Localizing Energy Through Nonlinearity and
Discreteness
Intrinsic localized modes have been theoretical
constructs for more than a decade. Only recently have they been
observed in physical systems as distinct as chargetransfer solids,
Josephson junctions, photonic structures, and micromechanical
oscillator arrays.
In solidstate physics, the phenomenon of localization is usually
perceived as arising from extrinsic disorder that breaks the
discrete translational invariance of the perfect crystal lattice.
Familiar examples include the localized vibrational phonon modes
around impurities or defects (such as atomic vacancies or
interstitial atoms) in crystals and Anderson localization of
electrons in disordered media.^{1}
The usual perception among solidstate researchers is that, in
perfect latticesthose free of extrinsic defectsphonons and
electrons exist only in extended, plane wave states. That notion
extends to any periodic structure, such as a photonic crystal or a
periodic array of optical waveguides. Such firmly entrenched
perceptions were severely jolted in the late 1980s by the discovery
that intrinsic localized modes^{2}
(ILMs), also known as discrete breathers^{3}
(DBs), are, in fact, typical excitations in perfectly periodic but
strongly nonlinear systems.
The past several years have seen this prediction confirmed by a
flood of experimental observations of ILMs in physical systems
ranging from electronic and magnetic solids, through microengineered
structures including Josephson junctions and optical waveguide
arrays, to laserinduced photonic crystals. Experimentalists are
currently hot on the trail of ILMs in BoseEinstein condensates
(BECs) and biopolymers. Hopes are high that these exotic excitations
will be useful in alloptical logic and switching devices and in
targeted breaking of chemical bonds, and will prove helpful to the
understanding of melting processes in solids and conformational
changes in biomolecules.
In this brief overview, we have attempted to capture the
excitement and to explain the essence of these remarkable nonlinear
excitations. We urge readers to consult some of the several
pioneering papers, recent reviews, and Web sites for more
details.^{2,
3}
Intuition and theory
A good working definition of ILMs (or DBs) is that they are
spatially localized, timeperiodic, and stable (or at least
longlived) excitations in spatially extended, perfectly periodic,
discrete systems. The existence of two distinct names for the same
phenomenon is an indication that separate historical paths led to
their discovery and provides key insights into the reasons for their
existence. A DB is a localized, oscillatory excitation that is
stabilized against decay by the discrete nature of the periodic
lattice. Box
1 discusses this path to DBs in more detail. An ILM is an
excitation that is localized in space by the intrinsic nonlinearity
of the medium, rather than by a defect or impurity. Box
2 reviews this path to ILMs.
By the early 1990s, researchers following these two paths had
converged on the insight that stable localized periodic modes,
whether called ILMs or DBs, were generic excitations in discrete
nonlinear systems, and that to study them systematically, one should
start with a system of uncoupled nonlinear oscillatorsthe
"anticontinuum limit"and treat the coupling as a weak
perturbation.
To pursue this insight, consider the simple problem of a diatomic
molecule, or dimer, modeled initially by a classical system of two
coupled anharmonic oscillators. First, imagine that the
interoscillator coupling is switched off; that leaves two
independent nonlinear oscillators. The nonlinearity of the
oscillators means that the frequency of their motion depends on the
amplitude or, equivalently, the input energy. In the case of the
familiar simple but nonlinear plane pendulum, for example, the
period varies from the small oscillation harmonic limit of
2π√(l/g), where l is the length of the
pendulum and g is the acceleration due to gravity, to
infinitely long as the amplitude of the pendulum's angle approaches
π. When the oscillators are completely uncoupled, we can form a
localized mode by exciting only one of the oscillators, and the
resulting frequency can fall anywhere in the range allowed by the
form of the anharmonicity of the individual oscillator.
Now consider exciting one oscillator strongly but the second one
only weakly so that most of the energy is initially localized at the
first oscillator. Because the frequencies depend on the amplitudes,
we can, in principle, choose amplitudes such that the frequencies of
each oscillator are irrationally related. For strictly
incommensurate frequencies, no possible resonances exist between any
of the oscillators' harmonics. If we now turn on the coupling
between the oscillators, intuition suggests that the transfer of
energy from one to the other must be very difficult, if even
possible.
That heuristic result can be formalized by the powerful
KolmogorovArnoldMoser (KAM) theorem of nonlinear dynamical
systems, which establishes that the incommensurate motions do remain
rigorously stable for sufficiently weak coupling and ensures that
the excitation energy remains localized on the first oscillator.
