ABSTRACT:

Recent results concerning the special role played by the KdV equation in the FPU problem are reported. In particular, it is shown that the Fourier-Galerkin truncation to N modes of a suitable 2N+2-periodic KdV equation coincides with the quasi-resonant normal form equations of a FPU problem with N degrees of freedom. The short time FPU phenomenology is then explained by means of the scaling group of the KdV equation and of simple estimates based on the classical Gagliardo-Nirenberg inequality. In particular, the dependence of the observed FPU dynamics on the phases of the initially excited modes is also explained.