PHYSICAL REVIEW E 84, 026605 (2011)

Discrete breathers in a nonlinear electric line: Modeling, computation, and experiment

F. Palmero,1,* L. Q. English,2 J. Cuevas,3 R. Carretero-Gonzalez,4,† and P. G. Kevrekidis5

1Nonlinear Physics Group, Escuela Tecnica Superior de Ingenierıa Informatica, Departamento de Fısica Aplicada I,Universidad de Sevilla, Avda. Reina Mercedes, s/n, E-41012 Sevilla, Spain

2Department of Physics and Astronomy, Dickinson College, Carlisle, Pennsylvania 17013, USA3Grupo de Fısica No Lineal, Departamento de Fısica Aplicada I, Escuela Politecnica Superior, Universidad de Sevilla,

C/ Virgen de Africa, 7, E-41011 Sevilla, Spain4Nonlinear Dynamical Systems Group, Department of Mathematics and Statistics and Computational Science Research Center,

San Diego State University, San Diego, California 92182-7720, USA5Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA

(Received 15 April 2011; published 5 August 2011)

We study experimentally and numerically the existence and stability properties of discrete breathers in aperiodic nonlinear electric line. The electric line is composed of single cell nodes, containing a varactor diodeand an inductor, coupled together in a periodic ring configuration through inductors and driven uniformly bya harmonic external voltage source. A simple model for each cell is proposed by using a nonlinear form forthe varactor characteristics through the current and capacitance dependence on the voltage. For an electricalline composed of 32 elements, we find the regions, in driver voltage and frequency, where n-peaked breathersolutions exist and characterize their stability. The results are compared to experimental measurements with goodquantitative agreement. We also examine the spontaneous formation of n-peaked breathers through modulationalinstability of the homogeneous steady state. The competition between different discrete breathers seeded bythe modulational instability eventually leads to stationary n-peaked solutions whose precise locations is seen tosensitively depend on the initial conditions.

DOI: 10.1103/PhysRevE.84.026605 PACS number(s): 05.45.Yv, 63.20.Pw, 63.20.Ry

I. INTRODUCTION

Nonlinear physics of discrete systems has witnessed enor-mous development in the past years. In particular, a greatdeal of attention has been paid to the existence and propertiesof intrinsic localized modes (ILMs), or discrete breathers,which result from the combination of nonlinearity and spatialdiscreteness. These spatially localized states have been ob-served in a wide variety of different systems [1,2]. They wereoriginally suggested as excitations of anharmonic nonlinearlattices [3], but the rigorous proof of their persistence undergeneral conditions [4] led to their investigation in a diverse hostof applications. These include, among others, antiferromagnets[5], charge-transfer solids [6], photonic crystals [7], supercon-ducting Josephson junctions [8], micromechanical cantileverarrays [9], granular crystals [10], and biopolymers [11]. Morerecently, the direct manipulation and control of such states hasbeen enabled through suitable experimental techniques [12].

Despite the tremendous strides made in this field, therelevant literature, nevertheless, often appears to be quitesharply divided between theory and experiment. Frequentlyexperimental studies do not capture the dynamics in enoughdetail to facilitate an exact comparison with theoretical studies.At other times, the theoretical models are not refined enough(or lack the inclusion of nontrivial experimental factorssome of which may be difficult to quantify precisely) tomake quantitative contact with the experimental results. Oursystem—a macroscopic electrical lattice in which solitonshave a time-honored history [13,14]—is, arguably, ideallysuited for this kind of cross comparison: the lattice dynamics

*palmero@us.es†[http://nlds.sdsu.edu]

can be measured fully in space and time, and the physicalproperties of individual unit cells of the lattice can becharacterized in enough detail to allow for the constructionof effective models.

In this paper we present a detailed study of discretebreathers in an electric lattice in which ILMs have beenexperimentally observed [15–17]. We propose a theoreticalmodel which allows us to systematically study their existence,stability and properties, and to compare our numerical findingswith experimental results. We demonstrate good agreement notonly at a qualitative but also at a quantitative level betweentheory and experiment. The presentation of our results isorganized as follows. In the next section we study the charac-teristics (intensity and capacitance curves versus voltage) ofthe varactor, the nonlinear circuit element, in order to developthe relevant model for the electrical unit cell. The resultsfor the single cell are validated through the comparison of itsresonance curves for different driving strengths. We also derivethe equations describing the entire electrical line. In Sec. IIIwe study the existence and stability properties of n-peakedbreathers for n = 1, 2, 3 in the driving frequency and voltageparameter space. The numerical results are compared to theexperimental data with good quantitative agreement. We alsobriefly study the spontaneous formation of n-peaked breathersfrom the modulational instability of the homogeneous steadystate. We observe that the location and number of the finalpeaks depends sensitively on the initial conditions. Finally,in Sec. IV, we conclude our manuscript and offer somesuggestions for possible avenues of further research.

