Fermi Pasta Ulam Lattice Research Articles

Branching patterns of wave trains in mass-in-mass lattices

We investigate the existence and branching patterns of wave trains in the mass-in-mass (MiM) lattice, which is a variant of the Fermi–Pasta–Ulam (FPU) lattice. In contrast to FPU lattice, we have to solve coupled advance-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction and invariant theory. We establish a link between the MiM lattice and the monatomic FPU lattice. That is, the monochromatic and bichromatic wave trains persist near $\mu =0$ in the nonresonance case and in the resonance case $p:q$ where $q$ is not an integer multiple of $p$ . Furthermore, we obtain the multiplicity of bichromatic wave trains in $p:q$ resonance where $q$ is an integer multiple of $p$ , based on the singular theorem.

Solitary Wave Transition from Low to High Energy in the Fermi–Pasta–Ulam Lattice

Solitary Wave Transition from Low to High Energy in the Fermi–Pasta–Ulam Lattice

Solitary waves on nonlocal Fermi–Pasta–Ulam lattices: Exponential localization

The paper is devoted to Fermi–Pasta–Ulam lattices with nonlocal interaction. Under natural assumptions we show that all solitary traveling waves are exponentially localized. We reduce the problem to a linear eigenvalue problem for certain integral operator. Then we use functional analysis tools and exponentially weighted spaces of functions to prove the localization of solitary waves.

Waves in strongly nonlinear discrete systems.

The paper presents the main steps in the development of the strongly nonlinear wave dynamics of discrete systems. The initial motivation was prompted by the challenges in the design of barriers to mitigate high-amplitude compression pulses caused by impact or explosion. But this area poses a fundamental mathematical and physical problem and should be considered as a natural step in developing strongly nonlinear wave dynamics. Strong nonlinearity results in a highly tunable behaviour and allows design of systems with properties ranging from a weakly nonlinear regime, similar to the classical case of the Fermi-Pasta-Ulam lattice, or to a non-classical case of sonic vacuum. Strongly nonlinear systems support periodic waves and one of the fascinating results was a discovery of a strongly nonlinear solitary wave in sonic vacuum (a limiting case of a periodic wave) with properties very different from the Korteweg de Vries solitary wave. Shock-like oscillating and monotonous stationary stress waves can also be supported if the system is dissipative. The paper discusses the main theoretical and experimental results, focusing on travelling waves and possible future developments in the area of strongly nonlinear metamaterials.This article is part of the theme issue 'Nonlinear energy transfer in dynamical and acoustical systems'.

Wave Propagation Analysis in a One – Dimensional Lattice with Long Range Interactions and Its Continuum Model

Fermi-Pasta-Ulam (FPU) lattice known as a nonlinear lattice includes nonlinear terms explicitly. Therefore, we can evaluate nonlinear effects of FPU lattice straightforwardly. It includes, however, only the nearest neighbor interactions. In some cases, long range interaction plays an important role in dynamical properties in real crystals. We take the second nearest neighbor interactions into account in FPU lattice. We derive the nonlinear wave equation (modified Kortweg-de Vries – type equation) from the nonlocal FPU lattice as a continuum limit. Linear dispersion relationship of the wave equation is discussed. We also find as exact solution of soliton in the nonlinear wave equation analytically. We reveal that the velocity of the soliton is related to the second nearest neighbor interactions. We analytically evaluate the temporal evolution of the soliton and estimate the spectrum of the solitary wave using two dimensional FFT in case of different interactions. The result shows that the second nearest neighbor interactions affects the appearance of the lean of peak.

Extensive packet excitations in FPU and Toda lattices

At low energies, the excitation of low-frequency packets of normal modes in the Fermi-Pasta-Ulam (FPU) and in the Toda model leads to exponentially localized energy profiles which resemble staircases and are identified by a slope σ that depends logarithmically on the specific energy . Such solutions are found to lie on stable lower-dimensional tori, named q-tori. At higher energies there is a sharp transition of the system's localization profile to a straight-line one, determined by an N-dependent slope of the form , d > 0. We find that the energy crossover between the two energy regimes decays as , which indicates that q-tori disappear in the thermodynamic limit. Furthermore, we focus on the times that such localization profiles are practically frozen and we find that these “stickiness times” can rapidly and accurately distinguish between a power-law and a stretched exponential dependence in .

