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The stability of periodic traveling wave solutions to dispersive PDEs withrespect to `arbitrary' perturbations is still widely open. The focus is puthere on stability with respect to perturbations of the same period as the wave,for KdV-like systems of one-dimensional Hamiltonian PDEs. Stability criteriaare derived and investigated first in a general abstract framework, and thenapplied to three basic examples that are very closely related, and ubiquitousin mathematical physics, namely, a quasilinear version of the generalizedKorteweg--de Vries equation (qKdV), and the Euler--Korteweg system in bothEulerian coordinates (EKE) and in mass Lagrangian coordinates (EKL). Thosecriteria consist of a necessary condition for spectral stability, and of asufficient condition for orbital stability. Both are expressed in terms of asingle function, the abbreviated action integral along the orbits of waves inthe phase plane, which is the counterpart of the solitary waves moment ofinstability introduced by Boussinesq. Regarding solitary waves, the celebratedGrillakis--Shatah--Strauss stability criteria amount to looking for the sign ofthe second derivative of the moment of instability with respect to the wavespeed. For periodic waves, the most striking results obtained here can besummarized as: an odd value for the difference between N -- the size of the PDEsystem -- and the negative signature of the Hessian of the action impliesspectral instability, whereas a negative signature of the same Hessian beingequal to N implies orbital stability. Since these stability criteria are merelyencoded by the negative signature of matrices, they can at least be checkednumerically. Various numerical experiments are presented, which clearlydiscriminate between stable cases and unstable cases for (qKdV), (EKE) and(EKL).

Co-periodic stability of periodic waves in some Hamiltonian PDEs