II. THEORETICAL SETUP

Our system consists on an electric line as represented inFig. 1. This line can be considered as a set of single cells, each

026605-11539-3755/2011/84(2)/026605(8) ©2011 American Physical Society

F. PALMERO et al. PHYSICAL REVIEW E 84, 026605 (2011)

L1

R

C

L2

Vn-1 Vn Vn+1

V(t)

FIG. 1. Schematic circuit diagram of the electrical transmissionline.

one composed of a varactor diode (NTE 618) and an inductorL2 = 330 μH, coupled through inductors L1 = 680 μH.Each unit cell or node is driven via a resistor, R = 10 k�,by a sinusoidal voltage source V (t) with amplitude Vd andfrequency f . In experiments a set of 32 elements has been used,with a periodic ring structure (the last element is connected tothe first one), and measurements of voltages Vn have beenrecorded. Related to the voltage source, we have consideredamplitudes from Vd = 1 V to Vd = 5 V and frequencies fromf = 200 kHz to f = 600 kHz.

In order to propose a set of equations to characterize theelectrical line, we have used circuit theory and Kirchhoff’srules; the main challenge has been to describe appropri-ately each element. In general, resistance and inductors areinherently imperfect impedance components, that is, theyhave series and parallel, reactive, capacitive, and resistiveelements. Moreover, due to the commercial nature of theelements, manufactured components are subject to toleranceintervals, and the resultant small spatial inhomogeneity in-troduces some additional uncertainty. We have quantifiedthe spatial inhomogeneity by separately measuring all latticecomponents. The diode capacitance was found to vary by 0.3%(standard deviation), whereas the inductors both exhibited a0.5% variation. Additional factors that may contribute slightlyto inhomogeneities are wire inductances as well as load andcontact resistances. The varactors (diodes) we use are typicallyintended for AM receiver electronics and tuning applications.As described later, we characterize this lattice element inmore detail, since it is the source of the nonlinearity inthe lattice. For such small spreads we consider the latticesufficiently homogeneous, so that the localization is controlledby the nonlinearity and not by disorder. This assumptionis validated by the fact that we do not see localizationin the linear regime (for low driver amplitudes). It is alsocorroborated a posteriori below by the direct comparisonof our numerical results in a homogeneous lattice with theexperimental observations in the very weakly heterogeneouscircuit.

As a guiding principle we are aiming to construct a modelwhich is as simple as possible, with a limited number ofparameters whose values are experimentally supported, butone which is still able to reproduce the main phenomenon,namely nonlinear localization and the formation of dis-crete breathers. With this balance in mind, we proceed asfollows.

In the range of frequencies studied it is a good approx-imation to describe the load resistance as a simple resistor,neglecting any capacitive or inductive contribution. Also,

0 2 4 60

2

4

6

8

10

V (Volt)

C(V

) (n

F) 0 2 4 6

−0.2

−0.1

0

V (Volt)

I D(V

) (A

)

−0.8 −0.6 −0.4 −0.2

10

20

30

V (Volt)

log(

|ID

(V)/

Is|)

FIG. 2. Experimental data (∗) and numerical approximationthereof (continuous line) corresponding to C(V ) and ID(V ) (linearand semilog plots) for the nonlinear varactor.

we have performed experimental measures of the varactorcharacteristics. This experimental data shows that it canbe modeled as a nonlinear resistance in parallel with anonlinear capacitance, where the nonlinear current ID(V ) isgiven by

ID(V ) = −Is exp(−βV ), (1)

where β = 38.8 V−1 and Is = 1.25 × 10−14 A (we considernegative voltage when the varactor is in direct polarization),and its capacitance as

C(V ) ={

Cv + Cw(V ′) + C(V ′)2 if V � Vc,

C0e−αV if V > Vc,

(2)

where V ′ = (V − Vc), C0 = 788 pF, α = 0.456 V−1, Cv =C0 exp(−αVc), Cw = −αCv (the capacitance and its firstderivative are continuous in V = Vc), C = 100 nF and Vc =−0.28 V. In Fig. 2 we present the experimental data and theircorresponding numerical approximations for ID(V ) and C(V ),where a good agreement between the two can be observed.

