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DISCRETE BREATHERS AND ENERGY LOCALIZATION IN NONLINEAR LATTICES Thierry Dauxois and Michel Peyrard Laboratoire de Physique de l’Ecole Normale Sup´e rieure de Lyon,CNRS URA 1325,46all´e e d’Italie,69007Lyon,France.(February 9,2008)Abstract We discuss the process by which energy,initially evenly distributed in a nonlinear lattice,can localize itlf into large amplitude excitations.We show that,the standard modulational instability mechanism,which can initiate the process by the formation of small amplitude breathers,is completed efficiently,in the prence of discreteness,by energy exchange mechanisms between the nonlinear excitations which favor systematically the growth of the larger ex-citations.The process is however lf regulated becau the large amplitude excitations are finally trapped by the Peierls-Nabarro potential.PACS numbers:63.10+a,3.40Kf,46.10+z
Typet using REVT E X
I.INTRODUCTION
Many physical phenomena involve some localization of energy in space.The formation of vortices in hydrodynamics,lf focusing in optics or plasmas,the formation of dislocations in solids under stress,lf trapping of energy in proteins,are well known examples.Following the original work by Anderson[1]disorder-induced localization has been widely studied,but, more recently,attention was attracted on the possibility to localize energy in an homogeneous system due to nonlinear effects.the process can become dramatic when it leads to collap in a plasma[2].In this paper we are interested in the process by which energy evenly distributed in such a system can concentrate itlf spontaneously into spatially localized nonlinear excitations.In some cas this evolution can lead to the formation of topological solitonlike excitations such as dislocations or ferroelectric or ferromagnetic domain walls. However,since there is an energy threshold for the creation of topological solitons,thefirst step of the evolution is the formation of breathers or envelope modes;we shall therefore focus our attention on such modes.
Nonlinear energy localization in continuous media has been extensively investigated since Benjamin and Feir[3]discovered the modulational instability of Stokes waves influids,but very little has been done in lattices although it would be of wide interest for solids or macromolecules.We want to point out here that,in a discrete lattice,nonlinear energy localization is very different from its counterpart in
a continuum medium.In particular,we show that,besides the familiar mechanism of modulational instability,which is itlf strongly modified by discreteness effects,there is an additional channel for energy concentration, which is specific to lattices,but is not nsitive to the details of the nonlinear lattice model which is considered.Therefore it appears as a very general process leading to localization of energy in a lattice.
Thefirst step toward the creation of localized excitations can be achieved through mod-ulational instability which exists in a lattice as well as in a continuum medium,although discreteness can drastically change the conditions for instability[4](e.g.,at small wave num-
bers a nonlinear carrier wave is unstable to all possible modulations of its amplitude as soon as the wave amplitude exceeds a certain threshold).However the maximum energy of the breathers created by modulational instability is bounded becau each breather collects the energy of the initial wave over the modulation lengthλso that its energy cannot exceed E max=λe where e is the energy density of the plane wave.Conquently,although modula-tional instability can lead to a strong increa in energy density in some parts of the system, it cannot create breathers with a total energy exceeding E max.For a given initial energy density,one can however go beyond this limit if one excitation can collect the energy of veral breathers created by modulational instability.Such a mec
hanism is not obrved in a continuum medium becau there the breathers generated by modulational instability are well approximated by solitons of the Nonlinear Schr¨o dinger(NLS)equation which can pass through each other without exchanging energy.On the contrary,when discreteness effects are prent,the energy of each excitation is not conrved in collisions,and,the important point is that the exchange tends to favor the growth of the larger excitation.In order to analyze the growth of the breathers in a lattice,we must therefore examine three of their properties:i)their stability,ii)their ability to move in the lattice,iii)the nature of their interactions.
