the optical branch is w+ (q = 0) = 2(I2 + J2 ). When q = π/a, there exists √ a frequency gap between the upper cutoff of the acoustic branch, ω− (q = π/a) = 2J2 and the √ lower cutoff of the optical branch, ω+ (q = π/a) = 2I2 so that the width of the frequency gap √ √ is ∆ω = 2I2 − J2 . With increasing α, one notices that the gap between the two frequencies
Jo
also increases. In Fig. 2(a), where α = 1.7889˚ A−1 , we have a gap ∆ω = 0.452t.u.−1 , while for ˚−1 (see Fig. 2(b)), the gap is ∆ω = 0.878t.u.−1 . Additionally, for α = 2.5˚ α = 2.2A A−1 , the frequency gap in Fig. 2(c) is ∆ω = 1.91t.u.−1 . In the frequency gap, no linear wave can propagate, but in the nonlinear context, some waves may propagate and their oscillatory frequency may lie in 9
Journal Pre-proof
the forbidden band of the continuum wave spectrum. The effects of nonlinearity and dispersion can be investigated by assuming trial solutions to
2 (2) 3 (3) un (t) = u(1) n,n + un,n + un,n =
∞ X j=1
of
Eqs. (9) in the form (j)
j un,n±1 ,
(11)
p ro
where is a small but finite parameter denoting the relative amplitude of the excitations and (j)
un,n = u(j) (ξn , τ ; φn ), ξn = (na − λt) and τ = 2 t are two multiple-scale variables which are known as slow variables. λ is a parameter to be determined by a solvability condition. The phase of the carrier wave φn = qnl − ωt is the fast variable and is considered fully discrete. (j)
Pr e-
Introducing solutions (11) into Eqs. (9) and comparing expressions in similar powers in lead (j)
to a hierarchy of equations about vn,n and wn,n (j = 1, 2, 3, ...) as follows:
∂2 (j) (j) (j) (j) + 2I2 vn,n − I2 wn,n + I2 wn,n−1 = Mn,n , ∂t2
with
(1) Mn,n = 0,
al
h 2 i ∂ 2 (1) ∂ (1) (1) (1) (1) 2 (1) vn,n − I2 a wn,n−1 + I3 wn,n − vn,n − wn,n−1 − vn,n , ∂t∂ξn ∂ξn 2 ∂2 a2 ∂ 2 (1) ∂ 2 (2) ∂ (2) 2 ∂ (3) (1) v − 2 +λ w + w vn,n + I2 −a Mn,n = 2λ ∂t∂ξn n,n ∂t∂τ ∂ξn2 ∂ξn n,n−1 2 ∂ξn2 n,n−1 ∂ (1) (1) (2) (1) (2) − 2I3 wn,n−1 − vn,n wn,n−1 − vn,n − a w ∂ξn n,n−1 # " 3 3 (1) (1) (2) (1) (1) (1) (1) (2) + I4 wn,n − vn,n + wn,n−1 − vn,n , − vn,n wn,n − vn,n + 2I3 wn,n and
Jo
.... ..
urn
(2) Mn,n = 2λ
∂2 (j) (j) (j) (j) , + 2J2 wn,n − J2 vn,n + J2 vn,n+1 = Nn,n 2 ∂t 10
(12)
(13a) (13b)
(13c)
(14)
Journal Pre-proof
with
p ro
of
(1) Nn,n = 0, (15a) h 2 i ∂2 ∂ (1) (1) (2) (1) (1) (1) 2 (1) Nn,n = 2λ wn,n + J2 a vn,n+1 + J3 vn,n − wn,n − vn,n+1 − wn,n , (15b) ∂t∂ξn ∂ξn 2 ∂2 ∂ (2) a2 ∂ 2 (1) ∂2 (2) 2 ∂ (1) (3) w − 2 +λ wn,n + J2 a v + v Nn,n = 2λ ∂t∂ξn n,n ∂t∂τ ∂ξn2 ∂ξn n,n+1 2 ∂ξn2 n,n+1 (2) ∂ (1) (1) (2) (1) (1) (2) (1) (2) + 2J3 vn,n − wn,n vn,n − wn,n − 2J3 vn,n+1 − wn,n vn,n+1 − wn,n − a v ∂ξn n,n+1 (15c) 3 (1) (1) (1) 3 (1) + J4 vn,n+1 − wn,n − wn,n + J4 vn,n
.... ..
Pr e-
The rest of the calculation procedure amounts in collecting terms in different powers of the small parameter . We should however keep in mind that we have obtained, in the linear approximation, the optical and acoustical frequencies, which implies that both of the modes should be considered in the treatment of the problem related to Eqs.(12)-(15).