Next consider embedding the two nonlinear oscillators in an
infinite chain of similar ones; that is, physically place the dimer
molecule in an infinite molecular crystal of similar dimers.^{4}
The following model is an example of such a system:
Here φ_{n}(t) represents the
displacement of a nonlinear "quartic" oscillator at lattice site
n, so that the equation represents an infinite
onedimensional array of anharmonic oscillators coupled to their
nearest neighbors with a coupling strength given by
1/(Δx)^{2}. The notation is a deliberate reminder
that our model includes a finite spacing between molecules; the
formal continuum limit is obtained by taking Δx → 0, in which
case the entire second term in the equation becomes
∂^{2}φ (x, t) / ∂ x^{2}.
As usual, we start by studying the system's small (linear)
oscillations. The local oscillator at each site corresponds to a
doublewell potential, with degenerate minima at φ_{n
}= ± 1. Expanding around the minimum at φ_{n
}= 1, we obtain the spectrum of the linear waves,
ω_{q}^{2 }= 2 +
(2/Δx)^{2} sin^{2}(q/2). The linear
spectrum consists of a band that is limited by two cutoff
frequencies, 2 < ω_{q}^{2} < 2 +
(2/Δx)^{2}, and is bounded above and below; the upper
cutoff arises solely from the effect of discreteness.
As with the plane pendulum, the
doublewell quartic oscillator has a frequency that, for small
oscillations around the minimum, decreases with increasing amplitude
of the motion. That means one can create a localized periodic
oscillation at a frequency ω_{b} lying below the
spectrum of linear oscillations, as shown in the bottom panels of figure
1. Setting just one of the nonlinear oscillators in motion at a
fairly small amplitude will do the job, so that its frequency is
just smaller than the smallest allowed linear frequency. Then if the
coupling between the sites is weak enoughΔx is large,
producing a very narrow bandnot only will the fundamental
frequency ω_{b} of this ILM be below the band of
allowed linear excitations, but all harmonics of
ω_{b} will be above the band. Hence, there will be no
possibility of a linear coupling to the extended modes, even in the
limit of an infinite system when the spectrum
ω_{q} becomes dense. This means that the ILM
cannot decay by emitting linear waves (that is, phonons) and is
hence linearly stable.
To understand the highenergy ILMs that also occur and are shown
in the top panels of figure
1, consider again the limit of large Δx, which creates a
weak coupling between the oscillators. If we set a single oscillator
in motion, but now with large amplitude, the quartic nature of the
potential implies that the frequency of the ILM will increase with
increasing amplitude. For large enough amplitude, the frequency will
move above the highest frequency in the very narrowbecause
Δx is largeband of linear excitations. For this
largeamplitude ILM, all higher harmonics lie above the band because
the fundamental frequency does, and so the excitation is linearly
stable.
The simple quartic model illustrates how the two critical
componentsnonlinearity and discretenesscan combine to make ILMs
possible. Nonlinearity allows strongly excited local modes to have a
fundamental frequency outside the spectrum of small oscillations,
and the finite extent of the spectrum in a discrete system allows
all higher harmonics of the ILM also to lie outside the linear
spectrum. The locations of these frequencies prevent the resonances
that, in general, destroy the continuum breathers discussed in box
1. This intuitive understanding of the origin of ILMs/DBs in
discrete nonlinear systems was presented in the pioneering paper of
Albert Sievers and Shozo Takeno in 1988.^{2}
Over the intervening years, both analytic and numerical studies
have explored the existence and properties of ILMs in a variety of
nonlinear mathematical models of physical systems. Robert MacKay and
Serge Aubry, for example, rigorously proved the existence of DBs in
networks of weakly coupled anharmonic oscillators.^{5}
Remarkably, their theorems are insensitive to the lattice dimension:
Unlike continuum breathers (see box
1), which occur only in highly constrained 1D systems, DBs are
equally common in two and three dimensions.
Additional analytic results established that ILMs occur in
oneparameter families, are dynamically stable with respect to
perturbations, and are structurally stable with respect to changes
of the equations of motion.^{3}
Combined analytic and numerical studies showed that ILMs act as
strong, frequencydependent scatterers of plane waves and can be
quantized. Theoretical studies by Ding Chen (Saclay) and his
collaborators gave an explicit algorithm for moving ILMs along the
lattice, and calculations of Michel Peyrard (ENS, Lyon) established
that ILMs can be generated from thermal fluctuations.^{6}
The Chen and Peyrard results suggest that ILMs may play critical
roles in the transport of energy and other dynamical properties of
nonlinear discrete systems, such as melting transitions in solids
and folding in polypeptide chains.
Experimental observations of ILMs
Much of the present excitement surrounding ILMs comes from
numerous recent experimental observations in physical systems. Those
systems range from solid state mixedvalence transition metal
complexes^{7}
and quasi1D antiferromagnetic chains,^{8}
through arrays of Josephson junctions^{9}
and micromechanical oscillators,^{10}
to optical waveguide systems^{11}
and 2D photonic structures.^{12}
We look briefly at the nature of ILMs in several of these distinct
systems.
