The anti-FPU problem

Delocalized nonlinear vibrational modes and discrete breathers in β-FPUT simple cubic lattice

The problem of finding various discrete breathers (DBs) in the β-Fermi-Pasta-Ulam-Tsingou simple cubic lattice is addressed. DBs are obtained by imposing localizing functions on delocalized nonlinear vibrational modes (DNVMs) having frequencies above the phonon spectrum of the lattice. Among 27 DNVMs with the wave vector at the boundary of the first Brillouin zone there are three satisfying this condition. Seven robust DBs of different symmetries are found using this approach.

The β Fermi-Pasta-Ulam-Tsingou recurrence problem

We perform a thorough investigation of the first Fermi-Pasta-Ulam-Tsingou (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≡Eβ(N+1). Our numerics also reveal that for small |S|, T_r is linear in S with positive slope for both positive and negative β. For large |S|, T_r is proportional to |S|^-1/2 for both positive and negative β but with different multiplicative constants. We numerically study the continuum limit and find that the recurrence time closely follows the |S|^-1/2 scaling and can be interpreted in terms of solitons, as in the case of the KdV equation for the α chain. The difference in the multiplicative factors between positive and negative β arises from soliton-kink interactions that exist only in the negative β case. We complement our numerical results with analytical considerations in the nearly linear regime (small |S|) and in the highly nonlinear regime (large |S|). For the former, we extend previous results using a shifted-frequency perturbation theory and find a closed form for T_r that depends only on S. In the latter regime, we show that T_r∝|S|^-1/2 is predicted by the soliton theory in the continuum limit. We then investigate the existence of the FPUT recurrences and show that their disappearance surprisingly depends only on Eβ for large N, not S. Finally, we end by discussing the striking differences in the amount of energy mixing between positive and negative β and offer some remarks on the thermodynamic limit.

Interplay between modulational instability and self-trapping of wavepackets in nonlinear discrete lattices

We investigate the modulational instability of uniform wavepackets governed by the discrete nonlinear Schrodinger equation in finite linear chains and square lattices. We show that, while the critical nonlinear coupling χMI above which modulational instability occurs remains finite in square lattices, it decays as 1/L in linear chains. In square lattices, there is a direct transition between the regime of stable uniform wavefunctions and the regime of asymptotically localized solutions with stationary probability distributions. On the other hand, there is an intermediate regime in linear chains for which the wavefunction dynamics develops complex breathing patterns. We analytically compute the critical nonlinear strengths for modulational instability in both lattices, as well as the characteristic time τ governing the exponential increase of perturbations in the vicinity of the transition. We unveil that the interplay between modulational instability and self-trapping phenomena is responsible for the distinct wavefunction dynamics in linear and square lattices.

Stability of nonlinear normal modes in the Fermi-Pasta-Ulam β chain in the thermodynamic limit

All possible symmetry-determined nonlinear normal modes (also called simple periodic orbits, one-mode solutions, etc.) in both hard and soft Fermi-Pasta-Ulam β chains are discussed. A general method for studying their stability in the thermodynamic limit as well as its application for each of the above nonlinear normal modes are presented.

Limiting phase trajectories and the origin of energy localization in nonlinear oscillatory chains

We demonstrate that the modulation instability of the zone-boundary mode in a finite (periodic) Fermi-Pasta-Ulam chain is the necessary but not sufficient condition for the efficient energy transfer by localized excitations. This transfer results from the exclusion of complete energy exchange between spatially different parts of the chain, and the excitation level corresponding to that turns out to be twice more than threshold of zone-boundary mode's instability. To obtain this result one needs in far going extension of the beating concept to a wide class of finite oscillatory chains. In turn, such an extension leads to description of energy exchange and transition to energy localization and transfer in terms of effective particles and limiting phase trajectories. The effective particles appear naturally when the frequency spectrum crowding ensures the resonance interaction between zone-boundary and two nearby nonlinear normal modes, but there are no additional resonances. We show that the limiting phase trajectories corresponding to the most intensive energy exchange between effective particles can be considered as an alternative to nonlinear normal modes, which describe the stationary process.

Discrete breathers in one-dimensional diatomic granular crystals

We report the experimental observation of modulational instability and discrete breathers in a one-dimensional diatomic granular crystal composed of compressed elastic beads that interact via Hertzian contact. We first characterize their effective linear spectrum both theoretically and experimentally. We then illustrate theoretically and numerically the modulational instability of the lower edge of the optical band. This leads to the dynamical formation of long-lived breather structures, whose families of solutions we compute throughout the linear spectral gap. Finally, we experimentally observe the manifestation of the modulational instability and the resulting generation of localized breathing modes with quantitative characteristics that agree with our numerical results.

Long-range energy transfer in proteins

Proteins are large and complex molecular machines. In order to perform their function, most of them need energy, e.g. either in the form of a photon, as in the case of the visual pigment rhodopsin, or through the breaking of a chemical bond, as in the presence of adenosine triphosphate (ATP). Such energy, in turn, has to be transmitted to specific locations, often several tens of A away from where it is initially released. Here we show, within the framework of a coarse-grained nonlinear network model, that energy in a protein can jump from site to site with high yields, covering in many instances remarkably large distances. Following single-site excitations, few specific sites are targeted, systematically within the stiffest regions. Such energy transfers mark the spontaneous formation of a localized mode of nonlinear origin at the destination site, which acts as an efficient energy-accumulating center. Interestingly, yields are found to be optimum for excitation energies in the range of biologically relevant ones.

Modulational instability of zone boundary mode and band edge modes in nonlinear diatomic lattices

We analyze the modulational instability of the zone boundary mode (ZBM) and the band edge modes (BEMs) in a one-dimensional nonlinear diatomic lattice and obtain rigorous results. Some numerical calculations of modulational instability in these modes are presented. These results indicate that the modulational instability of the BEMs leads to excitation of the discrete breathers (DBs) in the band gap, while that of the ZBM leads to excitation of the DBs above the phonon band.

Stochastic resonance in the Fermi-Pasta-Ulam chain

We consider a damped beta-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, signaling 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 semicontinuum approximation of the model. Stochastic resonance is shown to be generated by transitions between these states.

Stability properties of the N/4 (pi/2-mode) one-mode nonlinear solution of the Fermi-Pasta-Ulam-beta system

We present a detailed numerical and analytical study of the stability properties of the N/4 (pi/2-mode) one-mode nonlinear solution of the Fermi-Pasta-Ulam-beta system. The numerical analysis is made as a function of the number N of the particles of the system and of the product lambda=epsilonbeta , where epsilon is the energy density and beta is the parameter characterizing the nonlinearity. It is shown that, both for beta>0 and beta<0 , the instability threshold value |lambda(t)(N)| converges, with increasing N , to the same value 2pi(2)(3N(2)) , that for beta>0 |lambda(t)N(2)| is a decreasing function of N as in the pi-mode, whereas, for beta<0 , it is an increasing one. The asymptotic behavior of |lambda(t)| for large values of N is analytically obtained in both cases with a Floquet analysis of the stability.

Quasiperiodic and chaotic discrete breathers in a parametrically driven system without linear dispersion

We study a one-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers (DB's) can be explicitly constructed by an exact separation of their time and space dependence. Introducing parametric periodic driving, we first show how a variety of such DB's can be obtained by selecting spatial profiles from the homoclinic orbits of an invertible map and combining them with initial conditions chosen from the Poincaré surface of section of a simple Duffing's equation. Placing then our initial conditions at the center of the islands of a major resonance, we demonstrate how the corresponding DB can be stabilized by varying the amplitude of the driving. We thus discover around elliptic points a large region of quasiperiodic breathers, which are stable for very long times. Starting with initial conditions close to the elliptic point at the origin, we find that as we approach the main chaotic layer, a quasiperiodic breather either destabilizes by delocalization or turns into a chaotic breather, with an evidently broadbanded Fourier spectrum before it collapses. For some breather profiles stable quasiperiodic breathers exist all the way to the separatrix of the Duffing equation, indicating the presence of large regions of tori around the DB solution in the multidimensional phase space. We argue that these strong localization phenomena are due to the absence of phonon resonances, as there are no linear dispersion terms in our lattices. We also show, however, that these phenomena persist in more realistic physical models, in which weak linear dispersion is included in the equations of motion, with a sufficiently small coefficient.

Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries

We present a theorem that allows one to simplify the linear stability analysis of periodic and quasiperiodic nonlinear regimes in N-particle mechanical systems with different kinds of discrete symmetry. This theorem suggests a decomposition of the linearized system arising in the standard stability analysis into a number of subsystems whose dimensions can be considerably less than the dimension of the full system. As an example of such a simplification, we discuss the stability of bushes of modes (invariant manifolds) for the Fermi-Pasta-Ulam chains and prove another theorem about the maximal dimension of the above-mentioned subsystems.

Measures of chaos and equipartition in integrable and nonintegrable lattices

We have simulated numerically the behavior of the one-dimensional, periodic FPU-alpha and Toda lattices to optical and acoustic initial excitations of small--but finite and large amplitudes. For the small-through-intermediate amplitudes (small initial energy per particle) we find nearly recurrent solutions, where the acoustic result is due to the appearance of solitons and where the optical result is due to the appearance of localized breather-like packets. For large amplitudes, we find complex-but-regular behavior for the Toda lattice and "stochastic" or chaotic behaviors for the alpha lattice. We have used the well-known diagnostics: Localization parameter; Lyapounov exponent, and slope of a linear fit to linear normal mode energy spectra. Space-time diagrams of local particle energy and a wave-related quantity, a discretized Riemann invariant are also shown. The discretized Riemann invariants of the alpha lattice reveal soliton and near-soliton properties for acoustic excitations. Except for the localization parameter, there is a clear separation in behaviors at long-time between integrable and nonintegrable systems.

Fermi-Pasta-Ulam, solitons and the fabric of nonlinear and computational science: history, synergetics, and visiometrics

This paper is mostly a history of the early years of nonlinear and computational physics and mathematics. I trace how the counterintuitive result of near-recurrence to an initial condition in the first scientific digital computer simulation led to the discovery of the soliton in a later computer simulation. The 1955 report by Fermi, Pasta, and Ulam (FPU) described their simulation of a one-dimensional nonlinear lattice which did not show energy equipartition. The 1965 paper by Zabusky and Kruskalshowed that the Korteweg-de Vries (KdV) nonlinear partial differential equation, a long wavelength model of the alpha-lattice (or cubic nonlinearity), derived by Kruskal, gave quantitatively the same results obtained by FPU. In 1967, Zabusky and Deem showed that a localized short wavelength initial excitation (then called an "optical" and now a "zone-boundary mode" excitation ) of the alpha-lattice revealed "n-curve" coherent states. If the initial amplitude was sufficiently large energy equipartition followed in a short time. The work of Kruskal and Miura (KM), Gardner and Greene (GG), and myself led to the appreciation of the infinity of denumerable invariants (conservation laws) for Hamiltonian systems and to a procedure by GGKM in 1967 for solving KdV exactly. The nonlinear science field exponentiated in diversity of linkages (as described in Appendix A). Included were pure and applied mathematics and all branches of basic and applied physics, including the first nonhydrodynamic application to optical solitons, as described in a brief essay (Appendix B) by Hasegawa. The growth was also manifest in the number of meetings held and institutes founded, as described briefly in Appendix D. Physicists and mathematicians in Japan, USA, and USSR (in the latter two, people associated with plasma physics) contributed to the diversification of the nonlinear paradigm which continues worldwide to the present. The last part of the paper (and Appendix C) discuss visiometrics: the visualization and quantification of simulation data, e.g., projection to lower dimensions, to facilitate understanding of nonlinear phenomena for modeling and prediction (or design). Finally, I present some recent developments that are linked to my early work by: Dritschel (vortex dynamics via contour dynamics/surgery in two and three dimensions); Friedland (pattern formation by synchronization in Hamiltonian nonlinear wave, vortex, plasma, systems, etc.); and the author ("n-curve" states and energy equipartition in a FPU lattice).

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.