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nonlinear physics of discrete systems has witnessed enormous development in the past years . in particular ,
a great deal of interest has been paid to the existence and properties of intrinsic localized modes ( ilms ) , or discrete breathers , which result from the combination of nonlinearity and spatial discreteness .
these spatially localized states have been observed in a wide variety of different systems @xcite .
they were originally suggested as excitations of anharmonic nonlinear lattices @xcite , but the rigorous proof of their persistence under general conditions  @xcite led to their investigation in a diverse host of applications .
these include , among others , antiferromagnets  @xcite , charge - transfer solids  @xcite , photonic crystals  @xcite , superconducting josephson junctions  @xcite , micromechanical cantilever arrays  @xcite , granular crystals  @xcite and biopolymers  @xcite .
more recently , the direct manipulation and control of such states has been enabled through suitable experimental techniques @xcite .    despite the tremendous strides made in this field , the relevant literature
, nevertheless , often appears to be quite sharply divided between theory and experiment .
frequently , experimental studies do not capture the dynamics in enough detail 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 factors some of which may be difficult to quantify precisely ) to make quantitative contact with the experimental results .
our system a macroscopic electrical lattice in which solitons have a time - honored history  @xcite is , arguably , ideally suited for this kind of cross - comparison : the lattice dynamics can be measured fully in space and time , and the physical properties of individual unit cells of the lattice can be characterized in enough detail to allow for the construction of effective models .    in this paper , we present a detailed study of discrete breathers in an electric lattice in which ilms have been experimentally observed @xcite .
we propose a theoretical model which allows us to systematically study their existence , stability and properties , and to compare our numerical findings with experimental results .
we demonstrate good agreement not only at a qualitative but also at a quantitative level between theory and experiment .
the presentation of our results is organized as follows . in the next section
we study the characteristics ( intensity and capacitance curves versus voltage ) of the varactor , the nonlinear circuit element , in order to develop the relevant model for the electrical unit cell .
the results for the single cell are validated through the comparison of its resonance curves for different driving strengths .
we also derive the equations describing the entire electrical line . in sec .
[ sec : numexp ] we study the existence and stability properties of @xmath0-peaked breathers for @xmath1 in the driving frequency and voltage parameter space .
the numerical results are compared to the experimental data with good quantitative agreement .
we also briefly study the spontaneous formation of @xmath0-peaked breathers from the modulational instability of the homogeneous steady state .
we observe that the location and number of the final peaks depends sensitively on the initial conditions . finally , in sec .
[ sec : conclu ] , we conclude our manuscript and offer some suggestions for possible avenues of further research .
our system consists on an electric line as represented in fig .  [ line ] .
this line can be considered as a set of single cells , each one composed of a varactor diode ( nte 618 ) and an inductor @xmath2  @xmath3h , coupled through inductors @xmath4  @xmath3h . each unit cell or node
is driven via a resistor , @xmath5  k@xmath6 , by a sinusoidal voltage source @xmath7 with amplitude @xmath8 and frequency @xmath9 . in experiments a set of 32 elements have been used , with a periodic ring structure ( the last element is connected to the first one ) , and measurements of voltages @xmath10 have been recorded .
related to the voltage source , we have considered amplitudes from @xmath11  v to @xmath12  v and frequencies from @xmath13  khz to @xmath14  khz .        ) and numerical approximation thereof ( continuous line ) corresponding to @xmath15 and @xmath16 ( linear and semi  log plots ) for the nonlinear varactor . ]    in order to propose a set of equations to characterize the electrical line , we have used circuit theory and kirchhoff s rules ; the main challenge has been to describe appropriately each element . in general , resistance and inductors
are inherently imperfect impedance components , i.e. , they have series and parallel , reactive , capacitive and resistive elements . moreover , due to the commercial nature of the elements , manufactured components are subject to tolerance intervals , and the resultant small spatial inhomogeneity introduces some additional uncertainty .
we have quantified the spatial inhomogeneity by separately measuring all lattice components .
the diode capacitance was found to vary by 0.3 percent ( standard deviation ) , whereas the inductors both exhibited a 0.5 percent variation .
