Intrinsic Localized Modes Research Articles

Quantization of intrinsic localized modes: Effective linear lattice method

A working method is proposed to quantize intrinsic localized modes (ILMs) in lattices with hard quartic anharmonicity in the framework of the rotating-wave approximation. This is done by reducing a nonlinear eigenvalue problem to a linear one by averaging slowly varying reduced effective force constants over frequencies, enabling us to quantize the ILM in a straightforward manner. Such an effective linear lattice (ELL) method is first applied to an analytically tractable d-dimensional cubic lattice to show that the concept of the ELL holds exactly in the strong localization limit. Next, general lattice models are investigated to achieve quantization of the ILM in an approximate manner. The obtained analytical results are tested by solving numerically a model one-dimensional lattice to show that phase-space trajectory of an ILM-bearing atom is of elliptic type with finite but small width. The numerical result confirms the validity of the ELL leading to its semi-classical quantization. On the other hand, orbits of all the remaining atoms exhibits complex non-periodic trajectory to which a direct application of the semi-classical quantization rule appears impossible.

Linear local modes induced by intrinsic localized modes in a monatomic chain

A theory is developed to describe the effect of an intrinsic localized mode (ILM) on small vibrations in a monatomic chain with hard quartic anharmonicity. One prediction is the appearance in the chain of linear local modes nearby the ILM. To check this result, MD calculations of vibrations under strong local excitation are carried through with high precision. The results fully confirm the prediction.

NONLINEAR EXCITATIONS IN STRONGLY-COUPLED PLASMA LATTICES: ENVELOPE SOLITONS, KINKS AND INTRINSIC LOCALIZED MODES

Ensembles of charged particles (plasmas) are a highly complex form of matter, most often modeled as a many-body system characterized by weak inter-particle interactions (electrostatic coupling). However, strongly-coupled plasma configurations have recently been produced in laboratory, either by creating ultra-cold plasmas confined in a trap or by manipulating dusty plasmas in discharge experiments. In this paper, the nonlinear aspects involved in the motion of charged dust grains in a one-dimensional plasma monolayer (crystal) are discussed. Different types of collective excitations are reviewed, and characteristics and conditions for their occurrence in dusty plasma crystals are discussed, in a quasi-continuum approximation. Dust crystals are shown to support nonlinear kink-shaped supersonic solitary longitudinal excitations, as well as modulated envelope localized modes associated with longitudinal and transverse vibrations. Furthermore, the possibility for intrinsic localized modes (ILMs) — Discrete Breathers (DBs) — to occur is investigated, from first principles. The effect of mode-coupling is also briefly considered. The relation to previous results on atomic chains, and also to experimental results on strongly-coupled dust layers in gas discharge plasmas, is briefly discussed.

Interaction of intrinsic localized modes with structures in anharmonic lattice systems

The dynamics of intrinsic localized modes (ILMs) in the system with periodic structures are investigated. In homogeneous lattice systems ILMs interact randomly and concentrate to a single large localization. We show that this energy concentration is suppressed by introducing periodic structures in the system. The dynamics of ILMs in the structures can be classified by comparing the size of structures with the characteristic size of the ILMs. These facts can be explained by introducing mean free time of ILMs against collisions of other ILMs or structures.

Self-consistent theory of intrinsic localized modes: Application to monatomic chain

A theory of intrinsic localized modes (ILMs) in anharmonic lattices is developed, which allows one to reduce the original nonlinear problem to a linear problem of small variations of the mode. This enables us to apply the Lifshitz method of the perturbed phonon dynamics for the calculations of ILMs. In order to check the theory, the ILMs in monatomic chain are considered. A comparison of the results with the corresponding molecular dynamics calculations shows an excellent agreement.

Simulations of defect-free coupled-resonator optical waveguides constructed in 12-fold quasiperiodic photonic crystals

In this paper we propose a defect free coupled-resonator optical waveguide (CROW) that is constructed based on the intrinsic localized modes in a 12-fold quasiperiodic photonic crystal (QPC). Different from the usual CROW created by introducing a serial of point defects in a periodic photonic crystal, the special CROW is fabricated by modifying the 12-fold QPC without destroying its basic cells. Our simulation results indicate that the guiding efficiency of the CROW is high. Furthermore, compared with the CROW constructed in a periodic photonic crystal, the maximal value of the group velocity of electromagnetic waves propagating in this kind of CROW can be greatly reduced.

