Compact envelope dark solitary wave in a discrete nonlinear electrical transmission line

We introduce an extended nonlinear Schrödinger (ENLS) equation describing the dynamics of modulated waves in a nonlinear discrete electrical transmission line (NLTL) with nonlinear dispersion. We show that this equation admits envelope dark solitary wave with compact support, with width and speed independent of the amplitude, as a solution. Analytical criteria of existence and stability of this solution are derived. In particular, we show that the modulated compact wave may exist in the NLTL depending on the frequency range of the chosen carrier wave, for physically realistic parameters. The stability of compact dark solitary wave is confirmed by numerical simulations of this ENLS equation and the exact equations of the network.     

Deterministic escape of a dimer over an anharmonic potential barrier

In the present paper we consider the deterministic escape dynamics of a dimer from a metastable state over an anharmonic potential barrier. The underlying dynamics is conservative and noiseless and thus, the allocated energy has to suffice for barrier crossing. The two particles comprising the dimer are coupled through a spring. Their motion takes place in a two-dimensional plane. Each of the two constituents for itself is unable to escape, but as the outcome of strongly chaotic coupled dynamics the two particles exchange energy in such a way that eventually exit from the domain of attraction may be promoted. We calculate the corresponding critical dimer configuration as the transition state and its associated activation energy vital for barrier crossing. It is found that there exists a bounded region in the parameter space where a fast escape entailed by chaotic dynamics is observed. Interestingly, outside this region the system can show Fermi resonance which, however turns out to impede fast escape.     

Anomalous thermostat and intraband discrete breathers

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. The coupling between both parts is bilinear. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation with memory kernels and noise term for the anharmonic coordinates . For zero temperature, i.e. for , we prove that the support of the Fourier transform of and of the time averaged velocity-velocity correlation functions of the anharmonic system cannot overlap. As a consequence, the asymptotic solutions can be constant, periodic, quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory with frequency we find that the energy E_T transferred to the harmonic system up to time T is proportional to T^α. If equals one of the phonon frequencies ω_ν, it is α=2. We prove that there is a zero measure set L such that for in its full measure complement R?L, it is α=0, i.e. there is no energy dissipation. Under certain conditions L contains a subset L^′ such that for the dissipation rate is nonzero and may be subdissipative (0≤α<1) or superdissipative (1<α≤2), compared to ordinary dissipation (α=1). Consequently, the harmonic bath does act as an anomalous thermostat, in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation, forcing convergence to equilibrium. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency exist for all in a Cantor set C(k) of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing for , related to . For the small denominators do not lead to divergencies such that is a smooth and bounded function in t.     

Stability change of intrinsic localized mode in finite nonlinear coupled oscillators

Intrinsic localized mode (ILM) is spatially localized and temporally periodic oscillation in nonlinear coupled oscillators. We numerically investigate the dynamical stability of ILMs in a microcantilever array, in which ILMs were experimentally observed by Sato et al. It is found that the stability change of ILMs is due to the ratio in nonlinear potentials. This phenomenon also occurs in the array without harmonic potentials. Consequently, the stability of ILMs substantially depends on the ratio in nonlinear on-site and inter-site potentials.     

Resonance effects in nonlinear lattices

We study a class of one-dimensional nonlinear lattices with nearest-neighbour interactions described by a potential of the binomial type. This potential contains a free parameter which can be chosen to reproduce a variety of models, such as the Toda, the Fermi-Pasta-Ulam and the Coulomb-like lattices. Carrying out essentially numerical experiments, the effects of soliton propagation on a lattice with defects are investigated. In particular, the properties of the localized mode, generated by the propagation of the soliton through the defect, are discussed with respect to the defect mass and the potential parameter, in the light of a simple theoretical model. Furthermore, an interesting phenomenon is observed: the amplitude of the speed of the mass defect shows a sequel of resonance peaks in terms of the mass defect. The positions of these peaks appear to be independent of the potential parameter. Received 16 August 1999 and Received in final form 3 February 2000     

Blind Extraction of Chaotic Signals by Using the Fast Independent Component Analysis Algorithm

We report the results of using the fast independent component analysis （FastICA） algorithm to realize Mind extraction of chaotic signals. Two cases are taken into consideration： namely, the mixture is noiseless or contaminated by noise. Pre-whitening is employed to reduce the effect of noise before using the FastICA algorithm. The correlation coefficient criterion is adopted to evaluate the performance, and the success rate is defined as a new criterion to indicate the performance with respect to noise or different mixing matrices. Simulation results show that the FastICA algorithm can extract the chaotic signals effectively. The impact of noise, the length of a signal frame, the number of sources and the number of observed mixtures on the performance is investigated in detail It is also shown that regarding a noise as an independent source is not always correct.     

