Wave scattering by discrete breathers

S. Flach, A. E. Miroshnichenko and M. V. Fistul†Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany

(September 23, 2002)

We present a theoretical study of linear wave scattering in one-dimensional nonlinear lattices byintrinsic spatially localized dynamic excitations or discrete breathers. These states appear in variousnonlinear systems and present a time-periodic localized scattering potential for plane waves. Weconsider the case of elastic one-channel scattering, when the frequencies of incoming and transmittedwaves coincide, but the breather provides with additional spatially localized ac channels whosepresence may lead to various interference patterns. The dependence of the transmission coefficienton the wave number q and the breather frequency Ωb is studied for different types of breathers:acoustic and optical breathers, and rotobreathers. We identify several typical scattering setups wherethe internal time dependence of the breather is of crucial importance for the observed transmissionproperties.

05.45.-a, 42.25.Bs, 05.60.Cd

Discrete breathers are generic solutions ofclassical Hamiltonian lattices. They are time-periodic and spatially localized. Of specialinterest is then the understanding of scatter-ing of small-amplitude plane waves by discretebreathers. As the breather presents a time-periodic scattering potential, incoming wave fre-quencies will be shifted by multiples of thebreather frequency, leading to the considerationof a multi-channel scattering problem. We willstudy the case when only the incoming chan-nel is conducting, while all other channels areclosed. Focussing on the one-dimensional latticecase we find strong interference and resonance ef-fects in various classes of breathers. Our resultson acoustic chains provide with new insights forthe related problem of anomalous heat conductiv-ity. We pay specific attention to the possibility ofperfect reflection, which is linked to a Fano-likeresonance due to the coupling of bound states andextended waves.

I. INTRODUCTION

The problem of wave propagation through media withvarious inhomogeneities has been a complex issue of con-stant interest and appears in different areas of physics.Particular examples are acoustic and electromagneticwave propagation in various disordered media [1,2], tun-neling of electrons in solids [3] and electron transportthrough quantum (molecular) wires [4,5]. In many casesof interest the conductivity (electron transport) [3,5,6]

†Present address: Physikalisches Institut III, UniversitätErlangen-Nürnberg, D-91058, Erlangen, Germany

and the heat conductivity (phonon transport) [7–10] aredetermined through the wave scattering by spatial inho-mogeneities. Of particular interest is the wave propaga-tion in one-dimensional systems where interference effectsmay be strongly enhanced.

In most of the studies wave scattering by static lo-calized inhomogeneities has been considered [10]. Morerecently the scattering by generic time-dependent poten-tials has received strong attention [11–13]. This is due tothe possibility to generate various time-dependent scat-tering potentials artificially e.g. in the presence of laserbeams or microwave radiation. Several interesting effectssuch as giant enhancement of tunneling [11,12] and Fanoresonances [14–17] have been found.

It is a well established fact that various nonlinear spa-tially discrete systems can support different types of exci-tations, namely, propagating linear waves (phonons) andtime-periodic spatially localized excitations called dis-crete breather states (DB) [18–20]. The origin of the lat-ter localized states is not the presence of disorder butrather the peculiar interplay between the nonlinearityand discreteness of the lattice. While the nonlinearityprovides with an amplitude-dependent tunability of oscil-lation or rotation frequencies of DBs, Ωb, the spatial dis-creteness of the system leads to finite upper bounds of thefrequency spectrum of small amplitude waves ωq. It al-lows to escape resonances of all multiples of the breatherfrequency Ωb with ωq. Note here, that nonlinear dis-crete lattices admit different types of DBs depending onthe spectrum of linear waves propagating in the lattice,i. e. acoustic breathers and rotobreathers (Fig. 1a and1b), optical breathers (Fig. 1c), etc. Such properties ofDBs as their frequency dependent localization length andthe stability of DBs with respect to small amplitude per-turbations have been widely studied. DBs have been ob-served in experiments covering such diverse fields as non-linear optics [21], interacting Josephson junctions [22,23]magnetic systems [24] and lattice dynamics of crystals

1

[25].

FIG. 1. A schematic representation of different types ofdiscrete breathers: a) acoustic breather; b) acoustic roto-breather; c) optical breather.

Although DBs present complex dynamical objects, inmany cases experimental measurements can be well un-derstood by using certain time-averaged properties ofDBs [26,27]. Thus, a natural question appears whetherthe internal breather dynamics is of crucial importance tounderstand the lattice dynamics in the presence of DBs.In this paper we study the propagation of small ampli-tude plane waves in one-dimensional nonlinear lattices inthe presence of a DB and obtain that the internal dynam-ics of the DB may lead to a drastical increase or decreaseof wave transmission as compared to the time-averagedsetup. Thus the wave scattering by DBs is interestingboth as some spectroscopical tool to study DB propertiesand as a way to control the wave transmission (conduc-tivity) by varying the DB state. Finally our studies areof use for the general understanding of wave scatteringby time-periodic potentials.

First successful attempts to describe the variety of phe-nomena arising from wave scattering by DBs have beenperformed some time ago [28–31,17]. While a numberof results have been obtained, we are far from a fulldescription of the complexity of possible scattering out-comes. This concerns the wave scattering in systems withacoustic spectra ωq, the importance of the internal DBdynamics including the comparison between rotational

and vibrational excitations, the dependence of the wavepropagation on the DB energy, and the single channelelastic versus the two channel inelastic scattering cases[29]. Here we will address these problems but restrictourselves to the elastic scattering case.

The paper is organized as follows: in Section II wepresent the general formalism and describe analytical andnovel numerical methods used to analyze the wave scat-tering by DBs, in Sections III, IV and V the dependenciesof the transmission coefficient on the breather frequencyΩb and the wave number q for different types of breathers(acoustic breathers and rotobreathers, optical breathers,see Fig. 1) are obtained, and finally a discussion is pro-vided in Section VI.

II. GENERAL FORMALISM

To proceed we will consider one-dimensional nonlinearlattices with nearest neighbor interaction, optional on-site (substrate) potential, and with one degree of freedomper lattice site. Both the increasing of the interactionrange and the extension to more than the one degree offreedom per lattice site are not of crucial importance.

The dynamics of the system is characterized by time-dependent coordinates xn(t) and the class of Hamiltoni-ans considered here reads

H =∑n

(ẋ2n2

+ V [xn] + W [xn − xn−1])

. (1)

Here V [x] is an optional on-site (substrate) potential andW [x] is the nearest neighbor interaction. The equationsof motion become

ẍn = −W ′[xn − xn−1] + W ′[xn+1 − xn] − V ′[xn] . (2)

Without loss of generality we take V [0] = W [0] = V ′[0] =W ′[0] = 0 and V ′′[0] ≥ 0, W ′′[0] > 0. This Hamiltoniansupports the excitation of small amplitude linear waveswith the frequency spectrum

ω2q = V′′[0] + 4W ′′[0] sin2

(q2

), (3)

with q being the wave number.Time-dependent spatially localized solutions (DBs) ex-

ist for different types of potentials V [x] and W [x], al-though at least one of the two functions (V [x] and/orW [x]) has to be anharmonic. DB solutions are charac-terized by being time-periodic x̂n(t+Tb) = x̂n(t) and spa-tially localized x̂|n|→∞ → 0 (except systems with V = 0where x̂n→±∞ → ±c with c possibly being nonzero). Ifthe Hamiltonian H is invariant under a finite translation(rotation) of any xn → xn+λ, then discrete rotobreathers(DRB) may exist [32]. These excitations are character-ized by one or several sites in the breather center evolv-ing in a rotational state x̂0(t + Tb) = x̂0(t) + λ, while

2

outside this center the lattice is governed again by timeperiodic spatially localized oscillations. The breather fre-quency Ωb = 2π/Tb can generally take any values pro-vided kΩb �= ωq for all nonzero integer k. As ω2q is ananalytic function of q, DBs are exponentially localizedon the lattice.