3.1
Acoustic Mode Excitations
al
For the acoustic mode, Eqs. (12) and (14)are rewritten as
urn
∂2 (j) (j) (j) (j) ˆ n,m Lw = J2 Mn,n + Mn,n+1 + + 2J Nn,n , 2 ∂t2 ∂2 (j) (j) (j) (j) + 2I v = I w + w 2 2 n,n−1 + Mn,n , n,n n,n ∂t2
(16a) (16b)
ˆ is defined by where the operator L ∂2
∂t2
+ 2I2
Jo
(j) ˆ n,m Lw =
∂ 2
∂t2
(j) (j) (j) (j) . + 2J2 wn,n − I2 J2 wn,n+1 + wn,n−1 + 2wn,n
(17)
Eqs. (16a) and (16b) correspond to their acoustic frequency expressions. The order j = 1 leads to linear equations, and solutions are found as −
−
(1) (1) wn,n = A− (ξn , τ )eiφn + c.c. and vn,n = BA− (ξn , τ )eiφn + c.c.,
11
(18)
Journal Pre-proof
where A− (ξn , τ ) is an envelope function to be determined later, φ− n = qna − ω− (q)t, and B = −iqa
) − I2 ω(1+e . The frequency ω− (q) given by (10) is related to the acoustic dynamical regime. 2 −2I 2 −
I2 J2 a sin(qa) 2 ), ω− (2I2 +2J2 −2ω−
of
− For the case j = 2, a solvability condition is obtained as λ = vg− = − ∂ω = ∂q
which is the group velocity, upon which solutions are obtained, after making the change of variable
p ro
ξn = ξn− = (na − vg− t), as −
(2) (2) = B0 wn,n = C1 A2− (ξn , τ )e2iφn + c.c. and vn,n
− − ∂ A− eiφn + B1 A2− e2iφn + c.c., − ∂ξn
(19)
J2 )2 − 4I2 J2 sin2 (qa)}1/2 .
Pr e-
2 −2I ) 2 −2I ) −(2iλω− I2 (1+e−iqa ))+I2 ae−iqa (ω− F1 J2 (1+e2iqa )−G1 (4ω− 2 2 C1 I2 (1+e−iqa )2 +F1 where B0 = , B = , with C = 2 2 2 2 2 2 [2q]) , 1 1 (ω− −2I2 )2 4ω− −2I2 (4ω− −ωq− [2q])(4ω− −ωq+ h i h i 2 F1 = I3 (1 − B)2 − (e−iqa − B)2 , G1 = J3 (B − 1)2 − (B eiqa − 1)2 , and 2ωq± = (I2 + J2 ) ± {(I2 +
At the third order j = 3, the solvability condition leads to the evolution equation for A− (ξn , τ ) in the form i
∂A− 1 ∂ ∂ + P− − − A− + Q− |A− |2 A− = 0, ∂τ 2 ∂ξn ∂ξn
al
with the coefficients P− and Q− being given by
(20)
Jo
urn
2 2 2 − 2J2 ) − 2I2 J2 − 2I2 )(ω− 2(2I2 + 2J2 − 4ω− )(Vg− )2 − a2 (ω− P− = , 2 2ω− (2I2 + 2J2 − 2ω− ) h i (21) 2 I3 (σ1 − σ2 ) + J3 (α1 − α2 ) − 3β1 + β2 Q− = , 2 2ω− (2I2 + 2J2 − 2ω− ) h i h i n h iqa 2 ∗ ∗ iqa −iqa iqa and α1 = B1 J2 (1 + e ) (1 − B ) , α2 = J2 B (1 + e ) + (1 + e ) , β1 = I4 J2 (1 + e ) 1 − 2 2 2 2 io nh i o 2 B (1−B)+ eiqa −B (e−iqa −B) , β2 = J4 B −1 (B −1)+ eiqa B −1 (eiqa B −1) (ω− −2I2 ) , h i h i σ1 = J2 B1 (1 + 2B ∗ )(1 − e2iqa ) , and σ2 = J2 C1 2 sin(qa) + B ∗ (e−2iqa − 1) .
