A. Hoffman, C. E. Wayne
We prove the existence of asymptotic two-soliton states in the Fermi-Pasta-Ulam model with general interaction potential. That is, we exhibit solutions whose difference in $\ell^2$ from the linear superposition of two solitary waves goes to zero as time goes to infinity.
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By combining results of Mizumachi on the stability of solitons for the Toda lattice with a simple rescaling and a careful control of the KdV limit we give a simple proof that small amplitude, long-wavelength solitary waves in the Fermi-Pasta-Ulam (FPU) model are linearly stable and hence by the results of Friesecke and Pego that they are also nonlinearly, asymptotically stable.
The Fermi-Pasta-Ulam problem was one of the first computational experiments. It has stirred the physics community since, and resisted a simple solution for half a century. The combination of straightforward simulations, efficient computational schemes for finding periodic orbits, and analytical estimates allows us to achieve significant progress. Recent results on $q$-breathers, which are time-periodic solutions that are localized in the space of normal modes of a lattice and maximize the energy at a certain mode number, are discussed, together with their relation to the Fermi-Pasta-Ulam probl…
The Fermi-Pasta-Ulam (FPU) paradox consists of the nonequipartition 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. Tr…
Bushes of normal modes represent the exact excitations in nonlinear physical systems with discrete symmetries [Physica D117 (1998) 43]. The present paper is the continuation of our previous paper [Physica D166 (2002) 208], where these dynamical objects of a new type were discussed for the monoatomic nonlinear chains. Here, we develop a simple crystallographic method for finding bushes in nonlinear chains and investigate stability of one-dimensional and two-dimensional vibrational bushes for both FPU-alpha and FPU-beta models, in particular, of those revealed recently in [Physica D175 (2003) 31…
The dynamics of energy relaxation in thermalized one- and two-dimensional arrays with nonlinear interactions depend in detail on the interactions and, in some cases, on dimensionality. We describe and explain these differences for arrays of the Fermi-Pasta-Ulam type. In particular, we focus on the roles of harmonic contributions to the interactions and of breathers in the relaxation process.
This work presents asymptotic solutions to a singularly-perturbed, period-2 FPUT lattice and uses exponential asymptotics to examine `nanoptera', which are nonlocal solitary waves with constant-amplitude, exponentially small wave trains which appear behind the wave front. Using an exponential asymptotic approach, this work isolates the exponentially small oscillations, and demonstrates that they appear as special curves in the analytically-continued solution, known as `Stokes curves' are crossed. By isolating these the asymptotic form of these oscillations, it is shown that there are special m…
In this paper, we study the smooth transition from solitons to solitary waves in localization, relation between energy and velocity, propagation and scattering property in the Fermi-Pasta-Ulam lattice analytically and numerically. A soliton is a very stable solitary wave that retains its permanent structure after interacting with other solitary waves. A soliton exists when the energy is small, and it becomes a solitary wave when the energy increases to the threshold. The transition could help to understand the distinctly different heat conduction behaviors of the Fermi-Pasta-Ulam lattice at lo…
Nonlinear normal mode solutions of the $β$-FPUT chain with fixed boundaries are presented in terms of the Jacobi sn function. Exact solutions for the two particle chain are found for arbitrary linear and nonlinear coupling strengths. Solutions for the N-body chain are found for purely nonlinear couplings. Three distinct solution types presented: a linear analogue, a chaotic amplitude mapping, and a localized nonlinear mode. The relaxation of perturbed modes are also explored using $l_{1}$-regularized least squares regression to estimate the free energy functional near the nonlinear normal mode…
Fermi, Pasta and Ulam observed, that the excitation of a low frequency normal mode in a nonlinear acoustic chain leads to localization in normal mode space on large time scales. Fast equipartition (and thus complete delocalization) in the Fermi-Pasta-Ulam chain is restored if relevant intensive control parameters exceed certain threshold values. We compare recent results on periodic orbits (in the localization regime) and resonant normal forms (in a weak delocalization regime), and relate them to various resonance overlap criteria. We show that the approaches quantitatively agree in their esti…
The Fermi-Pasta-Ulam (FPU) pioneering numerical experiment played a major role in the history of computer simulation because it introduced this concept for the first time. Moreover, it raised a puzzling question which was answered more than 10 years later. After an introduction to this problem, we briefly review its history and then suggest some simple numerical experiments, with a provided Matlab code, to study various aspects of the ``FPU'' problem.
