For Hamiltonian systems of PDEs the stability of periodic waves is encoded by the Hessian of an action integral, as shown in earlier work. This talk will deal with two asymptotic regimes, namely for waves of small amplitude and for waves of long wavelength. In both cases stability criteria can be investigated analytically, thanks to the asymptotic expansions of the Hessian of the action and their special structure. The stability results thus obtained apply to various models of mathematical physics for nonlinear dispersive waves.