In order to discuss the points quantitatively,let us,in afirst step,examine a specific model.We consider a chain of harmonically coupled particles situated at positions u n and submitted to the substrate potential
V(u n)=ω2d u2n3 ,(1) whereω2d is a parameter which measures the amplitude of the substrate potential,and therefore controls discreteness.We will be interested in motions inside the potential well (u<1).This potential can be viewed as a medium amplitude expansion of any asymmetric potential around a minimum.It can for instance reprent the expansion of a Mor potential in a nonlinear model for DNA denaturation[5]or the expression around a minimum of the well knownφ4po
tential[6].The hamiltonian of the model is
H= n 12(u n−u n−1)2+V(u n) .(2) The existence and stability of breathers in nonlinear Klein-Gordon models has been the subject of many investigations[6]and is not yet completely understood.However,we have shown that,provided that discreteness is strong enough,extremely stable large amplitude breathers can exist in such a model[5].They can be obtained with the Green’s function method introduced by Sievers and Takeno[7]for intrinsic localized modes in lattices with anharmonic coupling.The role of discreteness to stabilize the breathers can be understood if one starts from the“anti-integrable”limit where the on-site nonlinear oscillators are decoupled and then turns on a coupling which remains weak with respect to the on-site potential[8].Thus discrete breathers are sufficiently stable to have a long lifetime which gives them sufficient time to interact,provided that they can move in the lattice.This point is not as trivial as it might en.
II.PEIERLS-NABARRO BARRIER FOR A BREATHER The trapping effect of the discreteness is well known for topological solitonlike excitations and has been extensively investigated in the context of dislocation theory[9].In a lattice,a kink cannot move freely.The minimum energy barrier which must be overcome to translate the kink by one lattice period is known as the Peierls-Nabarro(PN)barrier,E PN.It can be calculated by evaluating the energy of a static kink as a function of its position in the latt
ice.For the various models which have been investigated,two extremal values are generally obtained when the kink is exactly situated on a lattice site(centered solution)or when it is in the middle between two sites(non-centered solution).
forsureFor a discrete breather very little is known,although the PN barrier has been shown to exist[6].One of the difficulties is that the breather is a two-parameter solution.While for a kink,the PN barrier depends only on the model parameters,for a breather it depends also upon its amplitude(or frequency).This amplitude dependence
is crucial for our analysis becau we are interested in the growth of breathers.As they increa in amplitude,the PN barrier that they feel changes.The definition of the Peierls barrier itlf is not as simple for a breather as for a kink.In principle,its value can be obtained by monitoring the breather as it is translated along one lattice constant.While for a kink the path followed by the particles in the multidimensional pha space of the system can be obtained by minimizing the energy while the position of the central particle is constrained in all intermediate states,in the ca of the breather,the path in the pha space is not a minimum energy path but a succession of saddle points.The energy of a kink which is exactly centered on a site or in the middle between two sites is defined without ambiguity.For a breather with a given frequency when it is centered on a site,there is no obvious con
straint which impos that it should have the same frequency when it is situated in the middle between two sites.We have ud,as a working definition of the PN barrier for a breather the difference between the energies of a centered and a non-centered breather with the same frequency.This definition gives results which agree with the obrvations of the breather motion made by molecular dynamics simulations,but the notion of PN barrier for a breather will require further analysis.
A.Large amplitude breathers
To calculate the PN-barrier,we have to compare the energy between two cas:the breathers is centered on a particle or between particles.In the previous paper[5],we focud our study in thefirst ca,but we can easily extend the method to the cond one. The procedure is the following:we look for stationary-mode solutions by putting
汉字转换u n=
∞
i=0φi n cos(iωb t),(3)
whereωb is the eigenfrequency of the breather andφi n are time independent amplitude oh the i th mode.Inrting the ansatz(3)in the dimensionless equation of motion,we t the coefficients of cos(iωb t)equal to each other,retaining only thefirst three terms.We obtain:
自学英语教材ω2dφ0n−[φ0n+1+φ0n−1−2φ0n]
=ω2d[φ0n2+
φ1n2+φ2n2
2
](4c) Then,invoking the Green’s functions for the linear left-hand sides,we get a t of simultane-ous nonlinear eigenvalue equations determining the eigenfrequencyωb and the eigenfunctions φi n:
φ0n= m G(n−m,0)[φ0m2+φ1m2+φ2m2
2
](5c) where the Lattice Green’s functions have the following expression:
G(n,ωb)=ω2d
ω2d−ω2b+2[1−cos(q)]
.(6)
For solving this system,the procedure requires more care than in the centered ca[5], to avoid the problem of instability of this mode.Indeed,since the position at the top of the PN barrier is intrinsically unstable,regardless of other possible caus of instability,even starting with a symmetrical initial condition,the results show that in all cas the breather moves so that the center reaches the bottom of the he solution converges toward the more stable breather.To prevent this tendency we cho to impo the symmetry and calculate the solution for only a half of the chain,the cond half being know by symmetry:
