Chopping Time of the FPU $${\alpha }$$-Model

We study, both numerically and analytically, the time needed to observe the breaking of an FPU \(\alpha \)-chain in two or more pieces, starting from an unbroken configuration at a given temperature. It is found that such a “chopping” time is given by a formula that, at low temperatures, is of the Arrhenius–Kramers form, so that the chain does not break up on an observable time-scale. The result explains why the study of the FPU problem is meaningful also in the ill-posed case of the \(\alpha \)-model.     

q-Breathers in finite two- and three-dimensional nonlinear acoustic lattices

In their celebrated experiment, Fermi, Pasta, and Ulam (FPU) [Los Alamos Report No. LA-1940, 1955] observed that in simple one-dimensional nonlinear atomic chains the energy must not always be equally shared among the modes. Recently, it was shown that exact and stable time-periodic orbits, coined q-breathers (QBs), localize the mode energy in normal mode space in an exponential way, and account for many aspects of the FPU problem. Here we take the problem into more physically important cases of two- and three-dimensional acoustic lattices to find existence and principally different features of QBs. By use of perturbation theory and numerical calculations we obtain that the localization and stability of QBs are enhanced with increasing system size in higher lattice dimensions opposite to their one-dimensional analogues.     

Modulational instability in isolated and driven Fermi–Pasta–Ulam lattices

We present a detailed analysis of the modulational instability of the zone-boundary mode for one and higher-dimensional Fermi–Pasta–Ulam (FPU) lattices. The growth of the instability is followed by a process of relaxation to equipartition, which we have called the Anti-FPU problem because the energy is initially fed into the highest frequency part of the spectrum, while in the original FPU problem low frequency excitations of the lattice were considered. This relaxation process leads to the formation of chaotic breathers in both one and two space dimensions. The system then relaxes to energy equipartition, on time scales that increase as the energy density is decreased. We supplement this study by considering the nonconservative case, where the FPU lattice is homogeneously driven at high frequencies. Standing and travelling nonlinear waves and solitonic patterns are detected in this case. Finally we investigate the dynamics of the FPU chain when one end is driven at a frequency located above the zone boundary. We show that this excitation stimulates nonlinear bandgap transmission effects.     

Korteweg-de Vries equation and energy sharing in Fermi-Pasta-Ulam

We address the problem of equipartition in a long Fermi-Pasta-Ulam (FPU) chain. After giving a precise relation between FPU and Korteweg-de Vries we use the latter equation to show that, corresponding to initial data a la Fermi, the time average of the energy on the kth mode decreases exponentially with kN. The result persists in the thermodynamic limit.     

The anti-FPU problem

We present a detailed analysis of the modulational instability of the zone-boundary mode for one and higher-dimensional Fermi-Pasta-Ulam (FPU) lattices. Following this instability, a process of relaxation to equipartition takes place, which we have called the Anti-FPU problem because the energy is initially fed into the highest frequency part of the spectrum, at variance with the original FPU problem (low frequency excitations of the lattice). This process leads to the formation of chaotic breathers in both one and two dimensions. Finally, the system relaxes to energy equipartition on time scales which increase as the energy density is decreased. We show that breathers formed when cooling the lattice at the edges, starting from a random initial state, bear strong qualitative similarities with chaotic breathers.     

Equilibration and universal heat conduction in fermi-pasta-ulam chains

It is shown numerically that for Fermi-Pasta-Ulam (FPU) chains with alternating masses and heat baths at slightly different temperatures at the ends, the local temperature (LT) on small scales behaves paradoxically in steady state. This expands the long established problem of equilibration of FPU chains. A well-behaved LT appears to be achieved for equal mass chains; the thermal conductivity is shown to diverge with chain length N as N(1/3), relevant for the much debated question of the universality of one-dimensional heat conduction. The reason why earlier simulations have obtained systematically higher exponents is explained.     

Nonlinear normal modes and quanta in the β Fermi-Pasta-Ulam lattice

We enumerate simple periodic orbits for the well-known Fermi-Pasta-Ulam (FPU) problem. Using these solutions as simple modes for the problem, we construct quantum solutions of the corresponding problem. These studies present a natural generalization of the concept of phonon in the nonlinear realm.     

