We have used multiple-time-scales perturbation theory as well as the numerical methods of molecular dynamics and spectral energy density (SED) to investigate the phonon band structure of a two-chain model with second-order anharmonic interactions. We show that when one chain is linear and the other is nonlinear, the two-chain model exhibits a nonlinear resonance near a critical wave number due to mode self-interaction. The nonlinear resonance enables wave number-dependent interband energy transfer. We have also shown that there exist nonlinear modes within the spectral gap separating the lower and upper branches of the phonon band structure. These modes result from three phonon interactions between a phonon belonging to the nonlinear branch and two phonons lying on the lower branch. This phenomenon offers a mechanism for phonon splitting.

References

1.
Vakakis
,
A. F.
,
Gendelman
,
O. V.
,
Bergman
,
L. A.
,
McFarland
,
D. M.
,
Kerschen
,
G.
, and
Lee
,
Y. S.
,
2009
,
Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems
(Solid Mechanics and Its Application),
Springer
, Dordrecht,
The Netherlands
.
2.
Vakakis
,
A. F.
, and
Rand
,
R. H.
,
2004
, “
Nonlinear Dynamics of a System of Coupled Oscillators With Essential Stiffness Nonlinearity
,”
Int. J. Nonlinear Mech.
,
39
(
7
), p.
1079
.
3.
Gendelman
,
O. V.
,
Sapsis
,
T.
,
Vakakis
,
A. F.
, and
Bergman
,
L. A.
,
2011
, “
Enhanced Passive Targeted Energy Transfer in Strongly Nonlinear Mechanical Oscillators
,”
J. Sound Vib.
,
330
(
1
), pp.
1
8
.
4.
Gendelman
,
O. V.
,
Manevitch
,
L. I.
,
Vakakis
,
A. F.
, and
M'Closkey
,
R.
,
2001
, “
Energy Pumping in Nonlinear Mechanical Oscillators: Part I-Dynamics of the Underlying Hamiltonian System
,”
Trans. ASME
,
68
(
1
), pp.
34
41
.
5.
Kerschen
,
G.
,
Vakakis
,
A. F.
,
Lee
,
Y. S.
,
McFarland
,
D. M.
,
Kowtko
,
J. J.
, and
Bergman
,
L. A.
,
2005
, “
Energy Transfer in a System of Two Coupled Oscillators With Essential Nonlinearity: 1:1 Resonance Manifold and Transient Bridging Orbits
,”
Nonlinear Dyn.
,
42
(
3
), pp.
283
303
.
6.
Laxalde
,
D.
,
Thouverez
,
F.
, and
Simou
,
J.-J.
,
2006
, “
Dynamics of a Linear Oscillator Connected to a Small Strongly Nonlinear Hysteretic Absorber
,”
Int. J. Nonlinear Mech.
,
41
(
8
), pp.
969
978
.
7.
Panagopoulos
,
P. N.
,
Vakakis
,
A. F.
, and
Tsakirtzis
,
S.
,
2004
, “
Transient Resonant Interactions of Finite Linear Chains With Essential Nonlinear End Attachments Leading to Passive Energy Pumping
,”
Int. J. Solids Struct.
,
41
(
22
), pp.
6505
6528
.
8.
Vakakis
,
A. F.
,
Manevitch
,
L. I.
,
Gendelman
,
O.
, and
Bergman
,
L.
,
2003
, “
Dynamics of Linear Discrete Systems Connected to Local Essential Nonlinear Attachments
,”
J. Sound Vib.
,
264
(
3
), pp.
559
577
.
9.
Starosvetsky
,
Y.
,
Hasan
,
M. A.
,
Vakakis
,
A. F.
, and
Manevitch
,
L. I.
,
2012
, “
Strongly Nonlinear Beat Phenomena and Energy Exchanges in Weakly Coupled Granular Chains on Elastic Foundations
,”
SIAM J. Appl. Math.
,
72
(
1
), pp.
337
361
.
10.
Seidel
,
A.
,
Lin
,
H. H.
, and
Lee
,
D. H.
,
2005
, “
Phonons in Hubbard Ladders Studied Within the Framework of the One-Loop Renormalization
,”
Phys. Rev. B
,
71
(22), p.
