  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.