Numerical simulation of the nonlinear dynamics of harmonically driven Riesz-fractional extensions of the Fermi–Pasta–Ulam chains

In this work, we introduce a spatially discrete model that is a modification of the well-known α-Fermi–Pasta–Ulam chain with damping. The system is perturbed at one end by a harmonic disturbance irradiating at a frequency in the forbidden band-gap of the classical regime, and a nonlocal coupling between the oscillators is considered using discrete Riesz fractional derivatives. We propose fully discrete expressions to approximate an energy functional of the system, and we use them to calculate the total energy of fractional chains over a relatively long period of time [Fract. Diff. Appl. 4 (2004) 153–162]. The approach is thoroughly tested in the case of local couplings against known qualitative results, including simulations of the process of nonlinear recurrence in the traditional chains of anharmonic oscillators. As an application, we provide evidence that the process of supratransmission is present in spatially discrete Fermi–Pasta–Ulam lattices with Riesz fractional derivatives in space. Moreover, we perform numerical experiments for small and large amplitudes of the harmonic disturbance. In either case, we establish the dependency of the critical amplitude at which supratransmission begins as a function of the driving frequency. Our results are in good agreement with the analytic predictions for the classical Fermi–Pasta–Ulam chain.

Breather solutions for inhomogeneous FPU models using Birkhoff normal forms

We present results on spatially localized oscillations in some inhomogeneous nonlinear lattices of Fermi–Pasta–Ulam (FPU) type derived from phenomenological nonlinear elastic network models proposed to study localized protein vibrations. The main feature of the FPU lattices we consider is that the number of interacting neighbors varies from site to site, and we see numerically that this spatial inhomogeneity leads to spatially localized normal modes in the linearized problem. This property is seen in 1-D models, and in a 3-D model with a geometry obtained from protein data. The spectral analysis of these examples suggests some non-resonance assumptions that we use to show the existence of invariant subspaces of spatially localized solutions in quartic Birkhoff normal forms of the FPU systems. The invariant subspaces have an additional symmetry and this fact allows us to compute periodic orbits of the quartic normal form in a relatively simple way.

Nonlocal continuum analysis of a nonlinear uniaxial elastic lattice system under non-uniform axial load

The static behavior of the Fermi-Pasta-Ulam (FPU) axial chain under distributed loading is examined. The FPU system examined in the paper is a nonlinear elastic lattice with linear and quadratic spring interaction. A dimensionless parameter controls the possible loss of convexity of the associated quadratic and cubic energy. Exact analytical solutions based on Hurwitz zeta functions are developed in presence of linear static loading. It is shown that this nonlinear lattice possesses scale effects and possible localization properties in the absence of energy convexity. A continuous approach is then developed to capture the main phenomena observed regarding the discrete axial problem. The associated continuum is built from a continualization procedure that is mainly based on the asymptotic expansion of the difference operators involved in the lattice problem. This associated continuum is an enriched gradient-based or nonlocal axial medium. A Taylor-based and a rational differential method are both considered in the continualization procedures to approximate the FPU lattice response. The Padé approximant used in the continualization procedure fits the response of the discrete system efficiently, even in the vicinity of the limit load when the non-convex FPU energy is examined. It is concluded that the FPU lattice system behaves as a nonlocal axial system in dynamic but also static loading.

Semiclassical intrinsic localized modes in the β-Fermi–Pasta–Ulam lattice model with first- and second-nearest neighbor interactions

We present an analytical study on intrinsic localized modes (ILMs) in the quantum [Formula: see text]-Fermi–Pasta–Ulam lattice model with first- and second-nearest neighbor interactions by means of the semiclassical approach. We quantize the lattice model Hamiltonian by introducing vibron creation and annihilation operators, and retaining only number conserving terms. The coherent state representation is considered as the basic representation of the quantum lattice system. In order to obtain the ILM solutions, we adopt the multiple scales method combined with a quasidiscreteness approximation. It is found that, when the system parameters satisfy [Formula: see text], at the Brillouin zone (BZ) boundary, a bright ILM occurs above the top of the harmonic wave frequency band. While for [Formula: see text], our results indicate that at wave number [Formula: see text] a bright ILM occurs above the top of the harmonic wave frequency band and at the BZ boundary, the system support a dark intrinsic localized resonant mode.