A. The linearized problem

To study the scattering of small amplitude plane wavesby a DB we linearize the equations of motion (2) arounda breather solution xn(t) = x̂n(t) + �n(t):

�̈n = −W ′′[x̂n(t) − x̂n−1(t)](�n − �n−1)+W ′′[x̂n+1(t) − x̂n(t)](�n+1 − �n) − V ′′[x̂n(t)]�n . (4)

This is a set of coupled linear differential equations withtime periodic coefficients of period Tb. Note that thesecoefficients are determined by the given DB solutionx̂n(t).

Eq.(4) determines the linear stability of the breatherthrough the spectral properties of the Floquet matrix,[20,29] which is given by a map over one breather period(

��(Tb)�̇�(Tb)

)= F

(��(0)�̇�(0)

), (5)

where �� ≡ (..., �n−1, �n, �n+1, ...). For marginally sta-ble breathers all eigenvalues of the symplectic matrix Fwill be located on the unit circle and can be written aseiθ. The corresponding eigenvectors are the solutions ofEq.(4), and fulfill the Bloch condition

�n(t + Tb) = e−iθt/Tb�n(t) . (6)

Because the DB solution is exponentially localized on thelattice, equation (4) will reduce to the usual small am-plitude wave equation far away from the breather cen-ter. Thus, only a finite number of Floquet eigenvectorsare spatially localized, while an infinite number of them(for an infinite lattice) are delocalized and these solu-tions correspond to plane waves with the spectrum (3)and the Floquet phases θ = ±ωqTb. The remaining Flo-quet eigenvalues correspond to local Floquet eigenvectorsand are separated from the plane wave spectrum on theunit circle. Note here, that two eigenvectors with the de-generated eigenvalue eiθ = + 1 always exist and reflectperturbations tangent to the breather family of solutions.

As a consequence of the Bloch condition (6) any spa-tially extended Floquet eigenvector can be represented inthe form

�n(t) =∞∑

k=−∞enkei(ωq+kΩb)t. (7)

What happens if we send a plane wave with frequencyωq to the DB? We will deal with the case of one-channel

scattering as for any k �= 0 and any q′, ωq′ �= ωq + kΩb.This condition determines that all channels with nonzerok are ’closed’, i.e. the spatial amplitudes enk are localizedin space. Note here, that the frequencies of transmittedand reflected waves are the same and coincide with thethe frequency of the incoming wave, since they all belongto the only open channel with k = 0.

For harmonic interaction potentials W it was shown inRef. [29] that the momentum

J = −W ′′[0]〈Im�∗n�n−1〉 (8)

is independent of the lattice site. Here the averagingis meant with respect to time. In a similar way it isstraightforward to show that the quantity

J̃ = −〈W ′′[x̂n(t) − x̂n−1(t)]Im�∗n�n−1〉 (9)

is independent of the lattice site. Since the breather so-lution x̂n(t) is spatially localized, at large distances fromthe breather we find again that the momentum J is in-dependent of the lattice site. Especially we find that it isthe same for n → ±∞. Following Ref. [29] we concludethat the one-channel scattering case is elastic, i.e. the en-ergy of an incoming wave equals the sum of the energiesof the outgoing (reflected and transmitted) waves.

Despite the fact that we will study the one-channelelastic scattering, the scattering potential of DB is time-periodic. The above mentioned ’closed’ channels are ac-tive inside the breather core, i. e. the frequency of linearwaves in a finite area around the breather center canchange due to the interaction with the DB (see Fig.2).

ωq

ωqωq ωq − Ωb

ωq − 2Ωb

ωq − 3Ωb

ωq + Ωb

ωq + 2Ωb

ωq + 3Ωb

FIG. 2. Schematic representation of the one-channel scat-tering of a wave by a discrete breather.

Thus one of the questions to be answered below is toidentify the cases when the well-known scattering by atime-averaged (static) potential is not sufficient to de-scribe the actual results of wave scattering by discretebreathers. This implies that interference effects through

3

local interactions between the active closed channels maysubstantially change the scattering results as comparedto a time-averaged scattering potential.

Assuming that the breather is located around the cen-tral site n = 0, the one-channel scattering problem canbe written as

�n(t) = AIe−i(ωqt+qn) + ARe−i(ωqt−qn), n < 0�n(t) = AT e−i(ωqt−qn) , n > 0 (10)

for

|n| � sup[ξb(0), ξb(ωq)] (11)

where ξb(ω) = sup ξ(ω + kΩb) and

sinh2 ξ(ν)/2 = V′′[0]−ν24W ′′[0]

, |ν| < ωq(0)cosh2 ξ(ν)/2 = ν

2−V ′′[0]4W ′′[0] , |ν| > ωq(π)

(12)

ξ measures the characteristic inverse localization lengthof a closed channel at frequency ν (note that for ν = kΩbthe localization length is that of the breather itself, seealso [29]). The incoming wave has amplitude AI , andthe reflected and transmitted wave amplitudes are givenby AR and AT respectively. The transmission coefficienttq = |AT /AI |2.

For further considerations we will use the notation ofthe Bloch functions ζn(t) defined as

�n(t) = ζn(t)e−iωqt. (13)

In order to estimate the relative strength of closedchannels with k �= 0 we expand the time-periodic co-efficients of (4) in a Fourier series with respect to time:

W ′′[x̂n(t) − x̂n−1(t)] =∞∑

k=−∞wn,keikΩbt , (14)

V ′′[x̂n(t)] =∞∑

k=−∞vn,keikΩbt . (15)

We consider first the case of a strongly localized opticalbreather located at site n = 0 with W (y) = c2y

2. Takinginto account a single closed channel with some value ofk, inserting (14),(15) and (7) into (4) and excluding enk,the relative strength sk of the closed channel to the openone will be given by

sk =

∣∣∣∣∣ v20,k

(v0,0 − 1)(v0,0 + 2c(1 − ηk) − (ωq + kΩb)2)

∣∣∣∣∣ ,(16)

where the relative amplitude

ηk = ±e1,ke0,k

= e−ξ(ωq+kΩb) (17)

has positive sign for ωq + kΩb located inside the phonongap and negative sign otherwise. For sk 1 we do notexpect any significant contribution from the given closedchannel, while sk ≥ 1 indicate a strong influence of theclosed channel on the scattering process. Note that theexpression

√v00 + 2c(1 − ηk) in the denominator of (16)

is just a frequency ωL of a local phonon mode of the time-averaged scattering potential. For spatially discrete sys-tems these local phonon modes may be located insideor outside of the phonon gap. The denominator of (16)may vanish for certain wave numbers, which would im-ply a resonance-like enhancement of the closed channelcontribution (for certain wavenumbers q). As such a res-onant enhancement of a closed channel amplitude acts asa huge effective scattering potential to the open channel,for these cases we expect a resonant suppression of trans-mission. Thus, qualitatively such a complete suppressionof transmission (Fano-like resonance [14,16]) is explainedby a resonant interaction of the propagating phonon withthe specific local phonon mode. However in order toquantitatively analyze this effect the renormalization ofthe value of ωL due to all nonresonant processes, has tobe taken into account. It will be done below for a par-ticular case of optical breathers by making use of theGreen function method. We obtain that although therenormalization of ωL is rather small it may become im-portant especially as the width of a phonon band is small,W ′′ 1.

In a similar way we proceed for the estimation of theclosed channel contribution of acoustic breathers. Weobtain the following expression for the relative strengthrk

rk =

∣∣∣∣∣ w20,k

(w0,0)(w0,0 − (ωq + kΩb)2)

∣∣∣∣∣ . (18)Thus, we again find a resonant suppression of transmis-sion when the presence of acoustic DB leads to a localincrease in the nearest neighbor interaction and to anappearance of a corresponding local phonon mode. How-ever, this case is much more involved as compared tothe optical DB case, and Eq. (18) may serve only asa qualitative tool to check whether a closed channel isstrongly contributing to the transmission or not. Wewill instead provide with a more quantitative analysisfor the cases considered, based on the particular acousticbreather properties.

B. Numerical scheme

To compute numerically the transmission coefficient wehave developed a scheme which generalizes the one givenin Ref. [29] (which relies on a spatial reflection symmetryof the breather and thus of the scattering potential). Atvariance with Ref. [29] our scheme is capable of dealing

4

with any (perhaps spatially nonsymmetric) time-periodicscattering potential.

We look for solutions of Eq.(4) which correspond tozeroes of the operator

G(��(0), �̇�(0)) =(

��(0)�̇�(0)

)− eiωqTb

(��(Tb)�̇�(Tb)

)(19)

on a lattice with 2N + 1 sites labeled −N, (−N +1), ...,−1, 0, 1, ...(N − 1), N . The incoming wave is fedfrom the left, and the transmitted wave is leaving thesystem to the right. The boundary condition at the rightend is �N+1 = e−iωqt, which implies that the transmit-ted wave will have amplitude 1. With a given boundarycondition at the left end �−N−1 = (A + iB)e−iωq t, whereA and B are real numbers, we may find the zeroes of(19) using a standard Newton routine. Due to the lin-earity of the equations of motion in � an arbitrary initialguess and one Newton step are needed to converge to thezeroes. In practice due to roundoff errors an additionalNewton step may be required.

However with arbitrary A and B we will not realizethe scattering case (10,10) in general. This is due to thefact that all extended Floquet states of an infinite sys-tem are twofold degenerated because time reversal sym-metry holds far from the breather center. To succeed weadd a second Newton procedure which uses A and B asfree parameters such that the solution on site N becomes�N = e−iq−iωqt, ensuring that we realize a single trans-mitted traveling wave of amplitude one at the right end ofthe system. After the Newton procedures are completed,the transmission coefficient is then given by

tq =4 sin2 q

|(A + iB)e−iq − ζ−N |2. (20)

While the Bloch functions ζn are in general time-dependent close to the breather center, they will be time-independent complex numbers at large distance from thebreather. We note that the computation operates at themachine precision, and we obtain results which are sizeindependent, i.e. with the above described boundaryconditions we emulate an infinite system. The discretebreather solution itself has to be obtained beforehandusing standard numerical procedures [33].

The enumerator in (20) vanishes at the extremal valuesof ωq, i.e. at q = 0 and q = π. If the denominator is finiteat these values of q, the transmission will also vanish.This is indeed the generic case for a quadratic dependenceof the spectrum ωq on q around these points. Exceptionsare expected for acoustic chains (see next Section).

Another peculiar point is that if upon changing somecontrol parameter, e.g. the breather frequency, a local-ized Floquet eigenstate attaches to or disattaches fromthe extended Floquet spectrum, the transmission coeffi-cient will be exactly t = 1 for the q-value which corre-sponds to the edge of the spectrum ωq, i.e. q = 0 orq = π [29,30].

C. Green function method for a time-periodiclocalized scattering potential

We will also analyze the wave propagation throughDBs by making use of the Green function technique [34].This method is especially convenient as the scatteringpotential is localized in space. To apply this method toa particular case when the presence of DBs leads to theappearance of a localized time-dependent on-site scatter-ing potential, V ′′[x̂n(t)] − 1, we perform a time Fouriertransformation of Eq. (4) and obtain the equation forthe Green function Gωq(n1, n2) :

Gωq(n1, n2) = G0ωq(n1, n2)−

−∑m

G0ωq(n1, m)∫

dΩUΩ(m)Gωq+Ω(m, n2) , (21)

where G0ωq(n1, n2) is the Green function of the linearequations of motion in the absence of the DB. TheFourier transform of the localized scattering potentialUΩ(m) =

∫dteiΩt(V ′′[x̂m(t)]− 1) is determined by the

properties of the DB solution. Because the DBs are peri-odic solutions in time the potential UΩ(m) contains justthe harmonics of the breather frequency, Ω = kΩb.Moreover, as we will see later in the considered caseswe can take into account the harmonics with small val-ues of k = 0 ,±1, ± 2 only. Thus, the calculationof the Green function Gωq(n1, n2) can be represented ina diagrammatic form where terms which correspond toactive closed channels describe the local (virtual) absorp-tion and emission of phonons by the propagating phononin the presence of the DB (some of the typical diagramsare shown in Fig.3). Moreover, for one-channel scatter-ing the energy conservation law of absorbed and emittedphonons has to hold. In order to obtain the transmissioncoefficient tq we need also an expression for the Greenfunction G0ω(n1, n2) which reads

G0ω(n1, n2) = −∫

dq

2πeiq(n1−n2)

ω2 − ω2q. (22)

The Green function of the full problem (for one-channelscattering) has a similar form for large distances from thebreather center (n1 → −∞ and n2 → +∞):

Gωq (n1, n2) = − iDqeiq|n1−n2|

d(ω2q)/dq, (23)

where tq = |Dq |2.As an example for a static scattering potential which

is strongly localized on a single site (n = 0) we obtain

Gωq(n1, n2) =G0ωq (n1, n2)

1 + β0G0ωq(0, 0), (24)

where β0 is the strength of the static potential. Thetransmission coefficient t̃q in this case is given by

5

t̃q =(d(ω2q)/dq)2

β20 + (d(ω2q)/dq)2. (25)

We will use the Green function method in Section Vwhere the wave scattering by optical DBs (see Fig. 1c)will be presented. We show how most important dia-grams can be taken into account in the case of a time-periodic localized scattering potential.

= ...n0 n0 n0

n0

n0n0n0n0

n0 n0

n0

n0n0

n1n1 n1 n1

n1

n1

n1

n2n2 n2 n2

n2

n2

n2ωq

ωqωq

ωqωq

ωq

Ωb

ΩbΩb

Ωb

2Ωb 2Ωb

2Ωb

2Ωb

2Ωb2Ωb

2Ωb

2Ωb

ωq−2Ωb

ωq−2Ωb

ωq−2Ωbωq−2Ωb

ωq−2Ωb

ωq−4Ωb

+ ++ a)

b)

c)

d)

FIG. 3. Typical diagrams describing the interaction of thepropagating phonons with the time-periodic DB scatteringpotential: a) scattering by a time-average (static) potential;b) resonant scattering process; c) a process leading to therenormalization of a phonon local mode frequency; d) variouscomplex processes. Here, thin and thick solid lines presentthe Green functions in the absence of the DB and in the pres-ence of the time-average part of the DB scattering potential.The time-average part of the DB scattering potential is shownby a black circle, and the dashed lines show the absorption(emission) of phonons.

D. Time-averaged scattering potential

Since we are interested in understanding the impor-tance of the time dependence of the scattering potential,we will also compare the numerical results with thoseobtained by time-averaging the DB scattering potential.This time-averaging can be found numerically beforehandand Eq. (4) becomes

¨̃�n = −wn,0(�̃n − �̃n−1) +wn+1,0(�̃n+1 − �̃n) − vn,0�̃n. (26)

Because in this case all inhomogeneities are time-independent we can use the standard scattering matrixmethod. With �̃n(t) = ζ̃ne−iωqt Eq.(26) is rewritten as(

ζ̃n+1ζ̃n

)= Mn

(ζ̃nζ̃n−1

), (27)

with

Mn =

(1 + En+cn,n−1−ω

2q

cn+1,n− cn,n−1

cn+1,n

1 0

)(28)

where En = vn,0 and cn,n−1 = wn,0. It follows(ζ̃N+1ζ̃N

)= M

(ζ̃−Nζ̃−N−1

), (29)

with

M =−N∏i=N

Mi. (30)

The expression for the transmission coefficient t̃q forthe time averaged scattering potential is then given by[10]

t̃q =4 sin2 q

|M11(q)e−iq + M12(q) −M21(q) − M22(q)eiq |2(31)

where Mij are the four matrix elements of the 2×2 matrixM.

III. SCATTERING BY ACOUSTIC BREATHERS

In this Section we study the wave scattering by so-called acoustic breathers. The corresponding systems arecharacterized by a gapless spectrum ωq of propagatinglinear waves, and by a conservation of total mechanicalmomentum.

The generic choice for the potentials in the Eq. (2) isV (y) ≡ 0 and W (y) = 12y2 +

β3 y

3 + 14y4. This choice

leads to the well known Fermi-Pasta-Ulam system [35].We will first consider the case β = 0 which implies thepresence of a parity in the interaction potential W andcomment on the influence of parity violation for β �= 0later on. The dispersion relation of phonons is given by

|ωq| = 2 sinq

2(32)

A breather solution with frequency Ωb = 4.5 is shownin Fig.4.

6

-10 -8 -6 -4 -2 0 2 4 6 8 10n

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 πq

0

0.1

0.2

0.3

0.4

r k

k=2

k=4

x̂n(t

=0)

FIG. 4. Displacements of an acoustic breather with zerovelocities at t = 0 and Ωb = 4.5.Inset: Relative strength rk for the second and fourth closedchannels versus q.

The corresponding Fourier components of the scatter-ing potential are plotted in Fig.5.

-8 -6 -4 -2 0 2 4 6 8n

10-10

10-8

10-6

10-4

10-2

100

102

k=0k=2k=4k=6

|wn

,k|

FIG. 5. Fourier components wnk for different k versus nfor the breather in Fig.4.

Acoustic breathers can be found with frequenciesabove the phonon band |Ωb| > max |ωq|. However, herewe consider |Ωb| > 2 max |ωq| which implies |Ωb| > 4.This condition is needed to realize the one-channel scat-tering case of linear waves. For β = 0 the ac part ofthe interaction potential W has even harmonics only,and the frequency of a propagating phonon in a closedchannel 2Ωb ±ωq can not match the frequency of a localmode of the time-averaged scattering potential. Thus theclosed channels play no important role in the scatteringprocess. As a consequence the wave scattering by acousticbreathers is practically identical with the scattering by thetime-averaged potential. Indeed numerical computationsdo not show any relevant difference between the linear

wave propagation in the presence of time-dependent andthe static (time-averaged) potentials. Yet there is a num-ber of interesting features of the scattering which deserveto be exploited.

For V = 0 the lattice conserves the total mechanicalmomentum P =

∑n ẋn in addition to the energy. With-

out loss of generality we will choose P = 0 here. Fromthis conservation law it follows that the total mechanicalmomentum of the linearized problem Π =

∑n �̇n is also

conserved. This implies that for any solution �n(t) theshift �(t) + C is also a solution. In particular �n = C isa solution, which corresponds to a wave with q = 0 andfor which we have tq=0 = 1. Thus we conclude that thetransmission of waves by a breather for acoustic systemsat q = 0 is always perfect, no reflection occurs. This is insharp contrast to systems without conservation of totalmechanical momentum (see below).

The peculiar dependence of the transmission coefficienttq on the wave number q and the breather frequency Ωbis shown in Fig.6. The breather frequency is varyingover nearly three decades. We indeed observe perfecttransmission at q = 0, and zero transmission at q = π.However we also find that in the studied breather fre-quency range two rather narrow peaks around q = πappear with perfect transmission. These structures aredue to the detachment of localized Floquet eigenvectorsfrom the continuum of extended Floquet eigenstates. Asurprising result is shown in Fig.7. We plot the trans-mission tq at q = π/4 as a function of the breather fre-quency Ωb. We obtain plateaus and crossovers at certainbreather frequencies. The crossover positions clearly cor-relate with the appearance of localized Floquet states,which are traced through the perfect transmission closeto q = π in Fig.6. The dependence of the transmissionon q for the two observed plateaus is shown in the insetof Fig.7. The plateaus range over several decades in Ωb,and the q-dependence of tq is rather similar on differentplateaus.

To understand this feature we remind that largebreather frequencies imply that the breather energy andits amplitude in the breather center are large as well.Thus we may neglect the harmonic part of the interac-tion potential W inside the breather core. The result-ing amplitude distribution of the breather core is char-acterized by a superexponential decay x̂n(t) = AnG(t),An ∼ A3n−1, where G(t) is an oscillatory master function[36,37]. Due to the symmetry of the breather solutionthe spatial profile is described by A−n = −An+1 , n ≤ 0.The amplitudes An have been computed: A1 = 1,A2 � − 0.166, A3 � 4.796 10−4, A4 � − 1.15 10−11,and so on [35]. The overall amplitude of the oscillationsx̂n(t) is tuned by the amplitude of the master functionG(t).

The scattering is essentially described by the time-averaged scattering potential. This potential correspondsto a large increase in the nearest neighbour coupling

7

terms. In the breather center we find

〈W ′′[x̂1(t) − x̂0(t)]〉 = 1 + 3〈(x̂1(t) − x̂0(t))2〉 (33)

with

3〈(x̂1(t) − x̂0(t))2〉 ≡ X2 ≈23π

Ω2b (34)

(the details of the calculation are presented in AppendixA). Due to the large value of X for large Ωb it becomesevident that time-dependent corrections to the scatteringpotential are negligible. By making use of the distribu-tion of oscillation amplitudes in the lattice we then find

3〈(x̂n(t) − x̂n−1(t))2〉 ≈ A2nX2. (35)

The scattering on a plateau in Fig.7 is due to a finitenumber of matrices Mn which should be taken into ac-count in Eq. (30). Qualitatively the crossover thresholdcan be defined as

A2nX2 � 1 . (36)

Thus we obtain the two crossover frequencies Ωb1 = 13.1and Ωb2 = 4527, which match with the observed crossoverpositions. For 4 < Ωb < Ωb1 we need only 4 matricesMn, for Ωb1 < Ωb < Ωb2, 6 matrices Mn and so on.We also computed the transmission at q = π/4 usingthe corresponding reduced set of relevant matrices. Theobtained limiting values of tq are shown as a dashed linein Fig.7. We observe very good agreement between thepredicted plateau heights and the actual data obtainedin direct numerical simulations.

FIG. 6. The dependence of the transmission coefficient onq and Ωb.

101

102

103

1040

0.106

0.189

0.3

0.4

0.5

0.6

t 0.5 1.0 1.5 2.0 2.5π

q

0

0.2

0.4

0.6

0.8

1

t

ΩbFIG. 7. The dependence of the transmission coefficient on

Ωb for a particular value of q =π4 . The dashed lines show

the predicted plateau heights (see text). The inset shows thedependence of the transmission on q for the two observedplateau regions.

With the above said it becomes also transparent, whywe observe detachment of localized Floquet states in thecrossover region. The acoustic breather presents an ef-fective potential well for the propagating phonons. Thewidth of this well increases with the breather frequency,and the number of possible localized Floquet states alsoincreases. Such a periodic appearance of perfect trans-mission is similar to the quantum mechanical scatteringby a potential well in the presence of quasi-discrete levels[38].

Finally we checked the influence of β �= 0. With re-spect to the breather the presence of cubic terms in theinteraction potential leads to the generation of a kink-shaped dc lattice distortion. The corresponding scat-tering potential becomes asymmetric in space and oddharmonics in the ac scattering potential appear. Thisimmediately leads to the possibility of matching betweenthe frequency of a propagating phonon in a closed chan-nel Ωb −ωq and several local modes of the time-averagedscattering potential. Thus, a resonant suppression of atransmission appears in this case. Indeed, we observethis effect by direct numerical simulations as shown inFig.8, where two transmission zeros are found. We willdiscuss this effect in more detail for the case of opticalbreathers below.