We investigate numerically the existence and stability of higher-order recurrences (HoRs), including super-recurrences, super-super-recurrences, etc., in the alpha and beta Fermi-Pasta-Ulam-Tsingou (FPUT) lattices for initial conditions in the fundamental normal mode. Our results represent a considerable extension of the pioneering work of Tuck and Menzel on super-recurrences. For fixed lattice sizes, we observe and study apparent singularities in the periods of these HoRs, speculated to be caused by nonlinear resonances. Interestingly, these singularities depend very sensitively on the initia…
We consider the two-dimensional Fermi-Pasta-Ulam lattice with hexagonal honeycomb symmetry, which is a Hamiltonian system describing the evolution of a scalar-valued quantity subject to nearest neighbour interactions. Using multiple-scale analysis we reduce the governing lattice equations to a nonlinear Schrodinger (NLS) equation coupled to a second equation for an accompanying slow mode. Two cases in which the latter equation can be solved and so the system decoupled are considered in more detail: firstly, in the case of a symmetric potential, we derive the form of moving breathers. We find a…
We consider a two-dimensional Fermi-Pasta-Ulam (FPU) lattice with hexagonal symmetry. Using asymptotic methods based on small amplitude ansatz, at third order we obtain a reduction to a cubic nonlinear Schrodinger equation (NLS) for the breather envelope. However, this does not support stable soliton solutions, so we pursue a higher-order analysis yielding a generalised NLS, which includes known stabilising terms. We present numerical results which suggest that long-lived stationary and moving breathers are supported by the lattice. We find breather solutions which move in an arbitrary directi…
We show that even small dissipation can strongly affect the Fermi-Pasta-Ulam-Tsingou recurrence phenomenon. Taking the fully nonlinear stage of modulational instability as an experimentally accessible example, we show that the linear attenuation induces the symmetry of the recurrence to be broken through separatrix crossing occurring at multiple critical values of the attenuation. We provide experimental evidence for this phenomenon in a fiber optics experiment designed in such a way that the effective losses can be carefully tailored by techniques based on Raman amplification.
It is reported that the numerical simulations of the Fermi-Pasta-Ulam problem were performed by a young lady, Mary Tsingou. After 50 years of omission, it is time for a proper recognition of her decisive contribution to the first ever numerical experiment, central in the solitons and chaos theories, but also one of the very first out-of-equilibrium statistical mechanics study. Let us quote from now on the Fermi-Pasta-Ulam-Tsingou problem.
Quasi-continuum descriptions of rarefaction and dispersive shock waves in Fermi-Pasta-Ulam lattices with Hertzian potentials
本文研究了费米-帕斯塔-乌拉姆晶格中赫兹型势能的准连续模型，用于近似离散色散激波和稀疏波，并通过分析其自相似解与数值解的比较，验证了模型的有效性。
Localized patterns and dispersive structures in two-dimensional Fermi-Pasta-Ulam lattices
研究二维费米-帕斯塔-乌拉姆晶格中的局域模式和色散结构，通过降维推导修正KdV方程，利用其精确解建模，探索黎曼问题二维推广及KPII极限，比较数值动力学检验准连续长波渐近极限性能。
The spatiotemporal propagation of a momentum excitation on the finite Fermi-Pasta-Ulam lattices is investigated. The competition between the solitary wave and phonons gives rise to interesting propagation behaviors. For a moderate nonlinearity, the initially excited pulse may propagate coherently along the lattice for a long time in a solitary wave manner accompanied by phonon tails. The lifetime of the long-transient propagation state exhibits a sensitivity to the nonlinear parameter. The solitary wave decays exponentially during the final loss of stability, and the decay rate varying with th…
Discrete breathers are time-periodic, spatially localized solutions of the equations of motion for a system of classical degrees of freedom interacting on a lattice. Such solutions are investigated for a diatomic Fermi-Pasta-Ulam chain, i. e., a chain of alternate heavy and light masses coupled by anharmonic forces. For hard interaction potentials, discrete breathers in this model are known to exist either as ``optic breathers'' with frequencies above the optic band, or as ``acoustic breathers'' with frequencies in the gap between the acoustic and the optic band. In this paper, bifurcations be…
