  Motivated by the ongoing study of dispersive shock waves in non integrable
systems, we propose and analyze a set of wave parameters for periodic waves of
a large class of Hamiltonian partial differential systems -- including the
generalized Korteweg de Vries equations and the Euler-Korteweg systems -- that
are well-behaved in both the small amplitude and small wavelength limits. We
use this parametrization to determine fine asymptotic properties of the
associated modulation systems, including detailed descriptions of eigenmodes.
As a consequence, in the solitary wave limit we prove that modulational
instability is decided by the sign of the second derivative -- with respect to
speed, fixing the endstate -- of the Boussinesq moment of instability; and, in
the harmonic limit, we identify an explicit modulational instability index, of
Benjamin--Feir type.