8

0 0.5 1 1.5 2 2.5 πq

0

0.2

0.4

0.6

0.8

1

t

FIG. 8. The dependence of the transmission coefficient on qfor an acoustic breather with Ωb = 4.5 and β = 1. The dashedline is the result for the time-averaged scattering potential.Two transmission zeros are observed at corresponding valuesof q.

IV. SCATTERING BY ACOUSTICROTOBREATHERS

Remarkable differences to the results of the precedingSection are obtained if the interaction potential W is cho-sen to be a periodic one (the potential V is still zero inthis Section):

W (y) = 1 − cos(y). (37)

With such an interaction potential the nonlinear chainallows for the existence of rotobreathers (cf. Fig.1b). Inthe simplest case a rotobreather consists of one particlebeing in a rotating (whirling) state, while all others par-ticles with spatially decaying amplitudes from the centerof DB:

x̂0(t + Tb) = x̂0(t) + 2π,x̂n �=0(t + Tb) = x̂n �=0(t), (38)

x̂|n|→∞ → 0.

To excite such a rotobreather the central particle needs toovercome the potential barrier generated by its two near-est neighbors. The rotobreather energy is thus boundedfrom below by Eb > 4. At sufficiently large energiesEb � 4 the central particle will perform nearly free ro-tations x̂0(t) = Ωbt + δ(t) with δ(t + Tb) = δ(t) beinga small correction. The time-averaged off-diagonal hop-ping terms between site n = 0 and n = ±1 are then givenby

〈cos(x̂0(t) − x̂1(t))〉 ≈ −〈sin(Ωbt)(δ(t) − x̂1(t))〉 (39)

where x̂1(t) is also a small function. Thus the time-averaged scattering potential of a rotobreather presents a

huge barrier that cuts the chain into nearly noninteract-ing parts, contrary to the previous case of an acousticbreather, where the acoustic breather potential corre-sponds to a potential well. Therefore it becomes ex-tremely difficult for waves to penetrate across the ro-tobreather. The rotobreather solution with Ωb = 4.5 isshown in Fig.9.

-10 -8 -6 -4 -2 0 2 4 6 8 10n

10-10

10-8

10-6

10-4

10-2

100 xx

dx

0 1 2 πq

0

0.5

1

1.5

2

r k k=1

k=2

x̂n(t

=0)

,˙̂ x n

(t=

0)

FIG. 9. Displacements and velocities of an acoustic roto-breather at t = 0 with Ωb = 4.5.Inset: Relative strength rk for the first and second closedchannels versus q.

The corresponding Fourier components of the scatter-ing potential are shown in Fig.10.

-6 -4 -2 0 2 4 6n

10-10

10-8

10-6

10-4

10-2

100

k=0k=1k=2k=3k=4k=5

|wn

,k|

FIG. 10. Fourier components wnk for different k versus nfor the rotobreather in Fig.9.

In Fig.11 we show the q-dependence of the transmis-sion for a rotobreather with Ωb = 4.5 and compare itto the corresponding curve of an acoustic breather fromthe preceding section. We observe a dramatic decreaseof the transmission at all q-values for the rotobreather,

9

in agreement with the above analysis.To decide whether the time-dependence of the roto-

breather scattering potential is important or not, we firstmake a more precise computation for (39) (details of thecalculations are presented in Appendix B)

〈cos(x̂0(t) − x̂1(t))〉 ≈ −3

2Ω2b. (40)

The time-averaged scattering potential for large Ωb cor-responds to two neighboring weak links of strength (40)inserted in a linear acoustic chain. Three matrices Mnare enough to compute the transmission. Two of theminvolve matrix elements which are proportional to Ω2b .The elements of the product will thus contain terms pro-portional to Ω4b, and according to (31) the result is

t̃q ∼ Ω−8b . (41)

This is precisely what we also find from a numerical eval-uation of the transmission for the time-averaged scatter-ing potential in Fig.12. However, although the transmis-sion coefficient tq for the full time-dependent scatteringpotential also drastically decreases with q and Ωb, thedependence is weaker than for the case of time-averagerotobreather potential. It scales as

tq ∼ Ω−4b (42)

(see also Fig.12). The reason is that besides a weak staticlink the rotobreather scattering potential has an ac termat frequency Ωb of amplitude one (see Fig.10). Thus analternative route for the wave is to approach the roto-breather, to be excited into the first closed channel, topass the breather and to relax back into the open channel.The corresponding scattering process of ”virtual” absorp-tion and emission of phonons from the rotobreather canbe also represented by three matrices as it happens forthe dc analysis. However now instead of two weak linkswe have links of order one with a frequency change atsite n = 0 from ωq to ωq +Ωb. This occurs in exactly oneof the three matrices. Consequently the product matrixwill contain elements proportional to Ω2b and the trans-mission will scale as (42).

0 0.5 1 1.5 2 2.5 π0

0.2

0.4

0.6

0.8

1

t

qFIG. 11. The dependence of the transmission coefficient on

q for Ωb = 4.5. Data for acoustic rotobreather and acousticbreather are shown correspondingly by solid and dashed lines.

3 10 10210

-14

10-12

10-10

10-8

10-6

10-4

10-2

100

t

ΩbFIG. 12. The dependence of the transmission coefficient on

the frequency of the breather Ωb at fixed q = 0.21. Scatteringby ”real” rotobreather and a static part of DB are showncorrespondingly by solid and dashed lines.

We conclude this section by stressing that the wavescattering by an acoustic rotobreather is essentially re-lying on the time-dependence of the scattering potential.The rotobreather effectively cuts the chain in weakly in-teracting parts and thus hinders waves from propagationin a very strong way.

V. OPTICAL BREATHERS

In this Section we consider systems with a nonvan-ishing on-site potential V [xn] �= 0 (see Fig. 1c). Thedifference of such systems to acoustic models is the exis-tence of a gap in the spectrum of phonons |ωq=0| = V ′′[0].As a consequence the total mechanical momentum is notconserved, and the transmission coefficient now vanishes

10

not only at q = π but also at q = 0. An exceptionis the case when a localized Floquet eigenstate bifur-cates from the corresponding band edge for some specialparameters [29–31]. Because of the presence of a gapin the plane wave spectrum there are now two differentcases of interest - the breather frequency being locatedoutside the spectrum |Ωb| > max |ωq| or inside the gap|Ωb| < min |ωq|.

A. The case |Ωb| > max |ωq|.

Here we choose V (y) = 12y2 + 1

3y3 + 1

4y4 and W (y) =

c2y

2.In this case the spectrum of phonons is

ω2q = 1 + 4c sin2(

q

2) . (43)

The breather profile for Ωb = 1.5 and c = 0.05 is shownin Fig.13.

-10 -5 0 5 10n

0

0.5

1

0 1 2 πq

10-1

100

101

102

s k

k=2

k=1

x̂n(t

=0)

FIG. 13. The initial displacements of an optical breatherwith Ωb = 1.5 and c = 0.05 (velocities are zero).Inset: sk=1,2 versus q.

The corresponding Fourier components of the scatter-ing potential are plotted in Fig.14.

-10 -5 0 5 10n

10-10

10-8

10-6

10-4

10-2

100

k=0k=1k=2k=3k=4

|vn

,k|

FIG. 14. Fourier components vnk for different k versus nfor the breather in Fig.13.

We note that the DB scattering potential perturbs thediagonal terms and in this particular case (Ωb > ωq)presents a barrier for propagating phonons. For largebreather frequencies the breather is strongly localized,i.e. essentially only one central oscillator is excited. Thetime-averaged scattering potential then becomes a singlediagonal defect with a large strength β0 � Ω2b . It isstraightforward to observe that the transmission coeffi-cient will thus scale as (see Eq.(25))

t̃q ∼ Ω−4b . (44)

Due to the fact that the transmission vanishes exactly forboth q = 0 and q = π we conclude that for large breatherfrequencies transmission is suppressed in general.