With respect to the inductors, in the range of frequen-cies considered, capacitive effects are negligible, but theypossess a small dc ohmic resistance which is around 2 �.The inductors and the varactor are a source of damping inthe ac regime, and these contributions must be taken intoaccount. However, we have no manufacturer data related todissipation parameters, and it is difficult to measure themexperimentally. In order to introduce these effects, we willmodel dissipation phenomenologically by means of a globalterm given by a resistance Rl , which appears in each unitcell in parallel with L2; to determine its value, we havestudied experimentally a single element as shown in Fig. 3.In this way we will consider the inductors themselves as idealelements.

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DISCRETE BREATHERS IN A NONLINEAR ELECTRIC . . . PHYSICAL REVIEW E 84, 026605 (2011)

R

C L2 L1V(t)

V

FIG. 3. Single cell element model.

Using basic circuit theory, the single element is describedby the equations

dv

dτ= 1

c(v)

[cos(�τ )

RC0ω0− 1

ω0C0

(1

Rl

+ 1

R

)v + (y − iD)

],

(3)dy

dτ= −

(1 + L2

L1

)v, (4)

where dimensionless variables have been used: τ = ω0t ,iD = ID/(ω0C0Vd ), v = V/Vd , the dimensionless voltage atpoint A, c(v) = C(V )/C0, � = ω/ω0, and ω0 = 1/

√L2C0; y

represents the normalized current through the inductors.We can generate theoretical nonlinear resonance curves

and, comparing with experimental data, select the optimaldissipation parameter value Rl . Results are summarized inTable I, and the comparison between theoretical and exper-imental data is shown in Fig. 4. Also, we consider a smallfrequency shift of 18 kHz in numerical simulations to matchthe resonance curves. This effect may originate from somesmall capacitive and/or inductive contributions that we havenot previously taken into account.

Thus, with a consistent set of parameter values, and usingagain elementary circuit theory, we describe the full electricline of N coupled single cell elements by the following systemof coupled ordinary differential equations:

c(vn)dvn

dτ= yn − iD(vn) + cos(�τ )

RC0ω0−

(1

Rl

+ 1

R

)vn

ω0C0,

dyn

dτ= L2

L1(vn+1 + vn−1 − 2vn) − vn, (5)

where all magnitudes are in dimensionless units. Within thisnonlinear dynamical lattice model, our waveforms of interest,namely the discrete breathers, are calculated as fixed points ofthe map ⎡

⎢⎢⎢⎣

vn(0)dvn

dτ(0)

yn(0)dyn

dτ(0)

⎤⎥⎥⎥⎦ →

⎡⎢⎢⎢⎣

vn(T )dvn

dτ(T )

yn(T )dyn

dτ(T )

⎤⎥⎥⎥⎦ , (6)

where T = 1/f is the temporal period of the breather. In orderto study the linear stability of discrete breathers, we introducea small perturbation (ξn,ηn) to a given solution (vn0,yn0) of

TABLE I. Values of the resistance Rl corresponding to differentvoltage amplitudes for the driving source Vd .

Vd (V) 1 2 3 4 5Rl (�) 15 000 10 000 6000 5000 4500

250 300 350 400 450 500 550−1

−0.5

0

0.5

1

1.5

2

f (kHz)

V (

Vol

t)

FIG. 4. (Color online) Nonlinear resonance curves correspondingto a single element. Dots (orange/gray) correspond to experimentaldata while the continuous and dashed black lines correspond,respectively, to stable and unstable numerical solutions. From bottomto top: Vd = 1, 2, 3, 4, and 5 V.

Eq. (6) according to vn = vn0 + ξn, yn = yn0 + ηn. Then, theequations satisfied to first order by (ξn,ηn) are

c(vn0)dξn

dτ= ηn − (vn0; t)ξn,

(7)dηn

dτ= L2

L1(ξn+1 + ξn−1 − 2ξn) − ξn

with (vn0; t) being

(vn0; t)

= diD(vn0)

dvn0+

(1

Rl

+ 1

R

)1

ω0C0+ d ln[c(vn0)]

dvn0

×[yn0 − iD(vn0) + cos(�τ )

RC0ω0−

(1

Rl

+ 1

R

)vn0

ω0C0

].

