Modes In Nonlinear Lattices Research Articles

Existence of exponentially spatially localized breather solutions for lattices of nonlinearly coupled particles: Schauder’s fixed point theorem approach

The problem of showing the existence of localized modes in nonlinear lattices has attracted considerable efforts not only from the physical but also from the mathematical viewpoint where a rich variety of methods have been employed. In this paper, we prove that a fixed point theory approach based on the celebrated Schauder’s fixed point theorem may provide a general method to concisely establish not only the existence of localized structures but also a required rate of spatial localization. As a case study, we consider lattices of coupled particles with a nonlinear nearest neighbor interaction and prove the existence of exponentially spatially localized breathers exhibiting either even-parity or odd-parity symmetry under necessary non-resonant conditions accompanied with the proof of energy bounds of solutions.

Spherically localized discrete breathers in bcc metals V and Nb

Discrete breathers (DBs), also called intrinsic localized modes (ILMs), are spatially localized, large-amplitude vibrational modes in nonlinear lattices. Rod-like discrete breathers (DBs) have been reported in fcc Ni and bcc Nb by Haas et al. in 2011 based on molecular dynamics simulations. Here we use a general approach to find new type DBs in bcc V and Nb. In this approach, we firstly find the lattice symmetry dictated exact solutions to the equations of atomic motion in the form of delocalized nonlinear vibrational modes (DNVMs). Secondly, a localizing function with spherical symmetry is imposed over the DNVMs. Parameter of the localizing function is chosen such that the obtained DB has a long lifetime. The results presented in this work demonstrate that pure metals can support a variety of DBs. Interatomic potentials have a strong effect on the DB lifetime. Maximal DB lifetime in V is two orders of magnitude larger than that in Nb. The results of this study are interesting for the theory of DBs, and also they will help to better understand the impact of DBs on the physical properties of metals.

Molecular dynamics study of two-dimensional discrete breather in nickel

A discrete breather (DB) is a spatially localized vibrational mode of large amplitude in a defect-free anharmonic lattice. Generally, zero-dimensional DB is considered to be localized in all [Formula: see text] directions of the [Formula: see text]-dimensional lattice. However, the question of existence of DBs localized in [Formula: see text]–[Formula: see text] directions and delocalized in other [Formula: see text] directions remains open. In the present paper, for the first time, the case of [Formula: see text] and [Formula: see text] is considered by constructing a two-dimensional (2D) DB in the fcc nickel lattice using molecular dynamics methods. In order to excite such DB, one of the delocalized vibrational modes of the triangular lattice was used (the (111) plane in fcc crystal is a triangular lattice). All simulations were carried out at zero temperature. The investigated 2D DB demonstrates hard-type nonlinearity, when its oscillation frequency increases with increasing amplitude. The oscillation frequencies of the DB are above the upper edge of the phonon spectrum for nickel, which is 10.3[Formula: see text]THz. The maximum DB lifetime is found to be 9.5[Formula: see text]ps. The obtained results expand our understanding of diversity of nonlinear spatially localized vibrational modes in nonlinear lattices.

Discrete dissipative localized modes in nonlinear magnetic metamaterials

We analyze the existence, stability, and propagation of dissipative discrete localized modes in one- and two-dimensional nonlinear lattices composed of weakly coupled split-ring resonators (SRRs) excited by an external electromagnetic field. We employ the near-field interaction approach for describing quasi-static electric and magnetic interaction between the resonators, and demonstrate the crucial importance of the electric coupling, which can completely reverse the sign of the overall interaction between the resonators. We derive the effective nonlinear model and analyze the properties of nonlinear localized modes excited in one-and two-dimensional lattices. In particular, we study nonlinear magnetic domain walls (the so-called switching waves) separating two different states of nonlinear magnetization, and reveal the bistable dependence of the domain wall velocity on the external field. Then, we study two-dimensional localized modes in nonlinear lattices of SRRs and demonstrate that larger domains may experience modulational instability and splitting.

Strong asymmetry for surface modes in nonlinear lattices with long-range coupling

We analyze the formation of localized surface modes on a nonlinear cubic waveguide array in the presence of exponentially-decreasing long-range interactions. We find that the long-range coupling induces a strong asymmetry between the focusing and defocusing cases for the topology of the surface modes and also for the minimum power needed to generate them. In particular, for the defocusing case, there is an upper power threshold for exciting staggered modes, which depends strongly on the long-range coupling strength. The power threshold for dynamical excitation of surface modes increase (decrease) with the strength of long-range coupling for the focusing (defocusing) cases. These effects seem to be generic for discrete lattices with long-range interactions.

Nonlinear localization in thermalized lattices: application to DNA

In the last few years numerous studies have been devoted to intrinsic localized modes in nonlinear lattices because they provide examples of localization without disorder. The properties of “discrete breathers” as exact solutions of these nonlinear lattices are now well understood, but this is not the case for the properties of nonlinear localization and energy relaxation in thermalized lattices. In biological molecules such as DNA, where large amplitude nonlinear motions are essential for function, temporary deviations from energy equipartition could play an important role. After a brief introduction to intrinsic localized modes, we address the following questions: (a) does nonlinear localization survive in the presence of a thermal bath and how can we characterize it? (b) what is the origin of localization and how do discrete breathers contribute to it? (c) can we observe nonlinear localization in an experiment? The last point discusses recent results of an experiment performed on DNA, which suggest that the effects of nonlinear localization in a thermalized system may have already been observed.

