The foundations of weak turbulence theory is explored through its application to the (alpha) Fermi-Pasta-Ulam (FPU) model, a simple weakly nonlinear
dispersive system. A direct application of the standard kinetic equations would
miss interesting dynamics of the energy transfer process starting from a large-scale
excitation. This failure is traced to an enforcement of the exact resonance condition, whereas mathematically the resonance should be broadened due to the energy
transfer happening on large but finite time scales. By allowing for the broadened
resonance, a modified three-wave kinetic equation is derived for the FPU model.
This kinetic equation produces some correct scaling predictions about the statistical
dynamics of the FPU model, but does not model accurately the detailed evolution of
the energy spectrum. The reason for the failure seems not to be one of the previously
clarified reasons for breakdown in the weak turbulence theory.
Mathematics Subject Classification: Primary: 41A60,82C05,82C28, Secondary: 34E13, 37K60.