Dynamical energy equipartition of the Toda model with additional on-site potentials

We study the dynamical energy equipartition properties in the integrable Toda model with additional uniform or disordered on-site energies by extensive numerical simulations. The total energy is initially equidistributed among some of the lowest frequency linear modes. For the Toda model with uniform on-site potentials, the energy spectrum keeps its profile nearly unchanged in a relatively short time scale. On a much longer time scale, the energies of tail modes increase slowly with time. Energy equipartition is far away from being attached in our studied time scale. For the Toda model with disordered on-site potentials, the energy transfers continuously to the high frequency modes and eventually towards energy equipartition. We further perform a systematic study of the equipartition time teq depending on the energy density εand the nonlinear parameter α in the thermodynamic limit for the Toda model with disordered on-site potentials. We find teq∝(1/ε)~a(1/α)~b, where b ≈ 2 a. The values of a and b are increased when increasing the strengths of disordered on-site potentials or decreasing the number of initially excited modes.     

Optical nutation in the exciton range of spectrum

Optical nutation in the exciton range of spectrum is studied in the mean field approximation taking into account exciton-photon and elastic exciton-exciton interactions. It is shown that the features of nutation development are determined by the initial exciton and photon densities, the resonance detuning, the nonlinearity parameter, and the initial phase difference. For nonzero initial exciton and photon concentrations, three regimes of temporal evolution of excitons and photons exist: periodic conversion of excitons to photons and vice versa, aperiodic conversion of photons to excitons, and the rest regime. In the rest regime, the initial exciton and photon densities are nonzero and do not change with time. The oscillation amplitudes and periods of particle densities determined by the system parameters are found. The exciton self-trapping and photon trapping appearing in the system at threshold values of the nonlinearity parameter were predicted. As this parameter increases, the oscillation amplitudes of the exciton and photon densities sharply change at the critical value of the nonlinearity parameter. These two phenomena are shown to be caused by the elastic exciton-exciton interaction, resulting in the dynamic concentration shift of the exciton level.     

The two-stage dynamics in the Fermi-Pasta-Ulam problem: from regular to diffusive behavior

A numerical and analytical study of the relaxation to equilibrium of both the Fermi-Pasta-Ulam (FPU) α-model and the integrable Toda model, when the fundamental mode is initially excited, is reported. We show that the dynamics of both systems is almost identical on the short term, when the energies of the initially unexcited modes grow in geometric progression with time, through a secular avalanche process. At the end of this first stage of the dynamics, the time-averaged modal energy spectrum of the Toda system stabilizes to its final profile, well described, at low energy, by the spectrum of a q-breather. The Toda equilibrium state is clearly shown to describe well the long-living quasi-state of the FPU system. On the long term, the modal energy spectrum of the FPU system slowly detaches from the Toda one by a diffusive-like rising of the tail modes, and eventually reaches the equilibrium flat shape. We find a simple law describing the growth of tail modes, which enables us to estimate the time-scale to equipartition of the FPU system, even when, at small energies, it becomes unobservable.     

Two-dimensional generalized Toda lattice

The zero curvature representation is obtained for the two-dimensional generalized Toda lattices connected with semisimple Lie algebras. The reduction group and conservation laws are found and the mass spectrum is calculated.     

Trapping and fusion of solitons in a nonuniform toda lattice

In this paper we consider strongly localized solitons leading to high energy events in a Toda lattice with springs of distinct nonlinearity. The trapping and superposition of solitons at a single soft spring embedded in a hard host lattice is investigated numerically. We further show the generation of high energy solitons at an interface between a hard and a soft lattice. The behaviour of a soliton incident on an inhomogeneity is approached analytically. The interaction of solitons in an inhomogeneous Toda lattice is compared to that in a homogeneous one.     

Modulation instability in negative refractive metamaterials with exponential saturable nonlinearity and self-steepening effects

Modulation instability (MI) in negative refractive metamaterials with exponential saturable nonlinearity and the self-steeping effect is investigated by using the standard linear stability analysis and the Drude electromagnetic model. The dispersion relation, instability condition, cut-off frequency, gain spectrum, and their corresponding dimensionless expressions are analytically deduced and then calculated. The results show that, the self-steepening effect will suppress MI by decreasing the cut-off frequency and the maximum gain. And the larger the power density ratio is, more slowly the cut-off frequency and the maximum gain decrease with increase of the self-steepening parameter. Although the sign of the self-steepening parameter does not influence MI but its absolute value must be smaller than a certain value. Depending on different self-steepening parameters, the cut-off frequency and maximum gain may decrease or increase before decrease with increase of the power density ratio. In particular, for large self-steepening parameters, the power density ratio may have a threshold value. When the power density ratio is under this threshold, MI will disappear.     

