Kink dynamics in one-dimensional coupled map lattices

We examine the problem of the dynamics of interfaces in a one-dimensional space-time discrete dynamical system. Two different regimes are studied: the non-propagating and the propagating one. In the first case, after proving the existence of such solutions, we show how they can be described using Taylor expansions. The second situation deals with the assumption of a travelling wave to follow the kink propagation. Then a comparison with the corresponding continuous model is proposed. We find that these methods are useful in simple dynamical situations but their application to complex dynamical behaviour is not yet understood. (c) 1995 American Institute of Physics.     

Molecular dynamics simulations of one-dimensional Lennard-Jones systems

One-dimensional Lennard-Jones systems are investigated by molecular dynamics simulations. The full Lennard-Jones potential is compared to the repulsive Lennard-Jones potential. It is found that the pair correlation function and the normalized velocity autocorrelation function agree at high densities and high temperature. However, the diffusion coefficient indicates that the attractive potential introduces additional correlations into particle dynamics which are not reflected in the statics. These results are in agreement with three-dimensional studies.     

Multiple-quantum dynamics of one-dimensional nuclear spin systems in solids

Multiple-quantum spin dynamics is studied using analytic and numerical methods for one-dimensional finite linear chains and rings of nuclear spins 1/2 coupled by dipole-dipole interactions. An approximation of dipole-dipole interaction between nearest neighbors having the same constants is used to obtain exact expressions for the intensities of the multiple-quantum coherences in the spin systems studied, which are initially in thermal equilibrium and whose evolution is described by a two-spin two-quantum Hamiltonian. An approximation of nearest neighbors with arbitrary dipole-dipole interaction constants is used to establish a simple relationship between the multiple-quantum dynamics and the dynamics of spin systems with an XY Hamiltonian. Numerical methods are developed to calculate the intensities of multiple-quantum coherences in multiple-quantum NMR spectroscopy. The integral of motion is obtained to expand the matrix of the two-spin two-quantum Hamiltonian into two independent blocks. Using the nearest-neighbor approximation the Hamiltonian is factorized according to different values of the projection operator of the total spin momentum on the direction of the external magnetic field. Results of calculations of the multiple-quantum dynamics in linear chains of seven and eight nuclear spins and a six-spin ring are presented. It is shown that the evolution of the intensities of the lowest-order multiple-quantum coherences in linear chains is accurately described allowing for dipole-dipole interaction of nearest and next-nearest neighbors only. Numerical calculations are used to compare the contributions of nearest and remote spins to the intensities of the multiple-quantum coherences.     

Kink plateau dynamics in finite-size lubricant chains

We extend the study of velocity quantization phenomena recently found in the classical motion of an idealized 1D model solid lubricant - consisting of a harmonic chain interposed between two periodic sliding potentials [A. Vanossi, M. Manini, G. Divitini, G.E. Santoro, E. Tosatti, Phys. Rev. Lett. 97 (2006) 056101]. This quantization is due to one slider rigidly dragging the commensurate lattice of kinks that the chain forms with the other slider. In this follow-up work we consider finite-size chains rather than infinite chains. The finite-size (i) permits the development of robust velocity plateaus as a function of the lubricant stiffness, and (ii) allows an overall chain length re-adjustment which spontaneously promotes single-particle periodic oscillations. These periodic oscillations replace the quasi-periodic motion produced by general incommensurate periods of the sliders and the lubricant in the infinite-size model. Possible consequences of these results for some real systems are discussed.     

Adiabatic dynamics of one-dimensional classical Hamiltonian dissipative systems

A linearized plane pendulum with the slowly varying mass and length of string and the suspension point moving at a slowly varying speed is presented as an example of a simple 1D mechanical system described by the generalized harmonic oscillator equation, which is a basic model in discussion of the adiabatic dynamics and geometric phase. The expression for the pendulum geometric phase is obtained by three different methods. The pendulum is shown to be canonically equivalent to the damped harmonic oscillator. This supports the mathematical conclusion, not widely accepted in physical community, of no difference between the dissipative and Hamiltonian 1D systems.     

Multiple-quantum NMR spin dynamics of inhomogeneous one-dimensional systems in solids

Multiple-quantum NMR spin dynamics of inhomogeneous one-dimensional systems in solids is investigated by analytical and numerical methods. A fermion approach for MQ spin dynamics of one-dimensional inhomogeneous systems is developed in the approximation of the dipole–dipole interactions (DDI) of nearest neighbors. It is shown that only MQ coherences of the zeroth and plus/minus second orders appear in the approximation of the DDI of the nearest neighbors even in inhomogeneous one-dimensional systems. We also investigate MQ dynamics of inhomogeneous chains numerically. Intensities of MQ NMR coherences for a linear chain consisting of 3000 spins are calculated.     

Kink dynamics in the highly discrete sine-Gordon system

We study kink dynamics in a very discrete sine-Gordon system where the kink width is of the order of the lattice spacing. Numerical simulations exhibit new properties of kinks in this case: they lose the memory of their initial velocity and propagate preferentially at well-defined velocities which correspond to quasi-steady states, while a kink moving at other velocities suffers relatively high rates of radiation of small amplitude oscillations. When a small external driving force is applied to the system, the same velocities appear as plateus in the strongly nonlinear mobility of the kink. The energy radiated by the kink is calculated for a simple model that preserves the discrete character of the system, and the preferential velocities for the kink are obtained to good accuracy. Similar results may be expected to be valid for other discrete systems manifesting topological solitons. The numerical simulations reveal also new stable “multiple-kink” excitations which can propagate almost freely in extremely discrete systems where “ordinary” simple kinks are pinned to the lattice by discreteness. The stability of the “multiple-kinks” is discussed.     

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     

Efficient approximation of the dynamics of one-dimensional quantum spin systems

In this Letter we show that an arbitrarily good approximation to the propagator e(itH) for a 1D lattice of n quantum spins with Hamiltonian H may be obtained with polynomial computational resources in n and the error epsilon and exponential resources in |t|. Our proof makes use of the finitely correlated state or matrix product state formalism exploited by numerical renormalization group algorithms like the density matrix renormalization group. There are two immediate consequences of this result. The first is that Vidal's time-dependent density matrix renormalization group will require only polynomial resources to simulate 1D quantum spin systems for logarithmic |t|. The second consequence is that continuous-time 1D quantum circuits with logarithmic |t| can be simulated efficiently on a classical computer, despite the fact that, after discretization, such circuits are of polynomial depth.     

非线性环型腔反馈激光系统的动力学特性及其混沌控制

 以环型腔反馈激光系统为主,综述了非线性激光系统的混沌动力学特性;分析了延迟反馈方法控制混沌的原理和稳定性条件,实现了对多介质非线性激光系统中的混沌控制。同时概述了近年来非线性激光系统中混沌控制的最新进展,诸如空间小微扰法、偶然正比反馈技术等,讨论了混沌控制在提高激光器功率和性能、利用混沌进行秘密通讯和信息处理等方面的应用前景。     

非线性环型腔反馈激光系统的动力学特性及其混沌控制

以环型腔反馈激光系统为主,综述了非线性激光系统的混沌动力学特性;分析了延迟反馈方法控制混沌的原理和稳定性条件,实现了对多介质非线性激光系统中的混沌控制。同时概述了近年来非线性激光系统中混沌控制的最新进展,诸如空间小微扰法、偶然正比反馈技术等,讨论了混沌控制在提高激光器功率和性能、利用混沌进行秘密通讯和信息处理等方面的应用前景。     

Impurities potential effect on the charge density wave dynamics in one-dimensional systems