12
Journal Pre-proof
3.2
Optical mode
Here, we consider the high-frequency optical mode excitations and Eqs. (16a) and (16b) can be rewritten as
(j) Nn,n+1
(22a) (22b)
p ro
(j) Nn,n
of
∂2 (j) = I2 + + + 2J2 Mn,n , ∂t2 2 ∂ (j) (j) (j) (j) + 2J2 wn,n = J2 vn,n + vn,n+1 + Nn,n , 2 ∂t
ˆ (j) Lv n,n
with j = 1, 2, 3.... Through a procedure similar to the one used for the acoustic mode, we solve the equations for the optical mode in a sequence. The following substitution can then be made: (j)
(j)
(j)
(j)
Pr e-
a → −a, I2 → J2 , I3 → J3 , I4 → J4 , wn,n → vn,n , wn,n±1 → vn,n±1 . Additionally, Eqs. (12)-(13) (j)
(j)
(j)
(j)
can be transformed into Eqs. (14)-(15) through the permutations Mn,n → Nn,n , Nn,n → Mn,n , and (j)
(j)
Mn,n+1 → Nn,n−1 . The same procedure is followed and we get solutions for different orders. Linear (1)
(1)
wave equations are also obtained from the first order j = 1, where Mn,n = Nn,n = 0. This leads to the solutions
+ (1) (1) e + (ξn , τ ) eiφ+n + c.c, vn,n = A+ (ξn , τ )eiφn + c.c. and wn,n = BA
al
(23)
J2 (1+eiqa ) e where φ+ n = qna − ω+ t, B = − ω 2 −2J2 and A+ (ξ, τ ) an envelope function to be determined. ω+ (q) +
urn
is the linear optical frequency of the dispersion relation (10). For j = 2, the solvability condition gives λ = vg+ =
∂ω+ ∂q
=
I2 J2 a sin(qa) 2) ω+ (2I2 +2J2 −2ω+
and ξn = ξn+ =
(na − vg+ t), leading to the solutions
e1 = where B
Jo
(2) (2) e1 A2+ (ξn , τ )e2iφ+n + c.c. and wn,n e0 ∂ A+ eiφ+n + C e1 A2− e2iφ+n + c.c., vn,n =B =C ∂ξn e 1 (4ω 2 −2J2 ) Fe1 I2 (1+e2iqa )−G + 2 −ω 2 [2q])(4ω 2 −ω 2 [2q]) (4ω+ q− + q+
e0 = ,C
Fe0 2 −2J ω+ 2
e1 = and C
e1 2J2 (1+eiqa )2 +Fe1 B . 2 −2J 4ω+ 2
(24)
(3)
For j = 3, we get the third-order approximation equations. The solvability condition for vn,n
13
Journal Pre-proof
(3)
and wn,n yields the NLS equation for A+ (ξn+ , τ ) i
∂A+ 1 ∂ ∂ + P+ + + A+ + Q+ |A+ |2 A+ = 0, ∂τ 2 ∂ξn ∂ξn
of
where
(25)
Pr e-
p ro
2 2 2 − 2J2 ) − 2I2 J2 − 2I2 )(ω+ )(Vg− )2 − a2 (ω+ 2(2I2 + 2J2 − 4ω+ , P+ = 2 ) 2ω+ (2I2 + 2J2 − 2ω+ h i (26) 2 J3 (e σ1 − σ e2 ) + I3 (e α1 − α e2 ) − 3βe1 + βe2 Q+ = , 2 ) 2ω+ (2I2 + 2J2 − 2ω+ h i h i n h iqa 2 ∗ ∗ iqa −iqa iqa e e e and α e1 = B1 I2 (1 + e ) (1 − B ) , α e2 = I2 B (1 + e ) + (1 + e ) , β1 = J4 I2 (1 + e ) 1 − 2 2 2 2 o nh i io e (1− B)+ e eiqa − B e (e−iqa −B) , βe2 = I4 B e −1 (B −1)+ eiqa B e −1 (eiqa B e −1) (ω 2 −2J2 ) , B − h i h i e ∗ (e−2iqa − 1) . From these equations, e ∗ )(1 − e2iqa ) , and σ e1 (1 + 2B e2 = I2 C1 2 sin[qa] + B σ e1 = I2 B the envelope function A− (ξn− , τ ) and A+ (ξn+ , τ ) evolve according to the NLS equation in a unified form i
∂A± 1 ∂ ∂ + P± ± ± A± + Q± |A± |2 A± = 0 ∂τ 2 ∂ξn ∂ξn
(27)
al
whose coefficients P± and Q± depend on the CNT model parameters. In order to return to the
have
urn
± ± 2 original variables we let A± (ξn± , τ ) = (1/)F± (x± n , t) and since (na − Vg t) = xn and τ = t we
∂F± ∂ 2 F± i + P± ±2 + Q± |F± |2 F± = 0. ∂t ∂xn
(28)
4
Jo
Solutions for Eqs.(28) will be discussed in the next sections.
Nonlinear localized modes and gap solitons in CNTs
In general, solutions whose amplitudes are governed by Eq.(28) depends on the sign of the product P± × Q± . To this effect, let us assume plane solutions for Eq.(28) to be of the form F± (t, x± n) = 14
Journal Pre-proof
±
2
iQ± F0 t i(Kxn −Ω± t) , with K F0 (x± , and subjected to a small perturbation δF± (t, x± n )e n ) = (U0 + iV0 )e
and Ω± being, respectively, the wavenumber and frequency of the perturbation. The use of the 2 Kcr = (K P± ) 1 − 2 , K 2
2
q
2Q± . P±
p ro
Ω2±
of
standard calculations of MI leads to the nonlinear dispersion relation
with the critical wavenumber of the perturbation being Kcr = F0
(29)
For the plane wave to be
Pr e-
unstable under modulation Eq.(29) must satisfy the condition Ω2± < 0, which leads to the condition s 2Q± K < Kcr = F0 . (30) P± This clearly shows that for P± Q± > 0, the amplitude-modulated envelope is unstable. Therefore, depending on the sign of the product P± × Q± , there are two kinds of solutions which are the pulse soliton solution and the kink soliton solution. The pulse soliton solution is generally obtained for P± Q± > 0. The corresponding generalized expression for such a soliton is given by =
P± Q±
1/2
n o 1 ±2 ± ± ± 0 ± |P± |η0 t − φ0 η0 xn − xn − vg t exp i , 2
η0± sech
(31)
al
F± (x± n , t)
the general form F± (x± n , t)
=
urn
where η0± is a free constant. On the other hand, when P± Q± < 0, the dark soliton is obtained in
P± − Q±
1/2
n o 1 ±2 ± ± ± 0 ± |P± |η0 t − φ0 η0 xn − xn − vg t exp i . 2
η0± tanh
(32)
Jo
In what follows, regions of instability are discussed along with the subsequent analytical solutions for specific values of the wavenumber q falling into the forbidden/gap zone of the frequency spectrum. √ Usually, the instability is characterized by the MI growth rate Γ = −Ω2 , which is plotted for each of the studied cases.
