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Since its elaboration by Whitham, almost fifty years ago, modulation theoryhas been known to be closely related to the stability of periodic travelingwaves. However, it is only recently that this relationship has been elucidated,and that fully nonlinear results have been obtained. These only concerndissipative systems though: reaction-diffusion systems were first considered byDoelman, Sandstede, Scheel, and Schneider [Mem. Amer. Math. Soc. 2009], andviscous systems of conservation laws have been addressed by Johnson, Noble,Rodrigues, and Zumbrun [preprint 2012]. Here, only nondissipative models areconsidered, and a most basic question is investigated, namely the expected linkbetween the hyperbolicity of modulated equations and the spectral stability ofperiodic traveling waves to sideband perturbations. This is done first in anabstract Hamiltonian framework, which encompasses a number of dispersivemodels, in particular the well-known (generalized) Korteweg--de Vries equation,and the less known Euler--Korteweg system, in both Eulerian coordinates andLagrangian coordinates. The latter is itself an abstract framework for severalmodels arising in water waves theory, superfluidity, and quantum hydrodynamics.As regards its application to compressible capillary fluids, attention is paidhere to untangle the interplay between traveling waves/modulation equations inEulerian coordinates and those in Lagrangian coordinates. In the most generalsetting, it is proved that the hyperbolicity of modulated equations is indeednecessary for the spectral stability of periodic traveling waves. This extendsearlier results by Serre [Comm. Partial Differential Equations 2005], Oh andZumbrun [Arch. Ration. Mech. Anal. 2003], and Johnson, Zumbrun and Bronski[Phys. D 2010]. In addition, reduced necessary conditions are obtained in thesmall amplitude limit. Then numerical investigations are carried out for themodulated equations of the Euler--Korteweg system with two types of 'pressure'laws, namely the quadratic law of shallow water equations, and the nonmonotonevan der Waals pressure law. Both the evolutionarity and the hyperbolicity ofthe modulated equations are tested, and regions of modulational instability arethus exhibited.

Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids