Physica D 216 (2006) 62–70www.elsevier.com/locate/physd

Discrete nonlinear Schrodinger approximation of a mixedKlein–Gordon/Fermi–Pasta–Ulam chain: Modulational instability and a

statistical condition for creation of thermodynamic breathers

Magnus Johansson∗

Max Planck Institute for the Physics of Complex Systems, Nothnitzer Strasse 38, D-01187 Dresden, GermanyDepartment of Physics and Measurement Technology, Linkoping University, SE-581 83 Linkoping, Sweden

Available online 15 March 2006

Abstract

We analyze certain aspects of the classical dynamics of a one-dimensional discrete nonlinear Schrodinger model with inter-site as well ason-site nonlinearities. The equation is derived from a mixed Klein–Gordon/Fermi–Pasta–Ulam chain of anharmonic oscillators coupled withanharmonic inter-site potentials, and approximates the slow dynamics of the fundamental harmonic of discrete small-amplitude modulationalwaves. We give explicit analytical conditions for modulational instability of travelling plane waves, and find in particular that sufficiently stronginter-site nonlinearities may change the nature of the instabilities from long-wavelength to short-wavelength perturbations. Further, we describethermodynamic properties of the model using the grand-canonical ensemble to account for two conserved quantities: norm and Hamiltonian. Theavailable phase space is divided into two separated parts with qualitatively different properties in thermal equilibrium: one part correspondingto a normal thermalized state with exponentially small probabilities for large-amplitude excitations, and another part typically associated withthe formation of high-amplitude localized excitations, interacting with an infinite-temperature phonon bath. A modulationally unstable travellingwave may exhibit a transition from one region to the other as its amplitude is varied, and thus modulational instability is not a sufficient criterionfor the creation of persistent localized modes in thermal equilibrium. For pure on-site nonlinearities the created localized excitations are typicallypinned to particular lattice sites, while for significant inter-site nonlinearities they become mobile, in agreement with well-known properties oflocalized excitations in Fermi–Pasta–Ulam-type chains.c© 2006 Elsevier B.V. All rights reserved.

Keywords: Anharmonic oscillator chain; Modulational instability; Breather formation; Thermal equilibrium

1. Introduction

The recent interest in nonlinear localized modes inanharmonic lattices, discrete breathers, from many diversescientific fields has largely been inspired by the work and ideasof Serge Aubry; see, e.g., [1–10], and [11–13] for reviews.The conditions for having discrete breathers as exact localizedsolutions in generic anharmonic lattices are by now wellunderstood, in terms of the absence of resonances with linear(phonon) excitations. However, in spite of a large number of

∗ Corresponding address: Department of Physics and MeasurementTechnology, Linkoping University, SE-581 83 Linkoping, Sweden. Tel.: +4613 281227; fax: +46 13 132285.

E-mail address: mjn@ifm.liu.se.URL: www.ifm.liu.se/∼majoh.

0167-2789/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2005.12.021

works on the topic (see, e.g., [3,5,14–27]), are still there manyremaining questionmarks regarding under which conditionspersistent localized breathers may spontaneously form fromnonlocalized initial conditions, and regarding their existencein thermal equilibrium. Typically, these works have shownthat the spontaneous creation of breathers is associated withmodulational instability (MI) of plane waves. Small-amplitudebreathers may then coalesce and grow through inelasticbreather–breather and breather–phonon scattering processes,leaving a few (or one single) large-amplitude breather(s), whichfinally after very long times may disappear due to interactionswith phonons corresponding to the modulationally stable partof the spectrum. Although this scenario gives a reasonablyclear qualitative picture, a convincing quantitative analysis isstill lacking, as much for Klein–Gordon (KG) (anharmonic

M. Johansson / Physica D 216 (2006) 62–70 63

on-site potential) as for Fermi–Pasta–Ulam (FPU) (anharmonicinteraction potential) lattices.

On the other hand, for the discrete nonlinear Schrodinger(DNLS) model, more quantitative results are known. Firstly,going beyond linear scattering theory to higher order inthe perturbation expansion, analysis of interactions betweenbreathers (which for DNLS lack higher-harmonics) and small-amplitude phonons quantitatively confirmed the predictionthat phonons with different wave numbers may lead toeither growth or decay of the breather [6,28]. Secondly,due to its norm-conserving character, a sharp transition hasbeen identified [29,10,30], separating those initial conditions(of non-zero energy density) that do generate localizedbreathers after thermalization from those that do not. Usingstandard Gibbsian statistical mechanics for the grand-canonicalensemble, this transition was associated [29] with a phasetransition.

However, as is well known (see, e.g., [31,21,11,9]), theDNLS model can be derived as an approximation of the slowmodulational small-amplitude dynamics for a generic classof KG-lattices. Thus, under the appropriate conditions, weshould expect that the transition between breather-formingand non-breather-forming initial conditions should be visiblealso in the KG dynamics, although not necessarily as a strictdivision between different kinds of equilibrium states (we donot expect phase transitions in generic 1D KG-lattices) butrather appearing in transient processes which, however, maybe extremely long. Examples of this were shown in [9], andrecently the connection between the KG and DNLS dynamicswas further clarified [32] by deriving general approximateexpressions for the transition line in terms of direct propertiesof the KG initial state, and by showing that the KG quantitiescorresponding to the DNLS Hamiltonian and norm, althoughnot strictly conserved, indeed vary only over very large time-scales as long as the oscillation amplitudes are not too large(low-temperature regime).

Thus, understanding the breather-forming mechanisms inDNLS lattices appears to be an important step towards thefull understanding of the corresponding mechanisms in generalanharmonic lattices. For example, it was shown in [29] thatthe nonlinear phonon corresponding to the unstable bandedge, although it is modulationally unstable for all amplitudes,crosses the transition line at some critical amplitude andthus generates localized equilibrium breathers only below thisthreshold. So MI is apparently not a sufficient condition forbreather formation. In this paper, we aim at discussing thecorresponding processes for lattices of mixed KG/FPU-type, inthe framework of an extended DNLS equation with both on-siteand inter-site nonlinearities, which can be derived e.g. througha rotating-wave type approximation (cf. e.g. [15,33,34] for thepure FPU case). In Section 2 we give the derivation of thismodel, which we term the KG–FPU–DNLS equation, focusingfor simplicity on the particular case where both the on-site andinter-site anharmonic terms in the Hamiltonian are hard andquartic. In Section 3 we give analytical conditions for MI ofits travelling plane-wave solutions, in Section 4 we describeits statistical properties in the grand-canonical ensemble in an

analogous manner as for the pure on-site DNLS model in [29,32] and derive explicit conditions for the transition into thebreather-forming regime for particular initial conditions, and inSection 5 we make some concluding remarks.

2. The KG–FPU–DNLS model

The general form of the Hamiltonian for a mixed KG/FPU-chain can be written as follows:

H =

N∑n=1

[12

u2n + V (un)+ W (un+1 − un)

], (1)

where the general on-site potential V (u) for small-amplitudeoscillations can be expanded as

V (u) = ω20

u2

2+ α

u3

3+ β ′

u4

4+O(u5), (2)

and the inter-site potential W (u) as

W (u) = k2u2

2+ k3

u3

3+ k4

u4

4+O(u5). (3)

Restricting to small-amplitude oscillations in symmetricpotentials (α = k3 = 0), the equations of motion then takethe form (to O(u5))

un = −ω20un − β ′u3

n + k2(un+1 + un−1 − 2un)

+ k4

[(un+1 − un)

3+ (un−1 − un)

3]. (4)

Considering small-amplitude solutions un(t) with typicaloscillation amplitudes |un| ∼ ε, they can be formally expandedin a Fourier series as

un(t) =

∞∑p=−∞

a(p)n eipωbt , (5)

where ωb is close to some linear oscillation frequency and theFourier coefficients are slowly depending on time, a(p)n (ε2t).Due to exponential decay of the Fourier coefficients in p, theymust satisfy a(p)n ∼ ε p for p > 0, while a(0)n ∼ ε2. Moreover,a(p)n = a(−p)∗

n since un is real. Inserting (5) into (4) yields:∑p

[a(p)n + 2ipωba(p)n + (ω2

0 − p2ω2b)a

(p)n − k2(a

(p)n+1

+ a(p)n−1 − 2a(p)n )]

eipωbt+ β ′

[∑p

a(p)n ei pωbt

]3

− k4

[∑

p(a(p)n+1 − a(p)n )eipωbt

]3

+

[∑p(a(p)n−1 − a(p)n )eipωbt

]3 = 0 +O(ε5). (6)

Then, from (6) we immediately obtain a DNLS-like equation toO(ε5) by considering the fundamental harmonic p = 1:

64 M. Johansson / Physica D 216 (2006) 62–70

2iωba(1)n +

(ω2

0 − ω2b

)a(1)n − k2(a

(1)n+1 + a(1)n−1 − 2a(1)n )

+ 3β ′|a(1)n |

2a(1)n − 3k4

[∣∣∣a(1)n+1 − a(1)n

∣∣∣2(a(1)n+1 − a(1)n )

+

∣∣∣a(1)n−1 − a(1)n

∣∣∣2(a(1)n−1 − a(1)n )

]= 0. (7)

Assuming hard potentials (β ′, k2, k4 ≥ 0, ω2b ≥ ω2

0) andrescaling coefficients as

a(1)n = (−1)n√

2ωb

3β ′ + 6k4ψne

iω2

0−ω2b+2k2

2ωbt,

C =k2

2ωb≥ 0, Q =

k4

β ′ + 2k4, (8)

we may rewrite Eq. (7) in the form that we will term theKG–FPU–DNLS equation:

iψn + C(ψn+1 + ψn−1)+ |ψn|2ψn

+ Q[2ψn(|ψn+1|

2+ |ψn−1|

2)+ ψ∗n (ψ

2n+1 + ψ2

n−1)

+ 2|ψn|2(ψn+1 + ψn−1)+ ψ2

n (ψ∗

n+1 + ψ∗

n−1)

+ |ψn+1|2ψn+1 + |ψn−1|

2ψn−1

]= 0. (9)

(The coupling parameter C could be removed by furtherrescalings (e.g. [9]), but we keep it here for convenience.)Thus, the parameter Q, 0 ≤ Q ≤

12 , measures the relative

strength of inter-site and on-site anharmonicity in (4), withQ = 0 corresponding to pure on-site and Q =

12 to pure

inter-site anharmonicity. Eq. (9) appears as a special case ofa more general extended DNLS equation, which has beenintroduced recently [35–38] as a model for an array of opticalwaveguides embedded in a nonlinear medium; putting C = 1,it is equivalent to Eq. (2) in [38] with z = −t and K4 = K5 =

Q/2.Let us remark that, although formally Eq. (9) with Q =

12

can be used to also describe the pure FPU-model without on-site potential (ω0 = β ′

= 0), the assumption of separation oftime-scales used in its derivation breaks down when ωb → 0,and thus it cannot be expected to give a good description oflong-wavelength modes in the FPU model due to the acousticnature of the corresponding linear phonons. By contrast, in [15,33,34] ωb was chosen as the top phonon band frequency of thezone boundary mode, and in this case Eq. (9) with Q =

12

is expected to give a good approximation of short-wavelengthsmall-amplitude modes (such as localized breathers that havegenerally out-of-phase oscillations at neighboring sites for hardanharmonicity) also for the pure FPU case. Let us also forclarity remark that short-wavelength modes of the original(hard) KG–FPU model (4) correspond to long-wavelengthmodes of the (soft) KG–FPU–DNLS model (9) (and vice versa)through the staggering transformation in Eq. (8).

3. Modulational instability of the KG–FPU–DNLS model

Let us consider a general nonlinear travelling (plane) wavewith wave vector q , |q| ≤ π , which is an exact stationary

solution to Eq. (9) with frequency Λ of the form

ψn =√

aeiqneiΛt ,

Λ = a(1 + 2Q)+ 2 cos q [C + 2Qa(2 + cos q)] . (10)

The conditions for plane wave MI were obtained for themore general model in [38]. (For the simpler case of ordinaryDNLS, Q = 0, see also [31].) Analogously, perturbing theexact solution (10) as ψn =

[√a + ueiκn

+ v∗e−iκn]

ei(qn+Λt),inserting in (9) and keeping only linear terms in u and v,a linear eigenvalue problem with q and κ as parameters isobtained for the perturbation vector (u, v). The plane wavewith wave vector q is linearly stable if and only if thecorresponding eigenfrequenciesω± are real for all κ . Explicitly,the eigenfrequencies are obtained as (cf. Eqs. (9)–(10) in [38]):

ω± = 2 sin q sin κ [C + 4Qa(1 + cos q)]

±

√16F1sin2 κ

2(F2sin2 κ

2− aF3),

F1 = (C + 2Qa) cos q + 2Qa cos 2q,

F2 = (C + 6Qa) cos q + 2Qa(2cos2q + 1),

F3 =12

+ Q + 2Q cos q(2 + cos q). (11)

Since F3 ≥ 0 for all q when 0 ≤ Q ≤12 , the condition to

have MI for long-wavelength perturbations (κ → 0) is F1 > 0,which happens when

|q| < q(0)max

≡ arccos

14

√( C

2Qa+ 1

)2

+ 8 −C

2Qa− 1

. (12)

For small QaC , Eq. (12) gives q(0)max ≈

π2 −

2QaC →

π2 ,

QaC → 0, in agreement with the well-known result for

ordinary DNLS [31], while for large QaC we obtain q(0)max ≈

π3 +

C6√

3Qa→

π3 , Qa

C → ∞. So an effect of a non-

zero inter-site nonlinearity Q 6= 0 is to decrease the regionof plane waves which are unstable towards long-wavelengthperturbations. However, this does not necessarily imply thatwaves with |q| > q(0)max are linearly stable, since additionalregions of short-wavelength instabilities centered around κ = π

may develop. From (11) we see that this happens when F1 < 0and F2 − aF3 > 0, which gives the condition to have MI forshort-wavelength perturbations as

q(0)max < |q| < q(π)max

≡ arccos

12

√( C

2Qa+ 1

)2

− 2 +1Q

−C

2Qa− 1

. (13)

When aC is small, Eq. (13) gives q(π)max ≈

π2 −

aC (

12 − Q) →

π2 ,

aC → 0, while in the large-amplitude limit a

C → ∞ we get

M. Johansson / Physica D 216 (2006) 62–70 65

q(π)max → arccos[

12

(√1Q − 1 − 1

)]. Note that when Q =

12

(corresponding to pure inter-site anharmonicity in (4)), q(π)max =

π/2, independent of a/C . Comparing Eqs. (12) and (13), wesee that the condition q(0)max < q(π)max can be fulfilled only forsufficiently large inter-site nonlinearities, Q > Qmin(a/C),where Qmin is a monotonously increasing function of a/C withQmin(0) =

16 and Qmin(∞) =

15 . Thus, the former value is

the smallest value of Q for which short-wavelength MI maydevelop. One may show that no additional linear instabilitiescan occur for |q| > qmax ≡ max(q(0)max, q(π)max) and, sinceqmax <

π2 for all Q, 0 < Q < 1

2 , the simultaneous on-site andinter-site nonlinearity always increases the regime of linearlystable plane waves.

Let us finally also remark that an analysis of MI fortravelling waves in the pure FPU model within a rotating-waveapproximation was performed in [39], which also reportedadditional instabilities corresponding to short-wavelengthperturbations (relative to the original wave). However, animportant difference is that, for the FPU chain in fact, alltravelling waves were found to be linearly unstable, whichwas also confirmed by a Floquet analys of numerically exactsolutions in [40]. The failure of the DNLS approximationto describe these additional instabilities for |q| > π

2 is

related to the assumption that the term a(1)n in (6) can beneglected, which is not generally justified for a pure FPU chainwith long-wavelength oscillations (corresponding to short-wavelength modes in the DNLS model due to the staggeringtransformation in (8)) since their linear oscillation frequenciesωb are small. In addition, higher-harmonic resonances may alsoyield instabilities. In fact, also for the KG model regimes ofadditional instabilities, appearing due to finite values of ω2

0 andnot describable within the simple DNLS model, were discussedin [41].

4. Statistical mechanics of the KG–FPU–DNLS model

Analogously to the ordinary DNLS equation, Eq. (9) has twoconserved quantities: Hamiltonian H and excitation number(norm) N (cf. Ref. [36], Eqs. (7–9)). Expressed in canonicalaction-angle coordinates defined through the transformationψn =

√Aneiφn , the Hamiltonian for a chain of N sites can

be written as

H =

N∑n=1

{2C√

An An+1 cos(φn − φn+1)+A2

n

2

+ Q√

An An+1

[√An An+1 (2 + cos 2(φn − φn+1))

+ 2 (An + An+1) cos(φn − φn+1)] }

, (14)

and the norm

N =

N∑n=1

An . (15)

We first prove that the staggered stationary homogeneous plane-wave solution, Eq. (10) with q = π and a = N /N , minimizes

H at fixedN and N , for all 0 ≤ Q ≤12 . The minimum value for

the energy density is H(min)/N = −2CNN +

(12 − Q

) (NN

)2.

To prove this, we write, using (15),

H−H(min)≥ 2C

N∑n=1

(An −

√An An+1

)

+

N∑n=1

A2n

2−

(12

− Q

)1N

(N∑

n=1

An

)2

+ QN∑

n=1

[An An+1

(1 + 2cos2(φn − φn+1)

)+ 2

√An An+1 (An + An+1) cos(φn − φn+1)

]≥ Q

{N∑

n=1

[An + An+1 +√

An An+1(cos(φn − φn+1)

+ sin(φn − φn+1))][An + An+1

+√

An An+1(cos(φn − φn+1)

− sin(φn − φn+1))] −12

N∑n=1

(A2

n + A2n+1

)}

≥Q

2

N∑n=1

(√An+1 −

√An

)4≥ 0. (16)

For the second inequality in (16) the condition 0 ≤ Q ≤12 has

been used, and the third inequality is obtained by noting that thefunction attains its minimum value for φn+1 − φn = π . Noticealso that when 0 ≤ Q < 1

2 , H(min)

N is bounded from below as

a function of NN , with the global minimum H(min)

N = −C2

12 −Q

obtained for NN =C

12 −Q

.

Having established the existence of a ground state, wenow proceed in an analogous manner as in [29,32], usingstandard Gibbsian equilibrium statistical mechanics to predictmacroscopic average values for a thermalized state in thethermodynamic limit N → ∞, by treating the norm Nas being analogous to ‘number of particles’ in the grand-canonical ensemble. The grand-canonical partition function isthen defined as

Z =

∫∞

0

∫ 2π

0

N∏n=1

dφndAne−β(H+µN ), (17)

where β ≡ 1/T (in units of kB ≡ 1) and µ play the rolesof inverse temperature and chemical potential, respectively. Asin [32], our main focus here is to investigate the boundaryof the regime in (N ,H) parameter space where the Gibbsianapproach is valid, with well-defined chemical potential and(positive) temperature. As was first shown in Ref. [29], thisregime does not cover the full parameter space, but is boundedby the line defined by β = 0, µ = ∞, with βµ ≡ γ finite,describing the relation between energy and norm in the high-temperature limit. Since it is possible to continuously changeN andH by varying parameters of the initial conditions in sucha way that the boundary line is crossed, a transition will occur

66 M. Johansson / Physica D 216 (2006) 62–70

into a regime where persistent localized modes are created. Asargued in [29], this transition corresponds to a phase transitionin the sense that it is associated with a discontinuity of thegrand-canonical partition function. For finite lattices and time-scales, this was associated with a transition from a positive-temperature to a negative-temperature type behavior [29,32],arising due to the decrease of entropy with energy at highenergy densities in the microcanonical ensemble.

Using (14)–(15), we may write the partition function as

Z =

N∏n=1

∫∞

0dAn Iφn+1−φn (An, An+1)e

−βAn

(An2 +Q An+1+µ

),

(18)

with the integral over the phase variables

Iφ(An, An+1)

=

∫ 2π

0dφe

−2β√

An An+1

[C+Q

(An+An+1+

√An An+1 cosφ

)]cosφ

.

(19)

Since we are mainly interested in the regime around thetransition line β = 0, we evaluate the integrals close to thehigh-temperature limit β → 0+ by expanding in power seriesof β. Thus, expanding the exponential in (19), performing theintegration and inserting in (18) we obtain

Z = (2π)N∫

∞

0

N∏n=1

dAn

(1 − βQ An An+1 +O(β2)

)× e

−βAn

(An2 +Q An+1+µ

). (20)

Note that, in contrast to the ordinary DNLS equation (Q = 0)where the integral Iφ is simply a Bessel function [29,32] andcan be approximated by Iφ ≈ 2π +O(β2), here it is crucial toalso take into account its dependence on (An, An+1) due to thenonlinear nearest-neighbour interaction term appearing to firstorder in β.

The integrals in (20) may then be evaluated recursively, toO(β2). Assuming, for simplicity, rigid boundary conditions(which should not matter for large N ) and expanding theexponential in (20), we may write

Z = (2π)N∫

∞

0

N∏n=2

dAne−βµAn

(1 −

βA2n

2− 2βQ An An+1

)× I1(A2)+O(β2), (21)

with

I1(A2) =

∫∞

0dA1e−βµA1

(1 −

βA21

2− 2βQ A1 A2

)

=1βµ

[1 −

β

βµ

(2Q A2 +

1βµ

)].

Evaluating analogously, in turn, the integrals over all theremaining action variables A2, . . . , AN , for the partition

function in the high-temperature limit we finally obtain

Z = (2π)N 1

(βµ)N

[(1 −

β

(βµ)2

)N

−(N − 1)2Qβ

(βµ)2

]+O(β2). (22)

For small β, β = O(1/N ), and large N (N − 1 ≈ N ), thisreduces to lnZ ' N ln(2π) − N ln(βµ) −

N (2Q+1)β(βµ)2

, so that

we have, in the high-temperature limit for the average energy:

〈H〉 =

(µ

β

∂

∂µ−∂

∂β

)lnZ '

N (2Q + 1)

(βµ)2, (23)

and for the average norm

〈N 〉 = −1β

∂ lnZ∂µ

'N

βµ. (24)

Thus, the relation between the energy density h ≡〈H〉

N and the

norm density a ≡〈N 〉

N in the high-temperature limit is

h = (2Q + 1)a2. (25)

Note that the quantity γ = βµ indeed is well-defined and finitein the high-temperature limit for any non-zero norm density;γ ' 1/a, according to (24).

So we may conclude that, qualitatively, the statisticalproperties in thermal equilibrium for the KG–FPU–DNLSequation (9) with 0 < Q ≤

12 are similar to those of the

ordinary DNLS equation (Q = 0), and in analogy with [29,32] we should expect to find a localization transition at the linein (a, h) parameter space defined by (25) (cf. Fig. 1 of Ref. [29]for Q = 0). Thus, for any given norm density a, typicalinitial conditions with (Hamiltonian) energy density h smallerthan the critical value (25) are expected to thermalize (after‘sufficiently’ long times) according to a Gibbsian equilibriumdistribution at temperature T = 1/β and chemical potentialµ, with exponentially small probabilities for large-amplitudeexcitations. In this regime, there is a 1–1 correspondencebetween (a, h) and (β, µ), which may be found numericallyas illustrated for Q = 0 in Ref. [29], or even analytically in thesmall-amplitude limit a → 0 as shown in Ref. [30]. Numericalevidence that such a thermalization generally takes place (forQ = 0) after sufficiently long integration times were givenin [29,32].

On the other hand, for initial conditions with energydensity h larger than the critical value (25) this descriptionbreaks down, and typically one has found numerically that forpure on-site nonlinearities, persistent large-amplitude standingbreathers are created from initial conditions belonging tothis regime [29,30,32]. Some heuristic explanations for thisbehaviour have been given in [29,30,32]. Essentially, theseexplanations are building on the fact that, in the microcanonicalensemble (fixed N , H, and N ), the energy H also has anupper bound, which is realized for a localized breather solutionwhich is essentially unique [42], having an energy scaling withthe norm as H(max)

∝ N 2 for large N . Thus, the entropyS [∝ logarithm of number of microstates] must decrease for

M. Johansson / Physica D 216 (2006) 62–70 67

Fig. 1. Numerically obtained time evolution of (9) with Q = 0.5 and C = 1 over different time scales, with initial condition taken as a randomly perturbed (∼10−6)constant-amplitude solution (10) with q = 0 and a = 1. N = 1000; periodic boundary conditions.

large h (see e.g. Fig. 3 in Ref. [30] for an explicit illustrationin the limit of small a) and, since β = ∂S/∂H|N ,N <

0 in the microcanonical ensemble, the system will operatein a regime of negative-temperature type behaviour where itis statistically favourable to create large-amplitude localizedexcitations. However, the description in terms of negativetemperatures breaks down in the thermodynamic limit (N →

∞ at constant a and h) where H(max)→ ∞. For the

same reason, negative temperatures are not compatible withthe grand-canonical partition function (17) since the integralsdiverge. Instead, it was proposed [30], and to some extentnumerically confirmed [30,32], that above the transition line(25) the equilibrium state in the thermodynamic limit becomesdivided into two weakly interacting parts: one small-amplitudepart corresponding to fluctuations described by a normalGibbsian distribution at T = ∞, and one large-amplitude partcorresponding to localized breathers.

Let us now choose initial conditions specifically as planewaves (10), and investigate whether these families of solutionsmay cross the transition line (25) when varying the amplitude.For modulationally unstable waves, this is expected to tellus whether, after thermalization, these initial conditions willgenerate persistent high-amplitude localized modes or not.From (10) and (14), we immediately obtain the relation betweenthe Hamiltonian and norm densities for families of plane waveswith wave vector q as

h = 2Ca cos q + a2[

12

+ Q(2 + cos 2q + 4 cos q)

]. (26)

Comparing (25) and (26), we find the condition for a plane waveto be in the ‘normal’ Gibbsian positive-temperature regime as

a > a(c) ≡4C cos q

1 + 2Q(1 − 2cos2q − 4 cos q). (27)

From (27), we can make some interesting observationsregarding the effect of the inter-site nonlinearity. First, just asfor the ordinary DNLS case [30,32], for any q with |q| <

π/2 we always expect statistical localization for small enoughamplitudes a, while for π/2 < |q| < π one is alwaysin the normal thermalizing regime. However, the locationof the predicted threshold for |q| < π/2 depends on Q,and in particular there is a critical value Q = Q(c)(q) ≡

12(2cos2q+4 cos q−1)

where the denominator of (27) becomeszero, implying that the plane wave with wave vector q neverenters the ‘normal’ regime but remains in the localizationregime for arbitrarily large amplitudes when Q(c)(q) < Q ≤

12 .

