Related papers: Resonant normal form for even periodic FPU chains
In this paper we prove that near the equilibirum position any periodic FPU chain with an odd number of particles admits a Birkhoff normal form up to order 4, and we obtain an explicit formula of the Hessian of its Hamiltonian at the fixed…
This paper considers the famous Fermi-Pasta-Ulam chain with periodic boundary conditions and quartic nonlinearities. Due to special resonances and discrete symmetries, the Birkhoff normal form of this Hamiltonian system is completely…
We consider four masses in a circular configuration with nearest-neighbour interaction, generalizing the spatially periodic Fermi--Pasta--Ulam-chain where all masses are equal. We identify the mass ratios that produce the…
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…
The Fermi-Pasta-Ulam (FPU) chains of particles in \textit{thermal equilibrium} are studied from both wave-interaction and particle-interaction points of view. It is shown that, even in a strongly nonlinear regime, the chain in thermal…
The Fermi-Pasta-Ulam (FPU) system, initially introduced by Fermi for numerical simulations, models vibrating chains with fixed endpoints, where particles interact weakly, nonlinearly with their nearest neighbors. Contrary to the anticipated…
The dispersive interacting waves in Fermi-Pasta-Ulam (FPU) chains of particles in \textit{thermal equilibrium} are studied from both statistical and wave resonance perspectives. It is shown that, even in a strongly nonlinear regime, the…
We show that any analytically integrable Hamiltonian system near an equilibrium point admits a convergent Birkhoff normalization. The proof is based on a new, geometric approach to the problem.
The Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and $n$ particles admits a large group of discrete symmetries. The fixed point sets of these symmetries naturally form invariant symplectic manifolds that are investigated…
The purposes of this note are: 1) to propose a direct and "elementary" proof of the main result proved by Guillemin-Paul-Uribe [GPU], namely that the semi-classical spectrum near a global minimum of the classical Hamiltonian determines the…
The Fermi-Pasta-Ulam $\alpha$-model of harmonic oscillators with cubic anharmonic interactions is studied from a statistical mechanical point of view. Systems of N= 32 to 128 oscillators appear to be large enough to suggest statistical…
All possible symmetry-determined nonlinear normal modes (also called by simple periodic orbits, one-mode solutions etc.) in both hard and soft Fermi-Pasta-Ulam-$\beta$ chains are discussed. A general method for studying their stability in…
Upon initial excitation of a few normal modes the energy distribution among all modes of a nonlinear atomic chain (the Fermi-Pasta-Ulam model) exhibits exponential localization on large time scales. At the same time resonant anomalies…
We study the dynamics of the $(\alpha+\beta)$ Fermi-Pasta-Ulam-Tsingou lattice (FPUT lattice, for short) for an arbitrary number $N$ of interacting particles, in regimes of small enough nonlinearity so that a Birkhoff-Gustavson type of…
The Bertrand-Darboux integrability condition for a certain class of perturbed harmonic oscillators is studied from the viewpoint of the Birkhoff-Gustavson(BG)-normalization: By solving an inverse problem of the BG-normalization on computer…
After a brief comprehensive review of old and new results on the well known Fermi-Pasta-Ulam (FPU) conservative system of $N$ nonlinearly coupled oscillators, we present a compact linear mode representation of the Hamiltonian of the FPU…
We consider the semilinear harmonic oscillator $$i\psi_t=(-\Delta +\va{x}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d, t\in \R$$ where $M$ is a Hermite multiplier and $g$ a smooth function globally of order 3 at least. We…
The inhomogeneous Fermi-Pasta-Ulam chain is studied by identifying the mass ratios that produce prominent resonances. This is a technically complicated problem as we have to solve an inverse problem for the spectrum of the corresponding…
In this paper we construct a higher order expansion of the manifold of quasi unidirectional waves in the Fermi-Pasta-Ulam (FPU) chain. We also approximate the dynamics on this manifold. As perturbation parameter we use $h^2=1/n^2$, where…
In systems of N coupled anharmonic oscillators, exact resonant interactions play an important role in the energy exchange between normal modes. In the weakly nonlinear regime, those interactions may facilitate energy equipartition in…