To identify the orbital stability of the relevant solutions,a Floquet analysis can be performed. Then, the stabilityproperties are given by the spectrum of the Floquet operatorM (whose matrix representation is the monodromy) defined as⎡

⎢⎢⎢⎣

ξn(T )dξn

dτ(T )

ηn(T )dηn

dτ(T )

⎤⎥⎥⎥⎦ = M

⎡⎢⎢⎢⎣

ξn(0)dξn

dτ(0)

ηn(0)dηn

dτ(0)

⎤⎥⎥⎥⎦ . (8)

The 4N × 4N monodromy eigenvalues λ are called theFloquet multipliers. If the breather is stable, all theeigenvalues lie inside the unit circle.

III. NUMERICAL COMPUTATIONSAND EXPERIMENTAL RESULTS

Using Eq. (6) we have generated n-peak ILMs and deter-mined numerically their stability islands in (Vd , f ) parameterspace. We have found that there exist overlapping regionswhere two or several of these n-peak configurations exist andare stable. Therefore, the long-term dynamics in these regions

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F. PALMERO et al. PHYSICAL REVIEW E 84, 026605 (2011)

is chiefly dependent on initial conditions. However, determin-ing precisely the basins of attraction of each configuration isnot possible because of the high dimensionality of the problem.

Figure 5 shows the existence and stability diagrams forone-, two-, and three-peak breathers. The values of Rl areobtained through cubic interpolation from Table I. Notice thatthe lattice allows different configurations of solutions of n-peak breathers for n > 1 corresponding to the peaks centeredat different locations, but determining precisely the diagramscorresponding to all possible configurations is not possible

FIG. 5. (Color online) Existence islands for n-peak breathers.The white line denotes the threshold where the homogeneous statebecomes unstable (above the line). Black areas correspond to stablesolutions and orange (gray) areas to unstable solutions. (a) One-peakbreather (the homogeneous state threshold is located at the top borderof the areas), (b) a family of two-peak breather, and (c) a family ofthree-peak breathers. All data correspond to numerical simulations.

5 10 15 20 25 300

0.5

1

1.5

V

n

(V

olt)

Site−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Im( λ

)

Re( λ )

5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

V

n

(V

olt)

Site−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Im( λ

)

Re( λ )

FIG. 6. (Color online) Breather profiles (left column) and corre-sponding Floquet multiplier spectra (right column) for τ = 0, Vd =4 V, and f = 243.6 kHz. The top (bottom) panels correspond to theunstable (stable) one-site (intersite) breather. All data correspond tonumerical simulations.

because of the size of the lattice. Therefore, we have studiedonly breathers with peaks as far apart as possible.

Remarkably, breathers are generally robust for Vd � 3 V.Above that critical value, the considered solutions may becomeunstable in some “instability windows” (see orange/grayregions). Those instabilities, which are of exponential kind,typically lead an onsite breather profile to deform into a stableintersite breather waveform. We have analyzed in more detailthe dependence of those instabilities for one-peak breathers.The study for higher peaked structure is cumbersome dueto the increasing number of intersite structures that might arise.Figure 6 shows an example of one-peaked onsite and intersitebreathers together with their Floquet spectrum.

A detailed analysis shows that for 3 � Vd � 5 V, intersitebreathers are always stable, whereas onsite ones may beunstable (i.e., there is no stability exchange as it occursin Klein-Gordon lattices). However, for Vd � 2 V, intersitebreathers are always unstable, whereas onsite breathers arestable. Finally, for 2 V� Vd � 3 V, intersite breathers expe-rience instability windows, whereas their onsite counterpartsare stable. Figures 7–9 illustrate a typical set of relevant resultsfor the cases of Vd = 1.5 V, Vd = 2.5 V, and Vd = 4 V whichsummarize the different possible regimes as Vd is varied. Theunity crossings indicated by the red line (wherever relevant) inthe right panel of each of the figures mark the stability changesof the pertinent solutions.

We have also performed an analysis of the stability ofone-peak breathers for larger lattices (N = 101) and observethat the existence and stability range for onsite and intersitebreathers are not significantly altered.

