seminar and workshop on nonlinear lattice structure and dynamics

8/3/2019 Seminar and Workshop on Nonlinear Lattice Structure and Dynamics

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Seminar and Workshop on Nonlinear Lattice Structure and Dynamics

1st Focus Week: Nonlinear Deformation Waves, September 4-9, 2001

ABSTRACTS


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Soliton dynamics in damped and forced Boussinesq equations

E. Arevalo, Yu. Gaididei and F. G. Mertens

Physikalisches Institut, Universitat Bayreuth,D-95440 Bayreuth, Germany

We investigate the dynamics of a lattice soliton on a monatomic chain in the presenceof damping and external forces. We consider Stokes and hydrodynamical damping.In the quasi-continuum limit the discrete system leads to a damped and forcedBoussinesq equation. By using a multiple-scale perturbation expansion up to secondorder in the framework of the quasi-continuum limit we derive a general expressionfor the first-order velocity correction which improves previous results. We comparethe soliton position predicted by the theory with simulations carried out on thelevel of the monatomic chain system as well as on the level of the quasi-continuumlimit system. For this purpose we restrict ourselves to specific examples, namelypotentials with cubic and quartic anharmonicities as well as the truncated Morsepotential, without taking into account external forces. We find a good agreementwith respect to the numerical simulations for the hydrodynamical damping case.Moreover we clarify why the quasi-continuum limit is not a good approach for theStokes damping case.


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Exact description of motion and interaction of

discrete breathers in a system of coupled oscillators

Mikhail M. Bogdan

B.I.Verkin Institute for Low Temperature Physics and Engineeringof National Academy of Sciences of Ukraine,

47, Lenin Ave., 61103 Kharkov, Ukraine

Effects of the discrete breathers dynamics are studied in the framework of exactly-solvable one-dimensional differential-difference equations. Some of these equationsare interconverted and, in general, describe a system of nonlinear coupled oscillators.

The multibreather formula is found explicitly. By the use of the formula the mo-tion of a discrete breather as well as the interaction of a pair of colliding breathersare investigated in detail. Shifts of breather centers and phases of temporal oscil-lations under the collision are calculated as functions of velocities and frequenciesof breathers. Dynamical characteristics of the discrete breather, the energy, the

momentum, velocity, and frequency and their relationship are discussed.

Possible experimental realization of the discrete multibreathers dynamics is pro-posed, and the connection of the breather solutions with nonlinear deformationwaves in discrete and continuum models of solids is specified.

This research was supported partly by grant INTAS-99 no. 167.


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Solitary waves in an inhomogeneous rod

composed of a general hyperelastic material

Hui-Hui Dai and Yi Huo

Department of Mathematics, City University of Hong Kong,Kowloon, Hong Kong.

Email: [email protected]

We consider analytically the development of a soliton propagating along a circularrod composed of a general compressible hyperelastic material with variable crosssections and variable material density. The purpose is to provide analytical de-scriptions for the following two phenomena found respectively in numerical andperturbation studies: 1. Fission of a soliton. When a soliton moves from a partof the rod with thick cross sections to a part with thin cross sections, it will splitinto two or more solitons; 2. When a soliton propagates along a rod with slowlydecreasing radius, it will develop into a solitary wave with a shelf behind. By using a

nondimensionalization process and the reductive perturbation technique, we derive avariable-coefficient Korteweg-de Vries (vcKdV) equation as the model equation. Theinverse scattering transforms are used to study the vcKdV equation. By consideringthe associated isospectral problem, the phenomenon of soliton fission is successfullyexplained. We are able to provide the condition that exactly how many solitons willemerge when a single soliton moves from a thick section to a thin section. Then, byintroducing suitable variable transformations, we successfully manage to transformthe vcKdV equation into a cylindrical KdV equation. As a result, several exactbounded solutions in terms of Airy Function Ai and Bi are obtained. One of thesolutions has the shape of a solitary wave with a shelf behind. Thus, it providesan analytical description for the perturbation and experimental results in literature.

Comparisons are also made between the analytical solutions and numerical resultsand good agreement is found.Keywords: solitary waves; inhomogeneous rod.


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Soliton ensembles and the periodicity

in their interaction patterns

Juri Engelbrecht, Andrus Salupere, Pearu Peterson

Centre for Nonlinear StudiesInstitute of Cybernetics at Tallinn TU

Akadeemia 21, 12618 Tallinn, Estonia

The paper is focused on the long-time behaviour of the KdV-solitons emerging froman initial harmonic excitation. Although studied intensively since the seminal worksby Zabusky and Kruskal, the sharing of energy between the modes in nonlinear sys-tems for a long run is not completely understood. Here we rely on the following.The emergence of solitons from a harmonic excitation leads to a well-defined solitontrain only in the beginning of the process and later the groups of interacting solitons,i.e. soliton ensembles propagate. Due to the fluctuation of the equilibrium level,the smaller solitons may show up for short time intervals only. In order to distin-

guish them from permanently visible solitons, we have used the notion of virtualsolitons. It means that soliton ensembles involve visible and virtual solitons in theirinteraction process.

Consequently, at an arbitrary time there are actually two processes mixed up inpropagation: (i) emergence of solitons from an ensemble (at t = 0 described by aharmonic function) and (ii) the interaction of definite solitons or their ensembles. Asfar as the widths of emerging solitons are larger than the visible distances betweenmaxima of wave profiles, the solitons within ensembles are not fully separated andthe Lax types of interaction are not acceptable in their strict form. The interaction ofand within soliton ensembles is characterised by curved trajectories of wave extrema

and/or by discontinuous successive small jumps of trajectories. The analysis of waveextrema during the interaction of two single solitons and those within an ensembleis presented.

In the long run, due to the existence of left- and right-going solitons, the trajectoriesform certain patterns. The arc-like trajectories in the early stage of the process arelater, i.e., for t tR (tR is the recurrence time) clearly transformed into a stablerhombus-like pattern in the x t plane. The time-directed diagonal of this pattern2tP is related to the recurrence phenomenon but satisfies the inequality tP > tR. Ananalysis of patterns over the range of dispersion parameters is presented.