Biphonons in the [formula omitted]-Fermi–Pasta–Ulam model

Discrete breathers or intrinsic localized modes are nonlinear localized states that appear in several classical extended systems, such as for instance the Fermi–Paste–Ulam (FPU) model. In order to probe the quantum states that correspond to discrete breathers, we quantize the β -FPU model using boson quantization rules, retain only number conserving terms, and analyze the two-quanta sector of the model. For both attractive and repulsive nonlinearity, we find the occurrence of biphonons in two forms, on-site and nearest-neighbor site, and analyze their properties. We comment on the use of this model as a minimal model for extended molecular and biomolecular systems.

Intrinsic localized modes in photonic crystal waveguides composed of a periodic mixed array of linear and Kerr nonlinear dielectric media

A theoretical treatment is given of intrinsic localized modes in waveguides and in junctions of waveguides formed inside two-dimensional photonic crystals. Two-dimensional photonic crystals are periodic arrays of parallel-axed dielectric cylinders, and the electromagnetic modes of interest propagate in the plane perpendicular to the cylinder axes. Waveguides are formed inside the photonic crystal by replacement of a row of cylinders. The replacement cylinders are chosen so that the waveguide channels formed bind and conduct electromagnetic energy along the channel in the plane perpendicular to the cylinder axes. In the systems we study, the waveguide channels are composed of a periodic array of linear and Kerr nonlinear media in which both the linear and nonlinear dielectric materials differ from the linear dielectric materials forming the bulk photonic crystal. These waveguides are shown to support intrinsic localized modes. The intrinsic localized modes are studied by determining their scattering properties in waveguide systems. The transmission characteristics of a waveguide formed of linear dielectric media but containing a barrier that is a periodic array of linear and Kerr nonlinear dielectric is studied. Studies are also made of the transmission characteristics of waveguides of linear dielectric media that are connected at a waveguide junction that is composed of co-joined segments containing a periodic array of linear and Kerr nonlinear dielectric media. Transmission anomalies in these two types of waveguide geometries (i.e., barrier and junction geometries), associated with the development of intrinsic localized modes in the barrier or segments composing the waveguide junctions, are identified. This work is an extension of that presented in A.R. McGurn, G. Birkok. Phys. Rev. B 69 (2004) 235105, which treated the scattering characteristics of intrinsic localized modes in barriers and junctions formed solely of Kerr nonlinear dielectric media to treat other types of waveguides that can support intrinsic localized modes.

Interaction of an external impurity with the surface intrinsic mode in a Heisenberg chain

In this paper we have demonstrated the existence of surface intrinsic localized modes in one-dimensional classical spin systems. These modes have frequencies above the top of the magnon band and are more localized than the corresponding modes that were found in the lattice dynamics. By numerical methods we have studied the forward evolution in time. These entities are stable in time for many time steps. Also, we consider the interaction of these surface excitations with an external dipolar field. We have first considered the case in which the external dipole is parallel to the spins of the bulk. According to the distance of the impurity from the surface we find that at a particular distance a new surface mode appears, that we have called a mixed mode because it is formed by a localized state at the surface plus a plane wave. At large distances the dipole traps the surface mode, while at small distances the external impurity potential is so strong that all the spins are forced to be aligned in the $z$ direction. For the case in which the external magnetic dipole is antiparallel to the bulk spins there are no mixed modes and the external impurity is so strong that the surface modes are trapped at large distances, and the bulk modes are trapped at very short distances.

Colloquium: Nonlinear energy localization and its manipulation in micromechanical oscillator arrays

It has been known for some time that nonlinearity and discreteness play important roles in many branches of condensed-matter physics as evidenced by the appearance of domain walls, kinks, and solitons. A recent discovery is that localized dynamical energy in a perfect nonlinear lattice can be stabilized by the lattice discreteness. Intrinsic localized modes (ILMs) are the resulting signature. Their energy profiles resemble those of localized modes at defects in a harmonic lattice but, like solitons, they can move. ILMs have been observed in macroscopic arrays as diverse as coupled Josephson junctions, optical waveguides, two-dimensional nonlinear photonic crystals, and micromechanical cantilevers. Such dynamically driven localized modes are providing a new window into the underlying simplicity of nonequilibrium dynamics. This Colloquium surveys the studies of ILMs in micromechanical cantilever arrays. Because of the ease of sample fabrication with silicon technology and the intuitive optical visualization techniques, these experiments are able to demonstrate the general nature and properties of dynamical energy localization while at the same time providing new information on ILM generation, locking, pinning, and interaction with impurities.

A regularized model equation for discrete breathers in anharmonic lattices with symmetric nearest-neighbor potentials

We propose a regularized continuum model equation for describing discrete breathers or intrinsic localized modes in one-dimensional anharmonic lattices with symmetric nearest-neighbor potentials. Exact stationary breather solutions with purely hard quartic anharmonicity, as well as approximate stationary breather solutions in the general case, are found. The application of the multiple scales analysis indicates the movability of the small-amplitude breather solutions. The results of numerical simulations for the model equation fully support the analytical solutions. As regards the breather–breather collisions, the continuum model shares many common features with its discrete counterpart, which provides an opportunity to clarify the energy exchange mechanism for collisions between discrete breathers in lattices.

An Integrable Three Particle System Related to Intrinsic Localized Modes in Fermi–Pasta–Ulam-β Chain

In this paper, a ring of three particles interacting via nearest-neighbor harmonic and quartic potentials is investigated for the study of intrinsic localized modes (ILMs) in Fermi–Pasta–Ulam (FPU) atomic chain. In spite of the fact that above Hamiltonian system has been shown to be integrable by Hietaniata [Phys. Rep. 147 (1987) 87], Yoshida and Ramani [Physica D 30 (1988) 151], respectively, we approve its integrability in a more straightforward way. Moreover, we obtain exact periodic solutions in terms of the Jacobi elliptic functions for both the zero and nonzero third first integral. The significance of these findings lies in the fact that analytical stationary and moving ILMs solutions are obtained, and a close relationship between the movability of ILMs and the third first integral is clarified.

Localized Structures in Nonlinear Lattice Systems –Intrinsic Localized Mode– and Its Application to Dynamics of Atoms in Materials

We investigate nonlinear energy localization called intrinsic localized modes (ILMs) in two dimensional lattice systems for its application to the dynamics of materials in atomic scale. At first, we calculate quasi-one dimensional ILMs in two dimensional lattice systems with nearest and second-nearest ideal interaction potential. We find that ILM can exist in this system and that the structure of ILM can be affected by the structure of the system. Then we investigate energy localization in graphene sheet by molecular dynamic simulation. Energy localization with finite lifetime which is similar to ILM can appear in the system. These results indicate that ILMs can be excited in materials.

2115 2次元格子系における非線形局在構造(G01-1 計算力学(1),G01 計算力学)

Localized structures in anharmonic lattice systems called intrinsic localized modes (ILMs) are investigated numerically. ILMs are time-periodic and spatial-localized structures due to nonlinearity and discreteness of anharmonic lattices systems. We find that two types of ILMs exist in two dimensional square lattice systems in terms of structures: quasi-one dimensional ILMs and two dimensional ones. Adding to this, we find ILMs envelope and internal frequency are highly depend on lattice structures and interaction potential. We also show that quasi-one dimensional ILMs can move with keeping their localized structures.

Solitons and intrinsic localized modes in a one-dimensional antiferromagnetic chain

By use of the Hartree approximation and the method of multiple scales, we investigate quantum solitons and intrinsic localized modes in a one-dimensional antiferromagnetic chain. It is shown that there exist solitons of two different quantum frequency bands: i.e., magnetic optical solitons and acoustic solitons. At the boundary of the Brillouin zone, these solitons become quantum intrinsic localized modes: their quantum eigenfrequencies are below the bottom of the harmonic optical frequency band and above the top of the harmonic acoustic frequency band.

Controlled Switching of Intrinsic Localized Modes in a One-Dimensional Antiferromagnet

Nearly-steady-state locked intrinsic localized modes (ILMs) in the quasi-1D antiferromagnet (C(2)H(5)NH(3))(2)CuCl(4) are detected via four-wave mixing emission or the uniform mode absorption. Exploiting the long-time stability of these locked ILMs, repeatable nonlinear switching is observed by varying the sample temperature, and localized modes with various amplitudes are created by modulation of the microwave driver power. This steady-state ILM locking technique could be used to produce energy localization in other atomic lattices.

