  Partial differential equations endowed with a Hamiltonian structure, like the
Korteweg--de Vries equation and many other more or less classical models, are
known to admit rich families of periodic travelling waves. The stability theory
for these waves is still in its infancy though. The issue has been tackled by
various means. Of course, it is always possible to address stability from the
spectral point of view. However, the link with nonlinear stability -in fact,
\emph{orbital} stability, since we are dealing with space-invariant problems-,
is far from being straightforward when the best spectral stability we can
expect is a \emph{neutral} one. Indeed, because of the Hamiltonian structure,
the spectrum of the linearized equations cannot be bounded away from the
imaginary axis, even if we manage to deal with the point zero, which is always
present because of space invariance. Some other means make a crucial use of the
underlying structure. This is clearly the case for the variational approach,
which basically uses the Hamiltonian -or more precisely, a constrained
functional associated with the Hamiltonian and with other conserved quantities-
as a Lyapunov function. When it works, it is very powerful, since it gives a
straight path to orbital stability. An alternative is the modulational
approach, following the ideas developed by Whitham almost fifty years ago. The
main purpose here is to point out a few results, for KdV-like equations and
systems, that make the connection between these three approaches: spectral,
variational, and modulational.