15
p ro
of
Journal Pre-proof
0.5
Pr e-
Figure 3: The panels show the growth rate of MI versus the perturbation wavenumber K and the parameter α in the case of the upper cutoff mode, i.e., q = 0. Different values of D are considered such that panel (a) is plotted for D = 2.5 eV, panel (b) for D = 3.6 eV and panel (c) for D = 4.9 eV, with the rest of parameters being fixed as M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV. 500
0
1
D = 4 eV D = 4.932 eV D = 5.4 eV
400
0.5
300 200
-0.5
D = 2.5 eV D = 3.6 eV D = 4.9632 eV
0
Q-
0
-0.5
P-
Q+ P+
-1
P+
Q+
100
-100 -1.5
-300 D = 2.5 eV D = 3.6 eV D = 4.9632 eV
0
0.5
(a) 1
1.5
2
-500
(b)
1
1.5
urn
(Å-1 )
-1.5
-400
al
-2
-2.5
-1
-200
(Å-1 )
2
2.5
-2 0.6
(c) 0.8
1
1.2
1.4
1.6
1.8
(Å-1 )
Jo
Figure 4: Panel (a) shows the product P+ Q+ for the upper cutoff frequency, i.e., q = 0, versus the Morse parameter α, for different values of the parameter D. Panel (b) also represents P+ Q+ , but for the lower cutoff frequency, i.e., q = π/a. Panel (b) depicts the behaviors of the product P− Q− for the upper acoustical frequency, i.e., ω− (q = π/a), versus the parameter α and different values of D. The rest of the parameters are such that M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV.
16
2
Journal Pre-proof
4.1
The optical upper cutoff mode
This mode corresponds to the wavenumber value q = 0 (see Fig. 2), which leads to vg+ = 0, ω+ (q = J √ I3 (I2 −J2 )+I2 (3J4 +I4 ) 1+ I 2 a2√ I2 J2 + 2 √ 0) = 2I2 + 2J2 , xn = na = xn , P+ = and Q+ = . The 2(I2 +J2 )
(I2 +J2 )
of
4(I2 +J2 )
2(I2 +J2 )
instability growth rate related to this case is shown in Fig. 3 versus the perturbation wavenumber
p ro
K and the Morse parameter α. For small values of the valent coupling energy D, the instability region is large as shown in Fig. 3(a). For some high values of K, there are regions of α where the plane wave solution is stable. For high values of α, i.e., α = 1.5˚ A−1 , restricted values of K are expected to give rise to instability. When D increases to 3.6 eV (see Fig. 3(b)), the stability
Jo
urn
al
Pr e-
region broadens and this persists when D = 4.9632 eV. If the parameters fall into the detected
Figure 5: Space-time evolution of the upper optical cutoff breather (33a) and (33b). Panels (a1) and (b1) have been recorded for α = 1.25˚ A. Panels (a2)-(b2) have been obtained for α = 1.3˚ A, and ˚ panels (a3)-(b3) have been obtained for α = 1.35A, with D = 2.5eV, M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ A, and t = 0.2 eV. the presented solutions correspond to positive values of product P+ Q+ picked from Fig. 4. 17
Journal Pre-proof
4
8
2
v n (t)
2
v n (t)
w n (t)
0
v n (t)
w n (t)
6
w n (t)
1
4
0
2
-1
0
-2
-2
-6
-2
-8
-3
(a)
-10 -5
0
of
-4
(b)
-4 -5
5
0
n
5
(c) 0
5
n
p ro
n
-4 -5
Figure 6: Solutions of Fig. 5 at different instant: (a) t = 0, (b) t = 0.5 and (c) t = 0.7 with D = 2.5 eV, α = 1.25˚ A, M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ A, and t = 0.2 eV. zones of instability, the plane wave will be said to be unstable under modulation. Otherwise, the
Pr e-
plane wave solution will remain stable under modulation. In other words, the product P+ Q+ plays a fundamental role in finding solutions for the model under our study. Therefore, for this specific case, P+ Q+ is plotted in Fig. 4(a) and shows positive regions whose intervals depend on α. As in Fig. 4, increasing the value of D reduces regions of α where bright solitons are expected and the generalized solutions
urn
al
1/2 n o P+ vn (t) = sech η0+ (n − n0 ) a cos Ω1 t − φ+ 0 , Q+ 1/2 n o n o P+ + + + J2 1 − η0 a sech η0 (n − n0 ) a tanh η0 (n − n0 ) a wn (t) = − 2η0 I2 Q+ n o + × sech η0 (n − n0 ) a cos Ω1 t − φ+ 0 2η0+
(33a)
(33b)
exist, where Ω1 = 12 P+ (η0+ )2 − ω+ (q = 0), with η0+ being any real constant, and 0 ≤ φ+ 0 < 2π. Otherwise, when the condition for MI is not satisfied, i.e., P+ Q+ < 0, solutions vn (t) and wn (t) are 1/2 n o P+ tanh η0+ (n − n0 ) a cos Ω2 t − φ+ vn (t) = − 0 , Q+ 1/2 n o P+ + J2 wn (t) = − 2η0 − 1 + aη0+ sech2 η0+ (n − n0 ) a I2 Q+ n o + × tanh η0 (n − n0 ) a cos Ω1 t − φ+ 0 .
