Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system

Abstract

The stability of the one-mode nonlinear solutions of the Fermi-Pasta-Ulam - $β$ system is numerically investigated. No external perturbation is considered for the one-mode exact analytical solutions, the only perturbation being that introduced by computational errors in numerical integration of motion equations. The threshold energy for the excitation of the other normal modes and the dynamics of this excitation are studied as a function of the parameter $μ$ characterizing the nonlinearity, the energy density $ε$ and the number N of particles of the system. The achieved results confirm in part previous results, obtained with a linear analysis of the problem of the stability, and clarify the dynamics by which the one-mode exchanges energy with the other modes with increasing energy density. In a range of energy density near the threshold value and for various values of the number of particles N, the nonlinear one-mode exchanges energy with the other linear modes for a very short time, immediately recovering all its initial energy. This sort of recurrence is very similar to Fermi recurrences, even if in the Fermi recurrences the energy of the initially excited mode changes continuously and only periodically recovers its initial value. A tentative explanation of this intermittent behaviour, in terms of Floquet's theorem, is proposed.