Localized versus traveling waves in infinite anharmonic lattices (original) (raw)
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Localized modes in inhomogeneous one-dimensional anharmonic lattices
Physical Review B, 1994
We present numerical calculations for the determination of localized modes in one-dimensional finite chains of atoms with free ends containing harmonic and quartic anharmonic interactions. By adding step by step the quartic term we can follow the formation of even and odd localized modes arising from the highest harmonic frequency mode. We have studied the role of crystal inhomogeneity by introducing a modification of the fourth-order force constant between neighboring atoms at the center of the chain, where the localized mode has its maximum displacement. For large weakening of this force constant the localized mode develops a double-peaked structure, as has been found in the continuum limit. In the case of asymmetrical local inhomogeneity the localized mode remains stable and moves toward the atom with the inhomogeneity. We also show the existence of anharmonic surface modes localized at the end of the chain.
Improved models of dense anharmonic lattices
Physics Letters A, 2017
We present two improved quasi-continuous models of dense, strictly anharmonic chains. The direct expansion which includes the leading effect due to lattice dispersion, results in a Boussinesq-type PDE with a compacton as its basic solitary mode. Without increasing its complexity we improve the model by including additional terms in the expanded interparticle potential with the resulting compacton having a milder singularity at its edges. A particular care is applied to the Hertz potential due to its nonanalyticity. Since, however, the PDEs of both the basic and the improved model are ill posed, they are unsuitable for a study of chains dynamics. Using the bond length as a state variable we manipulate its dispersion and derive a well posed fourth order PDE.
Localized nonlinear, soliton-like waves in two-dimensional anharmonic lattices
Wave Motion, 2011
We discuss here nano-scale size localized wave excitations, which are intrinsic localized traveling modes in two-dimensional anharmonic crystal lattice systems. In particular, using different initial conditions of coordinates and momenta we search for the longest lasting excitations in triangular lattices. As most stable and longest lasting unaltered appear quasi-one-dimensional Toda-like solitons running in rectilinear chains along the main crystallographic axes of such lattices. Furthermore, by following the trace of high energetic excitations like in "bubble chamber" methodology (or in scanning tunneling microscopy) we show how such localized nonlinear waves appearing spontaneously in heated systems can be detected and followed in space-time.
Localized vibrations of homogeneous anharmonic chains
Physics Letters A, 1990
Communicated by V.M. Agranovich An intrinsic localized vibration (ILV) in homogeneous anharmonic lattices has recently been predicted theoretically and investigated in the limit of strong anharmonicity. The existence of these vibrations has been confirmed by molecular dynamic (MD) simulation. In the present paper ILVs have beeninvestigated in the crossover region of anhannonicity. The ILV frequency and shape function calculated within the rotating wave approximation (RWA) are well confirmed by MD simulation. The decay ofthe top phonon band mode into fairly localized ILVs in the anharmonic chain with third and quartic anharmonicity terms have been observed by the MD simulation. Phonon thermalization in the chain through the ILV decay has also been observed.
Intrinsic Localized Modes in the Bulk and at the Surface of Anharmonic Chains
Current Problems in Condensed Matter, 1998
Intrinsic localized vibrational modes in a diatomic anharmonic chain are discussed in detail. We consider a finite ͑even or odd͒ number of particles interacting with nearest-neighbor harmonic and cubic and quartic anharmonic potentials. For suitable values of the potential parameters we find two odd parity modes localized about light and heavy atoms, respectively, with frequencies above the top of the optical branch. In the gap there are localized modes arising from the top of the acoustic branch with odd parity if they are centered on a heavy atom and even parity if centered on a light atom. There are also two localized modes originating from the bottom of the optical branch. The odd parity mode is centered on a light atom, while the even parity mode is centered on a heavy atom. In addition, we have found two types of surface modes. One is the anharmonic version of the surface mode occurring in a finite diatomic harmonic chain with frequency in the gap, and the other is entirely due to the anharmonicity. The latter mode has its frequency above the top of the optical branch. A comparison with other work is given.