Quantization of [formula omitted]-Fermi–Pasta–Ulam lattice with nearest and next-nearest neighbor interactions

We quantize the β-Fermi–Pasta–Ulam (FPU) model with nearest and next-nearest neighbor interactions using a number conserving approximation and a numerically exact diagonalization method. Our numerical mean field bi-phonon spectrum shows excellent agreement with the analytic mean field results of Ivić and Tsironis (2006), 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 states by varying the system parameters. From the calculation of the spatial correlation function we then show that these multi-phonon bound states are particle like states with finite spatial correlation. Accordingly we identify these multi-phonon bound states as the quantum equivalent of the breather solutions of the corresponding classical FPU model. The four-phonon spectrum of the system is then obtained and its properties are studied. We then generalize the study to an extended range interaction and consider the quantization of the β-FPU model with next-nearest-neighbor interactions. We analyze the effect of the next-nearest-neighbor interactions on the eigenvalue spectrum and the correlation functions of the system.

Discrete breathers in honeycomb Fermi–Pasta–Ulam lattices

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 Schrödinger 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 an ellipticity criterion for the wavenumbers of the carrier wave, together with asymptotic estimates for the breather energy. The minimum energy threshold depends on the wavenumber of the breather. We find that this threshold is locally maximized by stationary breathers. Secondly, for an asymmetric potential we find stationary breathers, which, even with a quadratic nonlinearity generate no second harmonic component in the breather. Plots of all our findings show clear hexagonal symmetry as we would expect from our lattice structure. Finally, we compare the properties of stationary breathers in the square, triangular and honeycomb lattices.

Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials.

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 analyse 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 Korteweg-de Vries (KdV) equation and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with Hölder-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When [Formula: see text], we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile.

Low-dimensional [formula omitted]-tori in FPU lattices: Dynamics and localization properties

Recent studies on the Fermi–Pasta–Ulam (FPU) paradox, like the theory of q-breathers and the metastability scenario, dealing mostly with the energy localization properties in the FPU space of normal modes (q-space), motivated our first work on q-tori in the FPU problem (Christodoulidi et al., 2010) [19]. The q-tori are low-dimensional invariant tori hosting trajectories that present features relevant to the interpretation of FPU recurrences as well as the energy localization in q-space. The present paper is a continuation of our work in Christodoulidi et al. (2010) [19]. Our new results are: we extend a method of analytical computation of q-tori, using Poincaré–Lindstedt series, from the β to the α-FPU and we reach significantly higher expansion orders using an improved computer-algebraic program. We probe numerically the convergence properties as well as the level of precision of our computed series. We develop an additional algorithm in order to systematically locate values of the incommensurable frequencies used as an input in the PL series construction of q-tori corresponding to progressively higher values of the energy. We generalize a proposition proved in Christodoulidi et al. (2010) [19] regarding the so-called ‘sequence of propagation’ of an initial excitation in the PL series. We show by concrete examples how the latter interprets the localization patterns found in numerical simulations. We focus, in particular, on various types of extensive initial excitations that lead to q-tori solutions with exponentially localized profiles. Finally, we discuss the relation between q-tori, q-breathers (viewed as one-dimensional q-tori), and the so-called ‘FPU-trajectories’ invoked in the original study of the FPU problem.

Breathers in oscillator chains with Hertzian interactions

We prove nonexistence of breathers (spatially localized and time-periodic oscillations) for a class of Fermi–Pasta–Ulam lattices representing an uncompressed chain of beads interacting via Hertz’s contact forces. We then consider the setting in which an additional on-site potential is present, motivated by the Newton’s cradle under the effect of gravity. We show the existence of breathers in such systems, using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schrödinger (DpS) equation. From a spectral analysis, we determine breather stability and explain their translational motion under very weak perturbations. Numerical simulations demonstrate the excitation of traveling breathers from simple initial conditions corresponding to small perturbations at the first site of the chain. This regime is well described by the DpS equation, and is found to occur for physical parameter values in granular chains with stiff local oscillators. In addition, traveling breather propagation can be hindered or even suppressed in other parameter regimes. For soft on-site potentials, a part of the energy remains trapped near the boundary and forms a surface mode. For hard on-site potentials and large to moderate initial excitations, one observes a “boomeron”, i.e. a traveling breather displaying spontaneous direction-reversing motion. In addition, dispersion is significantly enhanced when a precompression is applied to the chain. Depending on parameters, this results either in the absence of traveling breather excitation on long time scales, or in the formation of a “nanopteron” characterized by a sizable wave train lying at both sides of the localized excitation.

