  We consider a wide class of model equations, able to describe wave
propagation in dispersive nonlinear media. The Korteweg-de Vries (KdV) equation
is derived in this general frame under some conditions, the physical meanings
of which are clarified. It is obtained as usual at leading order in some
multiscale expansion. The higher order terms in this expansion are studied
making use of a multi-time formalism and imposing the condition that the main
term satisfies the whole KdV hierarchy. The evolution of the higher order terms
with respect to the higher order time variables can be described through the
introduction of a linearized KdV hierarchy. This allows one to give an
expression of the higher order time derivatives that appear in the right hand
member of the perturbative expansion equations, to show that overall the higher
order terms do not produce any secularity and to prove that the formal
expansion contains only bounded terms.