A finite (periodic) FPU chain is chosen as a convenient point for investigating the energy exchange phenomenon in nonlinear oscillatory systems. As we have recently shown, this phenomenon may occur as a consequence of the resonant interaction between high-frequency nonlinear normal modes. This interaction determines both the complete energy exchange between different parts of the chain and the transition to energy localization in an excited group of particles. In the paper, we demonstrate that this mechanism can exist in realistic (asymmetric) models of atomic or molecular oscillatory chains. …
The Fermi-Pasta-Ulam (FPU) problem consists of the nonequipartition 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. Trajecto…
The lifetimes of localized nonlinear modes in both the $β$-Fermi-Pasta-Ulam-Tsingou ($β$-FPUT) chain and a cubic $β$-FPUT lattice are studied as functions of perturbation amplitude, and by extension, the relative strength of the nonlinear interactions compared to the linear part. We first recover the well known result that localized nonlinear excitations (LNEs) produced by a bond squeeze can be reduced to an approximate two-frequency solution and then show that the nonlinear term in the potential can lead to the production of secondary frequencies within the phonon band. This can affect the st…
We investigate the spontaneous growth of noise that accompanies the nonlinear evolution of seeded modulation instability into Fermi-Pasta-Ulam recurrence. Results from the Floquet linear stability analysis of periodic solutions of the three-wave truncation are compared with full numerical solutions of the nonlinear Schrödinger equation. The predicted initial stage of noise growth is in good agreement with simulations, and is expected to provide further insight in the subsequent dynamics of the field evolution after recurrence breakup.
Currently, effective phonons (renormalized or interacting phonons) rather than solitary waves (for short, solitons) are regarded as the energy carriers in nonlinear lattices. In this work, by using the approximate soliton solutions of the corresponding equations of motion and adopting the Boltzmann distribution for these solitons, the average velocities of solitons are obtained and are compared with the sound velocities of energy transfer. Excellent agreements with the numerical results and the predictions of other existing theories are shown in both the symmetric Fermi-Pasta-Ulam-$β$ lattices…
The spatiotemporal propagation behavior of a solitary wave is investigated on a Fermi-Pasta-Ulam ring. We observe the emergence of a cnoidal wave excited by the solitary wave. The cnoidal wave may coexist with the solitary wave for a long time associated with the periodic exchange of energy between these two nonlinear waves. The module of the cnoidal wave, which is considered as an indicator of the nonlinearity, is found to oscillate with the same period of the energy exchange. After the stage of coexistence, the interaction between these two nonlinear waves leads to the destruction of the cno…
We visualize the Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in a classical Heisenberg ferromagnetic (HF) spin chain by exploiting its gauge eq uivalence to the nonlinear Schrödinger equation (NLSE). We discuss two types of spatially periodic breather excitations in the spin chain, that are associated with: (I) Akhmediev breather, and (II) Galilean transformed Akhmediev breather. The recurrence in the former is exact in the sense that the initial and final states are identical. In the later, the spin chain undergoes an additional global rotation during the rec urrence process, which makes the i…
We consider a damped $β$-Fermi-Pasta-Ulam chain, driven at one boundary subjected to stochastic noise. It is shown that, for a fixed driving amplitude and frequency, increasing the noise intensity, the system's energy resonantly responds to the modulating frequency of the forcing signal. Multiple peaks appear in the signal to noise ratio, signalling the phenomenon of stochastic resonance. The presence of multiple peaks is explained by the existence of many stable and metastable states that are found when solving this boundary value problem for a semi-continuum approximation of the model. Stoch…
Periodic Orbits in Fermi-Pasta-Ulam-Tsingou Systems