Fermi acceleration revisited

Mappings that have been used to describe the Fermi acceleration mechanism are examined. It is shown that results which appear to be contradictory are due to differences in the mapping equations. For those mappings that can be locally approximated by the standard mapping, the value of the nonlinear parameter of the standard mapping, for which the last isolating KAM surface exists, can be used to predict the loss of KAM stability with action for the more general mappings. Previous results of the variation in the density distribution in the stochastic region of the phase space, averaged over phases, is shown to be consistent with the ergodic hypothesis. Fine scale structure of the mappings is found to be model dependent. The standard mapping is a member of a class of mappings which retains some KAM trajectories at arbitrarily large nonlinearity. This feature is not generic to a wider class of mappings discussed in this paper. The stability of two-iteration fixed points are discussed in detail, including the bifurcation sequence for one type of mapping.     

The Toda Equations and the Gromov–Witten Theory of the Riemann Sphere

Consequences of the Toda equations arising from the conjectural matrix model for the Riemann sphere are investigated. The Toda equations determine the Gromov–Witten descendent potential (including all genera) of the Riemann sphere from the degree 0 part. Degree 0 series computations via Hodge integrals then lead to higher-degree predictions by the Toda equations. First, closed series forms for all 1-point invariants of all genera and degrees are given. Second, degree 1 invariants are investigated with new applications to Hodge integrals. Third, a differential equation for the generating function of the classical simple Hurwitz numbers (in all genera and degrees) is found – the first such equation. All these results depend upon the conjectural Toda equations. Finally, proofs of the Toda equations in genus 0 and 1 are given.     

不公度相及公度-

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不公度相及公度—不公度相变：一维情形

本文通过ＦｒｅｎｋｅｌＫｏｎｔｏｒｏｖａ（ＦＫ）模型评述了凝聚态物理中的一个重要课题：不公度相和公度不公度（ＣＩ）相变。论述主要集中在一维情形,对不公度相和ＣＩ相变的概念、物理图象、产生机制、研究背景、研究目标和意义进行了必要的介绍;概述了早期、近期和现代研究过程中一些典型的基本理论和方法;对标准ＦＫ模型及原子间势为凸函数的ＦＫ模型、经典非凸ＦＫ模型、ＦＫ模型的量子效应、一维双原子链ＦＫ模型、ＦＫ模型向二维和三维的推广、ＣＩ相变的有限温度效应等专题的研究进展作了概述,并指出了一些有待进一步探讨的课题。     

不公度相及公度-不公度相变:一维情形

本文通过Ｆｒｅｎｋｅｌ Ｋｏｎｔｏｒｏｖａ（ＦＫ）模型评述了凝聚态物理中的一个重要课题：不公度相和公度 不公度（ＣＩ）相变。论述主要集中在一维情形,对不公度相和ＣＩ相变的概念、物理图象、产生机制、研究背景、研究目标和意义进行了必要的介绍;概述了早期、近期和现代研究过程中一些典型的基本理论和方法;对标准ＦＫ模型及原子间势为凸函数的ＦＫ模型、经典非凸ＦＫ模型、ＦＫ模型的量子效应、一维双原子链ＦＫ模型、ＦＫ模型向二维和三维的推广、ＣＩ相变的有限温度效应等专题的研究进展作了概述,并指出了一些有待进一步探讨的课题。     

光场及原子-光场耦合的非线性对腔内原子辐射谱的影响

研究了包括任意的光场非线性和原子光场非线性依赖强度耦合J-C(Jaynes-Cummings)模型中单原子的辐射谱,分析了上述非线性对辐射谱的影响.结果表明,原子的辐射谱强烈地依赖于光场的非线性项R(a⁺a)及原子光场非线性耦合因子f(a⁺a)的形式,其谱结构与标准J-C模型中原子的辐射谱明显不同.当腔内充满的非线性介质使得f(a⁺a)=1/(a⁺a)^1/2时,对于任意的初始光场态,辐射谱总是显示 关键词： 辐射谱 非线性耦合     

Bäcklund transformation for the first flows of the relativistic Toda hierarchy and associated properties

In this communication we study a class of one parameter dependent auto-Bäcklund transformations for the first flow of the relativistic Toda lattice and also a variant of the usual Toda lattice equation. It is shown that starting from the Hamiltonian formalism such transformations are canonical in nature with a well defined generating function. The notion of spectrality is also analyzed and the separation variables are explicitly constructed.     