In this work, we present in the weak pinning case the numerical simulation results of the one-dimensional deformable charge density wave (CDW) properties considering the potential amplitude fluctuations effect generated by different impurity types randomly distributed in the lattice. When the electric field approaches threshold value ET, the static equilibrium characteristic time τ and the polarization PCDW become large and seem to diverge at critical field Ecr from below ET following a power law [1−(E/Ecr)]^−α where α is an impurity dependant critical exponent. This divergence indicates that the CDW depinning can be described in terms of a dynamical critical phenomena, where the critical field Ecr plays the role of a transition temperature as in ordinary phase transitions. In agreement with several experimental results, we show that the electric current density JCDW and electric conductivity σCDW follow respectively a power law β[(E/ET)−1] and (ET/E)ν[(E/ET)−1] where β and ν are critical exponents. This results are analogous to these obtained in the case of one impurity type.     

Non-equilibrium dynamics of one-dimensional infinite particle systems with a hard-core interaction

An infinite system of Newton's equation of motion is considered for one-dimensional particles interacting by a finite-range hard-core potential of singularity like an inverse power of distance between the hard cores. Existence of limiting solutions is proved for initial configurations of finite specific energy and the semigroup of motion is constructed if energy fluctuations near infinity increase only as a small power of distance from the origin. In this case uniqueness of solutions is also proved and the solution is a weakly continuous function of initial data. The allowed set of initial configurations carries a wide class of probability measures including Gibbsian fields with different potentials. In the absence of hard cores limiting solutions are constructed for initial configurations with a logarithmic order of energy and density fluctuations.     

Vibration suppression in belt-driven servo systems containing uncertain nonlinear dynamics

This paper proposes a vibration suppression strategy for belt-driven servo systems subject to uncertain nonlinear dynamics and external disturbances. The function approximation technique is applied to estimate the uncertainties that are further covered by sliding-based design. The closed loop stability is justified with Lyapunov-like method to ensure ultimately uniformly bounded performance of the output error. Simulation cases show that the proposed strategy can stabilize the closed loop system with effective suppression of vibration regardless of various uncertain nonlinear dynamics and external disturbances.     

Pulse propagation in one-dimensional nonlinear chains

Summary Several numerical computations are carried out to investigate the effects of the propagation of pulses (solitons and quasi-solitons) in one-dimensional nonlinear chains. Numerical results concerning the Toda lattice and a Coulomb-like lattice are discussed and compared. We consider lattices both with and without defects.

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Riassunto Si studiano gli effetti della propagazione di solitoni e di quasi solitoni in reticoli non lineari unidimensionali usando procedure di calcolo numerico. Si considerano il reticolo di Toda ed un reticolo di tipo Coulomb sia in presenza che in assenza di difetti.

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Резюме Используя процедуру численных вычислеий, исследуются эффекты распространения импульсов (солитонов или квази-солитонов) в одномерных нелинейных цепях. Обсуждаются и сравниваются численные редультаты, относящиеся к решетке тода и решетке кулоновского типа. Мы рассматриваем решетки с и вез дефектов.

     

Mixed-state dynamics in one-dimensional quantum lattice systems: a time-dependent superoperator renormalization algorithm

We present an algorithm to study mixed-state dynamics in one-dimensional quantum lattice systems. The algorithm can be used, e.g., to construct thermal states or to simulate real time evolution given by a generic master equation. Its two main ingredients are (i) a superoperator renormalization scheme to efficiently describe the state of the system and (ii) the time evolving block decimation technique to efficiently update the state during a time evolution. The computational cost of a simulation increases significantly with the amount of correlations between subsystems, but it otherwise depends only linearly on the system size. We present simulations involving quantum spins and fermions in one spatial dimension.     

On the dynamics of nonlinear systems with input constraints

In this work we deal with the dynamical analysis of nonlinear systems with input constraints. A characterization of the domain of attraction of the region of controllability of an equilibrium point under bounded control is provided and the concept of regions of invariance within such domains of attraction is introduced and characterized. The concepts and results are illustrated through case studies on chemical reactor models. (c) 1999 American Institute of Physics.