Coupling of cutoff modes in nonlinear lattices

The coupling of cutoff modes in nonlinear lattices is considered. It is shown that a novel class of lattice solitons is possible in the form of a coupled soliton-soliton or a coupled kink-soliton pair through a cross-phase modulation. Exact soliton solutions of coupled nonlinear amplitude equations for describing such a type of interacting nonlinear localized mode in lattices are presented and a macro-lattice experiment for their observation is suggested. Analytical results based on a quasi-discreteness approach are checked by numerical simulations.

Multiple states of intrinsic localized modes

In the framework of the continuum approximation, localized modes in nonlinear lattices ~''intrinsic localized modes'' or ''discrete breathers'' ! are described by the nonlinear Schrodinger ~NLS! equation. We go beyond this approximation and analyze what kind of qualitatively new effects can be introduced by discreteness. Taking into account the higher-order linear and nonlinear dispersion terms in the NLS equation derived from a lattice model, we predict the existence of bound states of intrinsic localized excitations. These bound states of nonlinear localized modes are also found numerically for a discrete chain with linear and nonlinear cubic interparticle interaction. @S0163-1829~98!05133-9#

Asymmetric impurity modes in nonlinear lattices.

We analyze spatially localized large-amplitude oscillations in the vicinity of an isotopic impurity. We show that, in addition to a symmetric nonlinear impurity mode found earlier [see, e.g., Yu. S. Kivshar, Phys. Lett. A 161, 80 (1991)], there exists an asymmetric impurity mode which appears as a result of a bifurcation from the symmetric mode. The structure of this asymmetric impurity mode is calculated analytically in the three-particle approximation and its stability is demonstrated by direct numerical simulations.

Standing localized modes in nonlinear lattices.

The theory of standing localized modes in discrete nonlinear lattices is presented. We start from a rather general model describing a chain of particles subjected to an external (on-site) potential with cubic and quartic nonlinearities (the so-called Klein-Gordon model), and, using the approximation based on the discrete nonlinear Schro$iuml---dinger equation, derive a system of two coupled nonlinear equations for slowly varying envelopes of two counterpropagating waves of the same frequency. We show that spatially localized modes exist in the frequency--wave number domain where the lattice displays modulational instability; two families of localized modes are found for this case as separatrix solutions of the effective equations for the wave envelopes. When the nonlinear plane wave in the lattice is stable to small modulations of its amplitude, nonlinear localized modes appear as dark solitons associated with the so-called extended modulational instability. These localized modes may be treated as domain walls or kinks connecting two standing plane-wave modes of the similar structure. We investigate analytically and numerically the special family of such localized solutions that, in the vicinity of the zero-dispersion point, cover exactly the case of the so-called self-induced gap solitons recently introduced by Kivshar [Phys. Rev. Lett. 70, 3055 (1993)]. Application of the theory to the case of parametrically driven damped lattices is also briefly discussed, and it is mentioned that some of the solutions considered in the present paper may be extended to include the case of localized modes in driven damped lattices, provided the mode frequency and amplitude are fixed by the parameters of the external parameters of the external parametric ac force.

Interrelation between the stability of extended normal modes and the existence of intrinsic localized modes in nonlinear lattices with realistic potentials.

Previous theoretical studies and molecular dynamics simulations indicate that the anharmonic version of a zone-boundary-mode phonon in a periodic one-dimensional lattice with nearest-neighbor quartic and quadratic interactions is unstable, and that this instability can lead to the production of intrinsic localized modes. We show here that such an instability occurs when nearest-neighbor cubic anharmonicity is added, but that the zone boundary mode is stabilized when the cubic anharmonicity becomes sufficiently large. A direct connection is established between the existence of this instability and the existence of intrinsic localized modes. Furthermore, our analysis reveals the existence of a second type of zone-boundary-mode instability, which is not related to intrinsic localized modes. This ``period-doubling'' instability is also found to occur in one-dimensional lattices with realistic potentials, such as Lennard-Jones, Morse, and Born-Mayer, whereas the instability related to intrinsic localized modes does not occur for these potentials, owing to their inclusion of higher-order anharmonicity. Likewise, intrinsic localized modes are not found in direct numerical searches for these cases, and we conclude that they do not exist in monatomic lattices with these potentials.

Moving localized modes in nonlinear lattices.

An analytical approach based on the perturbed discrete Ablowitz-Ladik equation is applied to investigate intrinsic localized modes for two different models of one-dimensional anharmonic lattices, namely, for a chain with nonlinear interatomic interaction and a chain with nonlinear on-site potential. It is shown that the motion of the localized modes is strongly affected by an effective periodic (Peierls-Nabarro) potential, but for the former model moving localized modes may still exist in a wide region of the mode parameters, whereas for the latter one they will be always captured by the lattice discreteness if the amplitude of the mode exceeds a certain threshold value.