Nonlocal integrable partners to generalized MKdV and two-dimensional Toda lattice equation in the formalism of a dressing method with quantized spectral parameter

Two new hierarchies, MILW₂ and a two-dimensional nonlocal Toda lattice are constructed. The characteristic property of the first one is the connection with the ILW₂ hierarchy by means ofgl(2) Miura transformation. On the other hand, MILW₂ equations turn out to be symmetry equations for a two-dimensional nonlocal Toda lattice. A new version of the dressing technique with quantized spectral parameter is proposed.     

Influences of tissue composition and structural features of biological media on the ultrasonic nonlinearity parameter

The Acoustic nonlinearity parameter is an important parameter innonlinear acoustics.In this article,the nonlinearity parameter B/A of normaland eight kinds of pathological porcine liver tissues were measured by finiteamplitude insert-substitution method.The mixture law for nonlinearityparameter is used to analyze and predict the volume fractions of the compo-nents in a given tissue.It was found that the nonlinearity parameter is sensitiveto the pathological forms of biological tissues and the values of B/A dependon the tissue composition and structural features.     

Experimental and theoretical study of harmonic generation at contacting interface

The second harmonic generation behavior of a contacting interface has been evaluated experimentally and discussed theoretically in the light of a nonlinear interface model. Two aluminum blocks were mated together to constitute a contact interface and subjected to normal compressive loading. A 5 MHz longitudinal toneburst wave was sent to the interface in the normal direction and the transmitted wave was recorded, from which the fundamental and the second harmonic components were extracted. A nonlinearity parameter was obtained as the ratio of the second harmonic amplitude to the squared fundamental amplitude. From the measured contact pressure dependence of the transmitted fundamental amplitude, the linear and the second-order interfacial stiffness parameters were identified, which enabled the evaluation of the nonlinearity parameter based on the theoretical model. The theoretical contact pressure dependence of the nonlinearity parameter was found to be in good qualitative agreement with the experimental results.     

The q-Deformed SU(2) Quantum Antiferromagnetic Model

In this paper we set up a q-analogous Holstein- Primakoff transformation (HPT) and obtain the.q-deformed spin-wave solutions. We apply a mean-field treatment that is arranged to preserve some nonlinearity of the Hamiltonian, and the results turn out to be interesting,e.g., a gapful spectrum of the elementary excitation spectrum is found, which is different from the spin-wave results for the conventionid model, and the specific heat is therefore modified greatly. The properties of the Néel state and the ground state are investigated.     

Integrable Systems and Two-Dimensional Gravity from Conformally Reduced WZNW Model

Imposing constraints with an integer ordering on WZNW model a large series of conformal invariant integrable systems will result. In this letter, a general approach for imposing the first and the second class constraints based on an arbitrary grading scheme of the Lie algebras of the WZNW groups is presented. The first order constraints correspond to integrable systems containing super Toda and conformal affine Toda systems as examples and are related to two-dimensional induced gravity, whilst the second order constraints correspond to supersymmetric-like integrable systems containing super Toda and conformal affine super Toda systems (for super WZNW groups) and are conjectured to be related to twodimensional induced supergravity.     

Frequency analysis of tick quotes on foreign currency markets and the double-threshold agent model

Power spectrum densities for the number of tick quotes per minute (market activity) on three currency markets (USD/JPY, EUR/USD, and JPY/EUR) are analyzed for periods from January 2000 to December 2000. We find some peaks on the power spectrum densities at a few minutes. We develop the double-threshold agent model and confirm that the corresponding periodicity can be observed for the activity of this model even though market participants perceive common weaker periodic information than threshold for decision-making of them. This model is numerically performed and theoretically investigated by utilizing the mean-field approximation. We propose a hypothesis that the periodicities found on the power spectrum densities can be observed due to nonlinearity and diversity of market participants.     

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