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. Using both direct numerical computations and a simplified asymptotic model of the oscillator chain, the so-called discrete p-Schr\"odinger (DpS) equation, we show the existence of discrete breathers and study their spectral properties and mobility. Due to the fully nonlinear character of Hertzian interactions, breathers are found to be much more localized than in classical nonlinear lattices and their motion occurs with less dispersion. In addition, we study numerically the excitation of a traveling breather after an impact at one end of a semi-infinite chain. This case is well described by the DpS equation when local oscillations are faster than binary collisions, a situation occuring e.g. in chains of stiff cantilevers decorated by spherical beads. When a hard anharmonic part is added to the local potential, a new type of traveling breather emerges, showing spontaneous direction-reversing in a spatially homogeneous system. Finally, the interaction of a moving breather with a point defect is also considered in the cradle system. Almost total breather reflections are observed at sufficiently high defect sizes, suggesting potential applications of such systems as shock wave reflectors

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

Geometry and dynamics in Hamiltonian lattices

E. Fermi, J. Pasta and S. Ulam introduced the Fermi-Pasta-Ulam lattice in the 1950s as a classical mechanical model for a mono-atomic crystal or a one-dimensional continuum. The model consisted of a discrete number of equal point masses that interact with their nearest neighbours only. On the basis of statistical mechanics, they expected that when the interparticle forces were anharmonic, the lattice would reach a thermal equilibrium. This means that averaged over time, its initial energy would be equipartitioned among all its Fourier modes. The famous computer experiment that they performed in Los Alamos in 1955, was intended to investigate in what manner and at what time-scale the equipartitioning would take place. The result was astonishing: the lattice did not come close to thermal equilibrium, but behaved more or less quasi-periodically. Only when the initial energy was larger than a certain threshold, did the lattice seem to `thermalise'. This paradox is nowadays known as the `Fermi-Pasta-Ulam problem'. The observations of Fermi, Pasta and Ulam were a great impulse for nonlinear dynamics. One possible explanation for the quasi-periodic behaviour of the FPU system, is based on the Kolmogorov-Arnol'd-Moser theorem, which states that most of the invariant Lagrangean tori of a Liouville integrable Hamiltonian system survive under small perturbations. It is required though for the KAM theorem that the integrable system satisfies a nondegeneracy condition. Unfortunately, it has for a long time been unclear how the Fermi-Pasta-Ulam lattice can be viewed as a perturbation of a nondegenerate integrable system. Nishida in 1971 and Sanders in 1977, investigated a Birkhoff normal form for the FPU lattice. Under the assumption of a rather strong nonresonance condition on the linear frequencies $\omega_k = 2 \sin (k\pi/n)$ of the lattice, they showed that this Birkhoff normal form is integrable and nondegenerate. This means that the KAM theorem can indeed be applied. The problem is of course that their required nonresonance condition is usually not met. This leaves a large gap in the proofs. The new idea in this thesis is to incorporate the discrete symmetries of the lattice in the argument. It turns out that this enables us to show that the nonresonance condition of Nishida and Sanders is not needed: every resonance is overruled by a symmetry. Hence the Birkhoff normal form is integrable and this proves the applicability of the KAM theorem. Moreover, much attention is paid to the analysis of exact and approximate solutions of the lattice equations. New exact solutions and invariant manifolds are found in the fixed point sets of the symmetries of the lattice. Also, the Birkhoff normal reveals approximate integrals and solutions in the low energy domain of the phase space. The analysis makes use of invariant theory and singular reduction. One of the conclusions is that the lattice with an even number of particles contains travelling wave solutions that change their direction. Moreover the integrable normal form contains nontrivial monodromy, meaning that it does not allow global action-angle variable

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 Cantor set of tori with monodromy near a focus-focus singularity

We write down an asymptotic expression for action coordinates in an integrable Hamiltonian system with a focus-focus equilibrium. From the singularity in the actions we deduce that the Arnol'd determinant grows infinitely large near the pinched torus. Moreover, we prove that it is possible to globally parametrise the Liouville tori by their frequencies. If one perturbs this integrable system, then the KAM tori form a Whitney smooth family: they can be smoothly interpolated by a torus bundle that is diffeomorphic to the bundle of Liouville tori of the unperturbed integrable system. As is well-known, this bundle of Liouville tori is not trivial. Our result implies that the KAM tori have monodromy. In semi-classical quantum mechanics, quantisation rules select sequences of KAM tori that correspond to quantum levels. Hence a global labeling of quantum levels by two quantum numbers is not possible.Comment: 11 pages, 2 figure

Dedication to Professor Michael Tribelsky

Professor Tribelsky's accomplishments are highly appreciated by the international community. The best indications of this are the high citation rates of his publications, and the numerous awards and titles he has received. He has made numerous fundamental contributions to an extremely broad area of physics and mathematics, including (but not limited to) quantum solid-state physics, various problems in light–matter interaction, liquid crystals, physical hydrodynamics, nonlinear waves, pattern formation in nonequilibrium systems and transition to chaos, bifurcation and probability theory, and even predictions of the dynamics of actual market prices. This book presents several extensions of his results, based on his inspiring publications

Near-integrability of periodic Klein-Gordon lattices

In this paper we study the Klein-Gordon (KG) lattice with periodic boundary conditions. It is an $N$ degrees of freedom Hamiltonian system with linear inter-site forces and nonlinear on-site potential, which here is taken to be of the $\phi^4$ form. First, we prove that the system in consideration is non-integrable in Liuville sense. The proof is based on the Morales-Ramis theory. Next, we deal with the resonant Birkhoff normal form of the KG Hamiltonian, truncated to order four. Due to the choice of potential, the periodic KG lattice shares the same set of discrete symmetries as the periodic Fermi-Pasta-Ulam (FPU) chain. Then we show that the above normal form is integrable. To do this we utilize the results of B. Rink on FPU chains. If $N$ is odd this integrable normal form turns out to be KAM nondegenerate Hamiltonian. This implies the existence of many invariant tori at low-energy level in the dynamics of the periodic KG lattice, on which the motion is quasi-periodic. We also prove that the KG lattice with Dirichlet boundary conditions (that is, with fixed endpoints) admits an integrable, KAM nondegenerated normal forth order form, which in turn shows that almost all low-energetic solutions of KG lattice with fixed endpoints are quasi-periodic.Comment: arXiv admin note: text overlap with arXiv:1710.0413

Construction and numerical simulation of a two-dimensional analogue to the KdV equation

Arising from an investigation in Hydrodynamics, the Korteweg-de Vries equation demonstrates existence of nonlinear waves that resume their profile after interaction. In this thesis, the classical equations governing wave motion are the starting point for the development of an analogue of the KdV that describes the evolution of a wave surface. The resulting partial differential equation is non-linear and third order in two spatial variables. The linear and and non-linear parts of this equation are analyzed separately. A variant of the method of stationary phase is used to study the linear third order terms, and it is found that the non-linear part equates to the non-viscous Burger's equation. Numerical methods are also used to investigate behavior of wave shapes. We find initial conditions that behave in a manner similar to those of the KdV in that the waves are nonlinear but retain their shape after interaction. These include all solutions of the KdV, but also some "lump" initial conditions

Stability of traveling waves in a driven Frenkel–Kontorova model

In this work we revisit a classical problem of traveling waves in a damped Frenkel–Kontorova lattice driven by a constant external force. We compute these solutions as fixed points of a nonlinear map and obtain the corresponding kinetic relation between the driving force and the velocity of the wave for different values of the damping coefficient. We show that the kinetic curve can become non-monotone at small velocities, due to resonances with linear modes, and also at large velocities where the kinetic relation becomes multivalued. Exploring the spectral stability of the obtained waveforms, we identify, at the level of numerical accuracy of our computations, a precise criterion for instability of the traveling wave solutions: monotonically decreasing portions of the kinetic curve always bear an unstable eigendirection. We discuss why the validity of this criterion in the dissipative setting is a rather remarkable feature offering connections to the Hamiltonian variant of the model and of lattice traveling waves more generally. Our stability results are corroborated by direct numerical simulations which also reveal the possible outcomes of dynamical instabilities.AEI/FEDER, (UE) MAT2016-79866-