22050
.
11.
Peyrard
,
M.
, and
Bishop
,
A. R.
,
1989
, “
Statistical Mechanics of a Nonlinear Model for DNA Denaturation
,”
Phys. Rev. Lett.
,
62
(
23
), pp.
2755
2758
.
12.
Bender
,
C. M.
, and
Orszag
,
S. A.
,
1999
,
Advanced Mathematical Methods for Scientists and Engineers I, Asymptotic Methods and Perturbation Theory
,
Springer-Verlag
,
New York
.
13.
Kevorkian
,
J.
, and
Cole
,
J. D.
,
1996
,
Scale and Singular Perturbation Methods
,
Springer-Verlag
,
New York
.
14.
Belhaq
,
M.
,
Clerc
,
R. L.
, and
Hartmann
,
C.
,
1988
, “
Multiple Scales Methods for Finding Invariant Solutions of Two Dimensional Maps and Application
,”
Mech. Res. Commun.
,
15
(
6
), p.
361
.
15.
Maccari
,
A.
,
1999
, “
A Perturbation Method for Nonlinear Two Dimensional Maps
,”
Nonlinear Dyn.
,
19
(
4
), pp.
295
312
.
16.
van Horssen
,
W. T.
, and
ter Brake
,
M. C.
,
2009
, “
On the Multiple Scales Perturbation Method for Difference Equations
,”
Nonlinear Dyn.
,
55
(
4
), pp.
401
418
.
17.
Helleman
,
R. H. G.
, and
Montroll
,
E. W.
,
1974
, “
On a Nonlinear Perturbation Theory Without Secular Terms
,”
Physica
,
74
(
1
), pp.
22
74
.
18.
Lee
,
P. S.
,
Lee
,
Y. C.
, and
Chang
,
C. T.
,
1973
, “
Multiple-Time-Scale Analysis of Spontaneous Radiation Processes. I. One- and Two-Particle Systems
,”
Phys. Rev. A
,
8
(
4
), p.
1722
.
19.
Khoo
,
I. C.
, and
Wang
,
Y. K.
,
1976
, “
Multiple Time Scale Analysis of an Anharmonic Crystal
,”
J. Math. Phys.
,
17
(
2
), p.
222
.
20.
Swinteck
,
N.
,
Muralidharan
,
K.
, and
Deymier
,
P. A.
,
2013
, “
Phonon Scattering in One-Dimensional Anharmonic Crystals and Superlattices: Analytical and Numerical Study
,”
ASME J. Vib. Acoust.
,
135
(
4
), p.
041016
.
21.
Überall
,
H.
,
1992
,
Acoustic Resonance Scattering
,
Gordon and Breach Science Publishers
,
Philadelphia, PA
, Chap. 4.
22.
Rapaport
,
D. C.
,
1995
,
The Art of Molecular Dynamics Simulation
,
Cambridge University Press
,
Cambridge, UK
.
23.
Thomas
,
J. A.
,
Turney
,
J. E.
,
Iutzi
,
R. M.
,
Amon
,
C. H.
, and
McGaughey
,
A. J. H.
,
2010
, “
Predicting Phonon Dispersion Relations and Lifetimes From the Spectral Energy Density
,”
Phys. Rev. B
,
81
(8), p.
091411
.
24.
Dot
,
A.
,
Borne
,
A.
,
Boulanger
,
B.
,
Segonds
,
P.
,
Félix
,
C.
,
Bencheikh
,
K.
, and
Levenson
,
J. A.
,
2012
, “
Energetic and Spectral Properties of Triple Photon Down Conversion in a Phase-Matched KTiOPO4 Crystal
,”
Opt. Lett.
,
37
(
12
), p.
2334
.
25.
Boitier
,
F.
,
Orieux
,
A.
,
Autebert
,
C.
,
Lemaître
,
A.
,
Galopin
,
E.
,
Manquest
,
C.
,
Sirtori
,
C.
,
Favero
,
I.
,
Leo
,
G.
, and
Ducci
,
S.
,
2014
, “
Electrically Injected Photon-Pair Source at Room Temperature
,”
Phys. Rev. Lett.
,
112
(
18
), p.
183901
.
You do not currently have access to this content.