EXACT-SOLUTIONS IN THE FPU OSCILLATOR CHAIN

After a brief comprehensive review of old and new results on the well- known Fermi-Pasta-Ulam (FPU) conservative system of N nonlinearly coup led oscillators, we present a compact linear mode representation of th e Hamiltonian of the FPU system with quartic nonlinearity and periodic boundary conditions, with explicitly computed mode coupling coefficie nts. The core of the paper is the proof of the existence of one- and t wo-mode exact solutions, physically representing nonlinear standing an d travelling waves of small wavelength whose explicit lattice represen tations are obtained, and which are valid also as N --> infinity. More over, and more generally, we show the presence of multi-mode invariant submanifolds. The full mode-space stability problem of the anharmonic zone boundary mode is solved, showing that this mode becomes unstable through a mechanism of the modulational Benjamin-Feir type. In the th ermodynamic limit the mode is always unstable but with instability gro wth rate linearly vanishing with energy density. The physical signific ance of these solutions and of their stability properties, with respec t to the previously much more studied equipartition problem for long w avelength initial excitations, is briefly discussed.