Direction-Reversing Traveling Waves in the Even Fermi-Pasta-Ulam Lattice

Summary. This paper considers the famous Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and quartic nonlinearities. Due to special resonances and discrete symmetries, the Birkhoff normal form of this Hamiltonian system is Liouville integrable. The normal form equations can easily be solved if the number of particles in the lattice is odd, but if the number of particles is even, several nontrivial phenomena occur. In the latter case we observe that the phase space of the normal form is decomposed in invariant subspaces that describe the interaction between the Fourier modes with wave number j and the Fourier modes with wave number n / 2-j . We study how the level sets of the integrals of the normal form foliate these invariant subspaces. The integrable foliations turn out to be singular and the method of singular reduction shows that the normal form has invariant pinched tori and monodromy. Monodromy is an obstruction to the existence of global action-angle variables. The pinched tori are interpreted as homoclinic and heteroclinic connections between traveling waves. Thus we discover a class of solutions of the normal form which can be described as direction-reversing traveling waves. The relation between the FPU lattice and its Birkhoff normal form can be understood from KAM theory and approximation theory. This explains why we observe the impact of the direction-reversing traveling waves numerically as a relaxation oscillation in the original FPU system.     

Intrinsic localized modes and nonlinear impurity modes in curved Fermi-Pasta-Ulam chain

We explore the nature of intrinsic localized modes (ILMs) in a curved Fermi-Pasta-Ulam (FPU) chain and the effects of geometry and second-neighbor interaction on the localization and movability properties of such modes. We determine analytically the structure of the localized modes induced by an isotopic light-mass impurity in this chain. We further demonstrate that a nonlinear impurity mode may be treated as a bound state of an ILM with the impurity.      

Phonon relaxation and heat conduction in one-dimensional Fermiben Pastaben Ulam β lattices by molecular dynamics simulations

The phonon relaxation and heat conduction in one-dimensional Fermi-Pasta-Ulam (FPU) β lattices are studied by using molecular dynamics simulations. The phonon relaxation rate, which dominates the length dependence of the FPU β lattice, is first calculated from the energy autocorrelation function for different modes at various temperatures through equilibrium molecular dynamics simulations. We find that the relaxation rate as a function of wave number k is proportional to k^1.688, which leads to a N^0.41 divergence of the thermal conductivity in the framework of Green-Kubo relation. This is also in good agreement with the data obtained by non-equilibrium molecular dynamics simulations which estimate the length dependence exponent of the thermal conductivity as 0.415. Our results confirm the N^2/5 divergence in one-dimensional FPU β lattices. The effects of the heat flux on the thermal conductivity are also studied by imposing different temperature differences on the two ends of the lattices. We find that the thermal conductivity is insensitive to the heat flux under our simulation conditions. It implies that the linear response theory is applicable towards the heat conduction in one-dimensional FPU β lattices.     

一维Klein-Gordon/Fermi-Pasta-Ulam混合原子链中非线性参数作用的研究

采用推广的旋转平面波近似对一维非线性Klein-Gordon/Fermi-Pasta-Ulam混合原子链的运动方程进行简化,数值求解得到该系统中存在的离散呼吸子解.研究了系统中各非线性参数对该振动模的直流、一阶简谐项和二阶简谐项三个分量的对称性的影响以及对系统中局域模的影响. 关键词： Klein-Gordon/Fermi-Pasta-Ulam混合原子链 推广的旋转平面波近似 离散呼吸子 对称性     

Energy sharing induced by the nonlinear interaction

Strong energy sharing is shown by numerically investigating coupled multi-component Bose–Einstein condensates(BECs) with a harmonic trap to simulate the Fermi–Pasta–Ulam model(FPU). For two-component BECs, the energy exchanging between each part, from regular, quantum beating to complete energy sharing, is explored by simulating their Husimi distributions, the time evolution of energies and the statistical entropy. Meanwhile, in the three-component case, a more complex energy sharing behavior is reported and a strong energy sharing is found.     

Fermi-Pasta-Ulam β 格点链系统能量载流子研究

Fermi-Pasta-Ulam (FPU) β格点链中能量输运的载流子是孤子还是声子一直存在较多的争议. 本文通过单脉冲方法, 明确了一个能量波包在该格点链系统中从声子波包转变成为孤子波包的条件, 即波包能量达到一定阈值. 基于纯四次势链的声子真空效应, 构造了由FPU-β链与纯四次势链构成的双段链系统. 通过对比研究双段链系统和单段FPU-β链中的热流, 发现低温下声子是FPU-β链中能量的主要载流子, 而随着温度的升高孤子逐步取代声子成为能量的主要载流子. 关键词： Fermi-Pasta-Ulam格点链 声子 孤子 热传导     

Computer Simulation Study of Thermal Conduction in 1D Chains of Anharmonic Oscillators

In this work thermal conduction in one-dimensional(1D) chains of anharmonic oscillators are studied using computer simulation.The temperature profile,heat flux and thermal conductivity are investigated for chain length N = 100,200,400,800 and 1600.In the computer simulation anharmonicity is introduced due to Fermi-Pasta-Ulam-β(FPU-β) model.For substrate interaction,an onsite potential due to Frenkel-Kontorova(FK) model has been used.Numerical simulations demonstrate that temperature gradient scales behave as N-1 linearly with the relation J = 0.1765/N.For the thermal conductivity K,KN to N obey the linear relation of the type KN = 0.8805N.It is shown that thermal transport is dependent on phonon-phonon interaction as well as phonon-lattice interaction.The thermal conductivity increases linearly with increase inanharmonicity and predicts relation κ = 0.133 + 0.804β.It is also concluded that for higher value of the strength of the onsite potential system tends to a thermal insulator.     

Nearly separable behavior of Fermi-Pasta-Ulam chains through the stochasticity threshold

For the periodic Fermi-Pasta-Ulam chain with quartic potential we prove the relation p _k ² _T (1+) _k ² q _k ² , i.e., the proportionality, already at early timesT, between averaged kinetic and harmonic energies of each mode. The factor depends on the parameters of the model, but not on the mode and the number of degrees of freedom. It grows with the anharmonic strength from the value =0 of the harmonic limit (virial theorem) up to few units for the system much above the threshold. In the stochastic regime the time necessary to reduce the fluctuations ink to a fixed percentage remains at least one order of magnitude smaller than the time necessary to reach a similar level of equipartition. The persistence of such a behavior even above the stochasticity threshold clarifies a number of previous numerical results on the relaxation to equilibrium: e.g., the existence of several time scales and the relevance of the harmonic frequency spectrum. The difficulties in the numerical simulation of the thermodynamic limit are also discussed.CNR-INFM.     

On the numerical integration of FPU-like systems

This paper concerns the numerical integration of systems of harmonic oscillators coupled by nonlinear terms, like the common FPU models. We show that the most used integration algorithm, namely leap-frog, behaves very gently with such models, preserving in a beautiful way some peculiar features which are known to be very important in the dynamics, in particular the “selection rules” which regulate the interaction among normal modes. This explains why leap-frog, in spite of being a low order algorithm, behaves so well, as numerical experimentalists always observed. At the same time, we show how the algorithm can be improved by introducing, at a low cost, a “counterterm” which eliminates the dominant numerical error.