To corroborate the numerical picture, we have also per-formed an analogous experimental analysis. To determine ex-perimentally the stability islands, we have chosen a particularvoltage Vd and, starting at low frequencies (homogeneousstate), and increasing f adiabatically, we can reach different

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DISCRETE BREATHERS IN A NONLINEAR ELECTRIC . . . PHYSICAL REVIEW E 84, 026605 (2011)

on-site breather

272 274 276 278 2800

1

2

3

arg(

λ)

f (kHz)272 274 276 278 280

0.4

0.6

0.8

1

|λ|

f (kHz)

intersite breather

272 274 276 278 2800

1

2

3

arg(

λ)

f (kHz)272 274 276 278 280

0

0.5

1

1.5

|λ|

f (kHz)

FIG. 7. (Color online) Dependence of the Floquet multipliers onthe frequency for N = 32 cells and Vd = 1.5 V. The top (bottom)row of panels corresponds to the onsite (intersite) breather. The leftand right column of panels depicts, respectively, the argument andmagnitude of the Floquet multipliers. The horizontal (red) line inthe spectra depicts the stability threshold. All data correspond tonumerical simulations.

regions. Going up and down adiabatically we are able to exam-ine regions of coexistence between different n-peak breathers.In particular, regions corresponding to one-peak, two-peak,and three-peak breathers have been detected experimentally.The three-peak region is bounded from above by the four- andfive-peak areas, which we did not track. We have determinedthe boundaries by doing up-sweeps and down-sweeps, withsignificant hysteresis phenomena.

on-site breather

258 260 262 2640

1

2

3

arg(

λ)

f (kHz)258 260 262 264

0.2

0.4

0.6

0.8

1

|λ|

f (kHz)

intersite breather

258 260 262 2640

1

2

3

arg(

λ)

f (kHz)258 260 262 264

0.2

0.4

0.6

0.8

1

1.2

|λ|

f (kHz)

FIG. 8. (Color online) Same as Fig. 7 but for Vd = 2.5 V.

on-site breather

243 244 245 2460

1

2

3

arg(

λ)

f (kHz)243 244 245 2460

0.5

1

1.5

|λ|

f (kHz)intersite breather

243 244 245 2460

1

2

3

arg(

λ)

f (kHz)243 244 245 2460

0.2

0.4

0.6

0.8

1

|λ|

f (kHz)

FIG. 9. (Color online) Same as Fig. 7 but for Vd = 4 V.

It should be mentioned that for the interior of the one-peakregion, we demonstrated experimentally that the discretebreather and ILM can be centered at any node in the latticeand survive there. This verification implies that the experimen-tal lattice—despite its inherent component variability—doesdisplay a sufficient degree of spatial homogeneity for thebasic localization phenomenon to be considered (discrete)translationally invariant. In practice, we employed an impurityin the form of an external inductor physically touching aL2 lattice inductor to make the ILM hop to that impuritysite because of the existence of an impurity mode localizedaround the impurity site (see Fig. 10). Upon removing theimpurity, the ILM would then persist at that site. We believe

2.5

2.0

1.5

1.0

0.5

0.0

V n (

Vol

t)

24222018161412Site

FIG. 10. (Color online) The ILM can exist at any site of the lattice;here f = 253 kHz and Vd = 4 V. We demonstrate that the ILM can bemade to jump to the neighboring site upon the temporary creation ofan impurity there. The left-most profile (centered at site 16) turns intothe dotted (red) trace when an external inductor is placed in the directvicinity of the L2 inductor at site 17. Upon removal of this externalinductor, the ILM persists at site 17 (solid black trace). This processcan be repeated to move the ILM further to the right, as shown. Alldata correspond to experiments.

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F. PALMERO et al. PHYSICAL REVIEW E 84, 026605 (2011)

FIG. 11. (Color online) Existence regions of one-peaked (toppanel) and a family of three-peaked (bottom panel) breathersobtained numerically (black areas correspond to stable solutions andorange/gray one to unstable solutions) as compared to experimentaldata identifying the range of observations of the corresponding typeof states (circles). The theoretical data are displaced by a +7 kHzfrequency offset.

that this technique could prove extremely valuable toward theguidance and manipulation (essentially, at will) of the ILMsin this system. The ability to controllably move localizedenergy around the lattice could, for instance, facilitate futureexperiments on breather-breather interactions. Moreover, notonly can ILMs be shepherded in this manner, but we canalso use the same technique to “seed” an ILM at a locationof our choice; this is possible for driving frequencies justbelow those for which the modulational instability sponta-neously produces localization patterns. In this way we cangenerate breathers at any time and any location in the lattice,depending only on where and when the impurity is brieflyintroduced.