Time scale for energy equipartition in a two-dimensional FPU model

The FPU problem, i.e., the problem of energy equipartition among normal modes in a weakly nonlinear lattice, is here studied in dimension two, more precisely in a model with triangular cell and nearest-neighbors Lennard-Jones interaction. The number n of degrees of freedom ranges from 182 to 6338. Energy is initially equidistributed among a small number n(0) of low frequency modes, with n(0) proportional to n. We study numerically the time evolution of the so-called spectral entropy and the related "effective number" n(eff) of degrees of freedom involved in the dynamics; in this (rather typical) way we can estimate, for each n and each specific energy (energy per degree of freedom) epsilon, the time scale T(n)(epsilon) for energy equipartition. Numerical results indicate that in the thermodynamic limit the equipartition times are short: more precisely, for large n at fixed epsilon we find a limit curve T(infinity)(epsilon), and T(infinity) grows only as epsilon(-1) for small epsilon. Larger equipartition times are obtained by lowering epsilon, at fixed n, below a crossover value epsilon(c)(n). However, epsilon(c) appears to vanish by increasing n (faster than 1n), and the total energy E=nepsilon, rather than epsilon, appears to be the relevant variable when n is large and epsilon相似文献   

Localized waves in nonlinear oscillator chains

This paper reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Fermi-Pasta-Ulam (FPU) lattice. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For FPU lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.     

The Fermi-Pasta-Ulam problem: fifty years of progress

A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with its suggested resolutions and its relation to other physical problems. We focus on the ideas and concepts that have become the core of modern nonlinear mechanics, in their historical perspective. Starting from the first numerical results of FPU, both theoretical and numerical findings are discussed in close connection with the problems of ergodicity, integrability, chaos and stability of motion. New directions related to the Bose-Einstein condensation and quantum systems of interacting Bose-particles are also considered.     

The Fermi-Pasta-Ulam recurrence and related phenomena for 1<Emphasis Type="Italic">D</Emphasis> shallow-water waves in a finite basin

Different regimes of the Fermi-Pasta-Ulam (FPU) recurrence are simulated numerically for fully nonlinear “one-dimensional” potential water waves in a finite-depth flume between two vertical walls. In such systems, the FPU recurrence is closely related to the dynamics of coherent structures approximately corresponding to solitons of the integrable Boussinesq system. A simplest periodic solution of the Boussinesq model, describing a single soliton between the walls, is presented in analytic form in terms of the elliptic Jacobi functions. In the numerical experiments, it is observed that depending on the number of solitons in the flume and their parameters, the FPU recurrence can occur in a simple or complicated manner, or be practically absent. For comparison, the nonlinear dynamics of potential water waves over nonuniform beds is simulated, with initial states taken in the form of several pairs of colliding solitons. With a mild-slope bed profile, a typical phenomenon in the course of evolution is the appearance of relatively high (rogue) waves, while for random, relatively short-correlated bed profiles it is either the appearance of tall waves or the formation of sharp crests at moderate-height waves.     

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.     

Discrete breathers in Fermi-Pasta-Ulam lattices

We study the properties of spatially localized and time-periodic excitations--discrete breathers--in Fermi-Pasta-Ulam (FPU) chains. We provide a detailed analysis of their spatial profiles and stability properties. We especially demonstrate that the Page mode is linearly stable for symmetric FPU potentials. A resonant interaction between a localized and delocalized perturbations causes weak but finite strength instabilities for asymmetric FPU potentials. This interaction induces Fano resonances for plane waves scattered by the breather. Finally we analyze the interplay between energy thresholds for breathers in the presence of strongly asymmetric FPU potentials and the corresponding profiles of the low-frequency limit of breather families.     

The Fermi-Pasta-Ulam Problem and Its Underlying Integrable Dynamics

This paper is devoted to a numerical study of the familiar α+β FPU model. Precisely, we here discuss, revisit and combine together two main ideas on the subject: (i) In the system, at small specific energy ε=E/N, two well separated time-scales are present: in the former one a kind of metastable state is produced, while in the second much larger one, such an intermediate state evolves and reaches statistical equilibrium. (ii) FPU should be interpreted as a perturbed Toda model, rather than (as is typical) as a linear model perturbed by nonlinear terms. In the view we here present and support, the former time scale is the one in which FPU is essentially integrable, its dynamics being almost indistinguishable from the Toda dynamics: the Toda actions stay constant for FPU too (while the usual linear normal modes do not), the angles fill their almost invariant torus, and nothing else happens. The second time scale is instead the one in which the Toda actions significantly evolve, and statistical equilibrium is possible. We study both FPU-like initial states, in which only a few degrees of freedom are excited, and generic initial states extracted randomly from an (approximated) microcanonical distribution. The study is based on a close comparison between the behavior of FPU and Toda in various situations. The main technical novelty is the study of the correlation functions of the Toda constants of motion in the FPU dynamics; such a study allows us to provide a good definition of the equilibrium time τ, i.e. of the second time scale, for generic initial data. Our investigation shows that τ is stable in the thermodynamic limit, i.e. the limit of large N at fixed ε, and that by reducing ε (ideally, the temperature), τ approximately grows following a power law τ～ε ^?a , with a=5/2.     

