Modulated equations of Hamiltonian PDEs and dispersive shocks

Abstract

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 generalized Korteweg de Vries equations and the Euler-Korteweg systems -- that are well-behaved in both the small amplitude and small wavelength limits. We use this parametrization 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.