针对FPUT悖论中的热化机制问题，研究了q呼吸子谱中的共振现象，揭示其源于特定频率重叠，会产生复合周期轨道，且这类共振不存在于可积系统中。
Instabilities are common phenomena frequently observed in nature, sometimes leading to unexpected catastrophes and disasters in seemingly normal conditions. The simplest form of instability in a distributed system is its response to a harmonic modulation. Such instability has special names in various branches of physics and is generally known as modulation instability (MI). The MI is tightly related to Fermi-Pasta-Ulam (FPU) recurrence since breather solutions of the nonlinear Schrödinger equation (NLSE) are known to accurately describe growth and decay of modulationally unstable waves in cons…
Recurrence in a periodically driven and weakly damped Fermi-Pasta-Ulam-Tsingou chain
研究发现弱阻尼周期驱动alpha-FPUT链中存在类似FPUT的复发现象，通过长期时间尺度上的低频模态能量交换，揭示了驱动开放多模系统中新型相干非线性动力学。
We report the observation of the thermalization of the Fermi-Pasta-Ulam-Tsingou recurrence process in optical fibers. We show the transition from a reversible regime to an irreversible one, revealing a spectrally thermalized state. To do so, we actively compensate the fiber loss to make the observation of several recurrences possible. We inject into the fiber a combination of three coherent continuous waves, which we call Fourier modes, and a random noise. We enhance the noise-driven modulation instability process against the coherent one by boosting the input noise power level to speed up the…
By invoking Bogoliubov's spectrum, we show that for the nonlinear Schrodinger equation, the modulation instability (MI) of its n = 1 Fourier mode on a finite background automatically triggers a further cascading instability, forcing all the higher modes to grow exponentially in locked-step with the n = 1 mode. This fundamental insight, the enslavement of all higher modes to the n = 1 mode, explains the formation of a triangular-shaped spectrum which generates the Akhmediev breather, predicts its formation time analytically from the initial modulation amplitude, and shows that the Fermi-Pasta-…
We show the relevance of the dispersive analogue of the shock waves in the FPU dynamics. In particular we give strict numerical evidences that metastable states emerging from low frequency initial excitations, are indeed constituted by dispersive shock waves travelling through the chain. Relevant characteristics of the metastable states, such as their frequency extension and their time scale of formation, are correctly obtained within this framework, using the underlying continuum model, the KdV equation.
We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schrödinger (NLS) equation with an isotropic harmonic oscillator potential. The dynamics in this regime is dominated by resonant interactions between quartets of linear normal modes, accurately captured by the corresponding resonant Hamiltonian system. In the framework of this system, we identify Fermi-Pasta-Ulam-like recurrence phenomena, whereby the normal-mode spectrum passes in close proximity of the initial configuration, and two-mode states with time-independent mode amplitude spectra that tran…
The Fermi-Pasta-Ulam (FPU) paradox was observed fifty years ago. The surprising finding was a localization of energy in the reciprocal q-space of a model with discrete translational invariance, despite the presence of interaction between extended normal modes. Thirty three years later Takeno, Kisoda and Sievers reported on the observation of energy localization in real space for the same class of FPU models, which is as surprising since these excitations, called discrete breathers or intrinsic localized modes, violate the underlying discrete translational symmetry of the model. The past decade…
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…
Recurrence recovery in heterogeneous Fermi--Pasta--Ulam--Tsingou systems
针对异质 FPUT 振子系统中能量递归失效的问题，本文提出利用结构化异质性恢复递归，并发现阵列中心对称性可能是关键来源。
In 1955, Fermi, Pasta, Ulam, and Tsingou reported recurrence over time of energy between modes in a one-dimensional array of nonlinear oscillators. Subsequently, there have been myriad numerical experiments using homogenous FPUT arrays, which consist of chains of ideal, nonlinearly-coupled oscillators. However, inherent variations --- e.g., due to manufacturing tolerance --- introduce heterogeneity into the parameters of any physical system. We demonstrate that such tolerances degrade the observance of recurrence, often leading to complete loss in moderately sized arrays. We numerically simula…
