  Generalizing ideas of MacKay, and MacKay and Saffman, a necessary condition
for the presence of high-frequency ( i.e., not modulational) instabilities of
small-amplitude periodic solutions of Hamiltonian partial differential
equations is presented, entirely in terms of the Hamiltonian of the linearized
problem. With the exception of a Krein signature calculation, the theory is
completely phrased in terms of the dispersion relation of the linear problem.
The general theory changes as the Poisson structure of the Hamiltonian partial
differential equation is changed. Two important cases of such Poisson
structures are worked out in full generality. An example not fitting these two
important cases is presented as well, using a candidate Boussinesq-Whitham
equation.