Physical review. B, Condensed matter, 1992
Computer simulations show that in a one-dimensional lattice both even and odd anharmonic localized modes can move with constant velocity. For nearest-neighbor forces described by a harmonic plus hard quartic potential, the dispersion relation cu(k) has been calculated for both types of modes. Numerical experiments show that, in general, moving modes with a near-Gaussian excitation envelope occur in parts of co(k) space, with this region becoming more restricted as the local-mode frequency increases.
Nonlinear vector waves of a flexural mode in a chain model of atomic particles
Communications in Nonlinear Science and Numerical Simulation, 2015
Flexural transverse waves in an anharmonic chain of atoms is considered and the nonlinear vector equation for the phonon modes in the long-wave approximation is derived taking into account the weak dispersion effects. Particular cases of the equation derived are discussed; among them the vector mKdV equation (Gorbacheva and Ostrovsky, 1983) [12], as well as the new model vector equations dubbed here the 'second order cubic Benjamin-Ono (socBO) equation' and 'nonlinear pseudo-diffusion equation'. Stationary solutions to the equation derived are studied and it is found in which cases physically reasonable periodic and solitary type solutions may exist. The simplest non-stationary interactions of solitary waves of different polarisation are studied by means of numerical simulation. A new interesting phenomenon is revealed when two solitons of the same or opposite polarities interact elastically, whereas the interaction of two solitons lying initially in the perpendicular planes is essentially inelastic resulting in the survival of only one soliton and destruction of another one.
The European Physical Journal B - Condensed Matter, 2002
Modulational instability of travelling plane waves is often considered as the first step in the formation of intrinsically localized modes (discrete breathers) in anharmonic lattices. Here, we consider an alternative mechanism for breather formation, originating in oscillatory instabilities of spatially periodic or quasiperiodic nonlinear standing waves (SWs). These SWs are constructed for Klein-Gordon or Discrete Nonlinear Schrödinger lattices as exact time periodic and time reversible multibreather solutions from the limit of uncoupled oscillators, and merge into harmonic SWs in the small-amplitude limit. Approaching the linear limit, all SWs with nontrivial wave vectors (0 < Q < π) become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. The dynamics resulting from these instabilities is found to be qualitatively different for wave vectors smaller than or larger than π/2, respectively. In one regime persisting breathers are found, while in the other regime the system thermalizes.
Transport of vibrational excitations in anharmonic lattices
Journal of Luminescence, 1998
The discussion is given on the effect of nonlinearity on the energy transport mechanisms in discrctc lattices. The spatiotcmporal evolution of classical self-localized modes in one-dimensional perfect and perturbed lattice is examined. The results of numerical simulations are presented for various types of perturbation. different temperatures. P-. Q-initial excitation. Comparative studies are made for highlighting the role played by quantum fluctuations in the stability of localized modes in nonlinear crystal lattices. (' 199X Elsevier Science B.V. All rights rcsercod.
Reprint of Nonlinear vector waves of a flexural mode in a chain model of atomic particles
Communications in Nonlinear Science and Numerical Simulation, 2015
Flexural transverse waves in an anharmonic chain of atoms is considered and the nonlinear vector equation for the phonon modes in the long-wave approximation is derived taking into account the weak dispersion effects. Particular cases of the equation derived are discussed; among them the vector mKdV equation (Gorbacheva and Ostrovsky, 1983) [12], as well as the new model vector equations dubbed here the 'second order cubic Benjamin-Ono (socBO) equation' and 'nonlinear pseudo-diffusion equation'. Stationary solutions to the equation derived are studied and it is found in which cases physically reasonable periodic and solitary type solutions may exist. The simplest non-stationary interactions of solitary waves of different polarisation are studied by means of numerical simulation. A new interesting phenomenon is revealed when two solitons of the same or opposite polarities interact elastically, whereas the interaction of two solitons lying initially in the perpendicular planes is essentially inelastic resulting in the survival of only one soliton and destruction of another one.