Counting discrete emission steps from intrinsic localized modes in a quasi-one-dimensional antiferromagnetic lattice

Intrinsic localized modes (ILMs) in a quasi-1D antiferromagnetic material ${({\mathrm{C}}_{2}{\mathrm{H}}_{5}\mathrm{N}{\mathrm{H}}_{3})}_{2}\mathrm{Cu}{\mathrm{Cl}}_{4}$ are counted by using a novel nonlinear energy magnetometer. The ILMs are produced by driving the uniform spin wave mode unstable with an intense microwave pulse. Subsequently a subset of these ILMs become captured by and locked to a cw driver so that their properties can be examined at a later time with a tunable cw low power probe source. Four-wave mixing is used to enhance the emission signal from the few large amplitude ILMs over that associated with the many small amplitude plane wave modes. A discrete step structure observed in the emission signal is identified with individual ILMs becoming unlocked from the driver. At most driver power and frequency settings the resulting emission step structure appears uniformly distributed; however, sometimes, nearby in parameter space, families of emission steps are evident as the driver frequency or power is varied. Two different experimental methods give consistent results for counting individual ILMs. Because of the discreteness in the emission both the size of an ILM and its energy can be estimated from these experiments. For the uniformly distributed case each ILM extends over $\ensuremath{\sim}42$ antiferromagnetic unit cells and has an energy value of $1.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}\phantom{\rule{0.3em}{0ex}}\mathrm{J}$ while for the case with families the ILM length becomes $\ensuremath{\sim}54$ antiferromagnetic unit cells with an energy of $1.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}\phantom{\rule{0.3em}{0ex}}\mathrm{J}$. An unresolved puzzle is that the emission step height does not depend on experimental parameters the way classical numerical simulations suggest.

Bloch modes and self-localized waveguides in nonlinear photonic crystals

We present a modeling technique that uses eigenmode expansion to simulate infinite periodic structures with Kerr nonlinearity. Using a unit cell with Bloch boundary conditions, our iterative algorithm efficiently calculates self-consistent two-dimensional Bloch modes. We show how it can be used to study the band structure of nonlinear photonic crystals and to gain rapid insight in the operation of devices. Furthermore, we present nonlinear transversely localized guided modes, which are kinds of gap solitons or intrinsic localized modes, that induce their own waveguide through a photonic crystal without linear defects.

Nonlinear lattice dynamics of Bose–Einstein condensates

The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine thermalization in nonmetallic solids and develop "experimental" techniques for studying nonlinear problems, continues to yield a wealth of results in the theory and applications of nonlinear Hamiltonian systems with many degrees of freedom. Inspired by the studies of this seminal model, solitary-wave dynamics in lattice dynamical systems have proven vitally important in a diverse range of physical problems-including energy relaxation in solids, denaturation of the DNA double strand, self-trapping of light in arrays of optical waveguides, and Bose-Einstein condensates (BECs) in optical lattices. BECs, in particular, due to their widely ranging and easily manipulated dynamical apparatuses-with one to three spatial dimensions, positive-to-negative tuning of the nonlinearity, one to multiple components, and numerous experimentally accessible external trapping potentials-provide one of the most fertile grounds for the analysis of solitary waves and their interactions. In this paper, we review recent research on BECs in the presence of deep periodic potentials, which can be reduced to nonlinear chains in appropriate circumstances. These reductions, in turn, exhibit many of the remarkable nonlinear structures (including solitons, intrinsic localized modes, and vortices) that lie at the heart of the nonlinear science research seeded by the FPU paradigm.

Specific solutions of the generalized Korteweg-de Vries equation with possible physical applications

Abstract Solutions for a type of generalized Korteweg-de Vries equation which should have physical impact will be determined here. These types of solutions should have applications in the study of intrinsic localized modes optical waveguide arrays and fluid dynamics. It is shown that trigonometric and hyperbolic solutions can be obtained by matching powers and coefficients of the independent terms in the equation after the assumed solution has been substituted. As well, solutions to the equation in terms of more complicated Jacobe elliptic functions are determined.