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We study the original $alpha$-Fermi-Pasta-Ulam (FPU) system with $N=16,32$ and $64$ masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave-wave interaction theory, i.e. we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the $alpha$-FPU equation of motion, we find that the first non trivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that for small amplitude random waves the time scale of such interactions is extremely large and it is of the order of $1/epsilon^8$, where $epsilon$ is the small parameter in the system. The wave-wave interaction theory is not based on any threshold: equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the {it Umklapp} (flip over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.
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 Fourier space. W
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 chain in therm
We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schr{o}dinger (NLS) equation with an isotropic harmonic oscillator potential. The dynamics in this regime is dominated by resonant interactions between qua
We consider a $pi$-mode solution of the Fermi-Pasta-Ulam $beta$ system. By perturbing it, we study the system as a function of the energy density from a regime where the solution is stable to a regime, where is unstable, first weakly and then strongl
The effect of discrete breathers (DBs) on macroscopic properties of the Fermi-Pasta-Ulam chain with symmetric and asymmetric potentials is investigated. The total to kinetic energy ratio (related to specific heat), stress (related to thermal expansio