  In this paper we construct a higher order expansion of the manifold of quasi
unidirectional waves in the Fermi-Pasta-Ulam (FPU) chain. We also approximate
the dynamics on this manifold. As perturbation parameter we use $h^2=1/n^2$,
where $n$ is the number of particles of the chain. It is well known that the
dynamics of quasi unidirectional waves is described to first order by the
Korteweg-de Vries (KdV) equation. Here we show that the dynamics to second
order is governed by a combination of the first two nontrivial equations in the
KdV hierarchy -- for any choice of parameters in the FPU potential. On the
other hand, we find that only if the parameters of the FPU potential satisfy a
condition, then a combination of the first three nontrivial equations in the
KdV hierarchy determines the dynamics of quasi unidirectional waves to third
order. The required condition is satisfied by the Toda chain. Our results
suggest why the close-to-integrable behavior of the FPU chain (the FPU paradox)
persists on a time scale longer than explained by the KdV approximation, and
also how a breakdown of integrability (detachment from the KdV hierarchy) may
responsible for the eventual thermalization of the system.