Jo
given by
2η0+
18
(34a)
(34b)
Journal Pre-proof
In Fig. 5, we have plotted solutions (33). The component vn is described by a single breather soliton, while wn displays features of double alternating breathers. Moreover, with increasing the
of
value of α, the temporal frequency of both solutions increases. We should also stress, from Fig. 6,
Pr e-
p ro
that the amplitude of vn is lower than that of wn .
Figure 7: The panels show the growth rate of MI versus the perturbation wavenumber K and the parameter α in the case of the lower cutoff mode, i.e., q = π/a. Different values of D are considered such that panel (a) is plotted for D = 3.2 eV, panel (b) for D = 4.932 eV and panel (c) for D = 5.4 eV, with the rest of parameters being fixed as M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV.
The optical lower cutoff mode
al
4.2
This case corresponds to q = π/a and we get ω+ (q = π/a) = √I4 . 2I2
2I2 , vg+ = 0, P+ =
2 √ a I2 J2 2 2I2 (J2 −I2 )
and
The MI features for the lower optical cutoff regime are summarized in Fig. 7, where the
urn
Q+ =
√
growth rate of MI is also plotted versus α and the perturbation wavenumber K. Obviously, MI is expected in this case only for small values of K, while the whole instability domain is very sensitive to the change in D. For D = 3.2 eV, the features of Fig. 7(a) show that very small values of α
Jo
do not support instability compared to its high values. Unstable intervals of α get reduced when D increases, as depicted in Figs. 7(b) and (c), with values 0.4932 eV and 5.4 eV. Consequently, the region of modulational stability gets expanded, giving more chance to the plane wave to keep its initial characteristics. However, the regions where Γ is positive strongly depend on the positive 19
Journal Pre-proof
sign of the product P+ Q+ which supports the general breather solutions
n
2
wn (t) = 2(−1) a
(η0+ )2
J2 2(I2 − J2 )
P+ Q+
P+ Q+
1/2
1/2
n o sech η0 (n − n0 ) a cos Ω2 t − φ+ 0 ,
(35a)
n o n o sech η0+ (n − n0 ) a tanh η0 (n − n0 ) a cos Ω1 t − φ+ 0 ,
of
vn (t) =
2(−1)n η0+
(35b)
p ro
where Ω2 = 21 P+ (η0+ )2 − ω+ (q = π/a). When the product P+ Q+ is negative, general solutions are given by
1/2 n o P+ vn (t) = − tanh η0 (n − n0 ) a cos Ω2 t − φ+ 0 , Q+ 1/2 n o J2 P+ n 2 + 2 wn (t) = 2(−1) a (η0 ) − sech2 η0+ (n − n0 ) a cos Ω1 t − φ+ 0 . 2(I2 − J2 ) Q+
Pr e-
2(−1)n η0+
(36a) (36b)
It is important to notice that the behavior of the product P+ Q+ depicted in Fig. 4(b) confirms the comportment of the growth rate of instability, and breather solitons are possible only for some values of α ≤ αcr . Solutions (35) expected from such regions of instability are depicted in Fig. 8, where vn shows single breathers and wn displays symmetrical double breathers. The amplitude of
al
vn is higher than that of wn and the temporal frequency of the obtained waves also increases with
4.3
urn
D.
The acoustic upper cutoff mode
In this case, as previously, q = π/a and vg− = 0, ω− =
√ 2J2 , P− =
a2 J2 √ 2 2J−2
and Q− =
J4 (I2 −J2 ) √ . 2I2 2J−2
In general, the growth rate of MI, that is plotted in Fig. 9, shows only one breast of instability.
