  We analyze Floquet theory as it applies to the stability and instability of
periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on
several examples of such PDEs, including the generalized KdV and BBM equations
(third order), the nonlinear Schr\"odinger and Boussinesq equations (fourth
order), and the Kawahara equation (fifth order).
  Our analysis reveals that the characteristic polynomial of the monodromy
matrix inherits symmetry from the underlying PDE, enabling us to determine the
essential spectrum along the imaginary axis and bifurcations of the spectrum
away from the axis, employing the Floquet discriminant. We present numerical
evidence to support our analytical findings.