A comparison between theoretical and experimental dataof one-peak and three-peak breathers is shown in Fig. 11.The precise width of any stability region is hard to matchbecause there exist regions of different peak numbers andregions of different families with the same number of peaksoverlap, and thus their competition prevents an absolutelydefinitive picture. Which one “wins” out appears to de-pend sensitively on small lattice impurities present in our(commercial) experimental elements. For instance, in theexperiment it looks as if the two-peak region is squeezed

Vd = 2 V Vd = 4 V

0 10 20 30

−0.5

0

0.5

1

1.5

2

2.5

Site

Vn(V

olt)

(a)

0 10 20 30

−0.5

0

0.5

1

1.5

2

2.5

Site

Vn(V

olt)

(d)

0 10 20 30

−0.5

0

0.5

1

1.5

2

2.5

Site

Vn(V

olt)

(b)

0 10 20 30

−0.5

0

0.5

1

1.5

2

2.5

Site

Vn(V

olt)

(e)

0 10 20 30

−0.5

0

0.5

1

1.5

2

2.5

Site

Vn(V

olt)

(c)

0 10 20 30

−0.5

0

0.5

1

1.5

2

2.5

Site

Vn(V

olt)

(f)

FIG. 12. Experimental (•) and numerical (◦) one-, two-, andthree-peak breather profiles for different driver voltage Vd andfrequencies values. The left (right) column of panels correspondsto a driver strength Vd = 2 V (Vd = 4 V). Each panel correspondto the following experimental (fexpt) and numerical (fnum) drivingfrequencies in kHz: (a) (fexpt,fnum) = (275,268), (b) (fexpt,fnum) =(276,269), (c) (fexpt,fnum) = (280,273), (d) (fexpt,fnum) = (248,244),(e) (fexpt,fnum) = (254,247), and (f) (fexpt,fnum) = (249,245).

in favor of the one-peak region. This might be the reasonwhy the experimental one-peak region is slightly wider athigher driver amplitudes than it appears in the correspondingtheoretical predictions. Nonetheless, the comparison betweenexperimental and theoretical existence regions depictedin Fig. 11 shows generally good qualitative (and evenquantitative) agreement in the context of the proposedmodel.

More detailed experimental results are shown in Fig. 12where we depict peak profiles at two different driver voltages.The profiles were taken at the times of largest peak voltageamplitude and lowest peak voltage amplitude. For Vd = 2 V(left column of panels) we see that, as the frequency is raisedfrom below, we cross from the one-peak region through thetwo-peak region and into the three-peak region. For the Vd =4 V case (right column of panels) the same sequence can beobserved when scanning in one frequency direction. In orderto illustrate both the hysteresis and the overlap between n-peakregions, we depict in Figs. 12(e) and 12(f) a situation wherethe two-peak solution occurs at a higher frequency than thethree-peaked one. The reason is that in Fig. 12(f) the three-peak solution was obtained at higher frequencies and then

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DISCRETE BREATHERS IN A NONLINEAR ELECTRIC . . . PHYSICAL REVIEW E 84, 026605 (2011)

adiabatically extended to lower ones, whereas in Fig. 12(e)the two-peak solution was obtained starting from the one-peakregion.

We show the eventual location of peaks in the breatherpattern (i.e., after the driver has been on for a long time).However, it is important to mention that the exact locationwhere the peaks eventually settle is sensitive to slight impu-rities in the lattice. We have noticed that when we turn onthe voltage source, at first we can observe a more sinusoidalpattern (corresponding to the most modulationally unstable k

value), but as the pattern reaches higher energy and becomesmore nonlinear, the peaks may shift and adjust themselves inthe lattice. As it can be observed, the peaks are not perfectlyequispaced in the lattice. This is due to the inhomogeneitiespresent in the experiment.