Jo
Very small values of the perturbation wavenumber are at the onset of unstable waves and this gets more pronounced with increasing D. The later also causes the region of instability to delocalize from high values of α to its small values. This corresponds of course to the behavior displayed by Fig. 4(c), where the product P− Q− is plotted versus the Morse potential parameter α. When its 20
Pr e-
p ro
of
Journal Pre-proof
Figure 8: Solutions vn (t) and wn (t) for the lower optical cutoff frequency, ω+ (q = π/a), given by Eqs.(35). Panels (aj)j=1,2,3 correspond to vn and panels (bj)j=1,2,3 picture wn . The first column (a1)-(b1) is plotted for α = 1.2˚ A−1 , (a2)-(b2) for α = 1.25˚ A−1 and (a3)-(b3) for α = 1.4˚ A−1 . These solutions are related to the case P+ Q+ > 0 of Fig. 4(b), with the remaining parameters being M = 12 amu, D = 4 eV,v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV.
al
values are positive, the interplay between nonlinear and dispersive effects will give rise to solitonic
by n
vn (t) =2(−1)
urn
structures, or breather, whose some exact expressions related to the model studied here are given
J2 a(η0− )2 I2
P− Q−
1/2
2(−1)n η0−
Jo
wn (t) =
n o n o sech η0− (n − n0 ) a tanh η0 (n − n0 ) a cos Ω3 t − φ− 0 ,
P− Q−
(37a)
1/2
n o sech η0− (n − n0 ) a cos Ω3 t − φ− 0 ,
(37b)
where Ω3 = 21 P− (η0− )2 − ω− (q = π/a). On the other hand, for negative values of the product P− Q− , the above solutions become n
vn (t) =2(−1)
J2 a(η0− )2 I2
1/2 n o P− 2 − − sech η0 (n − n0 ) a cos Ω1 t − φ− 0 , Q− 21
(38a)
p ro
of
Journal Pre-proof
Figure 9: The panels show the growth rate of MI versus the perturbation wavenumber K and the parameter α in the case of the acoustical upper cutoff mode, i.e., q = π/a. Different values of D are considered such that panel (a) is plotted for D = 2.5 eV, panel (b) for D = 3.6 eV and panel (c) for D = 4.932 eV, with the rest of parameters being fixed as M = 12 amu, v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV.
1/2
Pr e-
wn (t) =
2(−1)n η0−
P− − Q−
n o tanh η0− (n − n0 ) a cos Ω3 t − φ− 0 .
(38b)
When the MI is satisfied, values of parameters collected from Fig. 4 lead to the solutions (37), depicted in Fig. 10. Here, the component vn is described by symmetrical double breather, while wn oscillates in the form of single breathers. However, one notices in this particular case that the
al
amplitude of solitonic structures in lower panels are higher than those of the double breather. In Fig. 10, columns from left to right correspond to different values of the Morse parameter α. Like
urn
in all the previous cases, the number of solitons, with respect to time, increases with α. The study of breathers, and other types of localized excitations, has been done extensively in CNTs. Kinoshita et al. [47] studied the formation of DBs in zigzag and armchair CNTs, and concluded that due to their structure, breathers could be excited in zigzag CNTs and not in armchair
Jo
ones. Moreover, if existent, breathers and DBs were found to be unstable, and it was suggested that an intensive strain could suppress any wave instability [11]. Most of the factors that may affect DBs formation in systems like CNTs may arise from the highly nonlinear character of the interatomic potentials which, combined to dispersive effects, may bring about the formation of 22
Pr e-
p ro
of
Journal Pre-proof
Figure 10: Solutions (37) corresponding to the upper acoustical cutoff frequency ωq=π/a in the case P− Q− > 0 shown in Fig. 4(c). Panels (aj)j=1,2,3 represent vn (t), while (bj)j=1,2,3 give wn (t), where (a1)-(b1) are plotted for α = 1.4˚ A−1 , (a2)-(b2) for α = 1.7889˚ A−1 and (a3)-(b3) for α = 1.9˚ A−1 , the rest of parameters being M = 12 amu, D = 2.5 eV, v = 1.3143eV, ρ0 = 1.3143˚ Aand t = 0.2 eV. exotic wave patterns. Together with such effects, energy localization may take place, which is
al
the main essence and advantage of studying the formation of solitonic structures in CNTs. This
urn
was reported by Doi and Nakatani [48] who obtained long-alive vibrational modes via molecular dynamics simulations and showed that energy localization in CNTs is due to nonlinearity of atomic interactions. Besides, all the interaction considered here arise from the nonlinearity originating from the interaction between excited atoms and the breathers obtained from the various cutoff
Jo
modes seem to be inherent to the studied system while, as already observed by Shimada et al. [49], indubitably, stronger nonlinearity excites ILMs with both higher frequency and longer lifetime. As witnessed from Figs. 5, 8 and 10, the wave solutions obtained are very sensitive to the interatomic potential parameters and depending on their values, the energy they transport may be qualitatively
23
Journal Pre-proof
and quantitatively modified for specific nanotechnological purposes.
