The Fermi-Pasta-Ulam problem: 50 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.Comment: 48 pages, no figures, corrected and accepted for publicatio

Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation

We study nonlinear waves in Newton's cradle, a classical mechanical system consisting of a chain of beads attached to linear pendula and interacting nonlinearly via Hertz's contact forces. We formally derive a spatially discrete modulation equation, for small amplitude nonlinear waves consisting of slow modulations of time-periodic linear oscillations. The fully-nonlinear and unilateral interactions between beads yield a nonstandard modulation equation that we call the discrete p-Schroedinger (DpS) equation. It consists of a spatial discretization of a generalized Schroedinger equation with p-Laplacian, with fractional p>2 depending on the exponent of Hertz's contact force. We show that the DpS equation admits explicit periodic travelling wave solutions, and numerically find a plethora of standing wave solutions given by the orbits of a discrete map, in particular spatially localized breather solutions. Using a modified Lyapunov-Schmidt technique, we prove the existence of exact periodic travelling waves in the chain of beads, close to the small amplitude modulated waves given by the DpS equation. Using numerical simulations, we show that the DpS equation captures several other important features of the dynamics in the weakly nonlinear regime, namely modulational instabilities, the existence of static and travelling breathers, and repulsive or attractive interactions of these localized structures

Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations

We prove the most general theorem about spectral stability of multi-site breathers in the discrete Klein-Gordon equation with a small coupling constant. In the anti-continuum limit, multi-site breathers represent excited oscillations at different sites of the lattice separated by a number of "holes" (sites at rest). The theorem describes how the stability or instability of a multi-site breather depends on the phase difference and distance between the excited oscillators. Previously, only multi-site breathers with adjacent excited sites were considered within the first-order perturbation theory. We show that the stability of multi-site breathers with one-site holes change for large-amplitude oscillations in soft nonlinear potentials. We also discover and study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure

Traveling wave solutions for the FPU chain: a constructive approach

Traveling waves for the FPU chain are constructed by solving the associated equation for the spatial profile $u$ of the wave. We consider solutions whose derivatives $u'$ need not be small, may change sign several times, but decrease at least exponentially. Our method of proof is computer-assisted. Unlike other methods, it does not require that the FPU potential has an attractive (positive) quadratic term. But we currently need to restrict the size of that term. In particular, our solutions in the attractive case are all supersonic

Pattern formation and localization in the forced-damped FPU lattice

We study spatial pattern formation and energy localization in the dynamics of an anharmonic chain with quadratic and quartic intersite potential subject to an optical, sinusoidally oscillating field and a weak damping. The zone-boundary mode is stable and locked to the driving field below a critical forcing that we determine analytically using an approximate model which describes mode interactions. Above such a forcing, a standing modulated wave forms for driving frequencies below the band-edge, while a ``multibreather'' state develops at higher frequencies. Of the former, we give an explicit approximate analytical expression which compares well with numerical data. At higher forcing space-time chaotic patterns are observed.Comment: submitted to Phys.Rev.

Discrete Breathers in One- and Two-Dimensional Lattices

Discrete breathers are time-periodic and spatially localised exact solutions in translationally invariant nonlinear lattices. They are generic solutions, since only moderate conditions are required for their existence. Closed analytic forms for breather solutions are generally not known. We use asymptotic methods to determine both the properties and the approximate form of discrete breather solutions in various lattices. We find the conditions for which the one-dimensional FPU chain admits breather solutions, generalising a known result for stationary breathers to include moving breathers. These conditions are verified by numerical simulations. We show that the FPU chain with quartic interaction potential supports long-lived waveforms which are combinations of a breather and a kink. The amplitude of classical monotone kinks is shown to have a nonzero minimum, whereas the amplitude of breathing-kinks can be arbitrarily small. We consider a two-dimensional FPU lattice with square rotational symmetry. An analysis to third-order in the wave amplitude is inadequate, since this leads to a partial differential equation which does not admit stable soliton solutions for the breather envelope. We overcome this by extending the analysis to higher-order, obtaining a modified partial differential equation which includes known stabilising terms. From this, we determine regions of parameter space where breather solutions are expected. Our analytic results are supported by extensive numerical simulations, which suggest that the two-dimensional square FPU lattice supports long-lived stationary and moving breather modes. We find no restriction upon the direction in which breathers can travel through the lattice. Asymptotic estimates for the breather energy confirm that there is a minimum threshold energy which must be exceeded for breathers to exist in the two-dimensional lattice. We find similar results for a two-dimensional FPU lattice with hexagonal rotational symmetry

Wave Turbulence and thermalization in one-dimensional chains

One-dimensional chains are used as a fundamental model of condensed matter, and have constituted the starting point for key developments in nonlinear physics and complex systems. The pioneering work in this field was proposed by Fermi, Pasta, Ulam and Tsingou in the 50s in Los Alamos. An intense and fruitful mathematical and physical research followed during these last 70 years. Recently, a fresh look at the mechanisms of thermalization in such systems has been provided through the lens of the Wave Turbulence approach. In this review, we give a critical summary of the results obtained in this framework. We also present a series of open problems and challenges that future work needs to address.Comment: arXiv admin note: text overlap with arXiv:1811.05697 by other author

A uniqueness result for a simple superlinear eigenvalue problem

We study the eigenvalue problem for a superlinear convolution operator in the special case of bilinear constitutive laws and establish the existence and uniqueness of a one-parameter family of nonlinear eigenfunctions under a topological shape constraint. Our proof uses a nonlinear change of scalar parameters and applies Krein-Rutmann arguments to a linear substitute problem. We also present numerical simulations and discuss the asymptotics of two limiting cases.Comment: revised version with enhanced introduction; 21 pages, several figure