For the numerical results depicted in Fig. 12 we used aset of initials conditions based on the experimental data anddetermined the stationary state by letting the numerical profilessettle to a steady state. For the cases corresponding to Vd = 2 Vand Vd = 4 V, adding a small frequency offset �f ≈ 4–7 kHz,we observe, in general, a good agreement between numericsand experimental data. The mismatch between experimentsand theory, in particular the intersite distance peaks, can beattributed to the above mentioned factors. Furthermore, toreproduce precisely the experimental peak voltage is extremelydifficult because it corresponds to the voltage at resonanceand, therefore, even very small parameter changes can createlarge differences in the maximum amplitudes. Nevertheless,the quantitative agreement appears fairly good, especially forthe Vd = 4 V case.

IV. CONCLUSIONS

In this paper we have formulated a prototypical modelthat is able to describe the formation of nonlinear intrinsiclocalized modes (or discrete breathers) in an experimentalelectric line. This has been derived based on a combinationof the fairly accurate characterization of a single elementwithin the lattice (including its nonlinear resonance curvesand hysteretic behavior) and fundamental circuit theory inorder to properly couple the elements. Comparison betweentheory and experiments shows very good qualitative andeven good quantitative agreement between the two. Wecharacterized the regions of existence and stability of n-peakedbreathers for n = 1, 2, 3 and illustrated how transitions of thecoherent waveforms of one kind to those of another kind takeplace, rationalizing them on the basis of stability propertiesand their corresponding Floquet spectra. We also showedthat the precise number of peaks and their location in thelattice is fairly sensitive to initial conditions, a feature alsogenerally observed in the experiments where the potentialfor states with different numbers of peaks similarly mani-fests.

The level of interplay between theory and experimentachieved in this study is beneficial from a number of largerscale perspectives. First, it allows us to better investigate andidentify which aspects of the single-cell characteristics areessential for the emergence of the collective phenomenon oflocalization in the lattice. We see, for instance, that an exact

match of the diode properties is not necessary to capture theessence of self-localization in the lattice and, furthermore,to match the experimental ILM profiles very well. On theother hand, some diode characteristics were found to beessential for the manifestation of these nonlinear phenomena,such as the large curvature in the C(V ) curve around theorigin. Experimentally, too, we observe the disappearance ofILMs when the diodes are reverse biased too much and theC(V ) curvature diminishes. Second, the success in recoveringnumerically the main experimental observations demonstratesthat these lattice effects are really derivable from and containedwithin the basic circuit equations and do not rely on somehidden feature of the experimental system. This cements thevalue of the theoretical modeling as a useful tool to potentiallyexplore interesting and relevant phenomenologies within thissystem, including its response to different types of externaldrive and its generalizations to higher dimensional setups.Finally, while the experimental observation first motivatedthe numerical study, it then helped shape the model and tiedown relevant parameter values, the numerical and theoreticalstudy also uncovered aspects missed by the experiments,such as the stability of intersite breathers at larger driveramplitudes; more broadly, it places the empirical evidenceinto a broader dynamical-systems context. Conversely, theexperimental investigation can sometimes yield unexpectedresults that would have been difficult to predict a priori,such as the possibility of moving breathers upon a slightmodification of the unit cell [17]. The above constructiveinterplay and possibility for continuous feedback betweentheory and modeling and experiment is, arguably, one ofthe major advantages of the present nonlinear lattice systemand one of the fundamental contributions of the presentstudy.

Naturally, many directions of potential future researchstem from the fundamental modeling and computation basisexplored in the present manuscript. On the one hand, itwould be very interesting to attempt to understand thestability properties of the different breather states from amore mathematical perspective, although this may admittedlyprove a fairly difficult task. On the other hand, from themodeling and computation perspective in conjunction withexperimental progress, the present work paves the way forpotentially augmenting these systems into higher dimensionalsetups and attempting to realize discrete soliton as well as morecomplex discrete vortex states therein [2,18]. Such studies willbe deferred to future publications.

ACKNOWLEDGMENTS

F.P. and J.C. acknowledges sponsorship by the SpanishMICINN under Grant No. FIS2008-04848. R.C.G. gratefullyacknowledges the hospitality of the Grupo de Fısica NoLineal (GFNL, University of Sevilla, Spain) and support fromNSF-DMS-0806762, Plan Propio de la Universidad de Sevilla,Grant No. IAC09-I-4669 of Junta de Andalucia and Ministeriode Ciencia e Innovacion, Spain. P.G.K. acknowledges thesupport from NSF-DMS-0806762, NSF-CMMI-1000337, andfrom the Alexander von Humboldt, as well as the AlexanderS. Onassis Public Benefit Foundation.

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