Solids. The natural periodicity of solidstate materials
makes them obvious targets to search for ILMs. But their small
length scales and quantum effects make observing or visualizing the
excitations experimentally challenging. As discussed in box
3, one quantum manifestation of localization is a redshift in
the frequencies of manyphonon excited vibrational states. In recent
experiments, Basil Swanson (Los Alamos National Laboratory) and
colleagues and K. Kisoda (Wakayama University) found strong evidence
for the existence of ILMs in the chargetransfer solid PtCl by
measuring resonance Raman spectra.^{7}
The redshift in the resonances was obtained for up to seven
participating optical phonons, indicating the localized excitation
of PtCl bonds.
Using a phenomenological model, theorists including Konstantin
Kladko (Ingrian Networks) and Nikos Vulgarakis (University of Crete)
and their collaborators have reached quantitative agreement with
those experiments.^{7}
But challenges remain for theorists to develop a more microscopic
model and for experimentalists to measure explicitly the spatial
extent of the ILM. Raman spectroscopy measures only frequency
shifts, from which one infers localization.
In magnetic solids, one expects to
find localized spinwave modes that are the direct spin analogs of
the ILM phonon modes. Ulrich Schwarz (Cornell University) and his
collaborators^{8} studied the quasi1D biaxial
antiferromagnet
(C_{2}H_{5}NH_{3})_{2}CuCl_{4}
by driving it with a microwave pulse at the lowest antiferromagnetic
resonance frequency of 1.5 GHz. Highintensity microwave pulses
drive the spatially uniform antiferromagnetic resonance into a
nonlinear region, where it becomes unstable. According to numerical
simulations, the resonance decays into a broad spectrum of ILMs,
which correspond to localized spin states. Measuring the
timedelayed absorption spectra reveals the corresponding redshifts
together with observable lifetimes of ILMs up to milliseconds. As in
the case of the PtCl chains, explicit measurements of the spatial
localization are needed to confirm the expectations from
simulations.
Josephson ladders. Following theoretical suggestions of
Louis Floria (University of Zaragoza) and his collaborators, Enrique
Trias (MIT), Peter Binder (University of Erlangen), and their
collaborators have made some of the most visually striking
observations of DBs yet found.^{9}
Their experiments involved periodic structures consisting of an
annular array of coupled Josephson junctions (see figure
2).
The excitations arise from the spatially localized voltage drops
that occur at particular junctions, as a homogeneous DC bias current
threads the ladder. A few junctions are in the resistive state,
while the others are superconducting. The superconducting junctions
generate AC voltages due to their coupling to the resistive
junctions. The dynamical effects produce a variety of resonances and
hysteresis loops in currentvoltage characteristics of the ladder.
Optical waveguides and photonic crystals. Optical and
photonic systems have proven to be fertile grounds for the creation
and observation of ILMs. Indeed, one of the first experimental
confirmations of ILMs was the observation of discrete spatial
solitons excited in optical waveguide arrays.^{11}
The periodic structure of the arrays produces the discreteness
effects. The nonlinearity arises from the Kerr effect (the
dependence of the index of refraction on the intensity of the light
pulse), which in a bulk medium or slab waveguide can produce spatial
optical solitons that correspond to propagating selftrapped optical
beams. Theoretical analysis has established^{11}
that this system is well described by a variant of the discrete
nonlinear Schrödinger equation (DNLSE) discussed in box
2. The discrete variable n of box
2 represents the position of one of the waveguides and the
continuous variable (time in the equation in box
2) corresponds to the spatial coordinate along the waveguide.
In the experiment, the individual
waveguides are made from aluminum gallium arsenide and are 4 µm wide
and a few millimeters long. Arrays contain typically 4060
waveguides; the strength of the coupling between neighboring
waveguides is controlled by their spacing, which varies from 2 to 7
µm. Yaron Silberberg (Weizmann Institute) and his collaborators
injected light from a synchronously pumped laser into a single
waveguide on the input side of a 6mmlong sample and recorded the
light distribution registered at the output facets (see figure
3).^{11}
At low light power, the propagation is linear, and the light expands
over all waveguides at the output. As the input power increases
above a threshold, the width of the output distribution shrinks. And
at a power greater than 500 W, the highly localized nonlinear mode
confines the light to about three waveguides around the input
waveguide; that light output is the signature for an ILM, which, in
this context, has been called a discrete soliton.