additional factors that may contribute slightly to inhomogeneities are wire inductances as well as load and contact resistances .
the varactors ( diodes ) we use are typically intended for am receiver electronics and tuning applications . as described later
, we characterize this lattice element in more detail , since it is the source of the nonlinearity in the lattice .    as a guiding principle
, we are aiming to construct a model which is as simple as possible , with a limited number of parameters whose values are experimentally supported , but one which is still able to reproduce the main phenomenon , namely nonlinear localization and the formation of discrete breathers . with this balance in mind , we proceed as follows .    in the range of frequencies
studied it is a good approximation to describe the load resistance as a simple resistor , neglecting any capacitive or inductive contribution .
also , we have performed experimental measures of the varactor characteristics .
this experimental data shows that it can be modeled as a nonlinear resistance in parallel with a nonlinear capacitance , where the nonlinear current @xmath16 is given by @xmath17 where @xmath18  v@xmath19 and @xmath20 a ( we consider negative voltage when the varactor is in direct polarization ) , and its capacitance as @xmath21    c_0 e^{-\alpha v } & \mbox{if } \quad v > v_{c } , \end{matrix } \end{cases}\ ] ] where @xmath22 , @xmath23  pf , @xmath24  v@xmath19 , @xmath25 , @xmath26 ( the capacitance and its first derivative are continuous in @xmath27 ) , @xmath28  nf and @xmath29  v. in fig .  [ expe ]
we present the experimental data and their corresponding numerical approximations for @xmath16 and @xmath15 , where a good agreement between the two can be observed .    with respect to the inductors , in the range of frequencies considered ,
capacitive effects are negligible , but they possess a small dc ohmic resistance which is around @xmath30 . the inductors and the varactor are a source of damping in the ac regime , and these contributions must be taken into account .
however , we have no manufacturer data related to dissipation parameters , and it is difficult to measure them experimentally . in order to introduce these effects , we will model dissipation phenomenologically by means of a global term given by a resistance @xmath31 , which appears in each unit cell in parallel with @xmath32 ; to determine its value , we have studied experimentally a single element as shown in fig .
[ single ] . in this way
, we will consider the inductors themselves as ideal elements .        using basic circuit theory , the single element is described by the equations : @xmath33 , \nonumber \\[2.0ex ]   \frac{d y}{d \tau } & = & -\left(1+\frac{l_2}{l_1}\right ) v,\end{aligned}\ ] ] where dimensionless variables have been used : @xmath34 , @xmath35 , @xmath36 , the dimensionless voltage at point @xmath37 , @xmath38 , @xmath39 , and @xmath40 ; @xmath41 represents the normalized current through the inductors .
we can generate theoretical nonlinear resonance curves and , comparing with experimental data , select the optimal dissipation parameter value @xmath31 .
results are summarized in table [ rl ] , and the comparison between theoretical and experimental data is shown in fig .
[ ress ] . also , we consider a small frequency shift of @xmath42  khz in numerical simulations to match the resonance curves
. this effect may originate from some small capacitive and/or inductive contributions that we have not previously taken into account .
.values of the resistance @xmath31 corresponding to different voltage amplitudes for the driving source @xmath8 . [ cols="^,^,^,^,^,^,^,^",options="header " , ]     more detailed experimental results are shown in fig .
[ profiles ] where we depict peak profiles at two different driver voltages .
the profiles were taken at the times of largest peak voltage amplitude and lowest peak voltage amplitude . for @xmath43v (
left column of panels ) we see that , as the frequency is raised from below , we cross from the 1-peak region through the 2-peak region and into the 3-peak region . for the @xmath44  v case ( right column of panels )
the same sequence can be observed when scanning in one frequency direction . in order to illustrate both the hysteresis and the overlap between @xmath0-peak regions , we depict in the figure ( panels ( e ) and ( f ) ) a situation where the 2-peak solution occurs at a higher frequency than the 3-peaked one .
the reason is that in fig .
[ profiles](f ) , the 3-peak solution was obtained at higher frequencies and then adiabatically extended to lower ones , whereas in fig .