However, in the case of an optical DB the time-dependent part of the DB scattering potential (more pre-cisely its second harmonic, see Fig. 14 ) is also large asβ2 � Ω2b for large values of the breather frequency.Thus, the time-dependent part of the DB potential canbe rather important. Indeed, for the breather from Fig.13the obtained transmission as a function of q shows thatt̃q (see Fig.15, dashed line) is at least one order of mag-nitude larger than tq (see Fig.15, solid line). In additiontq shows a resonant minimum around q = 2.1 (see in-set). Let us use the estimation (16) for k = 1, 2. Theparameters are v0,0 = 3.5 , v0,1 = 0.69 , v0,2 = 3. Firstwe find that s1 ≈ 0.05 for all q implying that the k = 1closed channel does not participate in the transmissionprocess. At the same time s2 > 10 for all q and thus,the k = 2 closed channel strongly participates in thetransmission process. It is because the frequency of thepropagating phonon 2Ωb −ωq � 1.92 is close to a local-ized eigenmode of the time-averaged scattering potentialωL = 1.9. Notice here that the frequency of this localmode is above the phonon band. Thus we interprete thesuppression of transmission as a strong coupling betweenthe propagating wave and a particular localized mode ofthe time-averaged scattering potential, mediated by the

11

ac terms of the scattering potential (the k = 2 channel inthis case). However, the simple estimation based on Eq.(16) does not show the resonant suppression of transmis-sion for a particular value of q and therefore, in orderto quantitatively describe this effect we next apply theGreen function method.

0 0.5 1 1.5 2 2.5 π0

0.0005

0.001

0.0015

0.002

t

0 0.5 1 1.5 2 2.5 π10-12

10-10

10-8

10-6

10-4

t

q

qFIG. 15. The dependence of the transmission coefficient tq

on the wave number q. The optical breather frequency isΩb = 1.5 and the coupling c = 0.05. The results are shownfor the time-averaged DB scattering potential (dotted line);the time averaged part and the second harmonic of the DBscattering potential (dashed line); the full DB scattering po-tential (solid line).Inset: same with tq on a logarithmic scale. Note the resonantsuppression around q = 2.1.

By making use of the estimation (16) we take into ac-count in (21) the terms with k = 0 and k = ±2, i. e.the scattering on the static potential and the phononinteraction with the second harmonic of the frequency2Ωb. Moreover, we use the fact that the DB scatteringpotential is a strongly localized one, and obtain (the cor-responding diagram is shown in Fig. 3b)

Gωq(n1, n2) = G̃ωq(n1, n2)+

+β22G̃ωq (n1, 0)G̃(rn)−Ω+ωq(0, 0)Gωq(0, n2) , (45)

where Gωq (n1, n2) is determined by G̃ω(n1, n2) that is theGreen function of propagating phonons in the presence oftime-average DB potential (the corresponding diagram isshown in Fig. 3a):

G̃ω(n1, n2) = G0ω(n1, n2) − β0G0ω(n1, 0)G̃ω(0, n2) .

(46)

Moreover, the central part of Eq. (45), namelyG̃

(rn)−Ω+ωq(0, 0), has a resonant form:

G̃(rn)−Ω+ωq(0, 0) =

1ω2L − (Ω − ωq)2

, (47)

where the local phonon mode frequency ωL is mostly de-termined by a time-average scattering potential. How-ever, in the case as the phonon band is narrow (c 1)the renormalization of ωL due to the ac nonresonantprocesses has to be taken into account (the correspond-ing diagram is shown in Fig. 3c). The Eqs. (45) and(46) can be solved and we obtain

Gωq (0, n2) =G0ωq (0, n2)

1 + (β0 − β22G̃(rn)−Ω+ωq (0, 0))G

0ωq(0, 0)

.

(48)

Thus, we arrive at the expression (25) for the transmis-sion coefficient tq but with the renormalized wave numberdependent parameter β

β = β0 −β22

ω2L − (2Ωb − ωq)2. (49)

As the breather frequency is large (Ωb � 1) both trans-mission coefficients for the static DB scattering potentialand for the full time-periodic DB scattering potential,decrease with the breather frequency according to (44).However, the effective potential strength |β| is larger thanβ0 and correspondingly the transmission tq is smallercompared to the transmission on the static DB scatteringpotential. Indeed, we observed this behaviour by directnumerical simulations (see Fig. 16).

A most peculiar effect is that the presence of ac termin a scattering potential allows to tune the frequency ofa local mode and to obtain the resonant suppression oftransmission. Indeed by taking into account the type ofdiagrams shown in Fig. 3c we obtain for the renormalizedlocal mode frequency ω̃L

ω̃2L = ω2L +

β22(4Ωb − ωq)2 − ω2L

. (50)

This formula is valid in the limit β2 ≤ 2√

2Ωb [39]. Forthe particular case of the DB with frequency Ωb = 1.5and β2 = v02/2 = 1.5 we find that the resonant value ofq � 2 is remarkably close to the resonant suppression oftq around q = 2.1. Thus a strong dependence of ωL onthe amplitude of the ac part of DB scattering potential,and therefore, on the breather frequency, allows easily tochange a position of the resonance.

Note here, that taking into account the static part andthe second harmonic of the DB potential allows to ob-tain a good agreement with the direct numerical simu-lations of a scattering by the ”full” DB (compare solidand dashed lines in Fig. 15). It is also interesting tomention that the static part and the first harmonic ofthe DB potential (k = 1) lead to an increase of thetransmission coefficient compared to the scattering bythe time-averaged DB potential. It is just due to the in-terplay between the strengths of different channels of thescattering potential (see Eq. (49)).

12

Thus, we conclude that the presence of a closed channelallowing absorption and emission of phonons around thecenter of the breather leads to strong interference effectswhich are of destructive nature in the given example ofoptical breather.

100

101

10210

-10

10-8

10-6

10-4

10-2

t

ΩbFIG. 16. The dependence of the transmission coefficient tq

on the optical breather frequency Ωb for a particular value ofwave number q = 0.5. The dashed line corresponds to thescattering by a time-averaged DB scattering potential.

B. The case |Ωb| < min |ωq|

Here we choose the on-site potential in the form V (y) =12y

2− 13y3 (note here that the results do not change if thecubic term has a positive sign) and the interaction termW (y) = c

2y2. The breather profile for Ωb = 0.85 and

c = 0.15 is shown in Fig.17.

-10 -8 -6 -4 -2 0 2 4 6 8 10n

0

0.2

0.4

0.6

0.8

0 1 2 πq

0

2

4

6

s k

k=1

k=2

x̂n(0

)

FIG. 17. The initial displacements of an optical breatherwith Ωb = 0.85 and c = 0.15 (velocities are zero).Inset: sk=1,2 versus q.

The corresponding Fourier components of the scatter-ing potential are plotted in Fig.18.

-10 -8 -6 -4 -2 0 2 4 6 8 10n

10-4

10-3

10-2

10-1

100

k=0k=1k=2k=3k=4

|vn

,k|

FIG. 18. Fourier components vnk for different k versus nfor the breather in Fig.17.