Photonic crystals^{13}periodic
materials in which the propagation of photons of certain wavelengths
is forbiddenare another important optical system in which
theoretical studies have predicted the existence of ILMs.^{12}
The optical analog of semiconductors, these artificial crystalline
structures provide novel and unique ways of controlling many aspects
of electromagnetic radiation, including the exciting possibility of
lightinduced radiation controlthat is, the use of light to switch
and channel light. Two years ago, Sergei Mingaleev (now at the
University of Karlsruhe) and one of us (Kivshar) predicted the
nature of the ILMs that could occur in a composite photonic crystal
formed by a regular 2D lattice of rods of two different types of
semiconductors. That work awaits experimental confirmation. However,
experimental groups led by Mordechai Segev (Technion) and Zhigang
Chen (San Francisco State University) have recently used optical
induction in a homogeneous nonlinear medium to confirm the existence
of this type of ILM in an analog of a 2D nonlinear photonic crystal.
Segev's group used a photorefractive crystal with a strong
electrooptic anisotropy to create a lattice with a polarization in
the nonelectrooptic direction and orthogonal to that of a probe
beam. They observed the light localization in the form of 2D
discrete solitons. Similarly, Chen's group used a nonlinear lattice,
created by partially incoherent light, to observe strong
localization in the regime of large nonlinearities (see figure
4). Both sets of results illustrate the tremendous promise for
optically excited and controllable nonlinear localized states as
elements of future alloptical logic and switching devices.
The best is yet to be
Armed with the fundamental understanding of the origin of ILMs,
experimentalists are becoming increasingly proficient at
discoveringor creatingnew physical systems in which to study
ILMs and their properties. For instance, in a recent experiment
designed to explore the important issues of creation, transport,
mobility, and interactions of ILMs, a group led by Sievers
constructed an array of micromechanical oscillators.^{10}
Nearly identical initial experimental conditions led to ILMs located
at widely different sites, an important confirmation that ILMs are
in fact localized by intrinsic nonlinear effects rather than by
disorder or impurities.
The ability to create periodic optical lattices in which to trap
BECs immediately suggests the possibility of creating and observing
ILMs in those systems.^{14}
Analytic and computational studies of the dynamical phase diagram of
a dilute BEC trapped in a multiwell periodic potential reveal that,
in the framework of the GrossPitaevsky equation (the standard
equation used to study BECs), the dynamics are governed by a variant
of the DNLSE (see box
2). ILMs can therefore be created even if the BEC's interatomic
potential is repulsive. Recent experiments by Francesco Cataliotti
and colleagues strongly suggest that an increase of the BEC density
will lead to the generation of ILMs in these systems.
The possible roles of ILMs in
biopolymers have also been a focus of concerted theoretical and
experimental efforts.^{15}
For instance, the conformational changes and buckling of long
biopolymer molecules may occur in response to the excitation of
nonlinear localized modes. Conformational flexibility is a
fundamental property that differentiates polymers from small
molecules and gives rise to many of their remarkable properties. A
distinctive feature of biological polymers is the complex structure
of their elementary subunits; that structure can support longlived
nonlinear excitations. ILM excitations have been discussed in
connection with the storage and transport of the energy released
during adenosine triphosphate hydrolysis, with the local opening of
the DNA doublehelix, and with the buckling, folding, and collapse
of a biopolymer chain to a compact coil, as shown in figure
5. Theoretical results predict that these effects survive in the
presence of viscous damping. Thus, although it is speculative, this
application of ILMs may prove to be crucial in the kinetics of
conformational phase transitions of semiflexible biopolymers in
solutions.
The recent theoretical, numerical, and experimental results, and
the innovative concepts associated with the physics of ILMs have
shed considerable light on the complex dynamics, properties, and
functions of nonlinear discrete physical systemsfrom the nanoscale
to the macroscale. The study of such systems, and the ILMs they
support, underpins applications ranging from smart materials that
respond collectively to external stimuli in a coherent, tunable
fashion to lightinduced, alloptical networks. Only a few years
ago, ILMs were almost exclusively the province of theorists. Today,
the rapidly expanding list of experimental observations not only
establishes the ubiquity of intrinsic localized modes in nonlinear,
discrete physical systems but also generates exciting possibilities
for future applications both in fundamental science and in
technology. Clearly, for ILMs, the best is yet to be.
We have enjoyed many helpful discussions with our colleagues
and coauthors who contributed to the field discussed in this
article. In particular, we recognize Serge Aubry, Alan Bishop, Oleg
Braun, Zhigang Chen, Konstantin Kladko, Arnold Kosevich, Robert
MacKay, Sergei Mingaleev, Elena Ostrovskaya, Alexander Ovchinnikov,
Michel Peyrard, Al Scott, Mordechai Segev, Al Sievers, Yaron
Silberberg, Andrey Sukhorukov, Shozo Takeno, George Tsironis, and
Alexey Ustinov.
David Campbell is dean of
the College of Engineering at Boston University. Sergej
Flach is head of the visitors program at the Max Planck
Institute for the Physics of Complex Systems in Dresden, Germany.
Yuri Kivshar is head of the nonlinear physics group of
the Research School of Physical Sciences and Engineering at the
Australian National University in Canberra.
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2004 American Institute of Physics


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