Perturbation method for periodic solutions of nonlinear jerk equations

A Lindstedt-Poincaré type perturbation method with bookkeeping parameters is presented for determining accurate analytical approximate periodic solutions of some third-order (jerk) differential equations with cubic nonlinearities. In the process of the solution, higher-order approximate angular frequencies are obtained by Newton's method. A typical example is given to illustrate the effectiveness and simplicity of the proposed method.     

Effect of Nonlinearity on Scattering Dynamics of Solitary Waves

We discuss the effect of nonlinearity on the scattering dynamics of solitary waves. The pure nth power model with the interaction potential V （Х） = Х^n/n is present, which is a paradigm model in the study of solitary waves. The dependence of the scattering property on nonlinearity is closely related to the topological structures of the solitary waves. Moreover, for one of the four collision types, the rates of energy loss increase with the strength of nonlinearity and would reach 1 at n ≥ 10, which means that the two solitary waves would become of fragments completely after the collision.     

An Adaptive Denoising Algorithm for Noisy Chaotic Signals Based on Local Sparse Representation

An adaptive denoising algorithm based on local sparse representation （local SR） is proposed. The basic idea is applying SR locally to clusters of signals embedded in a high-dimensional space of delayed coordinates. The clusters of signals are represented by the sparse linear combinations of atoms depending on the nature of the signal. The algorithm is applied to noisy chaotic signals denoising for testing its performance. In comparison with recently reported leading alternative denoising algorithms such as kernel principle component analysis （Kernel PCA）, local independent component analysis （local ICA）, local PCA, and wavelet shrinkage （WS）, the proposed algorithm is more efficient.     

Two-dimensional discrete gap breathers in a two-dimensional discrete diatomic Klein-Gordon lattice

We study the existence and stability of two-dimensional discrete breathers in a two-dimensionai discrete diatomic Klein-Gordon lattice consisting of alternating light and heavy atoms, with nearest-neighbor harmonic coupling. Localized solutions to the corresponding nonlinear differential equations with frequencies inside the gap of the linear wave spectrum, i.e. two-dimensional gap breathers, are investigated numerically. The numerical results of the corresponding algebraic equations demonstrate the possibility of the existence of two-dimensional gap breathers with three types of symmetries, i.e., symmetric, twin-antisymmetric and single-antisymmetric. Their stability depends on the nonlinear on-site potential （soft or hard）, the interaction potential （attractive or repulsive） and the center of the two-dimensional gap breathers （on a light or a heavy atom）.     

Periodic, Quasiperiodic and Chaotic Discrete Breathers in a Parametrical Driven Two-Dimensional Discrete Klein-Gordon Lattice

We study a two-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the two-dimensional Klein-Gordon lattice with hard on-site potential. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers and chaotic discrete breathers by changing the amplitude of the driver.     

Multi-site Compact-Like Discrete Breather in Discrete One-Dimensional Monatomic Chains

Multi-site compact-like discrete breathers in discrete one-dimensional monatomic chains are investigated by discussing a generalized discrete one-dimensional monatomic model. We obtain that the two-site compact-like discrete breathers with codes σ = （0,..., 0, 1, 1, 0,..., 0）and codes σ= （0,..., 0, 1, -1, 0, ..., 0） can exist in discrete one-dimensional monatomic chain with quartic on-site and inter-site potentials. However, the former can only exist in hard quartic on-site potential and cannot exist in soft quartic on-site potential, whereas the latter is just reversed. A11 of the two-site Compact-like discrete breathers with codes σ = （0,..., 0, 1, 1, 0,..., 0） and σ （0,... ,0, 1, -1,0,... ,0} cannot exist in a pure K4 chain.     

Existence and Stability of Compact-Like Discrete Breather in Discrete One-Dimensional Monatomic Chains

Compact-like discrete breathers in discrete one-dimensional monatomic chains are investigated by discussing a generalized discrete one-dimensional monatomic model. It is proven that compact-like discrete breathers exist not only in soft Ф^4 potential but also in hard Ф^4 potential and K4 chains. The measurements of compact-like discrete breathers＇ core in soft and hard Ф^4 potential are determined by coupling parameter K4, while the measurements of compact-like discrete breathers＇ core in K4 chains are not related to coupling parameter K4. The stabilities of compact-like discrete breathers correlate closely to coupling parameter K4 and the boundary condition of lattice.     

Existence and Stability of Two-Dimensional Compact-Like Discrete Breathers in Discrete Two-Dimensional Monatomic Square Lattices

Two-dimensional compact-like discrete breathers in discrete two-dimensional monatomic square lattices are investigated by discussing a generalized discrete two-dimensional monatomic model. It is proven that the two- dimensional compact-like discrete breathers exist not only in two-dimensional soft Φ4 potentials but also in hard two-dimensional Φ4 potentials and pure two-dimensional K4 lattices. The measurements of the two-dimensional compact-like discrete breather cores in soft and hard two-dimensional Φ4 potential are determined by coupling parameter K4, while those in pure two-dimensional K4 lattices have no coupling with parameter K4. The stabilities of the two-dimensional compact-like discrete breathers correlate closely to the coupling parameter K4 and the boundary condition of lattices.     

