Modulated equations of Hamiltonian PDEs and dispersive shocks

TL;DR

This paper develops a new parametrization for periodic waves in Hamiltonian PDEs, enabling detailed analysis of dispersive shock waves and modulational instability in non-integrable systems like KdV and Euler-Korteweg.

Contribution

It introduces a wave parameter framework that captures asymptotic behaviors and instability criteria for a broad class of Hamiltonian PDEs, including new stability indices.

Findings

Modulational instability is linked to the second derivative of the Boussinesq moment of instability.

Explicit Benjamin--Feir type instability index is identified for harmonic limits.

Asymptotic properties of modulation systems are characterized in detail.

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.