Conclusion
of
5
The present paper was devoted to the investigation of gap envelope solitons, as consequence of MI,
p ro
in single-walled CNTs. As proposed by Savin and Kivshar [9, 10], when the radius of the nanotube is large, the dynamics of the system becomes similar to that of a one-dimensional diatomic lattice. This case was considered here and, under the linear approximation, two frequency modes have been detected: the acoustical and the optical frequency modes. In order to specifically study each of the
Pr e-
modes, the QDA has been used, and amplitude equations have been found to be NLS equations, whose the generalized criterion for MI has been derived. Values for the wavenumber related to the upper and lower cutoff frequencies have been used, and the MI gain has been plotted for each of the cases. To start, the optical upper cutoff regime has been considered, and solutions for the corresponding NLS equation have been discussed relatively to the MI gain. In general, solutions have been found as single and double asymmetric breathers, which have shown to be very sensitive
al
to the change in the Morse potential width α. Calculations have been extended to the optical lower cutoff mode and the acoustical upper cutoff regime. Most of the results have converged to
urn
the fact that the interaction potential parameters highly affect the parametric expansion of the growth rate of instability which reduces with increasing the value of the valent coupling energy D. We should stress that the valent coupling energy and the Morse potential width α appear
Jo
in nonlinear and linear terms. Their values then contribute to balance nonlinear and dispersive effects, the straightforward consequence being the emergence of modulated waves. In the studied model, due to such effects, the transverse and longitudinal degrees of freedom have been found to support single breathers and symmetric double breathers in the case of the optical lower cutoff
24
Journal Pre-proof
mode, and the inverse in the case of the upper acoustical cutoff regime. Important dynamical consequences of discrete breather have been addressed in other configurations of CNTs such as
of
zigzag CNTs, where the nonlinearity has been found to be strong in the optical mode because of the in-plane vibrations. The discrete breathers specific to zigzag CNTs are strongly dependent on
p ro
their structure and need to be characterized through MI. This is to be addressed in another work. Moreover, as said so far, when the longitudinal vibrations are considered, the Brenner’s potential can be simplified for the motion to be studied in a one-dimensional context, ie., after neglecting the radial degree of freedom [42]. It would be interesting to explore the dynamics of such a model when
nonlinear and dispersive effects.
Acknowledgements
Pr e-
both longitudinal and radial degrees of freedom are involved, which could bring about interesting
The work by CBT is supported by the Botswana International University of Science and Technology under the grant DVC/RDI/2/1/16I (25). CBT thanks the Kavli Institute for Theoretical
al
Physics (KITP), University of California Santa Barbara (USA), for invitation.
urn
References
[1] R. H. Baughman, A. A. Zakhidov and W. A. de Heer WA., Science 297 (2002) 787. [2] P. Avouris, Z. Chen and V. Perebeinos, Nat. Nanotechnol. 2 (2007) 605.
Jo
[3] M. S. Dresselhaus, G. Dresselhaus and P. Avouris, Carbon nanotubes: synthesis, structure, properties and applications. Berlin: Springer-Verlag (2001). [4] R. Narula, N. Bonini, N. Marzari and S. Reich, Phys. Rev. B 85 (2012) 115451.
25
Journal Pre-proof
[5] A. L. Kitt, V. M. Pereira, A. K. Swan and B. B. Goldberg, Phys. Rev. B 85 (2012) 115432. [6] P. P. Azar, N. Nafari and M. R. R. Tabar, Phys. Rev. B 83 (2011) 165434.
p ro
[8] A. V. Savin and Y. S. Kivshar, EPL 89 (2010) 46001.
of
[7] J. A. Baimova, S. V. Dmitriev, K. Zhou and A. V. Savin, Phys. Rev. B 86 (2012) 035427.
[9] A. V. Savin and Y. S. Kivshar, EPL 82 (2008) 66002.
[10] A. V. Savin and Y. S. Kivshar, Fiz. Nizk. Temp. 34 (2008) 695.
Pr e-
[11] Y. Doi and A. Nakatani, Lett. Mater. 6 (2016) 49
[12] W. Chen and D. L. Mills, Phys. Rev. Lett. 58 (1987) 160. [13] G. Huang, S.-Y. Lou and M. G. Velarde, Int. J. Bifurc. Chaos 6 (1996) 1775. [14] S. A. Kiselev and A. J. Sievers, Phys. Rev. B 55 (1997) 5755.
al
[15] G. Huang and B. Hu, Phys. Rev. B 57 (1998) 5746.
urn
[16] A. De Rossi, C. Conti and S. Trillo, Phys. Rev. Lett. 81 (1998) 85. [17] T. Iizuka and Y. S. Kivshar, Phys. Rev. E 59 (1999) 7148. [18] C. B. Tabi, H. P. F. Ekobena, A Mohamadou and T. C. Kofan´e, Phys. Scr. 83 (2011) 035802.
Jo
[19] C. B. Tabi, E. Tankou and A. Mohamadou, Chaos Solit. Fract. 95 (2017) 187. [20] B. Z. Essimbi and I.V. Barashenkov, J. Phys. Soc. Japan 71 (2002) 448. [21] B. Z. Essimbi and I.V. Barashenkov, J. Phys. D: Appl. Phys. 35 (2002) 1438.
26
Journal Pre-proof
[22] B. Z. Essimbi and D. J¨ager, J. Phys. D: Appl. Phys. 39 (2006) 390. [23] A. Mohamadou, B. E. Ayissi, and T. C. Kofan´e, Phys. Rev. E 74 (2006) 046604.
of
[24] C. G. L. Tiofack, F. II Ndzana, A. Mohamadou, and T. C. Kofan´e, Phys. Rev. E 97 (2018)
p ro
032204.