In this case the time-averaged DB scattering potentialcorresponds to a potential well in the diagonal terms, i.e. the parameter β0 in Eq. (46) has a negative sign.It leads again to the possibility to obtain a resonanceas the frequency of an active closed channel matchesthe frequency of a localized phonon mode located in thephonon gap. Moreover, at variance to the optical DBwith large frequency (preceding subsection), the num-ber of active closed channels may now increase substan-tially. This leads to a more complicated interference sce-nario between the open channel and several closed chan-nels. Indeed, the estimation of sk=1,2 shows that the firstchannel is strongly contributing, and the second closedchannel can not be neglected either. At variance to theprevious case for all propagating phonon frequencies thek = 1 closed channel has a much stronger contributionthan the k = 2 one. However, we expect a resonant cou-pling between the propagating wave and the local modethrough the k = 2 closed channel, as |ωq=0−2Ωb| = 0.7 isclose to a local eigenmode of the time-averaged scatter-ing potential with frequency ωL = ±0.78. Moreover, thenonresonant processes in a strong k = 1 channel allow torenormalize ωL similarly to the previous case of an opticalbreather with a large frequency. This resonant effect canbe analyzed by making use of Eqs. (25), (45)-(49). In-deed, due to the resonance with a localized phonon modethe Green function G̃(rn)−2Ωb+ωq (0, 0) � 1 for a particularwave number q0. In this case the parameter β goes toinfinity as the resonant condition, (2Ωb−ωq0)2 −ω2L = 0,is valid and correspondingly the transmission coefficientvanishes. [40]. As the phonon frequency deviates from theresonant condition, the parameter β decreases and cor-respondingly the transmission coefficient reaches a max-imum (see Fig. 19).

13

In Fig.19 we show that the time-averaged DB scatter-ing potential provides with perfect transmission at a cer-tain wave number due to the presence of a quasi-boundstate in the static scattering potential. At the same timethe transmission for the full dynamical problem shows amaximum value of 0.1, and an additional minimum int(q) with actually a full vanishing of transmission. Thesepatterns are entirely absent in the scattering by the time-averaged potential. Thus we again conclude that thepresence of active closed channels inside the breather coreleads to strong interference effects.

0 0.5 1 1.5 2 2.5 π0

0.2

0.4

0.6

0.8

1

t

0 0.5 1 1.5 2 2.5 π10-6

10-5

10-4

10-3

10-2

10-1

t

q

q

FIG. 19. The numerically calculated dependence of thetransmission coefficient tq (solid line) on q for the breatherfrequency Ωb = 0.85 and c = 0.15. The dashed line showsthe scattering by the time-averaged DB scattering potential.Inset: The same on a logarithmic scale.

VI. DISCUSSION

In this paper we considered the propagation of small-amplitude phonons through a nonlinear lattice in thepresence of various discrete breathers. In particular, weobtained the transmission coefficient tq for the cases ofacoustic and optical discrete breathers, and acoustic ro-tobreathers. We have shown that the presence of a DBleads to an effective scattering potential for phonons andthis potential contains both static (time-averaged) partsand time-dependent periodic parts where the first andsecond harmonics are most important. Moreover, thestrength and the width of the scattering potential aredetermined by the breather frequency Ωb.

The static part of the effective DB scattering potentialmay fully describe the scattering outcome if the closedchannels weakly contribute (acoustic breather with β =0). Equally the leading contribution to a finite trans-mission may come from a closed channel (acoustic roto-breather). A very interesting case is realized when a localmode of the time-averaged scattering potential resonates

with a closed channel (acoustic breather with β �= 0, op-tical breathers). This resonance leads to a full suppres-sion of transmission at some value of q. The nonresonantclosed channels renormalize the local mode frequency andthe corresponding position of the transmission zero in q.This effect has been discussed in several papers in relationto the Fano resonance [17,41]. A detailed explanation ofthe similarities and differences to the physics of a Fanoresonances is beyond the scope of this work and will bepublished separately. It is worthwhile to note here thatthe location of a zero transmission value in q is not re-lated to the presence of so-called quasi-bound Floquetstates, i.e. states which by parameter tuning are trans-formed from a localized state into one colliding (inter-acting) with the continous part of the Floquet spectrum.Instead a formally exact definition of a zero transmissionis given in [29] through the asymptotic phase propertiesof Floquet eigenstates far from the breather center.

The phonon scattering by inhomogeneities can be rel-evant with respect to the ongoing discussion about finitevs. infinite heat conductivity in acoustic chains with con-servation of total mechanical momentum Ref. [7–10,42].The heat conductivity κ mediated by noninteractingphonons is determined as

κ � L∫

dqTq(L) , (51)

where L is the size of the system. Here Tq(L) denotes thetransmission of the given system. Thus, obviously in theabsence of inhomogeneities the phonon heat conductiv-ity goes to infinity (κ � L) as the size of the system Lincreases. However, the situation becomes more complexin the presence of inhomogeneities. In the case, when theinhomogeneities are randomly distributed along the sys-tem, the number of inhomogeneities N = nL, where n isthe concentration of defects, and the total transmissionTq(L) � tNq with tq being the transmission through agiven defect. We obtain that the behaviour of heat con-ductivity is determined by the values of q where tq is closeto one. Thus, the problem of an infinite heat conductiv-ity in the limit of a large size L naturally appears in thesystems preserving the total mechanical momentum.

In particular we can apply this general considerationto the phonon propagation in the presence of DBs. Inthe most interesting case of acoustic rotobreathers weobtain that the transmission coefficient tq � 1 − αq2in the limit of small wave numbers. The coefficient α isdetermined by the frequency Ωb and the specific choiceof the acoustic rotobreather. Thus we argue that themere presence of acoustic rotobreathers does not lead toa finite heat conductivity and κ is still divergent in thelimit of large L as

κ �√

L

nα. (52)

14

Here we assume that the heat is carried by phonons,and did not take into account phonon-phonon interac-tions. Another conclusion drawn from our results foracoustic breathers versus acoustic rotobreathers is thatthe q-domain around q = 0 where the transmission isclose to one is by orders of magnitude smaller for acousticrotobreathers as compared to acoustic breathers. This re-gion is responsible for the quasiballistic transport of en-ergy by large wavelength phonons in the hydrodynamicregime and for the appearance of anomalous heat conduc-tivity. Numerical simulations for systems with acousticrotobreathers have to be performed on length and timescales which are thus also orders of magnitude larger thanthe corresponding simulations for standard FPU chains.

Our analysis can be also applied to the electromagneticwave propagation through various Josephson transmis-sion lines. Although these systems are intrinsically dis-sipative, the dissipation may be rather small, and thetransmission tq is still determined by the presence of in-homogeneities like rotobreathers [22,23,26,27] or dynamicedge states [43]. Moreover, in this particular case ofJosephson lattices the properties of DBs can be easilytuned experimentally by changing the external dc bias.Notice here that local phonon modes located on a ro-tobreather play an important role in various switchingprocesses [27]. A study of an electromagnetic wave prop-agation may allow a direct observation of this mode.

Finally, a similar analysis can be carried out for aSchrödinger equation in the presence of artificially ap-plied time-periodic perturbations [39]. This case is im-portant for the description of electron transport througha quantum wire in the presence of external electromag-netic fields.

AcknowledgementsWe thank S. Aubry, G. Casati, S. Kim and L. E. Reichlfor useful discussions. This work was supported by theDeutsche Forschungsgemeinschaft FL200/8-1. M. V. F.thank the Alexander von Humboldt Stiftung for partialsupporting this work.