Such a critical value of Q exists for all wave numbers q with

0 ≤ q ≤ arccos(√

2 − 1)

, and in particular Q(c)(q = 0) =

0.1.We illustrate these statistical predictions numerically

in Fig. 1 by performing direct numerical integration of(9), choosing as initial condition a (weakly perturbed)modulationally unstable plane wave with q = 0. The particularexample of Fig. 1 with Q = 0.5 shows a typical behaviour fora generic set of neighboring initial conditions and parametervalues in the regime of statistical localization, where, afterthe initial stage of modulational instability, a few localizedbreathers emerge, which interact inelastically and finally merge

68 M. Johansson / Physica D 216 (2006) 62–70

Fig. 2. (a) Time-averaged participation ratio versus time for the simulation shown in Fig. 1. (b) Average (over space and time) of the linear coupling part of h (i.e.,〈2C

√An An+1 cos(φn − φn+1)〉) versus time for the same simulation. The straight line is a fit with the function f (t) = 2.3t−0.26. (c) Normalized distribution

function p(An) at time t = 108 700 for the same simulation. The straight line is the T = ∞ prediction p(A) =1a e−A/a for the fluctuation part [29,32] (here

a = 1).

into one single surviving breather. A notable difference to thescenario for the ordinary DNLS with Q = 0 described in [29,30,32] is that, with pure on-site nonlinearity, the generatedbreathers, once they have grown sufficiently large, are stronglypinned to the lattice and can interact only very weakly witheach other, while the inter-site nonlinearity generally tends toenhance their mobility [36] and thus their interactions. Thisis a consequence of the fact that for the pure KG–DNLSmodel, the Peierls–Nabarro barrier, defined as the differencein energy between solutions of the same norm centered atand between lattice sites respectively, grows rapidly withamplitude [43], while, including inter-site nonlinearities, thePeierls–Nabarro barrier may become very small even for large-amplitude excitations [36]. Thus, while in the Q = 0 case, evenafter extremely long integration times, one does not observeone single localized excitation but rather some array of standingbreathers; here, for Q = 0.5, after t ∼ 104 we already find onlyone remaining large-amplitude breather which moves aroundthe lattice in a random-like fashion as it interacts with the small-amplitude ‘phonon’ fluctuations. Thus, the behaviour is moresimilar to that observed for the pure FPU-model over long butfinite time-ranges (e.g. [22]).

Let us finally discuss some complementary numericalresults, describing aspects of the evolution towards the thermalequilibrium in the breather-forming regime. In Fig. 2(a) weshow the time evolution of the so-called participation ratio,P =

N 2

N∑

n A2n, averaged over time, 〈P(t)〉 =

1t

∫ t0 P(t ′)dt ′,

for the same simulation as in Fig. 1. It is generally arguedthat P(t) gives an approximate measure of the fraction ofthe total number of sites participating in the main supportof the solution at a given time, with P = 1 for ahomogeneously extended solution, and P = 1/N for single-site localization. Thus, the continuous decrease of 〈P〉 indicatesthat the energy (and norm) is transfered continuously fromthe extended to the localized excitations as time evolves,apparently reaching an equilibrium state where 〈P〉 saturatesat a value slightly below 0.3. In Fig. 2(b) we show, forthe same simulation, the time evolution of the space-timeaverage of the linear coupling part of the Hamiltonian (14),〈h(coup)(t)〉 =

1t

∫ t0

1N

∑n2C

√An An+1 cos(φn − φn+1)dt ′.

Similarly as found in [32], 〈h(coup)〉 decreases towards zero

in the course of thermalization, which is consistent with the

conjecture that, for an infinite-temperature phonon bath, thereare no phase correlations between neighbouring sites. We notethat, for the simulation in Fig. 2(b), 〈h(coup)(t)〉 apparentlydecays approximately according to a power law with exponent∼0.26 for the considered time ranges. Finally, in Fig. 2(c)we show the normalized distribution function p(A) for thesquared amplitudes An at the time instant when the integrationin Fig. 1(d) ends. It is seen that the leftmost part, correspondingto the small-amplitude phonons, is well approximated by apure exponential decay, as predicted for an equilibrium stateat T = ∞ [29,32]. The two right-most points correspondto two neighbouring sites, where the localized breather hasits main support. Thus, in conclusion, the numerical resultspresented in Fig. 2 are consistent with the conjecture that ageneric initial condition belonging to the ‘non-normal’ regimeafter sufficiently long times thermalizes into a small-amplitudephonon part at infinite temperature, together with a localizedlarge-amplitude breather.

5. Conclusions

We derived an extended DNLS-like equation with both on-site and inter-site nonlinearity, the KG–FPU–DNLS model,as a rotating-wave type approximation for the slow small-amplitude dynamics of a chain of anharmonically coupledanharmonic oscillators. Performing the linear stability analysisof travelling plane waves analytically in the one-dimensionalKG–FPU–DNLS model, we found that the simultaneouspresence of on-site and inter-site nonlinearities generallyincreases the regime of linearly stable finite-amplitude planewaves, as compared to the pure on-site DNLS model.However, waves with wave number |q| < π/3 always remainunstable. We also found that, for sufficiently strong inter-sitenonlinearities, the waves may be unstable with respect to short-wavelength instead of long-wavelength perturbations, the latteralways being the case for the ordinary DNLS model.