[25] M.-R. Soumo Tchio, S. Abdoulkary and A. Mohamadou, Phys. Scr. 94 (2019) 035207 [26] S. A. El-Tantawy, N. A. El-Bedwehy and W. M. Moslem, J. Plasma Phys. 79 (2013) 1049.
Pr e-
[27] C. S. Panguetna, C. B. Tabi and T. C. Kofan´e, Commun. Nonlinear Sci. Numer. Simul. 55 (2018) 326.
[28] C. S. Panguetna, C. B. Tabi and T. C. Kofan´e, Phys Plasmas 24 (2017) 092114. [29] C. B. Tabi, C. S. Panguetna and T. C. Kofan´e, Physica B 545 (2018) 70.
al
[30] R. Y. Ondoua, C. B. Tabi, H. P. Ekobena, A. Mohamadou and T. C. Kofan´e, Eur. Phys. J. B 86 (2012) 374.
urn
[31] H. P. F. Ekobena, C. B. Tabi, A. Mohamadou and T. C. Kofan´e, J. Phys.: Condens. Matter 23 (2011) 375104.
[32] C. B. Tabi, J. C. F. Mimshe, H. P. F. Ekobena, A. Mohamadou and T. C. Kofan´e, Eur. Phys.
Jo
J. B 86 (2013) 374.
[33] J. C. F. Mimshe, C. B. Tabi, H. Edongue, H. P. F. Ekobena, A. Mohamadou and T. C. Kofan´e, Phys. Scr. 87 (2013) 025801. [34] S. E. Madiba, C. B. Tabi, H. P. F. Ekobena and T. C. Kofan´e, Physica A 514 (2019) 298. 27
Journal Pre-proof
[35] I. Ma¨ına, C. B. Tabi, A. Mohamadou, H. P. F. Ekobena and T. C. Kofan´e, Chaos 25 (2015) 043118.
of
[36] C. B. Tabi, I. Ma¨ına, A. Mohamadou, H. P. F. Ekobena and T. C. Kofan´e, Physica A 435 (2015) 1.
p ro
[37] A. S. Et´em´e, C. B. Tabi and A. Mohamadou, Commun. Nonl. Sci. Num. Simul. 43 (2017) 211. [38] C. B. Tabi, A. S. Et´em´e, A. Mohamadou and T. C. Kofan´e, Chaos Solitons and Fractals 123 (2019) 116.
Pr e-
[39] E. Wamba, A. Mohamadou, and T. C. Kofan´e, Phys. Rev. E 77 (2008) 046216. [40] R. Tamilthiruvalluvar, E. Wamba, S. Subramaniyan and K. Porsezian, Phys. Rev. E 99 (2019) 032202.
[41] I. A. Bhat, T. Mithun, B. A. Malomed, and K. Porsezian, Phys. Rev. A 92 (2015) 063606.
al
[42] E. Parasuraman, Wave Motion 83 (2018) 188.
urn
[43] V.V. Smirnov, L.I. Manevitch, M. Strozzi and F. Pellicano, Physica D 325 (2016) 113. [44] A.Tsurui, Prog. Theor. Phys. 48, 1190(1972) [45] R. Al-Jishi and G. Dresselhaus, Phys. Rev. B 26 (1982) 4514.
Jo
[46] T. Aizawa, R. Souda, S. Otani, Y. Ishizawa and C. Oshima, Phys. Rev. B 42 (1990) 11469. [47] Y. Kinoshita, Y. Yamayose, Y. Doi, A. Nakatani and T. Kitamura, Phys. Rev. E 77 (2008) 024307.
[48] Y. Doi and A. Nakatani, Procedia Eng. 10 (2011) 3393. 28
Journal Pre-proof
[49] T. Shimada, D Shirasaki, Y. Kinoshita, Y. Doi, A. Nakatani and T. Kitamura, Physica D 239
Jo
urn
al
Pr e-
p ro
of
(2010) 407.
29
Journal Pre-proof Highlights: • Nonlinear localized excitations are addressed in the framework of a one-dimensional diatomic carbon nanotube lattice. • Two frequency modes are studied through the multiple-scale approximation combined with the quasi-discrete approximation.
Jo
urn
al
Pr e-
p ro
of
• The modulational instability analysis is combined to the study of soliton solutions in the upper and lower optical cutoff modes, and in the upper acoustic cutoff regime.
Journal Pre-proof Conflicts of Interest Statement:
of
The authors of the manuscript “Modulational instability of gap solitons in single-walled carbon nanotube lattices”, submitted to Wave Motion, certify that they have NO affiliations with or involvement in any organization or entity with any financial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patent-licensing arrangements), or nonfinancial interest (such as personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript. On behalf of all the co-authors:
Department of Physics & Astronomy
p ro
Conrad Bertrand TABI, PhD Botswana International University of Science and Technology Private Mail Bag 16 Palapye, Botswana
Jo
urn
al
Pr e-
Email: [email protected] or [email protected]