Two-Dimensional Discrete Gap Breathers in a Two-Dimensional Diatomic β Fermi--Pasta--Ulam Lattice

We study the existence of two-dimensional discrete breathers in a two-dimensional face-centred square lattice consisting of alternating light and heavy atoms, with nearest-neighbour coupling containing quartic soft or hardnonlinearity. This study is focused on two-dimensional breathers with frequency in the gap that separates the acoustic and optical bands of the phonon spectrum. We demonstrate the possibility of existence of two-dimensional gap breathers by using the numerical method, the local anharmonicity approximation and the rotating wave approximation. We obtain six types of two-dimensional gap breathers, i.e., symmetric, mirror-symmetric and asymmetric, no matter whether the centre of the breather is on a light or a heavy atom.     

Surface defect linear modes in one-dimensional photonic lattices

We report on the existence of surface defect linear modes at an interface between the defect of one-dimensional photonic lattices and the uniform media. The interface defect can significantly affect the properties of linear modes. Such new type of modes exists in the first bandgap for positive defects; while they exist in the second bandgap for negative defects. Particularly, when a Gaussian beam, which is similar to the linear mode, is launched at the defect site, we find that the Gaussian beam can be strongly confined at defect site and robustness along longitudinal direction for a long distance. When launched at a small angle into the defect site, the Gaussian beam exhibits stable snake propagation.     

Internal mode dynamics in driven nonlinear Klein-Gordon systems

We study analytically and numerically the action of a constant force on the propagation of kinks in the φ⁴ and sine-Gordon systems, with and without dissipation. We specifically investigate the relation of the external force with the oscillations of the kink width due to excitation of its internal mode or quasimode. We demonstrate that both dc force and dissipation, either jointly or separately, damp the oscillations of the kink width. We further prove that, in contrast to earlier predictions, those oscillations can only arise if we use a distorted kink as initial condition for the evolution. Finally, we show that for the φ⁴ system the oscillations of the kink width come from the excitation of its internal mode, whereas in the sG equation they originate in the excitation of the lowest radiational modes and an internal mode induced by the discreteness of the numerical simulations. Received 6 June 2000     

Parametrically Driven Solitons in a Chain of Nonlinear Coupled Pendula with an Impurity

The system consisting of a chain of parametrically driven and damped nonlinear coupled pendula with a mass impurity is studied by means of a discrete version of the envelope function approach. An analogue of the parametrically driven damped nonlinear Schodinger equation with an impurity term is derived from the original lattice equation. Analytical solutions of impurity pinned high-frequency breathers and kinks are obtained. The results show that the mass impurity has striking influence on the high-frequency modes. In addition, we perform numerical simulations, showing that the light mass impurity has a stabilizing effect on the chain. The breathers seeding chaos in the homogeneous chain are pinned on a suitable light impurity to pull the chain from the chaotic state.     

Mobility and conductivity of ionic and bonded defects in hydrogen-bonded chains with nonlinear interactions

The proton conductivity and the mobility arising from motions of the ionic and bonded defects, in hydrogen-bonded molecular systems are investigated by means of the quantum mechanical method. Our two component model goes beyond the usual classical harmonic interaction by inclusion of a quartic interaction potential between the nearest-neighbor protons. Among the rich variety of soliton patterns obtained in this model, we focus our attention to compact kink (kinkon) solutions to calculate analytically, the mobility of the kinkon-antikinkon pair and the specific electrical-conductivity of the protons transfer in the hydrogen-bonded systems under an externally applied electrical-field through the dynamic equation of the kinkon-antikinkon pair. For ice, the mobility and the electrical conductivity of the proton transfer obtained are about 5.307×10^-7 m²  V^-1  s^-1 and 6.11×10^-4 Ω^-1 m^-1, respectively. The results obtained are in qualitative agreement with experimental data.     

Waves and patterns in ring lattices with delays

We investigate the existence and the stability of waves and phase locked states in rings of coupled oscillators with delayed interactions. Using center manifold reduction and the normal form method, we reduce the equation governing the dynamics of the whole network to an amplitude-phase model (i.e. a set of coupled ordinary differential equations describing the evolution of both the amplitudes and the phases of the oscillators). Then we prove the existence of traveling waves, in-phase and anti-phase locked oscillations, in both one-dimensional and two-dimensional lattices. The influence of the interaction strength and the number of oscillators is investigated, and the possible coexistence of waves and phase locked oscillations is shown.