We further analyzed the statistical properties of theKG–FPU–DNLS model, discussing its thermodynamic equi-librium state in the grand-canonical formalism. In a similarway as for the on-site DNLS model, we identified, when vary-ing the norm and energy densities of the initial states, a tran-sition from a ‘normal’ thermalizing state, characterized by a

M. Johansson / Physica D 216 (2006) 62–70 69

well-defined temperature and chemical potential and with ex-ponentially small probabilites for large-amplitude excitations,to a ‘localizing’ state, apparently approaching an equilibriumstate with simultaneous existence of localized breather(s) andan infinite-temperature phonon bath after long times. An impor-tant effect of the inter-site nonlinearity is to increase the mobil-ity of large-amplitude breathers, and thus we typically observethat the final state in the breather-forming regime does not con-tain an array of pinned breathers, as found after long times forthe on-site DNLS model, but rather one single mobile large-amplitude breather interacting with the small-amplitude fluc-tuations. When using modulationally unstable plane waves asinitial conditions, we found that statistical localization shouldalways occur for small enough amplitudes, but if the inter-sitenonlinearity is not too strong, ‘normal’ thermalization will ap-pear above some amplitude threshold. It would be very interest-ing to investigate to what extent these features can be observedin the original model (Eq. (4)) too.

Let us finally make some remarks about the connection toknown results for the statistical behaviour of (nonintegrable)continuum equations of nonlinear Schrodinger (NLS) type.In this case, there is a totally different behaviour in thefocusing and defocusing cases, respectively. For the focusingcase (see, e.g., [44] and references therein, and also [45,46] for related rigorous results), the statistically preferred(maximum entropy) state has been shown to consist of alocalized macroscopic soliton coupled with infinitesimally fine-scale fluctuations. The soliton, which is a minimizer of theHamiltonian for fixed norm, absorbs all the available norm,while the remainder of the energy is contained purely askinetic energy of the fluctuations, which are equipartitionedover wave numbers. This behaviour is similar to the behaviourof the DNLS models above the transition line (25) (−Hthen corresponds to the NLS Hamiltonian), except for thefact that the discreteness introduces an upper cut-off in wavenumber, making it impossible for the fluctuations to absorb allsuperfluous energy without also absorbing some norm. Also,similarly to the case for the DNLS models in the ‘negative-temperature’ regime above the transition line, the standardgrand-canonical ensemble is ill-defined since the Hamiltonianfor the focusing NLS is unbounded from below.

On the other hand, for the defocusing NLS (see, e.g., [47]and references therein), the Hamiltonian is bounded frombelow, and the equilibrium statistical mechanics is ‘standard’with equivalence between canonical and microcanonicalensembles etc. [45]. The resulting equilibrium distribution islorentzian in wave number, and no large-amplitude excitationsare created. Thus, this behaviour is qualitatively similar tothat of the DNLS models below the transition line (25). Itis important to note that there are no phase transitions inthe continuum NLS equations, neither in the focusing northe defocusing cases,1 in one dimension. Thus, the phase

1 Note however that, in three dimensions, there is a phase transition forthe defocusing NLS, corresponding to a Bose–Einstein type condensationin the homogeneous ground state at a critical non-zero temperature wherethe chemical potential vanishes [47]. Thus, this is a different kind of phasetransition to the localization transition discussed in this paper.

transition in the DNLS models is entirely due to discreteness,and may be interpreted as a transition from an essentiallydefocusing behaviour at low energies (corresponding tomainly anti-phase oscillations between neighboring sites) toan essentially focusing behaviour at high energies (mainly in-phase oscillations).

Acknowledgements

It is a pleasure to thank Serge Aubry for always being asource of inspiration, and for many enlightening discussionsduring the years. Also, this work was inspired, in severalaspects, by his ideas and suggestions. This work is acontinuation of previous works performed in collaboration withMichael Oster and Kim Ø. Rasmussen, and I thank themfor many fruitful discussions. I am also grateful to MichaelOster for confirming the calculations in Section 3. This workwas performed while visiting the Max Planck Institute forthe Physics of Complex System, Dresden, and I thank them,and in particular Sergej Flach, for their hospitality. I thankAndrey Gorbach for valuable comments on the manuscript.I also benefitted from many lively discussions during mystay in Dresden with, among others, Joachim Brand, LarissaBrizhik, Sergej Flach, George Kalosakas, Larry Schulman, andWojtek Zakrzewski. Partial support from the Swedish ResearchCouncil is also acknowledged.

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