For numerical simulation, the pseudospectral method is used with different ODE

solvers for integration with respect to the time variable.


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A numerical study of finite amplitude wave propagation

in a nonlinear elastic tube

H. A. Erbay1 and V. H. Metin2

1Istanbul Technical University,Faculty of Science and Letters, Department of Mathematics,

Maslak 80626, Istanbul, Turkey2Isik University,

Faculty of Science and Letters, Department of Mathematics,Maslak 80670, Istanbul, Turkey

In this study, we consider wave propagation in a circular cylindrical hyperelastictube when the tube is subjected to dynamic extension. One end of the tube is fixedand the other end is subjected to a dynamic extension. The tube is considered as ahyperelastic, isotropic, homogeneous membrane. For the numerical solution of thegoverning equations we employ a second-order Godunov-type finite volume method.

Numerical results are given for various strain-energy functions which are widely usedto describe the behavior of rubber-like materials. The response of a compressiblemembrane is compared with that for an incompressible membrane.

The same problem has been studied by Tait and Zhong [1] for an incompressibleMooney-Rivlin elastic material. They have considered a numerical technique basedon the method of characteristics. In another study [2] Tait and Zhong have alsoconsidered the same tube and, in additon to the extension, imposed a dynamicaltwist at the moving end. We also discuss how the results of Tait and Zhong arerelated to those obtained in this study.

References

[1] R.J. Tait and J.L. Zhong, Wave propagation in a non-linear elastic tube. Bull.Tech. Univ. Istanbul 47, 127-150 (1994).

[2] R.J. Tait and J.L. Zhong, Dynamic extension and twist of a non-linear elastictube. Int. J. Non-linear Mech. 30, 887-898 (1995).


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Coupled higher-order nonlinear Schrodinger equations

in a generalized elastic solid

Irma Hacinliyan and Saadet Erbay

Department of Mathematics, Faculty of Science and Letters,Istanbul Technical University, 80626 Maslak, Istanbul, Turkey

In the present study, the nonlinear modulation of transverse waves propagatingin a generalized elastic solid is studied using a multi-scale expansion of quasi-monochromatic wave solutions. In particular, to include the higher-order nonlinearand dispersive effects in the evolution equations, higher-order perturbation equa-tions are considered, and it is shown that the modulation of two transverse wavesis governed by a pair of the coupled higher-order nonlinear Schrodinger (HONLS)equations. In the absence of one of the transverse waves, the coupled HONLS equa-tions reduce to the single HONLS equation which has been already obtained in thecontext of nonlinear optics. Some special solutions to the coupled HONLS equations

are also presented.


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Intrinsic localized modes: beyond the rotating wave approximation

A. Franchini, V. Bortolani and R.F. Wallis

Istituto Nazionale di Fisica della Materia, Unita di Modena eDipartimento di Fisica, Universita di Modena e Reggio Emilia,

Via Campi 213/A, 41100 Modena (Italy)

Institute for Surface and Interface Science and Department of Physics and Astronomy,University of California, Irvine, CA 92697 (USA)

The existence of the highly localized modes in anharmonic crystals is by now wellestablished [?, ?]. These modes have been termed intrinsic localized modes (ILMs),reflecting the fact that no external defects are needed for the creation. The firststudies [?, ?] showed the existence of ILMs above the top of the harmonic phononbranch for monoatomic one-dimensional lattices with harmonic and quartic anhar-monic interactions. In a diatomic chain with realistic potentials, however, Kiselev etal. [?] showed that ILMs arise in the gap between acoustic and optical branches. The

above treatments are based on the rotating wave approximation (RWA) in whichthe atomic displacements contain a static part and a vibrational part proportionalto cos(t), where is the frequency of the stationary localized mode. In this pre-sentation intrinsic localized modes in the gap of a diatomic chain with free ends arediscussed in detail by going beyond the rotating wave approximation. We includein the time dependence of the displacements terms up to cos(2t). We consider afinite chain of particles interacting with nearest neighbours interactions. We studyamplitudes of the intrinsic localized modes smaller than 0.25 A. In this range ofamplitudes the full potential can be well represented by an expansion in power ofthe displacements up to forth order terms. The case of a force constant model allowsus to simplify the problem. As a test case we consider a chain of LiI atoms. We

found intrinsic localized modes in the gap. The amplitudes of the first harmonicterm (cos(t)) are of even or odd parity, thereas the amplitudes of the static partand those of the second harmonic can have only even symmetry. The main results isthat the amplitudes associated with the second harmonic terms are two or three or-ders of magnitude less than those of the first harmonic. Furthermore the frequencyof the localized modes are modified by less than 1 % by the inclusion of the secondharmonic.

References

[1] Sievers A J and Takeno S 1988 Phys. Rev. Lett. 61 970

[2] Page J B 1990 Phys. Rev. B41 7835

[3] Wallis R F, Franchini A and Bortolani V 1994 Phys. Rev. B50 9851

[4] Kiselev S A, Bickham S R and Sievers A J 1994 Phys. Rev. B50 9135

[5] Franchini A, Bortolani V and Wallis R F 1996 Phys. Rev. B53 5420


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Necking instability waves in a stretched elastic plate

Yibin Fu1 and A. Ilichev2

1Department of Mathematics, Keele University, Staffordshire ST5 5BG, U.K.2Mathematical Steklov Institute, Gubkina str. 8, 117966 Moscow, Russia

When a solid rubber rod is stretched, the rod initially undergoes a uniform exten-sion and the radius of the rod remains constant along the axis. When the stretchingforce exceeds a certain critical value, the rod may develop a localized section wherethe radius varies along the axis and is smaller than in the rest of the rod. This phe-nomenon is known as a necking instability and has been observed in a number ofengineering situations involving rubber materials. Necking instability has been muchstudied in plasticity theory where necking is triggered off by the material becomingplastic (see, e.g., Hutchinson and Miles 1974). Limited studies have been carriedout to explain the necking phenomenon in the context of purely elastic deformationswhere necking becomes possible because of the intrinsic softening behaviour of cer-

tain rubber materials. See, e.g., Coleman (1983), Owen (1987) and Mielke (1991).All these studies except that of Mielke (1991) have been based on the analysis ofsimplified models. Recently, Scherzinger and Triantafyllidis (1998) presented an as-ymptotic analysis using the slenderness ratio

area/length as a small parameter.

None of these studies took dynamic effects into account.

In the present study, we analyze the simpler case of a stretched rubber plate in astate of plane strain. We believe that the same ideas can be applied to the moreinvolved case of a circular rod under stretching. Our analysis is based on the finiteelasticity theory and does not involve any approximations. Dynamic effects are alsoincorporated into our asymptotic analysis with relative ease. Thus, our concern iswith instability waves propagating in a plate stretched by a force that is close to acertain critical value. This is the value at which extensional waves have zero wavespeed, or equivalently, the load-stretch curve in a uni-axial tension test turns. Weshow that the amplitude of the necking waves satisfies the Boussinesq equation, andthat only waves with big enough speed are stable. In particular, we show that thelocalized static necking solution is unstable.

Some of the results reported here have already appeared in Chapter 10 of the bookedited by Fu and Ogden (2001).

References

1. Coleman, B.D. 1983 Necking and drawing in polymeric fibres under tension.Arch. Rat. Mech. Anal. 83, 115-137.

2. Fu, Y.B. and Ogden, R.W. (eds) 2001 Nonlinear Elasticity: Theory and Ap-plications. Cambridge University Press (LMS Lecture Note Series 283).

3. Hutchinson, J.W. and Miles, J.P. 1974 Bifurcation analysis of the onset ofnecking in an elastic/plastic cylinder under uniaxial tension. J. Mech. Phys.Solids 22, 61-71.

4. Mielke, A. 1991 Hamiltonian and Lagrangian Flows on Center Manifolds.

Berlin: Springer (Lecture Notes in Mathematics vol. 1489).


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5. Owen, N. 1987 Existence and stability of necking deformations for nonlinearlyelastic rods. Arch. Rat. Mech. Anal. 98, 357-384.

6. Scherzinger, W. and Triantafyllidis, N. 1998 Asymptotic analysis of stabilityfor prismatic solids under axial loads. J. Mech. Phys. Solids 46, 955-1007.


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Nonlinear Scholte waves: theory and experimental observations

V. Gusev1, C. Glorieux2 and K. Van de Rostyne2

1Laboratoire de Physique de lEtat Condense, UMR-CNRS 6087,Faculte des Sciences, Universite du Maine, 72085 Le Mans, France

2Laboratorium voor Akoestiek en Thermische Fysica,

Departement Natuurkunde, Katholieke Universiteit Leuven,Celestijnenlaan 200D, B-3001 Leuven, Belgium

First experimental observations of nonlinear Scholte waves are reported. In our caseof an interface of a hard solid with a liquid the Scholte wave propagates very muchlike a bulk compression wave in liquid. The observations could be qualitativelyfitted in the framework of the simple-wave equation with an attenuation term. Wehave noted that of two recently published theories of nonlinear Scholte waves (V.E.Gusev, W. Lauriks, J. Thoen, IEEE UFFC, 45, 170 (1998) and G.D. Meegan, M.F.Hamilton, Y.A. Ilinskii, E.A. Zabolotskaya, J. Acoust. Soc. Am. 106, 1712 (1999))

the former reproduces transformation of nonlinear Scholte waves into nonlinear bulkwaves in the limiting case of an infinitely hard solid, while the latter (based on theHamiltonian formalism) - does not. In relation to this observation, a possibility toderive evolution equations for nonlinear surface and interface waves, avoiding someof the constraining assumptions essential to the Hamiltonian formalism, is discussed.


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Acoustic pulses in media with quadratic hysteretic nonlinearity

Vitalyi Gusev

Laboratoire de Physique de lEtat Condense, UMR - CNRS 6087,Faculte des Sciences, Universite du Maine, 72085 Le Mans, France

Evolution equations for propagation of both unipolar and bipolar acoustic pulsesare derived by using hysteretic stress-strain relationships. Hysteretic stress-strainloops that incorporate quadratic nonlinearity are derived by applying the modelof Preisach - Mayergoyz space for the characterization of structural elements in amicro-inhomogeneous material. Exact solutions of the nonlinear evolution equationspredict broadening in time and reduction in amplitude of a unipolar finite-amplitudeacoustic pulse. In contrast with some earlier theoretical predictions, the transforma-tion of the pulse shape predicted here satisfies the law of momentum conservation(the equality of areas law in nonlinear acoustics of elastic materials). A bipolarpulse of nonzero momentum first transforms during its propagation into a unipolarpulse of the same duration. This process occurs in accordance with the momen-tum conservation law and without formation of shock fronts in the particle velocityprofile. Qualitative comparison with available results of the experiments is under-taken.


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Modeling of nonlinear surface acoustic waves

M. F. Hamilton, Yu. A. Ilinskii, and E. A. Zabolotskaya

Department of Mechanical EngineeringThe University of Texas at Austin

Austin, Texas. USA

Over the past decade a consistent theory based on Hamiltonian mechanics has beendeveloped for a wide variety of nonlinear surface acoustic waves. Our presentationreviews evolution equations and physical insight resulting from this body of work.The early studies were devoted to Rayleigh waves in isotropic solids, for whichevolution equations have been derived in three alternative forms. One is a set ofcoupled spectral equations. A second is an integral equation that permits separateconsideration of local and nonlocal nonlinearity. The third is a differential equationinvolving Hilbert transforms. Absorption, shock formation, beam diffraction, andtransient effects in pulses have been taken into account in analytical investigationsand numerical simulations based on these equations. More recent work has beendevoted to modeling other types of surface waves. Stoneley and Scholte interfacewaves in isotropic media are described by coupled spectral equations possessingthe same mathematical properties as the equations for Rayleigh waves. Evolutionequations for surface waves in crystalline and piezoelectric media reveal a widevariety of nonlinear wave profiles. Current work is focused on effects of dispersiondue to a thin layer attached to the substrate.


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Fracture of isotropic and anisotropic materials

by nonlinear surface acoustic waves

G. Lehmann, A.M. Lomonosov, and P. Hess

University of HeidelbergInstitute of Physical Chemistry

Im Neuenheimer Feld 25369120 Heidelberg, Germany

An important property of materials to be determined experimentally is their me-chanical strength. To study the fracture strength up to now mainly steady-statecracking experiments have been performed by loading the material with a constantforce. This tensile loading geometry generates a running crack starting at a pre-formed seed crack.

With laser-excited nonlinear SAW pulses it is possible to generate shocks withstresses that exceed the mechanical strength of covalently bonded materials, such as

silicon. During propagation elastic nonlinearity leads to the development of shockfronts and to transient fracture by impulsive loading without seed crack. Thus, frac-ture will occur along the weakest crystal planes. Indeed, a series of cracks extendingalong the 110 direction, perpendicular to pulse propagation could be detected byscanning force microscopy (SFM) for a strongly nonlinear SAW pulse propagatingalong the 112 direction on the Si(111) plane. These experiments indicate that insingle-crystal silicon impulsive fracture occurs mainly along the Si(111) plane withthe lowest bond density, the so-called cleavage plane. This behavior was also ob-served for SAW propagation along other silicon planes and directions in nearly allexperiments. In addition, fracture of isotropic fused quartz by nonlinear SAW pulses

was investigated.The development of microscopic models and continuum theories for impulsive frac-ture (e.g. by SAWs) is still at the beginning. The goal is to model the initiation anddevelopment of cracks as a function of the applied transient load. The fundamentalproblem is to understand the microscopic processes using macroscopically observedparameters. First results will be presented to explain the experimental finding thatfracture in silicon behaves different if on the Si(111) plane the direction of SAWpropagation is reversed from the 112 to the 112. This yields new insight intothe cracking process in anisotropic media. Another important piece of information,obtained by the focused ion beam (FIB) technique, is the penetration depth of SAWinduced cracks into the solid.

In real materials the theoretical strength may be orders of magnitude higher thanthe measured value. The reason for this behavior can be the microstructure, wheremicrodefects, microcracks, or dislocations determine the actual strength. It is as-sumed that the cleavage behavior observed in this work can be explained withoutresorting to the influence of defects, and thus the phenomena considered are essen-tially intrinsic to the silicon lattice.


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Standing wave instabilities, breather formation

and thermalization in Hamiltonian anharmonic lattices

Magnus Johansson1, Anna Maria Morgante2, Serge Aubry2,

and Georgios Kopidakis3

1Department of Physics and Measurement Technology, Linkoping University,S-581 83 Linkoping, Sweden

2Laboratoire Leon Brillouin (CEA-CNRS), CEA Saclay,F-91191 Gif-sur-Yvette Cedex, France

3Department of Physics, University of Crete,P.O. Box 2208, Heraklion, Crete, Greece

Modulational instability of travelling plane waves is a wellknown mechanism whichis generally accepted to constitute the first step in the formation of intrinsicallylocalized modes (discrete breathers) in anharmonic lattices. Here, we consider analternative mechanism for breather formation, originating in oscillatory instabilitiesof spatially periodic or quasiperiodic nonlinear standing waves (SWs). These SWsare constructed for Klein-Gordon or Discrete Nonlinear Schrodinger (DNLS) latticesas exact time periodic and time reversible multi-breather solutions from the (anticon-tinuous) limit of uncoupled oscillators, and merge into the standard harmonic SWsin the small-amplitude limit. Approaching the linear limit, all SWs with nontrivialwave vectors (0 < Q < ) become unstable through oscillatory instabilities, whichpersist for arbitrarily small amplitudes in infinite lattices. Investigating the dynam-ics resulting from these instabilities, we find two qualitatively different regimes forwave vectors smaller than or larger than /2, respectively. In one regime persistingbreathers are found, while in the other regime the system rapidly thermalizes.


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Nonlinear waves in a beam lattice model

Takuji Kawahara and Hiroaki Matsuoka

Department of Aeronautics and Astronautics,Graduate School of Engineering,

Kyoto University, Kyoto 606-8501, Japan

email: [email protected]

Fundamental wave behaviours are investigated numerically for a one-dimensionalnonlinear lattice model composed of a chain of masses connected by elastic beam.The system dealt with here is the same as that considered by A.Askar (1973) exceptfor the inclusion of quadratic nonlinearity and it is given as follows;

myn = 12

ka(n+1 n1) + (k + k1)(yn+1 + yn1 2yn)

+k2{(yn+1 yn)|yn+1 yn| (yn yn1)|yn yn1|},

jn = 12

ka(yn+1 yn1) 16

ka2(n+1 + n1 + 4n),

where yn and n denote displacement and rotation, m , j, and a represent mass,moment of inertia, and lattice constant, k is the coefficient of bending rigidity ofbeam, k1 and k2 are linear and nonlinear coefficients of restoring force for relativedisplacement of masses.

Evolutions of initial disturbances are pursued numerically to observe generationand propagation of nonlinear waves. Either the acoustic or the optical modes can begenerated. For long wave disturbances, a pair of stable acoustic solitons with positiveand negative amplitude are generated in strongly nonlinear regime. Meanwhile

for short wave disturbances, optical envelope solitons governed by the nonlinearSchrodinger equation are generated in weakly nonlinear regime. It is found thatthe couplings between displacement and rotation differ for the acoustic mode andthe optical mode, which is responsible for the different wave behaviours in the twomodes. For some specified parameters, three wave resonances amongst the acousticmodes and those between the acoustic and the optical modes are observed.


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Propagation of nonlinear waves and solitons in elastic plates

C. Eckl1, A.S. Kovalev2, A.P. Mayer1, and E.S. Sokolova2

1Institute for Theoretical Physics, University of Regensburg,93040 Regensburg, Germany

2B. Verkin Institute for Low Temperature Physics and Engineering,

310164 Kharkov, Ukraine

The nonlinear dynamics of elastic shear waves in a plate is investigated taking intoaccount their interaction with small amplitude sagittal components. The nonlinearevolution equation for the shear displacements is derived. This one-dimensionalintegro-differential equation is similar to the modified Boussinesq equation (MBE)but contains an additional nonlinear dispersion term arising from the interactionwith the sagittal displacements. As in the case of MBE the obtained nonlinearevolution equation admits soliton solutions but nonlinear dispersion modifies thestructure of such solitons. They tend to transform into exotic solitons (compactons

or peakons for different materials). The interaction with sagittal components leadsto the emission of Rayleigh - Lamb waves by a moving soliton but this emissionis small for small-amplitude solitons. The nonlinear dispersion relation is obtainedfor nonlinear periodic waves in the plate and the splitting of two branches of thespectrum (for shear and Rayleigh-Lamb waves) is demonstrated.

This work was partly supported by INTAS99 (grant No167).


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Stationary and envelope Rayleigh surface solitons

in the presence of surface nonlinearity

C. Eckl1, A.S. Kovalev2, A.P. Mayer1, and G.A. Maugin3


2B. Verkin Institute for Low Temperature Physics and Engineering,310164 Kharkov, Ukraine

3Laboratoire de Modelisation en Mecanique,Universite Pierre et Marie Curie, 75252 Paris, France

The propagation of nonlinear Rayleigh waves is investigated in a half-space of linearelastic medium coated with a thin film of nonlinear material. For this system, anonlinear evolution equation is derived which admits solutions for Rayleigh-typesolitons of stationary profile. This equation may be regarded as a special case in thewider class of evolution equations with a more general type of nonlocal nonlinearity.

Periodic pulse train solutions are computed. For a certain member of the classof nonlinear equations, several families of exact solitary wave solutions and theirassociated periodic stationary wave solutions are derived analytically. These infiniteseries of exact solutions represent surface solitons of stationary profile with differentaxial symmetry. The first soliton in the series is close to the Kadomtsev-Petviashvilisoliton. For the above-mentioned model system approximate solutions for Rayleigh-type envelope surface solitons were obtained as well. A new asymptotic method forthe problem of solving integro-differential nonlinear evolution equations is proposed.

The work was partly supported by INTAS99 (grant No 167).


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Surface acoustic solitons: experimental results

A.M. Lomonosov, A.P. Mayer, and P. Hess

University of HeidelbergInstitute of Physical Chemistry

Im Neuenheimer Feld 253

69120 Heidelberg, Germany

The evolution equation of nonlinear SAWs, supplemented with an additional termdescribing weak dispersion, admits solitary solutions. In this work we present ex-perimental observations of surface acoustic solitons in comparison with numericalsimulations in several particular cases. We consider the cases of normal and anom-alous dispersion in an isotropic medium possessing a purely imaginary and negativenonlinear parameter F(1/2), and the case of normal dispersion in an anisotropicmedium with complex-valued parameter F(1/2). As an example for an isotropicmedium we used fused quartz samples, coated either with a thin metal film in orderto obtain normal dispersion or with a TiN film, which provides anomalous disper-sion with respect to SAWs. In each case solitons were observed having shapes andpropagation velocities consistent with theoretical predictions. The case of complex-valued F(1/2) was realized in crystalline silicon in the 112 direction on the Si(111)plane. Normal dispersion was generated by a thin thermally grown oxide layer.

In these experiments we used laser-generated intense SAW pulses and an opticaldetection scheme with a two-point probe beam deflection setup measuring the pulseprofile at two different distances from the source. Efficient excitation of these non-linear waveforms was achieved by explosive evaporation of a thin highly absorbingcarbon suspension at the source region.

In the numerical simulations, a dispersion term was chosen that corresponds toa linear dependence of the phase velocity on frequency, which is typical for SAWdispersion in a thin layer. A comparison of experimental results with numericalsimulations showed that the applied theoretical model describes the evolution ofthe solitary pulses adequately, especially in the case of isotropic materials. Thisagreement made it possible to study the properties of solitons numerically based onthe experimentally determined pulse shapes, measured at the two probe locations.It turned out that the propagation velocity of the surface solitons deviates fromthe linear Rayleigh velocity according to the signs of the dispersion term and theimaginary part of F(1/2). The shape of the soliton depends strongly on the realpart of F(1/2). Its polarity is determined by the signs of both the dispersion term

and the real part of the nonlinear parameter.


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Multiwave nonlinear couplings

in

elastic structures

Gerard A. Maugin, D.A. Kovriguine and A.I. Potapov

Universite Pierre et Marie Curie, Laboratoire de Modelisation en Mecanique,Paris, France ([email protected])Mechanical Engineering Institute, Russian Academy of Science,

Nizhny-Novgorod, Russia.

Multiple-wave nonlinear resonance coupling is a known and well-studied phenom-enon in fluid mechanics (cf. A.D.D. Craig, Wave Interactions and Fluid Flows,C.U.P., 1985) and optics (D.F. Nelson, Electric, Optic and Acoustic Interactionsin Dielectrics, J. Wiley, 1979). Here three paradigmatic examples are examined insolid mechanics: nonlinear waves in a thin elastic rod (simple Bernoulli-Euler model;exhibiting a continuous spectrum), nonlinear oscillations in a circular elastic ring(exhibiting a discrete spectrum) , and nonlinear waves in a thin elastic plate (2Dexample). This allows one to show such phenomena as stress amplification and four-wave resonant interactions in a rod, triple and four-wave nonlinear couplings in thesecond case, and resonant triads in the plate case. In all the paper presents a studyof the hierarchy of instabilities in these three types of slender elastic structures.(Work within the INTAS Programme: 96-2370, 1997-2001).


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Surface aoustic solitary waves: theoretical predictions

C. Eckl1, P. Hess2, A.S. Kovalev1,3, A.M. Lomonosov2,4, and A.P. Mayer1


2Institute of Physical Chemistry, University of Heidelberg,

69120 Heidelberg, Germany3B. Verkin Institute for Low Temperature Physics and Engineering,

310164 Kharkov, Ukraine4General Physics Institute, 117942 Moscow, Russia

Propagation of nonlinear surface acoustic waves in a semi-infinite elastic medium isknown to be governed by an evolution equation that contains a nonlocal nonlinearityof second order. Modifications of the surface that introduce a length scale in thesystem, e.g. coating the surface with a thin film, give rise to an additional linear dis-persion term that can balance the nonlinearity. The dispersion term is usually that

of the Benjamin-Ono (BO) equation. In special cases, it may be of the Korteweg -de Vries (KdV) type. In the presence of such a dispersion term, the evolution equa-tion has solitary wave solutions that can be computed as limiting cases of stationaryperiodic waves. Properties of these solutions, including their depth structure, arediscussed, and their stability is tested in a linear stability analysis. Numerical sim-ulations reveal that pulse collisions are highly inelastic if the dispersion term is BOlike. For a linear dispersion term of KdV type, solitary pulses have been found tosurvive collisions with a comparatively small amount of radiation produced. A ten-tative explanation of this behaviour is given by establishing a connection with theBenjamin-Ono hierarchy. Characteristic differences between the two types of lineardispersion are also found in numerical simulations of pulse evolution with initialconditions close to solitary solutions.

Finally, the regime of small nonlinearity and strong dispersion is revisited, for whichthe existence of surface acoustic envelope solitons had been predicted long ago. Itis shown how resonant long-wave - short-wave interaction can influence propertiesof these envelope solitons.


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Effects of thermal noise on solitons in anharmonic chains

E. Arevalo, F. G. Mertens, and Y. Gaididei

Physics Institute, University of Bayreuth, D-95440 Bayreuth

We consider a one dimensional lattice with nearest neighbour interactions with cubic

anharmonicities which in the continuum limit leads to a Boussinesq equation whichhas soliton solutions. Including thermal noise and damping in the discrete equationsof motion we derive a stochastic Boussinesq equation.

Using a collective variable theory we obtain stochastic equations of motion for thesoliton from which the variance of the soliton position is calculated. The leadingterm is linear in time, the coefficient is the soliton diffusion constant. For largertimes higher powers in time become important. These results are confirmed byLangevin-dynamics simulations for the original, discrete model.


23/33

Non-dissipative diffusion of Toda solitons

E. Arevalo1, F. G. Mertens1, A. Bishop2, and Y. Gaididei1

1Physics Institute, University of Bayreuth, D-95440 Bayreuth2 Los Alamos National Laboratory, Los Alamos NM 87545, USA

We consider a soliton in a gas of phonons in a non-equilibrium situation, i. e. thesoliton is not thermalized. We derive the transport equations for the soliton diffusionand calculate the drift velocity and the diffusion constant by taking into account thespatial shifts of the soliton due to elastic scattering with the phonons. The results aregeneralized to solitons on anharmonic chains with arbitrary interaction potentialsand are compared with MD-simulations for such chains. Here the phonon bath isobtained by switching off noise and damping in Langevin-dynamics simulations aftera sufficiently long time.


24/33

Effects of local field singularities

for nonlinear elastic waves

V. G. Mozhaev

Faculty of Physics, Moscow State University,Moscow, 117234 Moscow, Russia

The simplest method to solve the nonlinear acoustics problems, which is widely usedfor the description of experimental observations, is the method of successive approx-imations. This method allows one to describe easily the first stage of the generationof second and higher harmonics of initially sinusoidal acoustic waves and to esti-mate the efficiency of nonlinear acoustic processes. As a quantitative measure of theefficiency of such processes, a so called nonlinear acoustic parameter is commonlyused. In homogeneous solids, this parameter depends on only elastic constants ofsecond and third order and it is usually less than 10. However, in inhomogeneoussolid media, a local concentration of acoustic fields can take place, that in its turn

results in local growth of the nonlinear parameter. Moreover, in inhomogeneoussolids, the situations may occur when the solutions of even the linear problems havelocal singularities of elastic strain fields. Such singularities are found, in particular,at crack tips, at dislocation cores, in point contacts of acute-angle spikes and sur-faces. If the initial singular elastic strains at these inhomogeneities are producedby incident harmonic acoustic waves, then taking into account the nonlinear elasticproperties of solids described by additional terms, which are nonlinear in strain,leads to divergence of the solutions for second and higher harmonics. Considerationof additional factors like finite radius of rounding of acute-angle spikes, nonlocalityof elastic properties or discreteness of crystalline lattice allows one to remove thesingularities of the solutions at the fundamental frequency and at its higher harmon-

ics. However, the essential local growth of nonlinear acoustic fields is still retainedin this case.

Similar effects may occur also in the case of local concentration of electrical fields atthe tips of needle electrodes and at the edges of plane electrodes that may result ingreat growth of efficiency of nonlinear acoustic wave generation by such electrodesin electrostrictive materials.

The problems of this type are intended to be discussed in the present talk eitherquantitatively (using the method of successive approximations) or qualitatively withspecial emphasis on the problem of nonlinear acoustic behavior of microcracks.


25/33

Nonlocality and nonlinearity in elastic surface waves

D. F. Parker

Department of Mathematics and Statistics, University of Edinburgh,The Kings Buildings, Edinburgh, EH9 3JZ, U.K.

Surface waves are examples of guided waves, so that a modal analysis is natural.For appreciable nonlinear interaction between modes, it is widely known that goodmatching of both phase and group velocities is necessary. However, for waves at thesurface of a homogeneous elastic half-space, isotropic or anisotropic, wavespeed isentirely independent of wavelength. In an arbitrarily chosen propagation direction,surface waves are entirely non-dispersive. Consequently, in a weakly nonlinear theoryall pairs of Fourier harmonics interact with both their sum and their differencefrequencies. The resulting spectral description involves interactive evolution of all(complex) Fourier amplitudes either in a sum (for periodic waveforms) or in anintegral (the general case).

The alternative directformulation, initiated by Hunter [1], describes the evolution ofthe surface slope through a quadratically nonlinear, but nonlocal, influence integral,generalizing that for the isotropic case [2]. Recent developments in the analysis ofthe kernel within this integral will be described, along with the incorporation intothe evolution equation of lateral spreading and diffraction effects.

1. J.K. Hunter (1989). Nonlinear surface waves, in Current Progress in Hy-perbolic Problems and Computations ed. W.B. Lindquist, Am. Math. Soc.Providence, R.I., 185-202.

2. M.F. Hamilton, Yu.A. Ilinsky and E.A. Zabolotskaya (1995). Local andnonlocal non-linearity in Rayleigh waves, J. Acoust. Soc. Am., 97, 882-890.


26/33

Four Lagrangian representations

for the Camassa-Holm equation

Maxim V. Pavlov

Loughborough [email protected]

The well-known hierarchy of Extended Harry-Dym Camassa-Holm equations hascouple of local Hamiltonian structures.

The Extended Harry-Dym equation

uy = (x 3x)H2u

= (2ux + ux)H3u

= (x 3x)u1/2 (1)

has Hamiltonian densities expressed via field variable u and its derivatives

H1 =

udx, H2 =udx, H3 = [u

5/2

u

2

x + u

1/2

]dx,... (2)

but Camassa-Holm equation is nonevolutionary equation

(1 2x)wt = 3wwx wwxxx 2wxwxx, (3)

where

u = w wxx, (4)

and Hamiltonian densities related with the same local Hamiltonian structures arepseudononlocal, it means that they can be determined as special solutions of someodinary differential equations. Complete description of all pseudononlocal conser-vation law densities exactly will be presented in this report. The most simplepseudononlocal Hamiltonian density is determined by momentum for the first Hamil-tonian structure

H =1

2

[w2 + w2x]dx.

Moreover, we will present for these local Hamiltonian structures couple of localLagrangian representations and other couple local Lagrangian representations forcouple neighbour nonlocal Hamiltonian structures too. These nonlocal Hamiltonianstructures will be described as well.

Thus, our claim is that the Camassa-Holm equation (and extended Harry-Dym too)has four different variational principles.


27/33

Nonlinear deformation waves in solids with microstructure

A.V. Porubov1,2 and F. Pastrone3

1A.F. Ioffe Physical Technical Institute of the Russian Academy of Sciences,St. Petersburg 194021, Russia

2Institute for High-Performance Computing and Data Bases,

P.O. box 71, St. Petersburg, 194291 Russia3Dipartimento di Matematica, Universita di Torino,

Via C. Alberto 10, 10123 Torino, Italy

The finite amplitude strain solitary waves evolution is studied in an elastic mediumwith microstructure when both macro- and microdissipation are taken into account.The important problem is in the lack of the data of the microstructure parameters,one can mention a few works where attempts to measure them were done. Strainwaves may help in the development a possible method of the microparameters esti-mation. Besides qualitative effects like dispersion of a wave, the shape, the amplitude

and the velocity of the strain wave carry an information about microstructure.The procedure is proposed to obtain the governing PDE for nonlinear longitudinalstrain waves in one dimensional case. It is obtained which microstructure featuresare responsible for the appearance of dispersion and dissipative terms in the equa-tion. The exact and asymptotic solutions are obtained. It is found the formation,propagation and attenuation/amplification of bell-shaped and kink-shaped waveswhose parameters are defined in an explicit form through the macro- and micropa-rameters of the elastic medium. The solutions allow to describe in an explicit formthe amplification of both types of the waves, as well as the selection of the solitarywave when its parameters tend to the finite values. The relationships between the

wave parameters define the thresholds that separate the parameters of the initialsolitary waves which will amplify or attenuate. The asymptotic solution describingbell-shaped solitary wave selection may explain transfer of the strain energy by themicrostructure.

This research has been supported by the INTAS under Grant 99-0167.


28/33

Nonlinear strain waves in micro-structured media.

Theory and experiments

A. I. Potapov

Mechanical Engineering Institute of the Russian Academy of SciencesNizhny Novgorod, Russia.

E-mail: apotapov@ sandy.ru

As high-frequency waves propagate in media with complex structure, for example,composite materials, polymers, liquid and molecular crystals, there occur the ef-fects which cannot be described by equations of the classic theory of elasticity [1-3].This necessitates development and investigation of new mathematical models of mi-crostructured media capable of representing internal degrees of freedom. A lot ofdynamical properties of oriented media, consisting of anisotropic molecules andhaving a lattice of the molecular type, can be revealed using a model of plane oscil-lations of dumb-bell-like particles (mechanical dipoles) [2]. Examples of such

media are layered and molecular crystals with complex lattice (KNO, NaNO, CH,CS), smectic liquid crystals, elastic ferroelectric crystals, polycrystalline materials,and composites. The models of this type contain new elastic moduli describing themedium microstructure that have to be measured empirically. Lack of informationabout the material constants in microelasticity is considered to be one of the mainfactors restraining the study of non-classical media models [5].

The study of wave processes in media with internal degrees of freedom showed, inparticular, that the presence of material microstructure leads to occurrence of dis-persion in longitudinal, shear and surface Rayleigh waves, and to possible existenceof a new type of elastic waves. In this paper phenomena of modulation instability

and nonlinear resonant interaction of longitudinal, transverse and microrotationalwaves in the Cosserat continuum liquid and layered crystals, are discussed. Theseproblems have no analogy in the classical theory of elasticity.

1. Maugin, G.A., Nonlinear Waves in Elastic Crystals, Oxford Univ. Press,Oxford, 1999.

2. Potapov A.I., Pavlov I.S., and Maugin G.A., Nonlinear wave interactions in1D crystals with complex lattice. Wave Motion, 1999, V. 29, pp.297-312.

3. Lisin V.B., Potapov A.I., Variational principle in mechanics of liquid crystals.Int J. Non-Linear Mechanics, 1997, V. 32, No 1, pp. 55-62.

4. Erofeyev, V.I., Potapov, A.I., Longitudinal strain waves in nonlinearly elasticmedia with coupled stresses. Int. J. Non-linear Mech, 1993, Vol. 28, pp.483-488.

5. Potapov A.I., Rodyushkin V.M., Experimental study of strain waves in mate-rials with a microstructure. Acoustical Phyth. 2001, V. 47, No3, pp.347-352.


29/33

How some phonon modes can affect the solitons?

Niurka Rodrguez Quintero

Departamento de Fsica Aplicada I (GFNL), Universidad de Sevilla,Ave. Reina Mercedes s/n, 41012, Sevilla, [email protected], http://merlin.us.es/ niurka/

The phonons (continuum spectrum) in the sine-Gordon (sG) equation represent theextended wave solutions, obtained from the corresponding linearized problem aroundthe soliton solution [?]. In general, phonons appear in this integrable model whenthe effect of the external fields is considered, either ac [?] or dc fields [?], etc. Dueto the effect of the discreteness and finite size the continuum spectrum becomes intodiscrete one and the eigenfunctions corresponding to the phonons satisfy certainspatial symmetry (some of them are odd functions, whereas the rest are even ones).We discuss how the odd and even phonon modes can affect the kink profile in thesG equation. As an example, we analyze the higher order resonance, which takeplace when the sG system is driven by an ac force (when the frequency of ac force is

1/2 of the phonons frequencies). By using a linear perturbation theory [?], we showthat the ac force is able to excite not all phonons, but only the odd ones [ ?, ?]. Inthis sense, we also discuss the non equivalence of the phonon modes for other kindsof perturbations and other systems.

References

[1] M. E. Fogel, S.E. Trullinger, A.R. Bishop and J.A. Krumhansl, Physical ReviewB 15, 1578 (1977).

[2] A.R. Bishop et al., Phys. Rev. Lett. 50, 1095 (1983). N. R. Quintero and A.Sanchez, Eur. Phys. J. B 6, 133 (1998).

[3] J.C. Ariyasu and A.R. Bishop, Phys. Rev. B 35, 3207 (1987); ibid. 39, 6409(1989).

[4] N.R. Quintero, A. Sanchez and F.G. Mertens, Phys. Rev. E 62 Rapid Comm.,R60 (2000).

[5] N.R. Quintero and Panayotis G. Kevrekidis submitted to Phys. Rev. E.


30/33

Phonon dispersion curves of SrFCl

Abdelhadi Sabry

Laboratory of Physics Matter and Microelectronics, Faculty of Sciences An Chock,BP. 5366, Maarif, 20100 Casablanca, Morocco

E-mail: [email protected]

Extensive Raman scattering studies of ionic layered SrFCl crystal have been carriedout in the literature. But, to the best of our knowledge there are no measured dis-persion curves explaining the vibrational properties of this PbFCl-type compound.Therefore, for a basic understanding of experimental observations in Raman scatter-ing, we have resorted to latticedynamical calculations. The long wavelength phononsare computed in the framework using the rigid shell model, which had been appliedpreviously to produce the phonon dispersion relations of BaFCl and BaFBr. Thismicroscopic model includes the long-range Coulomb interactions and the short-rangeinteractions and taking into account the electronic polarizability of constituent ions.The shell model parameters have been obtained in such way that a best fit of themeasured Raman frequencies has been achieved in the center of the first Brillouinzone. The values of relevant parameters are critically analyzed. A complete set oftransverse and longitudinal phonon branches in each of the five principal directions(100), (001), (110), (101) and (111) have been deduced.


31/33

Non-linear strain waves in a wave guide with dissipation

Alexander M. Samsonov

The Ioffe Physico-Technical Institute of the Russian Academy of Sciences,St.Petersburg, 194021 Russia

E-mail: [email protected]

Long non-linear solitary strain (density) waves or solitons may be of considerableimportance for to study of short-run and weak reversible (elastic) loading of mate-rials with radiation and/or wear resistance, usable in aerospace industry, in layerednuclear fusion targets fabrication etc. It was proved recently that an energy transferis possible for long distances by means of strain solitons without significant energylosses in 1D and 2D elastic wave guides, having strong dispersion and made ofmaterials with remarkable linear dissipation.

The non-linear quasi hyperbolic doubly dispersive equation (DDE) describes along non-linear wave evolution in a thin hyperelastic rod, and the coupled non-

linear equations were obtained for 2D wave guides like plates and disks. Numericaland physical experiments in soliton generation are considered. The generation ofdensity solitons was made in various elastic wave guides by means of laser pulseholographic interferometry. In particular, the solitary pulse length was observed tobe 5-7 times more then the cross section radius of a polystyrene rod. The worldpioneering experiments were performed to detect the soliton focusing in a narrowingrod and the soliton propagation in plates.

When the wave guide is embedded into another solid it results in a dissipation-liketerm in the DDE. Various exact travelling wave solutions to dissipative DDE canbe obtained, and for square and cubic non-linearity the exact solutions were found

in closed form in terms of the Weierstrass elliptic function.Various applications of strain solitons to solid state physics, material science andnondestructive testing are discussed.

The partial support of this study by INTAS Grant no. 99-0167 is gratefullyacknowledged.

References

A.M.Samsonov. Strain solitons in solids and how to construct them.

Chapman&Hall/CRC Press, London, Boca Raton, 248 pp.


32/33

Dynamics of solitary wave fronts

in isotropic two-dimensional lattices

Y. Zolotaryuk

Section for Mathematical Physics, IMM,Technical University of Denmark, DK-2800 Lyngby, Denmark

Fronts of solitary waves in an isotropic two-dimensional lattice which can propagatein different directions on the plane are found by using the pseudospectral method.This method allows us to obtain solitary wave solutions with very narrow profile, thethickness of which may contain a few atoms. Since these nonlinear waves are quitenarrow, details of lattice microstructure appear to be important for their motion.Particularly, the regime of their propagation qualitatively depends whether or notthe direction of their motion occurs along the lattice bonds. The stability of thesesolitary waves is investigated numerically by their interactions with vacancies andlattice edges. Propagation of solitary plane waves through finite lattice domains

with isotopic disorder is studied.


33/33

New explicit periodical solution

of the nearly-integrable extended Camassa-Holm equation

(nonlinear waves in elastic rods)

Sergej Zykov and Maxim Pavlov

Institute of Metal Physics, Nonlinear Mechanics Lab.Ekaterinburg 620219, GSP-170, RussiaLandau Institute for Theoretical Physics,

Russian Academy of Sciences, Moscow, Russia

Application of the proliferation scheme

for the Dodd-Bullough system:new integrable reductions of modified elliptic Toda lattice

Sergej Zykov

Institute of Metal Physics, Nonlinear Mechanics Lab.Ekaterinburg 620219, GSP-170, Russia

seminar and workshop on nonlinear lattice structure and dynamics

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