  Stability criteria have been derived and investigated in the last decades for
many kinds of periodic traveling wave solutions to Hamiltonian PDEs. They
turned out to depend in a crucial way on the negative signature of the Hessian
matrix of action integrals associated with those waves. In a previous paper
(Nonlinearity 2016), the authors addressed the characterization of stability of
periodic waves for a rather large class of Hamiltonian partial differential
equations that includes quasilinear generalizations of the Korteweg--de Vries
equation and dispersive perturbations of the Euler equations for compressible
fluids, either in Lagrangian or in Eulerian coordinates. They derived a
sufficient condition for orbital stability with respect to co-periodic
perturbations, and a necessary condition for spectral stability, both in terms
of the negative signature - or Morse index - of the Hessian matrix of the
action integral. Here the asymptotic behavior of this matrix is investigated in
two asymptotic regimes, namely for small amplitude waves and for waves
approaching a solitary wave as their wavelength goes to infinity. The special
structure of the matrices involved in the expansions makes possible to actually
compute the negative signature of the action Hessian both in the harmonic limit
and in the soliton limit. As a consequence, it is found that nondegenerate
small amplitude waves are orbitally stable with respect to co-periodic
perturbations in this framework. For waves of long wavelength, the negative
signature of the action Hessian is found to be exactly governed by the second
derivative with respect to the wave speed of the Boussinesq momentum associated
with the limiting solitary wave.
