  The stability of periodic traveling wave solutions to dispersive PDEs with
respect to `arbitrary' perturbations is still widely open. The focus is put
here on stability with respect to perturbations of the same period as the wave,
for KdV-like systems of one-dimensional Hamiltonian PDEs. Stability criteria
are derived and investigated first in a general abstract framework, and then
applied to three basic examples that are very closely related, and ubiquitous
in mathematical physics, namely, a quasilinear version of the generalized
Korteweg--de Vries equation (qKdV), and the Euler--Korteweg system in both
Eulerian coordinates (EKE) and in mass Lagrangian coordinates (EKL). Those
criteria consist of a necessary condition for spectral stability, and of a
sufficient condition for orbital stability. Both are expressed in terms of a
single function, the abbreviated action integral along the orbits of waves in
the phase plane, which is the counterpart of the solitary waves moment of
instability introduced by Boussinesq. Regarding solitary waves, the celebrated
Grillakis--Shatah--Strauss stability criteria amount to looking for the sign of
the second derivative of the moment of instability with respect to the wave
speed. For periodic waves, the most striking results obtained here can be
summarized as: an odd value for the difference between N -- the size of the PDE
system -- and the negative signature of the Hessian of the action implies
spectral instability, whereas a negative signature of the same Hessian being
equal to N implies orbital stability. Since these stability criteria are merely
encoded by the negative signature of matrices, they can at least be checked
numerically. Various numerical experiments are presented, which clearly
discriminate between stable cases and unstable cases for (qKdV), (EKE) and
(EKL).
