Modulated equations of Hamiltonian PDEs and dispersive shocks

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Motivated by the ongoing study of dispersive shock waves in non integrable systems, we propose and analyze a set of wave parameters for periodic waves of a large class of Hamiltonian partial differential systems—including the generalised Korteweg–de Vries equations and the Euler–Korteweg systems—that are well-behaved in both the small amplitude and large wavelength limits. We use this parametrisation to determine fine asymptotic properties of the associated modulation systems, including detailed descriptions of eigenmodes. As a consequence, in the solitary wave limit we prove that modulational instability is decided by the sign of the second derivative—with respect to speed, fixing the endstate—of the Boussinesq moment of instability; and, in the harmonic limit, we identify an explicit modulational instability index of Benjamin–Feir type.