Major advances in the theory of nonlinear excitations in discrete lattices occurred in the late 1980's and early 1990's and Professor Shozo Takeno was a key contributor. The discovery that some localized vibrations in perfectly periodic but strongly anharmonic lattices can be stabilized by the discreteness in a lattice of any dimension proved to be a conceptual and practical break though. Shozo called these localized excitations “intrinsic localized modes” (ILMs) emphasizing the fact that their formation involved no static disorder, in contrast to defect-induced localized modes, which had been studied for many decades. More recently these localized excitations also have been called “discrete breathers” (DBs) to emphasize their similarity to exact breather soliton solutions in nonlinear continuum theories. Such localized excitations have been observed in macroscopic arrays as diverse as coupled Josephson junctions, optical waveguides, two dimensional nonlinear photonic crystals, and micromechanical cantilevers. In solids ILMs have been observed for spin waves and also for vibrational excitations. So far in most experiments the system has been driven far from equilibrium. In this special section on Energy Localization and Waves in Nonlinear Lattices dedicated to Professor Shozo Takeno we present a small sampling of recent thrusts in this expanding field. The guest editor in chief would like to express his sincere thanks to the authors for their contributions. He also thanks the reviewers and the members of the guest editorial Committee, especially Dr. Y. Doi; the secretary of the special section, Dr. M. Kimura; and the editorial staff of the NOLTA Journal for their support in publishing this Special Section.
Research into defect lattice dynamics played an important roll in the development of our work on the local mode properties of anharmonic lattices. Using a historical perspective I show that the different theoretical and experimental studies of the impurity-induced absorption associated with the low frequency resonances in alkali halide crystals provided the kernel for our breakout into a new area.
I give a nutshell introduction to the exciting field of discrete breathers / intrinsic localized modes. I focus on how to obtain these states, how they are localized in space, and how they may interact with extended plane wave states. This work is dedicated to Shozo Takeno, one of the early pioneers in the field.
We discuss nonlinear dynamic models for the fluctuational opening of the base pairs in DNA and show that a standard model which is satisfactory for time-independent properties has to be improved to properly describe the time scales of the fluctuations. The existence of an energy barrier for the closing of the base pairs has to be taken into account. This introduces a model which sustains a new class of Intrinsically Localized Modes (ILMs). We investigate their properties numerically, and then consider two simplified versions of the improved DNA model allowing an analytical study of some properties of those ILMs. The models are different because the effective barrier necessary for the existence of this new class of ILMs is obtained either through the on-site potential or through the nonlinear stacking interaction, but they nevertheless sustain similar nonlinear localized excitations. An extension of the usual anti-continuum limit has to be introduced for the analysis, and relies on a continuation of localized equilibria from infinity.
We consider the discrete breathers in one-dimensional diatomic nonlinear oscillator chains. A discrete breather in the limit of zero mass ratio, i.e., the anti-continuous limit, consists of a finite number of in-phase or anti-phase excited light particles, separated by particles at rest. Existence of the discrete breathers is proved for small mass ratio by continuation from the anti-continuous limit. We prove that a discrete breather is linearly stable if it is continued from an anti-continuous solution consisting of a single excited particle or alternating anti-phase excited particles, otherwise it is linearly unstable, near the anti-continuous limit.
Intrinsic localized modes (ILMs) in two dimensional Fermi-Pasta-Ulam lattices are investigated. We consider in-plane vibrations of particles which have two degrees of freedom. We find two types of ILMs, quasi-one dimensional ILM and two dimensional ILM. Effect of interaction of second nearest neighbor lattices on stricture is also discussed.
We study linear and nonlinear dynamics of strained graphene lattices. We find the region of structural stability of flat graphene sheet in the space of two-component strain (εxx, εyy) with x and y axes oriented along the zigzag and armchair directions, respectively. We demonstrate that a gap in the phonon spectrum appears when graphene is strained by uniaxial load either along zigzag or along armchair direction, while no gap appears for hydrostatic loading. We find that discrete breathers (DBs) can be generated when graphene is uniaxially loaded in zigzag direction, and the DB frequency decreases with the amplitude revealing soft-type anharmonicity. An unusual feature of the DBs observed in this system is that they can have the frequencies within the phonon spectrum. This can be explained by the fact that DBs are polarized along the armchair direction, while the band of phonon spectrum containing DB frequency is occupied by the phonons having only out-of-plane atomic displacements. DB does not radiate energy because even large in-plane displacements of atoms in flat graphene are only weakly coupled with out-of-plane displacements.
The speed of a traveling intrinsic localized mode (ILM) in the acoustic spectrum of a micromechanical cantilever array is experimentally measured at high resolution as a function of the driving frequency. A repeating speed pattern is observed for chaotic and regular traveling ILMs between adjacent extended wave normal mode frequencies. The speed of a regular traveling ILM is almost the same as the plane wave dispersion group velocity at that frequency. Since ILM amplification only occurs during reflections at the ends of the array the phase matching condition for long time stability is greatly relaxed. A double humped distribution of speeds, found for chaotic ILMs, is shifted from the regular ILM nearly monochromatic speed value due to the modulational instability. Numerical simulations reproduce many of the experimental observations, demonstrating that intrinsic dynamical properties of the small array are being measured.
In this paper, an algorithm for an accurate matrix factorization based on Cholesky factorization for extremely ill-conditioned matrices is proposed. The Cholesky factorization is widely used for solving a system of linear equations whose coefficient matrix is symmetric and positive definite. However, it sometimes breaks down by the presence of an imaginary root due to the accumulation of rounding errors, even if the matrix is actually positive definite. To overcome this, a completely stable algorithm named inverse Cholesky factorization is investigated, which never breaks down as long as the matrix is symmetric and positive definite. The proposed algorithm consists of standard numerical algorithms and an accurate algorithm for dot products. Moreover, it is shown that the algorithm can also verify the positive definiteness of a given real symmetric matrix. Numerical results are presented for illustrating the performance of the proposed algorithms.