Asymptotic Stability of N-Solitary Waves of the FPU Lattices

We study stability of N-solitary wave solutions of the Fermi-Pasta-Ulam (FPU) lattice equation. Solitary wave solutions of the FPU lattice equation cannot be characterized as critical points of conservation laws due to the lack of infinitesimal invariance in the spatial variable. In place of standard variational arguments for Hamiltonian systems, we use an exponential stability property of the linearized FPU equation in a weighted space which is biased in the direction of motion. The dispersion of the linearized FPU equation balances the potential term for low frequencies, whereas the dispersion is superior for high frequencies.We approximate the low frequency part of a solution of the linearized FPU equation by a solution to the linearized Korteweg-de Vries (KdV) equation around an N-soliton solution. We prove an exponential stability property of the linearized KdV equation around N-solitons by using the linearized Backlund transformation and use the result to analyze the linearized FPU equation.

Quantized intrinsically localized modes of the Fermi–Pasta–Ulam lattice

We have examined the quantized n = 2 excitation spectrum of the Fermi–Pasta–Ulam lattice. The spectrum is composed of a resonance in the two-phonon continuum and a branch of infinitely long-lived excitations which splits off from the top of the two-phonon creation continuum. We calculate the zero-temperature limit of the many-body wavefunction and show that this mode corresponds to an intrinsically localized mode (ILM). In one dimension, we find that there is no lower threshold value of the repulsive interaction that must be exceeded if the ILM is to be formed. However, the spatial extent of the ILM wavefunction rapidly increases as the interaction is decreased to zero. The dispersion relation shows that the discrete n = 2 ILM and the resonance hybridize as the center of mass wavevector q is increased towards the zone boundary. The many-body wavefunction shows that as the zone boundary is approached, there is a destructive resonance which occurs between pairs of sites separated by an odd number of lattice spacings. We compare our theoretical results with the recent experimental observation of a discrete ILM in NaI by Manley et al.

N-Soliton States of the Fermi–Pasta–Ulam Lattices

In this paper, we prove existence and uniqueness of solutions to the Fermi–Pasta–Ulam lattice equation that converges to a sum of copropagating N solitary waves as $t\to\infty$ using a linear stability property of multisoliton states in an exponentially weighted space proven by Mizumachi [Asymptotic Stability of N-solitons of the FPU Lattices, preprint, http://arxiv.org/abs/0906.1320]. Counterpropagating two soliton states have been studied by [A. Hoffman and C. E. Wayne, J. Dynam. Differential Equations, 21 (2009), pp. 343–351].

Negative Differential Thermal Resistance in the System Consisting of Dissimilar Anharmonic Lattices

We investigate negative differential thermal resistance (NDTR) in a system consisting of two dissimilar anharmonic lattices exemplified by the Frenkel–Kontorova and Fermi–Pasta–Ulam lattices. Our results show that NDTR depends on the system parameters, such as the coupling constant, interface, and system size. The dependences of NDTR on these system parameters are studied. Possible applications of NDTR in nanoscale heat management and control are also discussed.

2-D modeling of dual-mode acoustic phonon excitation of a triangular nanoplate

In this theoretical work, we investigated coherent phonon excitation of a triangular nanoplate based on 2-D Fermi–Pasta–Ulam lattice model. Based on the two-temperature model commonly used in description of laser heating of metals, we considered two kinds of forces related to electronic and lattice stresses. Based on extensive simulation and analysis, we identified two major planar phonon modes, namely, a standing wave mode related to the triangle bisector and another mode corresponding to half of the side length. This work elucidates the roles of laser-induced electronic stress and lattice stress in controlling the initial phase and the amplitude ratio between these two phonon modes.