Soliton molecules in Fermi-Pasta-Ulam-Tsingou lattice: Gardner equation approach
本研究通过推导Gardner方程研究FPUT晶格的孤子分子，发现速度共振激发的台式孤子分子及其两类形态，验证其碰撞稳定性并分析子情况，数值验证结构存在，可推广至多领域应用。
We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order $α>1$. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyze the propagation of localized waves when $α$ is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic KdV e…
Dirac Solitons and Topological Edge States in the $β$-Fermi-Pasta-Ulam-Tsingou dimer lattice
研究了 $β$-FPUT 二聚格子中通过弱非线性约化得到的连续非线性 Dirac 系统的孤子与边缘态，解析推导了孤子轮廓和守恒律，解释了多种非线性边缘态的存在，并通过数值模拟验证了其稳定性。
Traveling and Dispersive Shock Waves in a Two-Dimensional Fermi-Pasta-Ulam-Tsingou Lattice
本文通过变分方法证明二维FPUT晶格中周期和孤立行波的存在，并数值研究准一维传播的色散冲击波，发现其形状依赖于传播方向而速度和振幅不依赖，与KdV近似在小跳跃高度下吻合。
In this paper we study the existence and linear stability of bright and dark breathers in one-dimensional FPU lattices. On the one hand, we test the range of validity of a recent breathers existence proof [G. James, {\em C. R. Acad. Sci. Paris}, 332, Ser. 1, pp. 581 (2001)] using numerical computations. Approximate analytical expressions for small amplitude bright and dark breathers are found to fit very well exact numerical solutions even far from the top of the phonon band. On the other hand, we study numerically large amplitude breathers non predicted in the above cited reference. In partic…
We perform a thorough investigation of the first FPUT recurrence in the $β$-FPUT chain for both positive and negative $β$. We show numerically that the rescaled FPUT recurrence time $T_{r}=t_{r}/(N+1)^{3}$ depends, for large $N$, only on the parameter $S\equiv Eβ(N+1)$. Our numerics also reveal that for small $\left|S\right|$, $T_{r}$ is linear in $S$ with positive slope for both positive and negative $β$. For large $\left|S\right|$, $T_{r}$ is proportional to $\left|S\right|^{-1/2}$ for both positive and negative $β$ but with different multiplicative constants. In the continuum limit, the $β$…
We quantize the β-Fermi-Pasta-Ulam (FPU) model with nearest and next-nearest neighbour interactions using a number conserving approximation and a numerical exact diagonalization method. Our numerical mean field bi-phonon spectrum shows excellent agreement with the analytic mean field results of Ivić and Tsironis ((2006) Physica D 216 200), except for the wave vector at the midpoint of the Brillouin zone. We then relax the mean field approximation and calculate the eigenvalue spectrum of the full Hamiltonian. We show the existence of multi-phonon bound states and analyze the properties of these…
Dispersive Fractalization in Linear and Nonlinear Fermi-Pasta-Ulam-Tsingou Lattices
本文解析与数值研究了周期性线性和非线性 FPUT 系统的色散分形化与量子化现象，发现线性模型在特定时间呈现分形与量子化解轮廓，非线性系统在弱非线性下类似但随非线性增强而消失。
The effect of discrete breathers (DBs) on macroscopic properties of the Fermi-Pasta-Ulam chain with symmetric and asymmetric potentials is investigated. The total to kinetic energy ratio (related to specific heat), stress (related to thermal expansion), and Young's modulus are monitored during the development of modulational instability of the zone boundary mode. The instability results in the formation of chaotic DBs followed by the transition to thermal equilibrium when DBs disappear due to energy radiation in the form of small-amplitude phonons. It is found that DBs reduce the specific heat…
On the role of 5-wave resonances in the nonlinear dynamics of the Fermi-Pasta-Ulam-Tsingou lattice
本文通过引入仅含共振相互作用的精确共振演化方程，研究了FPUT晶格在弱非线性下的动力学，发现当3整除N时，5波共振主导其超混沌行为。
We prove global existence and uniqueness of solutions of some important nonlinear lattices which include the Fermi-Pasta-Ulam (FPU) lattice. Our result shows (on a particular example) that the FPU lattice with high nonlinearity and its continuum limit display drastically different behaviour with respect to blow up phenomenon.
For the Fermi-Pasta-Ulam chain, an effective Hamiltonian is constructed, describing the motion of approximate, weakly localized discrete breathers traveling along the chain. The velocity of these moving and localized vibrations can be estimated analytically as the group velocity of the corresponding wave packet. The Peierls-Nabarro barrier is estimated for strongly localized discrete breathers.
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