APPENDIX A: TIME-AVERAGED SCATTERINGPOTENTIAL FOR ACOUSTIC BREATHER

We take into account the oscillations of two lattice sitesin the center of a DB only. These oscillations evolveexactly in antiphase with large amplitudes. The effectiveHamiltonian is written as

E =ẋ212

+ẋ222

+14(x41 + x

42 + (x1 − x2)4) (A1)

and with x1 = −x2, we arrive at the single oscillatorproblem with energy

E = ẋ2 +92x4. (A2)

The frequency of oscillation (the breather frequency) isgiven by

1Ωb

=2π

xm∫0

d x√E − 92x4

, (A3)

where xm = 4√

2E9 . After integration we obtain

Ωb =2π 4

√9E2

B(14, 1

2), (A4)

where B(x, y) is the B-function [44]. Next we computethe value of X =

√3〈x2〉. We can express 〈x2〉 in terms

of the energy

〈x2〉 = 2Ωbπ

xm∫0

x2d x√E − 92x4

. (A5)

After some algebra we find 〈x2〉 =√

2E9

B( 12 ,34 )

B( 14 ,12 )

and

X =

√23π

Ωb. (A6)

APPENDIX B: TIME-AVERAGED SCATTERINGPOTENTIAL FOR ACOUSTIC ROTOBREATHER

In order to calculate the time-averaged off-diagonalhopping terms we will take into account the central sitewhich is in a rotational state, denoted by φ, and two near-est neighbor oscillators (denoted by α1 and α2). Thus weconserve the total mechanical momentum. The energy ofsuch a system is

E = α̇2 +φ̇2

2+ 2(1 − cos(φ − α)). (B1)

The two equations of motion are given by

φ̈ = −2 sin(φ − α),α̈ = sin(φ − α). (B2)

Introducing new variables u = φ + 2α and w = φ− α werewrite the system (B2) as

ü = 0,ẅ = −3 sinw. (B3)

The relevant energy part is

Ẽ =ẇ2

2+ 3(1 − cos w). (B4)

15

Now we compute the average coupling between rotationaland oscillatory states for large frequencies

� = 〈cos w〉 = Ωb2π

√2Ẽ

2π∫0

cos w dw√1 − 3(1−cosw)

Ẽ

. (B5)

Because Ẽ ≈ Ω2b

2we find

� ≈ Ωb2π

√2Ẽ

2π∫0

(1 +3(1− cosw)

2Ẽ) cos w dw (B6)

which leads to

� = − 3Ωb4√

2Ẽ3/2(B7)

and finally to

� = − 32Ω2b

. (B8)

[1] P. Sheng, B. White, Z. Zhang, and G. Papanicolaou,Scattering and localization of classical waves in randommedia, World Scientific, Singapore, (1989).

[2] V. I. Klyatskin, A. I. Saichev, Sov. Phys. Usp. 35, 231(1992) [Usp. Fiz. Nauk 162, 161 (1992)].

[3] I. M. Lifshitz, S. A. Gredeskul, and L. A. Pastur, Intro-duction to the theory of disordered systems, Wiley, NewYork (1988).

[4] W. Zwerger, Theory of coherent transport in: Quantumtransport and dissipation, Wiley, New York (1988).

[5] S. A. Gurvitz, and Y. B. Levinson, Phys. Rev. B 47,10578 (1993).

[6] P. J. Price, Phys. Rev. B 48, 17301 (1993).[7] A. V. Savin, and O. V. Gendelman, Phys. Solid State 43,

355 (2001).[8] S. Lepri, R. Livi, and A. Politi, Phys. Rev. Lett. 78, 1896

(1997).[9] S. Lepri, R. Livi, and A. Politi, Europhys. Lett. 43, 271

(1998).[10] P. Tong, B. Li, and B. Hu, Phys. Rev. B 59, 8639 (1999)[11] B. I. Ivlev, and V. I. Melnikov, JETP Lett., 41 142

(1985).[12] S. V. Lempitskii, Sov. Phys. JETP, 67 2362 (1988).[13] P. Hänggi, Driven Quantum Systems, in: Quantum

transport and dissipation, Wiley, New York (1988).[14] U. Fano, Phys. Lett. 124, 1866 (1961).[15] J. U. Nöckel, and A. D. Stone, Phys. Rev. B 50, 17415

(1994).[16] C. S. Kim and A. M. Satanin, J.Phys.: Condens. Matter

10, 10587 (1998).[17] S. W. Kim, and S. Kim, Phys. Rev. B 63, 212301 (2001)

[18] A. J. Sievers, and J. B. Page, in Dynamical propertiesof solids VII phonon physics the cutting edge, Elsevier,Amsterdam, 1995.

[19] S. Aubry, Physica D, 103, 201 (1997)[20] S. Flach, and C. R. Willis, Phys. Rep. 295, 181 (1998).[21] H. S. Eisenberg, Y. Silberberg, R. Morandotti,

A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81,3383 (1998).

[22] E. Trias, J. J. Mazo, and T. P.Ornaldo, Phys. Rew. Lett.84, 741 (2000).

[23] B. Binder, D. Abraimov, A. V. Ustinov, S. Flach, andY. Zolotaryuk, Phys. Rew. Lett. 84, 745 (2000).

[24] U. T. Schwarz, L. Q. English, and A. J. Sievers, Phys.Rev. Lett. 83, 223 (1999)

[25] B. I. Swanson, J. A. Brozik, S. P. Love, G. F. Strouse,A. P. Shreve, A. R. Bishop, and W.-Z. Wang, Phys. Rev.Lett. 82, 3288 (1999)

[26] E. Trias, J. J. Mazo, A. Brinkman, and T. P. Orlando,Physica D 156, 98 (2001).

[27] A. E. Miroshnichenko, S. Flach, M. V. Fistul, Y.Zolotaryuk, and J. B. Page, Phys. Rev. E, 64, 066601(2001).

[28] S. Flach, and C. R. Willis, in: Nonlinear Excitations inBiomolecules, ed. by M. Peyrard, Editions de Physique,Springer-Verlag, 1995

[29] T. Cretegny, S. Aubry and S. Flach, Physica D 199, 73(1998).

[30] S. W. Kim and S. Kim, Physica D 141, 91 (2000).[31] S. Kim, C. Baesens, and R. S. MacKay, Phys. Rev. E 56,

R4955 (1997).[32] S. Takeno and M. Peyrard, Physica D 92, 140 (1996).[33] T. Cretegny and S. Aubry, Phys. Rev. B 55, R11929

(1997)[34] E.N. Economou, Green’s functions in quantum physics,

Berlin, Springer-Verlag, 1990.[35] J. L. Marin, and S. Aubry, Nonlinearity 9, 1501 (1996).[36] S. Flach, Phys. Rev. E 50, 3134 (1994)[37] B. Dey, M. Eleftheriou, S. Flach, and G. P. Tsironis,

Phys. Rev. E 65, 017601 (2001)[38] P.M. Morse and H. Feshbach, Methods of Theoretical

Physics, McGraw-Hill, New York, 1953.[39] Notice here, that a resonant suppression of phonon trans-

mission can be also realized in an ac driven harmoniclattice in the presence of a static scattering potential. Inthis case the expression (50) for the frequency of a localphonon mode is still valid with substitutions 2Ωb to ω,and β2 to A/2, where ω and A are correspondingly thefrequency and amplitude of an external ac drive. More-over, a similar resonant suppression of transmission canbe observed as an electron moving in a periodic poten-tial, scatters on a time-dependent potential. The partic-ular condition for this effect can be written in a simpleform: E ± ω − ẼL = 0, where E is the electron energy.The renormalized value of the local energy level ẼL inthe limit of small A (more precisely A �

√ω ) is given

as ẼL = EL ± A2

4ω, where EL is a local energy level of a

static potential.[40] In this particular case the k = 1 channel strongly con-

tributes. In order to calculate the renormalized value ofωL we have to take into account many different diagrams

16

of the kind shown in Fig. 3c. Although it can be done inthe same manner as in the case of a weak renormaliza-tion, the expression for ωL looses its transparency.

[41] A. Emmanouilidou and L. E. Reichl, Phys. Rev. A 65,033405 (2002); D. F. Martinez and L. E. Reichl, Phys.Rev. B 64, 245315 (2001).

[42] S. Lepri, R. Livi, A. Politi, cond-mat/0112193[43] M. V. Fistul and J. B. Page, Phys. Rev. E 64, 036609

(2001)[44] I. S. Gradshteyn, and I. M. Ryzhik, Table of integrals,

series, and products, Acad. Pr., New York, 2000

17