[ profiles](e ) the 2-peak solution was obtained starting from the 1-peak region .
we show the eventual location of peaks in the breather pattern ( i.e. after the driver has been on for a long time ) . however , it is important to mention that the exact location where the peaks eventually settle is sensitive to slight impurities in the lattice .
we have noticed that when we turn on the voltage source , at first we can observe a more sinusoidal pattern ( corresponding to the most modulationally unstable @xmath45-value ) , but as the pattern reaches higher energy and becomes more nonlinear , the peaks may shift and adjust themselves in the lattice .
as it can be observed , peaks are not perfectly equispaced in the lattice .
this is obviously due to the inhomogeneities and noise present in the experiment .    for the numerical results depicted in fig .
[ profiles ] we used a set of initials conditions based on the experimental data and determined the stationary state by letting the numerical profiles to settle to a steady state . for the cases corresponding to @xmath43
v and @xmath44  v , adding a small frequency offset @xmath46  khz , we observe , in general , a good agreement between numerics and experimental data .
the mismatch between experiments and theory , in particular the intersite distance peaks , can be attributed to the above mentioned factors .
furthermore , to reproduce precisely the experimental peak voltage is extremely difficult because it corresponds to the voltage at resonance and , therefore , even very small parameter changes can create large differences in the maximum amplitudes .
nevertheless , the quantitative agreement appears fairly good , especially for the @xmath44  v case .
in this paper we have formulated a prototypical model that is able to describe the formation of nonlinear intrinsic localized modes ( or discrete breathers ) in an experimental electric line .
this has been derived based on a combination of the fairly accurate characterization of a single element within the lattice ( including its nonlinear resonance curves and hysteretic behavior ) and fundamental circuit theory in order to properly couple the elements .
comparison between theory and experiments shows very good qualitative and even good quantitative agreement between the two .
we characterized the regions of existence and stability of @xmath0-peaked breathers for @xmath1 and illustrated how transitions of the coherent waveforms of one kind to those of another kind take place , rationalizing them on the basis of stability properties and their corresponding floquet spectra .
we also showed that the precise number of peaks and their location in the lattice is fairly sensitive to initial conditions , a feature also generally observed in the experiments where the potential for states with different numbers of peaks similarly manifests .
naturally , many directions of potential future research stem from the fundamental modeling and computation basis explored in the present manuscript . on the one hand , it would be very interesting to attempt to understand the stability properties of the different breather states from a more mathematical perspective , although this may admittedly prove a fairly difficult task .
on the other hand , from the modeling and computation perspective in conjunction with experimental progress , the present work paves the way for potentially augmenting these systems into higher dimensional setups and attempting to realize discrete soliton as well as more complex discrete vortex states therein  @xcite .
such studies will be deferred to future publications .
f.p . and j.c .
acknowledges sponsorship by the spanish micinn under grant no .
fis2008 - 04848 . r.c.g .  gratefully acknowledges the hospitality of the grupo de fsica no lineal ( gfnl , university of sevilla , spain ) and support from nsf - dms-0806762 , plan propio de la universidad de sevilla , grant no .
iac09-i-4669 of junta de andalucia and ministerio de ciencia e innovacin , spain .
p.g.k .  acknowledges the support from nsf - dms-0806762 , nsf - cmmi-1000337 and from the alexander von humboldt , as well as the alexander s. onassis public benefit foundation .
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we study experimentally and numerically the existence and stability properties of discrete breathers in a periodic nonlinear electric line . the electric line is composed of single cell nodes , containing a varactor diode and an inductor , coupled together in a periodic ring configuration through inductors and driven uniformly by a harmonic external voltage source . 
 a simple model for each cell is proposed by using a nonlinear form for the varactor characteristics through the current and capacitance dependence on the voltage . for an electrical line composed of 32 elements , we find the regions , in driver voltage and frequency , where @xmath0-peaked breather solutions exist and characterize their stability . 
 the results are compared to experimental measurements with good quantitative agreement . 
 we also examine the spontaneous formation of @xmath0-peaked breathers through modulational instability of the homogeneous steady state . the competition between different discrete breathers seeded by 
 the modulational instability eventually leads to stationary @xmath0-peaked solutions whose precise locations is seen to sensitively depend on the initial conditions .