Exact solutions in the FPU oscillator chain

Abstract

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 system with quartic nonlinearity and periodic boundary conditions, with explicitly computed mode coupling coefficients. The core of the paper is the proof of the existence of one-mode and two-mode exact solutions, physically representing nonlinear standing and travelling waves of small wavelength whose explicit lattice representations are obtained, and which are valid also as $N \rightarrow \infty$. Moreover, and more generally, we show the presence of multi-mode invariant submanifolds. Destabilization of these solutions by a parametric perturbation mechanism leads to the establishment of chaotic in time mode interaction channels, corresponding to the formation in phase space of bounded stochastic layers on submanifolds. The full mode-space stability problem of the $N/2$ zone-boundary mode is solved, showing that this mode becomes unstable through a mechanism of the modulational Benjamin-Feir type. In the thermodynamic limit the mode is always unstable but with instability growth rate linearly vanishing with energy density. The physical significance of these solutions and of their stability properties, with respect to the previously much more studied equipartition problem for long wavelength initial excitations, is briefly discussed.