Fermi–Pasta–Ulam Recurrence Interaction with Fluctuations of the Medium

It is shown in numerical study that the simultaneous influence of discrete and continuous random processes on the Fermi–Pasta–Ulam (FPU) recurrence dynamics in chains of vibrators having both fixed and open ends leads to a widening of the FPU spectrum and to its permanent evolution. These phenomena are much more markedly manifested in the chain with open ends. Taking into account that the FPU recurrence takes place in distributed oscillatory systems one of which is a laser, the present research can be promising in developing multimode lasers exhibiting the recurrence phenomena.     

Full Fermi–Pasta–Ulam Recurrence Interaction with Fluctuations of the Medium

It is shown in a numerical study that the simultaneous influence of discrete and continuous random processes on the full Fermi–Pasta–Ulam (FPU) recurrence dynamics in two parametrically coupled identical chains of vibrators with open ends and under different initial conditions leads to stabilization of the FPU spectrum. A greater influence on chains of gaussian noise as compared to discrete noise is revealed. An increase in the amplitudes of both noises by one order of magnitude results in considerable parameter changes in their FPU spectrum. The full FPU recurrence interaction in coupled chains with discrete and continuous random noise of the medium manifests itself in extremely low-frequency periodic processes in stochastic dynamics of the medium (with frequency two orders of magnitude lower than the lowest frequency in the initial conditions of the chains).     

q-Breathers and the Fermi-Pasta-Ulam problem

The Fermi-Pasta-Ulam (FPU) paradox consists of the non-equipartition of energy among normal modes of a weakly anharmonic atomic chain model. In the harmonic limit each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q. We continue normal modes from the harmonic limit into the FPU parameter regime and obtain persistence of these periodic orbits, termed here q-breathers (QB). They are characterized by time periodicity, exponential localization in the q-space of normal modes and linear stability up to a size-dependent threshold amplitude. Trajectories computed in the original FPU setting are perturbations around these exact QB solutions. The QB concept is applicable to other nonlinear lattices as well.     

Energy localization and transport in two-dimensional Fermi–Pasta–Ulam lattices

We investigate energy localization and transport in the form of discrete breathers and their movability in two-dimensional Fermi–Pasta–Ulam(FPU) lattices. We study the dynamics of the two-dimensional Fermi–Pasta–Ulam(FPU) lattice, incorporating the complicated effects of geometry, long-range interactions as well as nonlinear dispersion. We obtain several exact discrete breather(DB) solutions, such as the odd-parity and even-parity DBs, compact-like DBs and moving DBs for various geometries of the two-dimensional FPU chain. We show that DBs also exist in the same lattice in presence of next-nearest neighbour interaction. Large-amplitude exact subsonic travelling kink-soliton solutions are obtained in such a periodic chain in presence of long-range nonlinear dispersive interaction in the long-wavelength and weakly nonlinear limit. Such a two-dimensional FPU lattice admits finite amplitude nonlinear sinusoidal wave (NSW) solutions with short commensurate as well as incommensurate wavelengths for different geometries of the chain. The usefulness of these solutions for energy localization and transport in various physical systems are discussed.     

Detecting chaos,determining the dimensions of tori and predicting slow diffusion in Fermi–Pasta–Ulam lattices by the Generalized Alignment Index method

The recently introduced GALI method is used for rapidly detecting chaos, determining the dimensionality of regular motion and predicting slow diffusion in multi-dimensional Hamiltonian systems. We propose an efficient computation of the GALI_k indices, which represent volume elements of k randomly chosen deviation vectors from a given orbit, based on the Singular Value Decomposition (SVD) algorithm. We obtain theoretically and verify numerically asymptotic estimates of GALIs long-time behavior in the case of regular orbits lying on low-dimensional tori. The GALI_k indices are applied to rapidly detect chaotic oscillations, identify low-dimensional tori of Fermi–Pasta–Ulam (FPU) lattices at low energies and predict weak diffusion away from quasiperiodic motion, long before it is actually observed in the oscillations.     

理想弹塑性固体Riemann问题的解结构与初值条件

针对理想弹塑性固体材料的一维Riemann问题,在不考虑真空的情况下,讨论其所有可能存在的解结构,给出每一种解结构下对应的初值条件且证明该系列初值条件的完备性,即任意给定的物理量初值均有且只有一种解结构与之对应.基于该理论,在设计精确或近似理想弹塑性Riemann问题求解器时,可以依据初值条件对任意物理量初值直接判断其对应的解结构,从而提高求解器的精度和效率.数值试验验证了该系列初值条件的正确性和有效性.